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Enumerative problems in algebraic geometry motivated from physics Leigh, Oliver 2019

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ENUMERATIVE PROBLEMSin Algebraic Geometry Motivated from PhysicsbyOLIVER LEIGHA thesis submitted in partial fulfillmentof the requirements for the degree ofDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(MATHEMATICS)The University of British Columbia(VANCOUVER)JUNE 2019© Oliver Leigh, 2019iiThe following individuals certify that they have read, and recommend to the Facultyof Graduate and Postdoctoral Studies for acceptance, the dissertation entitled:Enumerative Problems in Algebraic Geometry Motivated from Physicssubmitted by Oliver Leigh in partial fulfillment of the requirements for the degree ofDoctor of Philosophy in Mathematics.Examining Committee:- Jim Bryan, Mathematics(Co-supervisor)- Kai Behrend, Mathematics(Supervisory Committee Member)- Joanna Karczmarek, Physics and Astronomy(University Examiner)- Christian Haesemeyer, Mathematics and Statistics, University of Melbourne(University Examiner)Additional Supervisory Committee Members:- Paul Norbury, Mathematics and Statistics, University of Melbourne(Co-supervisor)- Arun Ram, Mathematics and Statistics, University of Melbourne(Supervisory Committee Member)- Nora Ganter, Mathematics and Statistics, University of Melbourne(Supervisory Committee Member)iiiAbstractThis thesis contains two chapters which reflect the two main viewpoints of modernenumerative geometry.In chapter I we develop a theory for stable maps to curves with divisible ramification.For a fixed integer r > 0, we show that the condition of every ramification locusbeing divisible by r is equivalent to the existence of an rth root of a canonical sec-tion. We consider this condition in regards to both absolute and relative stable mapsand construct natural moduli spaces in these situations. We construct an analogue ofthe Fantechi-Pandharipande branch morphism and when the domain curves are genuszero we construct a virtual fundamental class. This theory is anticipated to have ap-plications to r-spin Hurwitz theory. In particular it is expected to provide a proof ofthe r-spin ELSV formula [SSZ’15, Conj. 1.4] when used with virtual localisation.In chapter II we further the study of the Donaldson-Thomas theory of the bananathreefolds which were recently discovered and studied in [Bryan’19]. These are smoothproper Calabi-Yau threefolds which are fibred by Abelian surfaces such that the sin-gular locus of a singular fibre is a non-normal toric curve known as a “banana con-figuration”. In [Bryan’19] the Donaldson-Thomas partition function for the rank 3sub-lattice generated by the banana configurations is calculated. In this chapter weprovide calculations with a view towards the rank 4 sub-lattice generated by a sectionand the banana configurations. We relate the findings to the Pandharipande-Thomastheory for a rational elliptic surface and present new Gopakumar-Vafa invariants forthe banana threefold.ivLay SummaryIn this thesis we use modern algebraic techniques to work on enumerative problemsthat are motivated by mathematical physics. The objects being counted are complexcurves which are surfaces that don’t have edges (e.g. spheres, donuts, etc.) with someextra structure. In string theory, these objects roughly translate to the path a vibratingstring would sweep out as it travels forward in time. We are interested in countingthe possible complex curves which can live within a given even-dimensional space. InChapter One, we develop the theory for a method of counting special sub-classes ofthese complex curves. We use this theory to provide a generalisation of the classicalconcept of Hurwitz numbers. In Chapter Two we provide an explicit computation of thenumber are complex curves that can live within a space called the banana threefold. Weshow that formulas obtained from these numbers have interesting properties related toprevious work.vPrefaceThis is to certify that this thesis comprises only my original work towards the Doctorof Philosophy in Mathematics and due acknowledgement has been made in the text toall other material used.This dissertation was originally formatted in accordance with the regulations of theUniversity of Melbourne and submitted in partial fulfillment of the requirements for aPhD degree awarded jointly by the University of Melbourne (lead university) and theUniversity of British Columbia. Different versions of this dissertation will exist in theinstitutional repositories of both institutions.Publication status of all chapters presented:I Chapter 1 : Submitted for publication on the 20th of February 2019.I Chapter 2: Unpublished material not submitted for publication.viDeclaration (The University of Melbourne)This is to certify that1. the thesis comprises only my original work towards the Doctor of Philosophy inMathematics except where indicated in the preface;2. due acknowledgement has been made in the text to all other material used; and3. the thesis is less than 100,000 words in length, exclusive of tables, maps, bibli-ographies and appendices.Oliver LeighviiPreface (The University of Melbourne)Publication status of all chapters presented:I Chapter 1 : Submitted for publication to “Transactions of the American Mathe-matical Society” on the 20th of February 2019.I Chapter 2: Unpublished material not submitted for publication.Note that these chapters are separate self-contained works. They do not refer to eachother and different notation is used in each.During the course of my candidature I received the following funding:I Research Assistantship: University of British Columbia. May 12th 2018 to com-pletion of degree.I Stanley M Grant Scholarship in Mathematics: University of British Columbia.Received November 8th, 2017.I Graduate Research Award in Pure Mathematics: University of British Columbia.Received October 27th, 2017.I Australian Government Research Training Program Scholarship: Full PhD fundingand stipend for the period January 1st 2017 to May 11th 2018. University ofMelbourne. Received January 1st, 2017.I Faculty of Science PhD Tuition Award : University of British Columbia. ReceivedAugust 31th, 2016.I Faculty of Science Travelling Scholarship: University of Melbourne. Received July18th, 2016.I Australian Postgraduate Award Scholarship for 2013-2015 : Full PhD funding andstipend for the period January 1st 2014 to December 31st 2016. University ofMelbourne. Received November 29th, 2013.I Research Higher Degree Studentship: Full PhD funding and stipend for the periodNovember 11th 2013 to December 31st 2013. University of Melbourne. ReceivedNovember 11th, 2013viiiContentsFront Matter iiAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vDeclaration (The University of Melbourne) . . . . . . . . . . . . . . . . . . . viPreface (The University of Melbourne) . . . . . . . . . . . . . . . . . . . . . . viiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiIntroduction 1I Stable Maps with Divisible Ramification 5Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Review of Stable Maps and r-Stable Curves . . . . . . . . . . . . . . . 91.1 Stable Maps and Relative Stable Maps . . . . . . . . . . . . . . 91.2 The Canonical Ramification Section . . . . . . . . . . . . . . . 111.3 r-Prestable curves . . . . . . . . . . . . . . . . . . . . . . . . . . 131.4 Line Bundles and their rth Roots on r-Twisted Curves . . . . . 142 Stable Maps with Roots of Ramification . . . . . . . . . . . . . . . . . . 152.1 r-Stable Maps with Roots of the Ramification Bundle . . . . . . 152.2 Power Map of Abelian Cone Stacks . . . . . . . . . . . . . . . . 162.3 Proof of Theorem A . . . . . . . . . . . . . . . . . . . . . . . . 183 Branching and Ramification of r-Stable Maps . . . . . . . . . . . . . . 193.1 Divisor Construction and the Branch Morphism for Stable Maps 193.2 A Branch Morphism for Maps with Divisible Ramification . . . 203.3 Special Loci of the Moduli Points . . . . . . . . . . . . . . . . . 234 Cotangent Complex ofM1/rg (X, d) . . . . . . . . . . . . . . . . . . . . 254.1 Perfect Relative Obstruction Theory . . . . . . . . . . . . . . . 26II DT Theory of the Banana 3-fold with Section Classes 331 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331.1 Donaldson-Thomas Partition Functions . . . . . . . . . . . . . . 331.2 Donaldson-Thomas Theory of Banana Threefolds . . . . . . . . 341.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381.4 Future . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382 Overview of the Computation . . . . . . . . . . . . . . . . . . . . . . . 392.1 Overview of the Method of Calculation . . . . . . . . . . . . . . 392.2 Review of Euler characteristic . . . . . . . . . . . . . . . . . . . 422.3 Pushing Forward to the Chow Variety . . . . . . . . . . . . . . 433 Parametrising Underlying 1-cycles . . . . . . . . . . . . . . . . . . . . . 443.1 Related Linear Systems in Rational Elliptic Surfaces . . . . . . 44ixx CONTENTS3.2 Curve Classes and 1-cycles in the Threefold . . . . . . . . . . . 453.3 Analysis of 1-cycles in Smooth Fibres of pr . . . . . . . . . . . 463.4 Analysis of 1-cycles in Singular Fibres of pr . . . . . . . . . . . 473.5 Parametrising 1-cycles . . . . . . . . . . . . . . . . . . . . . . . 494 Techniques for Calculating Euler Characteristic . . . . . . . . . . . . . 524.1 Quot Schemes and their Decomposition . . . . . . . . . . . . . 524.2 An Action on the Formal Neighbourhoods . . . . . . . . . . . . 524.3 Partitions and the topological vertex . . . . . . . . . . . . . . . 544.4 Partition Thickened Section, Fibre and Banana Curves . . . . . 554.5 Quot Schemes on C3 and the Topological Vertex . . . . . . . . 595 Euler Characteristic of the Fibres of the Chow Map . . . . . . . . . . . 605.1 Calculation for the class σ + (0, •, •) . . . . . . . . . . . . . . . 605.2 Preliminaries for classes of the form •σ + (i, j, •) . . . . . . . . 655.3 Calculation for the class •σ + (0, 0, •) . . . . . . . . . . . . . . 695.4 Calculation for the class •σ + (0, 1, •) . . . . . . . . . . . . . . 695.5 Calculation for the class •σ + (1, 1, •) . . . . . . . . . . . . . . 726 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.1 Connected Invariants and their Partition Functions . . . . . . . 856.2 Linear System in P1 × P1 . . . . . . . . . . . . . . . . . . . . . 866.3 Topological Vertex Formulas . . . . . . . . . . . . . . . . . . . . 89Bibliography 93List of FiguresI.1 Loci with ramification order 3 . . . . . . . . . . . . . . . . . . . . . . . 7II.1 The banana threefold . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34II.2 Natural divisors in the banana threefold . . . . . . . . . . . . . . . . . 35II.3 The banana configuration and the surface containing it . . . . . . . . . 36II.4 The process for reducing to partition thickened curves . . . . . . . . . 40II.5 Euler characteristic from subscheme decompositions . . . . . . . . . . 41II.6 Normalisation of the singular fibres . . . . . . . . . . . . . . . . . . . . 48II.7 A 3D partition asymptotic to((2, 1), (3, 2, 2), (1, 1, 1)). . . . . . . . . 55II.8 Subschemes in C2 and monomial ideals . . . . . . . . . . . . . . . . . . 56II.9 Decomposition of the Chow sub-scheme for vertical fibres . . . . . . . 61II.10 Cohen-Macaulay subschemes in the fibres of ρ• . . . . . . . . . . . . . 62II.11 Cohen-Macaulay subschemes in the fibres of η• . . . . . . . . . . . . . 66II.12 Linear system |f1 + f2| on P1 × P1. . . . . . . . . . . . . . . . . . . . . 87xixiiAcknowledgementsI would first and foremost like to thank my two supervisors Jim Bryan and Paul Nor-bury. I have been extremely fortunate to have two supervisors who are both greatmathematicians and great people.I would also like to thank the various mathematicians I have met through the years andthe useful conversations we have had. In particular I would like to thank Kai Behrend,Emily Clader, Barbara Fantechi, Felix Janda, Martijn Kool, Georg Oberdieck, StephenPietromonaco, Jørgen Rennemo, and Dustin Ross whose helpful conversations havecontributed to this work.Parts of this thesis were completed during visits to the Max Planck Institute for Math-ematics, Ludwig Maximilian University of Munich, the Bernoulli Center, the HenriPoincaré Institute and the Mathematical Sciences Research Institute. I would like tothank them for providing stimulating and welcoming work environments.Lastly, I would like to thank my family. Especially my parents who have made every-thing possible and my amazing wife who has made everything worthwhile.IntroductionThere have been links between geometry and physics for millennia. Indeed, manygreat discoveries from historical figures such as Newton, Maxwell and Einstein haveboth arisen from and strengthened these links. However, some areas of geometry havenot always played significant roles in these links. Enumerative geometry was one ofthese areas until relatively recently. The last thirty years have seen a rapid develop-ment and expansion of the links between enumeration of geometric structures andtheoretical physics, particularly in the area of string theory.This link can be intuitively described as follows: As a string moves around in space-time, it sweeps out a Riemann surface called a “worldsheet”. In complex geometry,this is one-dimensional, so we call it a curve. Counting curves, that live within aspacetime, gives information about interactions and probabilities of changing states.In studying curve enumeration, two main viewpoints have arisen:Gromov-Witten Theory: Curves are external with a map to the space, and areparameterised by the moduli space of stable maps,Mg(X,β).Donaldson-Thomas Theory: Curves are internal with structure coming from anembedding in the ambient space, and are parameterised by the Hilbert schemeHilbβ,1−g(X).We should note that these viewpoints are not completely separate and they have beenproven equivalent in many cases. However, the techniques employed can vary greatlybetween the two approaches. One key similarity between the two theories is that themoduli spaces involved are not equidimensional, and a “virtual fundamental class” isrequired for their definitions. This is an inherent technical issue which one must dealwith to use the theories.This thesis reflects the separation of these two viewpoints by having each of the twochapters devoted to one side. They are separate self-contained works, do not refer toeach other and have distinct notation.Spin Structures and Map EnumerationOne link between theoretical physics and enumerative geometry was proposed byWitten in 1991 during an investigation into two-dimensional quantum gravity. He con-jectured that certain curve counts would satisfy the KdV integrable hierarchy. Thisis a well known set of differential equations which possess soliton solutions. Witten’sconjecture was subsequently proven by Kontsevich with an ingenious use of combina-torial methods.12 IntroductionThere is another set of differential equations arising in soliton theory called the 2-Toda Hierarchy. In some ways it can be thought of as a more fundamental object thanthe KdV hierarchy. Okounkov and Pandharipande show in [OP] that this hierarchyhas solutions arising from a generalised form of Hurwitz numbers. Classically, Hur-witz numbers count maps from smooth curves to the complex projective line whereramification is specified to be simple. Okounkov and Pandharipande generalise thisdefinition using the representation theory of the symmetric group.Moreover, it has since been conjectured in [SSZ] that these generalised Hurwitz num-bers are actually natural intersections on the moduli space of curves with r-spin struc-ture. The form of this conjecture generalises the celebrated ELSV formula for classicalHurwitz numbers. However a proof of the formula and its underlying geometric mech-anism, has proved elusive.In Chapter I, a moduli space is introduced that gives a geometric interpretation of theobjects being counted by these generalised Hurwitz numbers. These are called stablemaps with divisible ramification. They are maps where the ramification number atevery point is divisible by r.The definition of such maps is clear for a smooth curve C . We simply specify that theramification divisor be divisible by r. However, when the curve is nodal this doesn’twork because the ramification divisor cannot be defined. For a morphism f : C → P1the ramification divisors is determined by the differential map df : f∗ΩP1 → ΩC . Thedivisor construction relies on the fact that ΩC in invertible when C is smooth andwhen C is nodal this is no longer the case.In Chapter I we overcome this using the observation that the ramification divisor isdefined by a canonical section δ : OC → ωC ⊗ f∗ω∨P1 and this section is still well de-fined when C is nodal. We then use the theory of r-spin structures to take an rth rootof this section. We show that this condition gives exactly the curves with ramificationorder divisible by r.There are three main results of Chapter I. The first is Theorem A, which shows that thespace described above is an appropriate space for enumerative study. Namely that it isa proper Deligne-Mumford stack. The theorem also gives a comparison between thisspace and the moduli space of stable maps which is the main moduli space studied inGromov-Witten theory.The second and third main results of Chapter I develop theory to allow enumerativestudy of this space. Theorem B gives an extension of the branching morphism of[FP] and an interpretation of the ramification properties of maps with nodal domains.A perfect obstruction theory for genus 0 is constructed in Theorem C which allowsintersection theory to be used on this space. When combined, the three main theoremsallow the definition and future study of the generalised Hurwitz problem in a geometricsetting.Enumeration of Subschemes in Calabi-Yau ThreefoldsMany conventional string theory models require ten real dimensions. These consist ofthe four usual dimensions, and six extra hidden “curled-up" ones coming from Calabi-Yau threefolds (three complex dimensions). This makes Calabi-Yau threefolds a naturalchoice for enumerative study.Introduction 3Even better, there are certain properties of Calabi-Yau threefolds that make the ex-pected dimension of the Hilbert scheme zero. This suggests that counting subschemesmay be related to the Euler characteristic. In fact, Behrend showed in [B1] that thevirtual curve counting theory known as Donaldson-Thomas theory is a weighted Eulercharacteristic.However, computing Donaldson-Thomas invariants is very hard. Even when we usethe Euler characteristic approach. In fact computing them for compact threefolds is sohard that the full Donaldson-Thomas theory is only known in computationally trivialcases. An example of this is the product of a K3 surface with an elliptic curve. Thegroup action of the elliptic curve extends to the Hilbert scheme making all the invari-ants trivial to compute. In non-trivial cases, there is not even a conjectural solution forthe full Donaldson-Thomas theory of a compact threefold. However, there are manybeautiful results that appear when we restrict our attention to subsets of the the fulltheory.These results will often not manifest themselves until one assembles the invariants intoa partition function. These are formal generating functions that store the enumerativeinvariants as coefficients of power series expansions. Partition functions will often haveproperties that are related to physical theories and modular forms. It is these prop-erties and connections that make a full Donaldson-Thomas partition function highlydesirable.Recent advances in techniques and the discoveries of new Calabi-Yau threefolds haveopened up new avenues for calculations in Donaldson-Thomas theory. One such tech-nique was recently introduced by Bryan and Kool in [BK] for studying local elliptic sur-faces. This method is extended in Chapter II to allow its use in a more general setting.The full generality in which this method can be used is currently unknown. However,it can certainly be used to study the Donaldson-Thomas theory of any Calabi-Yauthreefold when the curve classes can be understood and when the subschemes arelocally determined by monomial ideals.In Chapter II these methods are then used to provide new calculations for the “banana”Calabi-Yau threefold recently introduced in [Br]. This is a smooth proper Calabi-Yauthreefold which is fibred by Abelian surfaces such that the singular locus of a sin-gular fibre is a non-normal toric curve known as a “banana configuration”. In [Br]the Donaldson-Thomas partition function for the rank 3 sub-lattice generated by thebanana configurations is calculated. In Chapter II we provide calculations with a viewtowards the rank 4 sub-lattice generated by a section and the banana configurations.4 IntroductionChapter IThe Moduli Space of StableMaps with DivisibleRamificationIntroductionConsider a smooth curve X and the moduli space parameterising degree d mapsf : C → X where C is a smooth curve of genus g. This space is denoted byMg(X, d) and point a [f ] ∈Mg(X, d) has an associated exact sequence0 −→ f∗ΩX ⊗ Ω∨C δ∨−→ OC −→ ORf −→ 0 (I.1)where Rf is the ramification divisor. If [f ] is a generic point then Rf is the union ofdisjoint points on C . In other words, f has simple ramification everywhere.As an alternative, we consider a space M1/rg (X, d) where a generic point [f ] givesa ramification divisor of the form Rf = r · p1 + · · · + r · pm for disjoint pointsp1, . . . , pm ∈ C . Specifically, we defineM1/rg (X, d) as the following sub-moduli spaceofMg(X, d):M1/rg (X, d) ={ [f : C → X] ∈Mg(X, d) ∣∣∣ Rf = r ·D for some D ∈ Div(C) }/ ∼ .In this chapter we construct a natural compactification of M1/rg (X, d). We developthe enumerative geometry of this space by constructing a virtual fundamental class inthe case g = 0 and by constructing a branch morphism.The above construction of the ramification divisor relies on the domain curve C beingsmooth. This means that ΩC is locally free and that df : f∗ΩX → ΩC is injective.If C is allowed to be singular either of these may be false and we no longer havea straightforward definition of ramification. This leads us to rephrase the moduliproblem using rth roots of δ which is defined in (I.1). One can show thatM1/rg (X, d)is naturally isomorphic to:{[f :C→X] ∈Mg(X, d) ∣∣∣∣ There is a line bundle L on C , σ ∈ H0(L) andan isom. L⊗r e→ ωC ⊗ f∗ω∨X with e(σr) = δ.}/ ∼ .56 CHAPTER I. STABLE MAPS WITH DIVISIBLE RAMIFICATIONWe now have the moduli problem in a form which can be naturally compactified.First we note that for nodal domain curves there is a natural morphism ΩC → ωC .Here we have used standard notation for the sheaf of differentials and the dualisingsheaf noting that the latter is locally free. This is combined with the differential mapdf : f∗ΩX → ΩC to obtain a morphism which we denote byδ : OC −→ ωC ⊗ f∗ω∨X . (I.2)Definition 1 Denote byM1/rg (X, d) the moduli stack parameterising morphisms f :C → X where1. C is a genus g r-prestable curve (a stack such that the coarse space C isa prestable curve, where points mapping nodes of C are balanced r-orbifoldpoints, and Csm ∼= Csm);2. f is a morphism such that the induced morphism f : C → X on the coursespace is a stable map;3. there exists a line bundle L on C , an isomorphism e : L⊗r ∼→ ωC ⊗ f∗ω∨X , anda morphism σ : OC → L such that e(σr) = δ, where δ is defined in (I.2).Remark 1 Throughout the chapter we will also be considering the same moduli prob-lem in the context of stable maps relative to a point x ∈ X and a partition µ of d > 0.The moduli space of relative stable mapsMg(X,µ) generically parameterises mapswhere the pre-image of x is smooth and locally has monodromy given by µ. We willleave the specifics of this moduli problem until section 1.1, however all of the followingresults will hold whenM1/rg (X, d) is replaced byM1/rg (X,µ), and 2g−2−d(2gX−2)is replaced by 2g − 2 + l(µ) + |µ|(1− 2gX).Remark 2 The r-prestable curves in definition 1 arise naturally when taking rth rootsof line bundles on nodal curves [AJ, Ch1]. We review this in section 1.3.Theorem A M1/rg (X, d) is a proper DM stack. It is non-empty only when r divides2g − 2− d(2gX − 2). The natural forgetful mapχ :M1/rg (X, d) −→Mg(X, d)is both flat and of relative dimension 0 onto its image. It is an immersion when restrictedtoM1/rg (X, d).The image of χ has an explicit point-theoretic description. Let f : C → X be a stablemap and consider the locus in C where f is not étale. Following [V, GV] a connectedcomponent of this locus is called a special locus. A special locus will be one of thefollowing:(a) A smooth point of C where f is locally of the form z 7→ za+1 with a ∈ N.(b) A node of C such that on each branch f is locally like z 7→ zai with ai ∈ N.(c) A genus g′ component B of C where f |B is constant and on the branches of Cmeeting B the map f is locally of the form z 7→ zai with ai ∈ N.7(a) (b) (c)  1푔=Figure I.1: Loci with ramification order 3. (a) A smooth point where the map is locallylike z 7→ z3+1. (b) A node where the map is locally like z 7→ z2 on one branchand z 7→ z on the other. (c) A genus one component meeting its complement at anode, where the map is constant on the sub-curve and locally like z 7→ z2 on thecomplement.Note that a slightly different definition is used for the relative case (see remark 3.3.1).Now, following [V, GV] again, we define a ramification order (or sometimes simplyorder ) for each type of special locus by:(a) a. (b) a1 + a2. (c) 2g′ − 2 +∑(ai + 1).This gives us an extended concept of ramification. There is also an extended conceptof branching constructed in [FP] which agrees with the ramification order assigned tospecial loci. Specifically there is a well defined morphism of stacks which agrees withthe classical definition of branching on the smooth locus:br : Mg(X, d) −→ Sym2g−2−d(2gX−2)X.Theorem B The objects in M1/rg (X, d) have the following ramification and branchingproperties:1. The closed points in the image of τ : M1/rg (X, d) −→ Mg(X, d) are the closedpoints ofMg(X, d) with the property:“Every special locus of the associated map has order divisible by r”.2. There is a morphism of stacksbr :M1/rg (X, d) −→ Sym1r (2g−2−d(2gX−2))Xthat commutes with the branch morphism of [FP] via the diagramM1/rg (X, d) br //χSym1r (2g−2−d(2gX−2))X∆Mg(X, d) br // Sym2g−2−d(2gX−2)Xwhere ∆ is defined by∑i xi 7→∑i rxi.Just like for regular stable mapsMg(X, d), the smooth-domain locusM1/rg (X, d) canbe empty whileM1/rg (X, d) is non-empty. For explicit examples consider degree onemaps to P1 with g > 0.8 CHAPTER I. STABLE MAPS WITH DIVISIBLE RAMIFICATIONThe properties of M1/rg (X, d) can be quite different to those of Mg(X, d). For ex-ample, if we consider genus zero domains we have thatM0(X, d) is smooth, but ingeneralM1/r0 (X, d) is not. An explicit example of this isM1/30 (P1, 4), which is notsmooth as it contains components of dimensions 2 and 3. However, we do have theexistence of a virtual fundamental class for g = 0.Theorem C M1/r0 (P1, d) has a natural perfect obstruction theory giving a virtual fun-damental class of dimension 1r (2d− 2) = 1rvirdim(M0(P1, d)).The moduli space M1/rg (P1, µ) has expected applications to r-spin Hurwitz theory.For example, in genus 0 using both theorems B and C we have the following naturalintersection ∫[M1/r0 (P1,µ)]virbr∗H1r (l(µ)+|µ|−2) (I.3)where H is the hyperplane class in Sym1r (l(µ)+|µ|−2)P1 ∼= P 1r (l(µ)+|µ|−2). This is adirect analogue of the characterisation of simple Hurwitz numbers given in [FP, Prop.2]. This was the first step towards a proof via virtual localisation of the ELSV formula.After applying the virtual localisation techniques of [GP], (I.3) is expected to be relatedto the r-ELSV formula of [SSZ, BKLPS].In the case where r = l(µ) + |µ| − 2 the spaceM1/r0 (P1, µ) has virtual dimension 1.These spaces are characterised by having exactly one free special locus of order r. Inthis situation the intersections given in (I.3) are expected to have a direct relation tothe completion coefficients and one-point invariants of [OP]:∫[M0,1(P1,µ)]virψr1ev∗1 [pt].This chapter is structured as follows:Section 1: Review the necessary theory of stable maps, r-prestable curves and linebundles on twisted curves required for the construction ofM1/rg (X, d) and its relativeversion.Section 2: Extend the theory of roots of line bundles to the space of stable maps,construct the moduli spaceM1/rg (X, d) and then prove theorem A.Section 3: Consider properties ofM1/rg (X, d) related to branching and ramificationwhile proving theorem B.Section 4: Consider the cotangent complex of M1/rg (X, d) and related propertieswhile proving theorem C.Conventions All stacks and schemes are over C. By local picture we will mean thefollowing. Let f : X → Y and g : U → V be morphisms of stacks. The local pictureof f at x ∈ X is the same as the local picture of g at u ∈ U if:• There is an isomorphism between the strict henselization f sh : Xsh → Y sh of fat x and the strict henselization gsh : U sh → V sh of g at u.1. REVIEW OF STABLE MAPS AND r-STABLE CURVES 9Throughout the chapter we will consider both absolute and relative stable maps. Thetheory will be similar so we introduce the following simplifying notation.Notation:• M is eitherMg(X, d) orMg(X,µ) for g ≥ 0, d > 0 and µ a partition of d.• C →M is the associated universal curve.• IfM isMg(X, d) (resp. Mg(X,µ)) then M is Mg (resp. Mg,l(µ)).• The expected number of order r special loci in the generic case is denoted bym. When M = Mg(X, d) we have m = 1r (2g − 2 − d(2gX − 2)) and whenM =Mg(X,µ) we have m = 1r (2g − 2 + l(µ) + |µ|(1− 2gX)).• Throughout, the notation used for a space without r or 1r will carry through toanalogous spaces involving r or 1r . ForMg(X,µ) the two key spaces are:– M[r] =M [r]g (X,µ) and C[r] = C[r]g (X,µ) defined in 2.1.3.– M[ 1r ] =M[1/r]g (X,µ) and C[1r ] = C[1/r]g (X,µ) are defined in 2.3.1.3.And the associated spaces are:– Mr =M rg (X,µ) and Cr = Crg (X,µ) defined in 2.1.– M 1r ,E =M 1r ,Eg (X,µ) and C 1r ,E = C1r,Eg (X,µ) are defined in 2.1.1.– M 1r =M1/rg (X,µ) and C1/r = C1/rg (X,µ) are defined in 2.3.1.1.1 Review of Stable Maps and r-Stable Curves1.1 Stable Maps and Relative Stable MapsFor the rest of this chapter we will set X to be a non-singular projective curve. Recallthat a stable map f : C → X is a degree d morphism from a genus g prestable curveto X which has no infinitesimal automorphisms. We denote byMg(X, d) the modulistack of these objects. Specifically this is the groupoid containing the objects:ξ =(pi : C → S, f : C → X )where pi is a proper flat morphism and for each geometric point p ∈ S we havefp : Cp → X is a degree d genus g stable map to X . A morphism ξ1 → ξ2 inMg(X, d) between objects ξi = (pii : Ci → Si, fi : Ci → X) is a commutativediagram where the left square is cartesian:S1C1pi1oof1 // XS2 C2pi2oo f2 // XLet x be a geometric point of X and µ a partition of d > 0. As we mentioned inremark 1, we will also be considering the moduli problem in the case of stable maps10 CHAPTER I. STABLE MAPS WITH DIVISIBLE RAMIFICATIONrelative to (x, µ). We use the algebro-geometric definition of this moduli space and itsobstruction theory provided in [L1, L2].The goal of relative stable maps is to parameterise maps where the pre-image of x liesin the smooth locus of C and where the map has monodromy given by µ locally abovex. However, this condition will not give a compact space. The solution provided in[L1] is to allow the target to degenerate in a controlled manner by allowing X to sprouta chain of P1’s.Specifically we can define the nth degeneration X[i] inductively from X[0] := X by:• X[i+ 1] is given by the union X[i] ∪ P1 meeting at a node ni+1.• The node n1 is at x ∈ X . For i > 0 the node ni+1 is in the ith component ofX[i+ 1], i.e. the node is not in X[i− 1] ⊂ X[i+ 1].Then a degenerated target is a pair (T, t) where T = X[i] for some i ≥ 0 and t is ageometric point in the smooth locus of ith component of T .A genus g stable map to X relative to (µ, x) is given by(h : C −→ T, p : T −→ X, q1, . . . , ql(µ))where (C, qi) is a l(µ)-marked prestable curve, h is a genus g stable map sending qito t and p is a morphism sending t to x such that:1. There is an equality of divisors on C given by h−1(t) =∑µiqi.2. We have p|X is an isomorphism and p|T\X : T \X → {x} is constant.3. The pre-image of each node n of T is a union of nodes of C . At any such noden′ of C , the two branches of n′ map to the two branches of n, and their ordersof branching are the same.4. The data has finitely many automorphisms (recall, an automorphism is a a pairof isomorphisms a : C → C and b : T → T taking qi to qi and t to t such thath ◦ a = b ◦ h and p = p ◦ b).We denote by Mg(X,µ) the moduli stack of genus g stable maps relative to (µ, x).This is the groupoid containing the objects:ξ =( Cpi SqiWW,Tpi′ StWW, h : C → T, p : T → X)where pi and pi′ are flat proper morphisms, h is a morphism over S and for each geo-metric point z ∈ S we have ξz is a genus g stable map relative to (µ, x). Furthermore,we require that in a neighbourhood of a node of Cz mapping to a singularity of Tzwe can choose étale-local coordinates on S, C and T with charts of the form SpecR,SpecR[u, v]/(uv − a) and SpecR[x, y]/(xy − b) respectively such that the map isof the form x 7→ αuk and y 7→ αvk with α and β units. A morphism ξ1 → ξ2 inMg(X,µ) between two appropriately label objects is a pair of cartesian diagramsC1pi1 a′ // C2pi2S1a // S2T1pi′1 b′ // T2pi′2S1b // S2that are compatible with the other data (i.e. we have a′ ◦ q1,i = q2,i ◦ a, b′ ◦ t1 = t2 ◦ b,b′ ◦ h1 = h2 ◦ a′ and p1 = p2 ◦ b′).1. REVIEW OF STABLE MAPS AND r-STABLE CURVES 111.2 The Canonical Ramification SectionAs we saw in the introduction, for a moduli point [f ] ∈Mg(X, d) we have two naturalmorphismsf∗ωX −→ ΩC and ΩC −→ ωc (I.4)which we can combine into a single morphism δ : OC → ωC ⊗ f∗ω∨X . This morphismreflects the ramification properties of f which we will see in section 3. Hence, we willcall the bundle ωC ⊗ f∗ω∨X the ramification bundle of f .Considering the universal curve pi : Cg(X, d) → Mg(X, d). The above constructionstill holds for the universal stable map f : Cg(X, d) → X . We then have a universalsectionδ : OCg(X,d) −→ R (I.5)where we have denoted the universal ramification bundle R := ωpi ⊗ f∗ω∨X .For the above case of stable maps we are interested in a subspace where a genericpoint [f ] corresponds to a map f with a ramification divisor of the form Rf =r · z1 + · · · + r · zm for disjoint points z1, . . . , zm. However, the key concept ofrelative stable maps is that the ramification above a fixed point is determined by agiven divisor. The ramification is allowed to be free elsewhere.Hence for the relative case we will be interested in a subspace ofMg(X,µ) where ageneric [f ] corresponds to a map f with a ramification divisor of the form:Rf = Dµ + r · z1 + · · ·+ r · zmwhereDµ =∑(µi−1)qi is the ramification divisor supported at the points qi mappingto x ∈ X . So, we are interested in taking rth roots of a section of the bundleωC⊗f∗ω∨X⊗OC(−Dµ) ∼= ωlogC ⊗f∗(ωlogX )∨. The situation is slightly more complicatedbecause of the possibility of a degenerated target. So we consider a general genus gstable map to X relative to (µ, x) over S:ξ =( Cpi SqiWW,Tpi′ StWW, h : C → T, p : T → X).Now, letting q = q1 + · · · + ql(µ), we have three line bundles which we are interestedin:ωlogC/S = ωC/S(q), ωlogT/S = ωT/S(t) and ωlogX = ωX(x)and we make choices of morphisms defining the divisors q, t and x respectively:Dq : OC(−q)→ OC , Dt : OT (−t)→ OT and Dx : OX(−x)→ OX .(I.6)Now there is a unique choice of isomorphism p∗ωlogX∼→ ωlogT/S such that the followingdiagram commutesp∗ωXp∗Dx //p∗ωlogX∼=ωT/SDt // ωlogT/S12 CHAPTER I. STABLE MAPS WITH DIVISIBLE RAMIFICATIONwhere the left vertical morphism is the natural morphism coming from (I.4) applied top : T → X .After using the isomorphism p∗ωlogX∼→ ωlogT/S we are interested in a canonical mor-phism h∗ωlogT/S −→ ωlogC/S . The construction used in (I.5) breaks down here becausethere we used the fact that ΩX ∼= ωX is locally free. In general, ΩT/S  ωT/S andΩT/S is not locally free, because of the nodes on the degenerated target. However, theadmissibility condition allows us to define a morphism h∗ωT/S −→ ωC/S directly.Away from the nodes of T we can simply define the morphism in the usual way.Locally at the nodes we have that S = SpecR, T = SpecR[x, y]/(xy − ξ) andC = SpecR[u, v]/(uv − ζ) with the map h defined byH : R[x, y]/(xy − ξ) −→ R[u, v]/(uv − ζ)x 7−→ αuay 7−→ βvafor α and β units and with H(ξ) = αβζa. Also, locally we have that ωT/X and ωC/Xare generated bydx ∧ dy(xy − ξ) anddu ∧ dv(uv − ζ)respectively. Hence, we have a natural isomorphism locally defined by:d(H(x)) ∧ d(H(y))(H(x)H(y)− Φ(ξ)) = d(αua) ∧ d(βvb)(αβuava − αβξa) = du ∧ dv(uv − ζ) .Hence we have the following lemma.Lemma 1.2.1. Let T sm be the smooth locus of T relative to S and B = h−1(T sm). Thereis a canonical morphism δ˜ : h∗ωT/S −→ ωC/S such that:1. The restriction δ˜|B to is the usual morphism (h|B)∗ωT sm/S → ωC/S ,2. δ˜ is locally an isomorphism at the nodes of T .Now, we restrict the morphism h to the smooth locus of C over S and denote theseby hsm and Csm respectively. The morphism from lemma 1.2.1 restricted to Csm isinjective and is the divisor sequence for the ramification divisor. Both the divisors q1 +· · ·+ql(µ) and h−1(t) are in Csm. Now using the choices from (I.6) it is straightforwardto show that there is now a unique map δ˜ log making the following diagram commuteh∗ωT/Sh∗Dt //δ˜h∗ωlogT/Sδ˜ logωC/SDq // ωlogC/SNow, using the isomorphism p∗ωlogX∼→ ωlogT/S we have the canonical morphism whichwe desire:δlog : OC/S −→ ωlogC/S ⊗ f∗(ωlogT/S)∨. (I.7)The above construction immediately lends itself to a universal construction. Considerthe universal curve pi : Cg(X,µ) → Mg(X,µ), the universal degenerated target1. REVIEW OF STABLE MAPS AND r-STABLE CURVES 13pi′ : T → Mg(X,µ) with universal maps h : Cg(X,µ) → T and p : T → X , anduniversal sections qi : Mg(X,µ) → Cg(X,µ) and t : Mg(X,µ) → T . Then wemake, once and for all, choicesDq : OCg(X,µ)(−q)→ OCg(X,µ), Dt : OT (−t)→ OT and Dx : OX(−x)→ OX(I.8)which allows us to define the universal sectionδlog : OCg(X,µ) −→ Rlog(I.9)where we have denoted the universal ramification bundle Rlog := ωlogpi ⊗ f∗(ωlogpi′ )∨.1.3 r-Prestable curvesOur moduli problem requires the use of nodal curves where the nodes have a balancedr-orbifold structure. These curves are also called twisted curves and were introducedin [AV] to study stable maps where the target is a DM stack. They have since beenextensively studied in [ACV, O, AGV, FJR1, FJR2]. In this chapter we are interested inusing them in relation to taking rth roots of line bundles, which have been studied in[AJ, Ch1, Ch2].Definition 1.3.1. Let S be a scheme. An r-prestable curve over S of genus g with nmarkings is: ( CpiS,( CSxiOO )i∈{1,...,n})where1. pi is a proper flat morphism from a tame stack to a scheme;2. each xi is a section of pi that maps to the smooth locus of C ,3. the fibres of pi are purely one dimensional with at worst nodal singularities,4. the smooth locus Csm is an algebraic space,5. the coarse space pi : C → S with sections xi is a genus g, n-pointed prestablecurve (C, pi : C → S, (xi : S → C)i∈{1,...,n})6. the local picture at the nodes is given by [U/µr]→ T , where• T = SpecA, U = SpecA[z, w]/(zw− t) for some t ∈ A, and the action ofµr is given by (z, w) 7→ (ξrz, ξ−1r w).We denote the space parameterising r-prestable curves by Mrg,n. This space is shownto be a smooth proper stack in [Ch1]. There is a natural forgetful mapMrg,n −→Mg,nwhich maps an r-prestable curve to its coarse space. This map is flat and surjective ofdegree 1, but it is not an isomorphism. Its restriction to the boundary is degree 1r .One can also consider r-orbifold structure at smooth marked points as well. Specifi-cally we can include in the definition étale gerbes Xi → X which are closed sub-stacksof the smooth locus of the curve Xi ↪→ Csm. The local picture at an r-orbifold markedpoint is given by [V/µr]→ T , whereT = SpecA, V = SpecA[z], and the action of µr is given by z 7→ ξrz.14 CHAPTER I. STABLE MAPS WITH DIVISIBLE RAMIFICATION1.4 Line Bundles and their rth Roots on r-Twisted CurvesThe theory of rth roots on line bundles on prestable curves has origins related totheta characteristics [Co]. This led naturally to the study of r-spin structures studiedin [J1, J2] using torsion free sheaves for rth roots. r-spin structures were also studiedusing twisted curves in [AJ, Ch1, Ch2] and other methods in [CCC]. The results of[Ch1] will be of particular interest to us.Consider a family of r-prestable curves Cγ→ C pi→ S, and let E be a line bundle on Cpulled back from C with relative degree divisible by r. Define the following groupoidRootrC(E) containing as objects:(h : Z → S, L, e : Lr ∼→ EZ)where h is a morphism of schemes, L is a line bundle on CZ and e is an isomorphism.Theorem 1.4.1. [Ch1, Prop 3.7, Thm 3.9] In the situation above we have:1. For each geometric point p ∈ S, we have r2g roots of Ep.2. RootrC(OC) is a finite group stack.3. RootrC(E) is a finite torsor under RootrC(OC).4. RootrC(E)→ S is étale of degree r2g−1.1.4.2. Consider an r-prestable curve over C with an r-orbifold marked point p. Thelocal picture at the marking p is given by [(SpecC[z])/µr] where the action of µr isgiven by z 7→ ξrz. Consider a line bundle L supported at p. Then (the sheaf) L islocally generated at p by φ = zn for n ∈ Z and we have φ(ξrz) = ξkrφ(z) for somek ∈ Z/r. We call k the multiplicity of L at p.Similarly, at a node q the the local picture is given by [(SpecC[u, v]/uv)/µr], wherethe action of µr is given by (u, v) 7→ (ξru, ξ−1r v). So a line bundle L on C sup-ported at q is locally generated by ψ = un1 − vn2 for n1, n2 ∈ Z \ {0} such thatψ(ξru, ξrv) = ξarψ(u, v) for some a ∈ Z/r. In fact a is determined only up to achoice of branch. Hence we obtain a pair numbers a, b ∈ Z/r with either a = b = 0or a+ b = r. We call this pair the multiplicity of L at the node q.We can also consider an associated sheaf on the coarse space C . Locally the coarsespace is given by the invariant sections of the structure sheaf. Then the sheaf L := γ∗Lis similarly given by the locally invariant sections of L. When C is a smooth curve Lis a line bundle. However when C is singular, then L is only torsion free in general.Using these ideas one can easily show the following lemma is true.Lemma 1.4.3. Let C be a r-prestable curve over C, with n smooth orbifold points x1, . . . , xnand let β : C → C be the map forgetting the orbifold structure at the smooth points. Alsolet L be a line bundle on C with multiplicities a1, . . . , an at the the orbifold points and letD be the divisor∑aixi. Then for a section σ : OC → L there is a commuting diagramwhere the bottom row is the divisor sequence:OC(β∗σ)rOCβ∗(σr)0 // (β∗L)r // β∗(Lr) // OD // 02. STABLE MAPS WITH ROOTS OF RAMIFICATION 152 Stable Maps with Roots of RamificationSection 2 Notation: Recall the notation convention:• M is eitherMg(X, d) orMg(X,µ) for g ≥ 0, d > 0 and µ a partition of d.• C →M is the associated universal curve.• IfM isMg(X, d) (resp. Mg(X,µ)) then M is Mg (resp. Mg,l(µ)).• The expected number of order r special loci in the generic case is denoted bym. When M = Mg(X, d) we have m = 1r (2g − 2 − d(2gX − 2)) and whenM =Mg(X,µ) we have m = 1r (2g − 2 + l(µ) + |µ|(1− 2gX)).• Throughout, the notation used for a space without r or 1r will carry through toanalogous spaces involving r or 1r . ForMg(X,µ) the two key spaces are:– M[r] =M [r]g (X,µ) and C[r] = C[r]g (X,µ) defined in 2.1.3.– M[ 1r ] =M[1/r]g (X,µ) and C[1r ] = C[1/r]g (X,µ) are defined in 2.3.1.3.And the associated spaces are:– Mr =M rg (X,µ) and Cr = Crg (X,µ) defined in 2.1.– M 1r ,E =M 1r ,Eg (X,µ) and C 1r ,E = C1r,Eg (X,µ) are defined in 2.1.1.– M 1r =M1/rg (X,µ) and C1/r = C1/rg (X,µ) are defined in 2.3.1.1.2.1 r-Stable Maps with Roots of the Ramification BundleIn this subsection we will be considering the results of [Ch1] in the context of stablemaps. We will begin by considering stable maps where the domain curve is r-prestable.We call these r-stable maps. The moduli stack of these and its universal curve fit intothe two cartesian squares:Cr //γMr //MrC //M //MNow we will considering stable maps with an rth root of a line bundle. Let E be a linebundle on C of degree divisible by r and define the line bundle E on Cr by E := γ∗E .Definition 2.1.1. Denote by M 1r ,E the moduli stack of r-stable maps with roots of Ewhich contains families: (ξ, L, e : Lr ∼−→ Eξ)where1. ξ is a family of r-stable maps inM;2. L is a line bundle on Cξ ;3. e is an isomorphism of line bundles on Cξ .16 CHAPTER I. STABLE MAPS WITH DIVISIBLE RAMIFICATIONLemma 2.1.2. M 1r ,E has the following properties:1. M 1r ,E is a proper DM stack.2. When E is the trivial bundleM 1r ,OC →Mr is a finite group stack.3. The forgetful map M 1r ,E → Mr is a finite torsor under M 1r ,OC and is étale ofdegree r2g−1.Proof. Let a : S → Mr be the morphism of stacks defined by the family ξ ∈ Mr .Also let C = (Cr)ξ and E = (E)ξ . Then we have the following cartesian diagrams:RootrC(OC) //M 1r ,OCS //MrRootrC(E) //M 1r ,ES //MrThe lemma now follows from theorem 1.4.1.Definition 2.1.3. In the special case where E = R we callM 1r ,R the moduli spaceof r-stable maps with roots of the ramification bundle and denote it with the simplifyingnotation:M[r] :=M 1r ,R.2.2 Power Map of Abelian Cone StacksLet E be a line bundle on C of degree divisible by r and define the line bundle E onC[r] by E := γ∗E where γ : Cr → C is the map forgetting the r-orbifold structureof the curves. ConsiderM 1r ,E , the space of r-stable maps with roots of E defined in2.1.1 with universal curve pi : C 1r ,E →M 1r ,E , universal section s :M 1r ,E → C 1r ,E anduniversal rth root bundle L and isomorphism e : Lr ∼→ E .Definition 2.2.1. For a line bundle F on C 1r ,E , we define the the following notation:1. Totpi∗F := SpecM 1r ,E(Sym•R1pi∗(F∨ ⊗ ωpi))which contains objects:(ξ, σ : OC −→ Fξ)where ξ is an object ofM 1r ,E and C := (C 1r ,E)ξ (discussed in [CL, Prop 2.2] and[CLL, Thm 2.11]). Also, let α : Totpi∗F → M 1r ,E denote the natural forgetfulmap.2. ψ : CTotpi∗F → Totpi∗F is the universal curve and α̂ : CTotpi∗F → C1r ,E is thenatural forgetful map.3. TotF := SpecC 1r ,E(Sym• F∨)which contains objects:(ζ, λ : OS −→ s∗Fζ)where ζ is an object of C 1r ,E over S and s := sζ . Also, let αˇ : TotF → C 1r ,Edenote the natural forgetful map.Remark 2.2.2. Note that while we use the notation Totpi∗F , it is often the case thatpi∗F is not locally free. This space is called an abelian cone in [BF].2. STABLE MAPS WITH ROOTS OF RAMIFICATION 17Let ζ be a family in C 1r ,E over S with piζ = pi : C → S and s := sζ . There is anatural evaluation morphism e : CTotpi∗F → TotF defined bye :(ζ, σ : OC −→ Fξ)7−→(ζ, s∗σ : OS −→ s∗Fζ). (I.10)This gives the following commutative diagram where the left-most square is Cartesian:Totpi∗FαCTotpi∗Fψoo e //α̂TotFαˇM 1r ,E C 1r ,Epioo C 1r ,E(I.11)There are special cases when F = L and F = Lr and we have we have canonicalmaps:Definition 2.2.3. The rth power map overM 1r ,E is the map τ : Totpi∗L → Totpi∗Lrdefined by: (ξ, σ)7−→(ξ, σr)and the rth power map over C 1r ,E is the similarly defined map τˇ : TotL → TotLr .Out of the two maps defined here, τˇ is nicer. It is a fibre-wise r-fold cover of the totalspace of TotLr ramified at the zero section. However, τ is the map more directlyrelated to out moduli problem.Lemma 2.2.4. The rth power map overM 1r ,E is factors viaTotpi∗L τ //ϕ %%Totpi∗LrXj88where j is a closed immersion and ϕ is the quotient by the following action of Zr onTotpi∗L:ζr ·(ξ, σ)=(ξ, ζr · σ)Proof. We first show that the image of τ is a closed substack of Totpi∗Lr . Denote theclosed immersion defined by taking the graph of τ by i : Totpi∗L → Totpi∗Lr×M 1r ,ETotpi∗L. Then τ factors via:Totpi∗L  i //τ **Totpi∗Lr ×M 1r ,E Totpi∗Lpr1Totpi∗LrWe claim that pr1 is a closed map. To see this we let ψ : CTotpi∗Lr → Totpi∗Lr bethe universal family. Then we have the following abelian cone stack over Totpi∗Lrp : SpecTotpi∗Lr(Sym•R1ψ∗(ψ∗L∨ ⊗ ωψ))−→Totpi∗Lrwhich is isomorphic over Totpi∗Lr to the pullback:Totpi∗Lr ×M 1r ,E Totpi∗L∼ //pr1''SpecTotpi∗Lr(Sym•R1ψ∗(ψ∗L∨ ⊗ ωψ))puuTotpi∗Lr18 CHAPTER I. STABLE MAPS WITH DIVISIBLE RAMIFICATIONp is a closed map, so pr1 is also closed and im(τ) is a well defined closed substack. Itis clear that im(τ) is isomorphic to the quotient of Totpi∗L by the action of Zr .2.3 Proof of Theorem AIn this section we will prove theorem A about the properties of M1/r . We will alsoconsider related spaces that contain extra information which we will denote byM[1/r]andM(1/r). In particularM[1/r] will be the key space for study in the later sectionsof this chapter.Let pi : C[r] →M[r] be the universal curve ofM[r] and f : C[r] → X be the universalr-stable map and δ : OCr → R be the pullback by γ : Cr → C of canonical ramifica-tion section defined in (I.5) and (I.9). Where γ is the map which forgets the r-orbifoldstructure. Also let L be the universal rth root onM[r].Definition 2.3.1. The moduli spaces of stable maps with divisible ramification are:1. M1/r is the substack ofMr containing families ξ where there exists:(a) a line bundle L on C := (Cr)ξ ;(b) an isomorphism e : Lr∼→ Rξ ;(c) a morphism σ : OC → L;such that e(σr) = δξ .2. M(1/r) is the substack ofM[r] containing families ζ = (ξ, L, e) where ξ, L ande are as above and there exists a morphism σ : OC → L as above.3. M[1/r] is the substack of Totpi∗L containing families χ = (ξ, L, e, σ) where ξ,L, e and σ are as above.These three stacks are related by the following diagram where the horizontal arrowsare forgetful maps and the vertical arrows are inclusions.M[ 1r ]//M( 1r )//M 1rTotpi∗L //M[r] //MrAfter pulling back toM[r], the canonical ramification section δ : OCr → R and theuniversal rth root e : Lr ∼→ R define a natural inclusion:i′ : M[r] −→ Totpi∗Lrξ 7−→ ( ξ, e−1ξ (δξ) ). (I.12)M[1/r] now fits into the following cartesian diagram defining ν :M[ 1r ] i //νTotpi∗LτM[r]i′// Totpi∗Lr3. BRANCHING AND RAMIFICATION OF r-STABLE MAPS 19Lemma 2.2.4 shows thatM[1/r] is a proper DM stack. We have thatM(1/r) is thequotient ofM[1/r] by the action of Z/r, showingM(1/r) is a closed substack ofM[r].Also, since M[r] → Mr is proper we can define M1/r to be the closed substackof Mr coming from the image of M(1/r). Hence we have proved theorem A aftercomposing withMr →M, which is flat and proper.Note that the forgetful mapM[1/r] →M(1/r) is étale of degree r. However, the mapM(1/r) →M1/r is more complicated and in general not étale. There are cases wherethe map is étale such as when the genus is zero, then the map is degree 1/r. Foranother example, consider the spaceM(1/r)g (P1, 1) where the map is étale of degreer2g−1.3 Branching and Ramification of r-Stable MapsSection 3 Notation: Consider the universal objects ofM[1/r]:1. The universal curve, ρ : C[1/r] →M[1/r]2. The universal stable map, f : C[1/r] → X and F : C[1/r] → X ×M[1/r]3. The universal canonical section, δ : OC[1/r] → R4. The universal rth root of δ, (L, e : Lr ∼→ R,σ : OC[1/r] → L)For a family ξ over S inM[1/r] we will use the following notationC := C[1/r]ξ , ρ := ρξ , f := fξ , F := Fξ , δ := δξ , L := Lξ , e := eξ and σ := σξ .We denote the expected number of order r ramification loci in the generic case bym = 1r (2g− 2−d(2gX − 2)) in the caseM =Mg(X, d) and m = 1r (2g− 2 + l(µ) +|µ|(1 − 2gX)) in the caseM = Mg(X,µ). For a morphism of sheaves a : A → B,we will denote the associated complex in degree [−1, 0] by [a : A→ B].3.1 Divisor Construction and the Branch Morphism for Stable MapsAs we saw in the introduction and discussed in 1.2 the ramification divisor is not welldefined for stable maps. However it is possible to define a branch divisor using thecanonical ramification section defined in 1.2. To do this, we must first review a con-struction of Mumford [MFK, §5.3] which allows us to assign a Cartier divisor to certaincomplexes of sheaves.Let Z be a scheme and recall that a complex of sheaves E• is torsion if the supportof each Hi(E•) does not contain any of the associated points of Z . Let E• be afinite torsion complex of free sheaves on Z and let U ⊂ Z be the complement of⋃i SuppHi(E•). Then E•|U is exact and U contains all the associated points of Z .There are two ways to construct isomorphismsdetE•|U ∼−→ OU .1. κ: This is a canonical isomorphism which arises from the exactness of E•|U .2. Ψ: Which is from an explicit choice of isomorphism Ei∼−→ OU for each i.20 CHAPTER I. STABLE MAPS WITH DIVISIBLE RAMIFICATIONSo a choice of Ψ defines a section Ψ ◦ κ−1 ∈ H0(U,O∗U ). Also, it is shown in [FP,Lemma 1] that if U contains all the associated points of Z then a section of H0(U,O∗U )defines a canonical section λ of H0(Z,K∗). A different choice of Ψ amounts tomultiplication of λ by an element of H0(Z,O∗Z). In this way E• defines an elementof H0(Z,K∗/O∗Z). This construction also holds when E• is a perfect complex (i.e.locally isomorphic to a finite complex of locally free sheaves).Definition 3.1.1. Let E• be a perfect torsion complex. The divisor associated to E•is the divisor constructed above and is denoted by div(E•).The divisor construction has the following important properties.Lemma 3.1.2. [FP, Prop 1] Let E• be a perfect torsion complex.1. div(E•) depends only on the isomorphism class of E• in the derived category of Z .2. If F is a coherent sheaf admitting a finite free resolution F •, then we have div(F ) :=div(F •) is an effective divisor.3. If D is an effective divisor then div(OD) = D.4. The divisor construction is additive on distinguished triangles.5. If h : Z ′ → Z is a base change such that h∗E• is torsion, then we have div(h∗E•) =h∗div(E•).6. If L is a line bundle then div(E• ⊗ L) = div(E•).The divisor construction is used in [FP] to construct a morphismbr : M −→ Symm′X.where m′ is the virtual dimension ofM. In particular, if ζ ∈ M is a family of stablemaps over S and pr2 : X×S → S is the projection, then they show that the canonicalramification section (see 1.2) defines a pr2-relative effective Cartier divisor of degreem′:Bζ := div(R(Fζ)∗[OCζδζ−→ Rζ ]).Hence the map br is defined by ζ 7→ Bζ .Remark 3.1.3. In [FP] the relative caseM =Mg(X,µ) is not considered. However,the results and proofs required to define br work in this case when we use the sectionδ constructed in section 1.2 and m′ = 2g − 2 − d(2gX − 2) is replaced by m′ =2g − 2 + l(µ) + |µ|(1− 2gX).3.2 A Branch Morphism for Maps with Divisible RamificationWe will show in this section that a branch morphism can be constructed for stablemaps with divisible ramification. The role of the canonical ramification section will bereplaced by its universal rth root.Lemma 3.2.1. The direct image RF∗[OC σ−→ L] is a perfect torsion complex.Proof. Recall that F factors via the forgetful map to the coarse space F = F ◦γ whereF := (f, ρ). Also recall that γ∗ is an exact functor, so we have the quasi-isomorphismRF∗[OC σ−→ L] ∼= RF ∗[OCγ∗σ−→ γ∗L]. F has finite tor-dimension, so RF ∗[γ∗σ] isquasi-isomorphic to a finite complex of quasi-coherent sheaves on X × S flat over S.Denote this complex by E•.3. BRANCHING AND RAMIFICATION OF r-STABLE MAPS 21Perfect is a local property so we can assume that S = SpecA. Also, let pr1 : X ×S → X and pr2 : X × S → S be the natural projections. Thus we have thatM := pr∗1OX(1) is an ample line bundle on the fibres of pr2. Then for sufficientlylarge n we have for each Ei the following properties:1. Ai := ρ∗ρ∗(Ei ⊗Mn)⊗M -n is locally free.2. The natural map ai : Ai −→ Ei is surjective.Let Ki = ker ai and note that these sheaves are all flat over S. Hence, restricting tothe fibres of s ∈ S we have an exact sequence0 −→ (Ki)s −→ (Ai)s ai−→ (Ei)s −→ 0.We have pr2 is smooth of relative dimension 1 so any module on the fibres has ho-mological dimension at most 1. Thus showing that (Ki)s is locally free and henceKi is locally free. A finite complex of locally free sheaves quasi-isomorphic to E•can be constructed from the total complex associated to the double complex of theseresolutions.By [FP, Lemma 5] we can show RF∗[σ] is torsion on X × S by showing is when Sis a point. Define Y ⊂ C to be the locus where f is not étale and Z = f(Y ) ⊂ C .Note that Z is a finite collection of points in X . Define C˜ = C \ Y with inclu-sion j : C˜ → C and X˜ = X \ Z with inclusion i : X˜ → X . We also have thatY = Supp(kerσ) ∪ Supp(cokerσ) so [j∗σ] is exact. Letting f˜ = f |C˜ be the restric-tion we have i∗Rf∗[σ] = Rf˜∗[j∗σ]. Hence, i∗Rf∗[σ] is exact also, showing that thecohomology of Rf∗[σ] is supported on points.Lemma 3.2.2. Let S = SpecA be Noetherian and let E be a line bundle on C . Thereexists a ρ-relative line bundleM on C such that H0(C,M) and H0(C,M ⊗E) containsections which define injective morphisms.Proof. Let G be the bundle ωρ ⊗ f∗OX(3) which is an ample ρ-relative line bundleon C . Let n ∈ N be large enough that both GN and E ⊗GN are generated by globalsections. We claim that M := γ∗GN has the desired properties. To see this notethat C is quasi-compact and so has a finite number of associated points. A standardargument then shows that the subspaces of H0(C,GN ) and H0(C,E ⊗ GN ) whichare not injective morphisms will then have strictly lower dimension. Then consider theisomorphismγ∗ : H0(C,E ⊗M) ∼−→ H0(C,E ⊗GN )and consider the pre-image s of a regular section s ∈ H0(C,E ⊗GN ). γ∗ is an exactfunctor and γ∗K = 0 if and only if K = 0. Hence we have that s is injective if andonly if s is.Lemma 3.2.3. Let M˜ be a relative line bundle on C and an injective morphism s : OC →M˜ . Let D be the divisor on C defined by s∨. ThenD := div(RF∗[OD ⊗OC id⊗σk−→ OD ⊗ Lk]) = 0.Proof. We first show that D is an effective divisor on X × S by considering the casewhere S = SpecA. Note that the map forgetting the stack structure γ : C → C hasthe property that γ∗ is left exact. Also, OD is supported in relative dimension 0, so Dis given byD = div(F ∗[OD −→ γ∗(OD ⊗ Lk)]).22 CHAPTER I. STABLE MAPS WITH DIVISIBLE RAMIFICATIONWe have that i : D → C is a relative effective divisor on C with coarse spacej : D → C . The natural map φ : D → S is quasi-finite and proper so it is alsofinite. Thus D is affine which shows that (γ|D)∗(OD ⊗ Lk) is generated by globalsections and so has sections which give injective morphism.Let Φ be a section of (γ|D)∗(OD ⊗ Lk) giving rise to an injective morphism. Thensince the divisor construction is additive on distinguished triangles we have:D = div(F ∗j∗[Φ])Also, Φ is regular so it is injective and we have D = div(F ∗j∗coker Φ). Hence showingthat it is a relative effective Cartier divisor.The degree of a relative effective Cartier divisor for a smooth morphism is locallyconstant. Hence we can compute the degree at geometric points. We see that thedegree of (D)z is zero for geometric points z ∈ S.Corollary 3.2.4. There is an equality of divisorsdiv(RF∗[OC σk→ Lk]) = div(RF∗(L⊗ [OC σk→ Lk]))Proof. We have that div is additive on exact sequences. So, to show that two sequencesgive the same divisor, it will suffice to show that the cone of a morphism between thetwo complexes is the zero divisor.We have two distinguished triangles coming from injective sections ofM andM⊗L−1,where M is the line bundle from lemma 3.2.2:[σk]s1−→ [σk]⊗M −→ Cone(s1) −→ [σk][1][σk]⊗ L s2−→ [σk]⊗M −→ Cone(s2) −→ [σk][1]We saw in the lemma 3.2.3 that div(Cone(s1)) = div(Cone(s2)) = 0 which showsthat [σk] and [σk]⊗ L have the same divisor.Lemma 3.2.5. Let E• a→ G• b→ H• be morphisms in the derived category. Then there isa distinguished triangle:cone(a) −→ cone(b ◦ a) −→ cone(b) −→ cone(a)[1]Proof. The result follows immediately from the following commuting diagram withdistinguished triangles for rows and columns:E• id //aE• //b◦a0 //E•[1]a[1]G• b //H• //cone(b) //G•[1]cone(a) //cone(b ◦ a) //cone(b) //cone(a)[1]E•[1]id[1] // E•[1] // 0 // E•[2]3. BRANCHING AND RAMIFICATION OF r-STABLE MAPS 23Corollary 3.2.6. Let a : E → G and b : G→ H be morphisms of coherent sheaves. Thenthere is a distinguished triangle:[Ea−→ G ] −→ [E b◦a−→ H ] −→ [G b−→ H ] −→ [E a−→ G ][1]where [a], [b] and [b ◦ a] are considered to be in degree [−1, 0].Proof. The proof is immediate from lemma 3.2.5.Corollary 3.2.7. We have the equality of divisors on X × S:div(RF∗[OC σr−→ Lr])= r · div(RF∗[OC σ−→ L])Proof. From corollary 3.2.6 we have the distinguished triangle:[OC σ−→ L ] −→ [OC σn+1−→ Ln+1 ] −→ L⊗ [OC σn−→ Ln ] −→ [OC σ−→ L ][1].After applying corollary 3.2.4 this shows that div(RF∗[σn+1])= div(RF∗[σn])+div(RF∗[σ]). The result follows from the induction hypothesis.Corollary 3.2.8. (Theorem B) The divisor div(RF∗[σ]) is relative effective and the asso-ciated morphism bξ : S → Symm(X) defines a morphism of stacks:br : M[1/r] −→ Symm(X)ξ 7−→ bξ.which satisfies the following commutative diagram:M[1/r] br //χSymmX∆M br // SymrmXProof. We have a natural quasi-isomorphism [σr]∼→ [δ]. It is shown in [FP, 3.2] thatdiv(RF∗[δ]) is a relative effective divisor of degree rm. Hence, corollary 3.2.7 showsthat div(RF∗[σ]) is relative effective as well and is of degree m. Corollary 3.2.7 alsoshows that the given diagram is commutative.3.3 Special Loci of the Moduli PointsIn this subsection we will prove theorem B part 1 by considering the case whenS = SpecC and examining the ramification properties induced by the rth root con-dition.Following [V, GV] we will call a special loci a connected component where the mapf : C → X is not étale. Then each special locus is one of:1. A smooth point of C where f is locally of the form z 7→ za+1 with a ∈ N.2. A node of C such that on each branch f is locally of the form z 7→ zai withai ∈ N.3. A genus g component B of C where f |B is constant and on the branches of Cmeeting B the map f is locally of the form z 7→ zai with ai ∈ N.We can also define a ramification order to each type of locus by:24 CHAPTER I. STABLE MAPS WITH DIVISIBLE RAMIFICATION1. a. (2) a1 + a2. (3) 2gB − 2 +∑(ai + 1).Remark 3.3.1. We use a slightly different definition for stable maps relative to a pointx ∈ X . Let (h : C → T, p→ X) be over S = SpecC inM[1/r]g (X,µ) with f = p ◦ h.Then a special locus of f will be a connected component where the map h : C → Tis not étale and not in the pre-image of a node of x. Everything else is the same. Thisagrees with lemma 1.2.1 which shows that δ will be an isomorphism at pre-images ofnodes of T .We will show that the existence of an rth root of δ is equivalent to each of these specialloci having ramification order divisible by r.Suppose we have ξ ∈M1/r over S = SpecC. Then locally on the coarse space C foreach of the types of special loci δ : OC → Rξ is of the form:1. C[x]→ 1xaC[x] given by a 7→ axaxa .2. C[x, y]/(xy)→ 1xa1−xa2 C[x, y]/(xy) given by a 7→ axa1−xa2xa1−xa2 .3. At each node C[x, y]/(xy)→ 1xai−xbi C[x, y]/(xy) given by a 7→ a xaixai−xbi .3.3.2. The rth root condition σr = e(δ) forces there to be local roots for special lociof types 1 and 2. This forces the divisibility of the ramification order:For type 1: Locally we must have σ being of the form C[x]→ 1xa/rC[x] and thusr divides a.For type 2: Pulling back from the coarse space via γ we see that δ is of theform C[u, v]/(uv) → 1ua1r−va2rC[u, v]/(uv). Then taking the rth root we seethat σ is of the form C[u, v]/(uv)→ ζkrua1−ζrva2 C[u, v]/(uv) for some k ∈ Z/r.However, there are multiplicities e1 and e2 of L at the node with e1 + e2 = r ore1 = e2 = 0. Also we have ai = ei + nir. Hence, r divides a1 + a2.3.3.3. We now consider special loci of type 3. Suppose there is a genus g sub-curve Bof C where f |B is constant and on the branches of C meeting B, the map f is locallyof the form z 7→ zai with ai ∈ N.Let A = C \ B and α : A unionsq B → C be the partial normalisation of C separating thecontracted component B from A. Also, let pi be the pre-images of the nodes on A andqi the pre-images on B. Finally, let ai and bi be the multiplicities of L correspondingto the branches on the nodes on A and B respectively.Now e restricts to an isomorphism eB : (LB)r∼−→ (Rξ)B ∼= ωB(∑qi). We have amap g : B → B which forgets the orbifold structure at the points qi. Pushing forwardvia g we have the following isomorphism coming from lemma 1.4.3:eB : (LB)r ∼−→ ωB(∑qi −∑biqi)Hence, ωB(∑qi −∑biqi) must have degree divisible by r. Then r divides 2g − 2 +∑(1− bi) and also divides 2g − 2 +∑(ai + 1).Remark 3.3.4. To consider the relative case in 3.3.3 we must replace f : C → X byh : C → T . Everything else remains the same.4. COTANGENT COMPLEX OFM1/rg (X, d) 253.3.5. To finish the proof of the theorem we to show that such an rth root can beconstructed if the ramification loci are of the desired form. First we observe that thereis a tensor product decomposition of δ:δ = δsm ⊗ δn ⊗ δcnwhere δsm : OC → Rsm and δn : OC → Rn define the divisors of δ supported onthe smooth locus of C and nodes of C not meeting contracted components. Thenδcn : OC → Rcn is the unique section such that the above decomposition holds.After reversing the reasoning of 3.3.2 we have rth roots σsm : OC → Lsm andσn : OC → Ln of δsm and δn respectively. For the contracted components δcn the linebundle Rcn will locally be of the form 1uair−vbirC[u, v]/(uv) at the connecting nodesand δcn will be of the form 1 7→ uairuair−vbir . Then let Lcn be an rth root of Rcn whichis locally of the form 1uai−ζrvbi C[u, v]/(uv) at the connecting nodes and σcn will beof the form 1 7→ uaiuai−vbi and identical to δcn elsewhere.Hence we have proved theorem B part 1.4 Cotangent Complex ofM1/rg (X, d)Section 4 Notation: Recall the notation convention. The following diagram showsthe relationships between the relevant spaces. It is commutative and many of thesquares are cartesian.Totpi∗Lr CTotpi∗Lr TotLrM[r] C[r] C[r]Totpi∗L CTotpi∗L TotLM[r] C[r] C[r]M[r] C[r]M[ 1r ] C[ 1r ]ρψ epiβ β̂ βˇijpiϕ e′piα α̂ αˇi′ j′ν ν̂τ τ̂ τˇff′(I.13)Here pi and ρ are the universal curves of their respective spaces. The maps i and i′are the natural inclusions defined by definition 2.3.1.3 and equation (I.12) respectively.The maps ϕ and ψ are the universal curves defined in 2.2.1 with e and e′ being thenatural evaluation maps defined by equation (I.10). The power maps τ and τˇ aredefined in 2.2.3 and τ̂ is the pullback by ϕ of τ . The maps α, α̂, αˇ, β, β̂ and βˇ arethe natural projection maps. The maps j and j′ are pullbacks of i and i′ by ψ and ϕrespectively. Lastly, we also define the maps f := i ◦ e and f′ := i′ ◦ e′.We denote the expected number of special loci of order r in the generic case bym = 1r (2g− 2−d(2gX − 2)) in the caseM =Mg(X, d) and m = 1r (2g− 2 + l(µ) +|µ|(1− 2gX)) in the caseM =Mg(X,µ).26 CHAPTER I. STABLE MAPS WITH DIVISIBLE RAMIFICATION4.1 Perfect Relative Obstruction TheoryRecall, that for a proper representable Gorenstein morphism a : X → Y of relativedimension n with relative dualising sheaf ωa and any complexes F• ∈ D(X ) andG• ∈ D(Y) one has the following functorial isomorphism coming from Serre duality(see for example [BBH, eq. C.12]):HomD(X )(Ra∗F•,G•) ∼−→ HomD(Y)(F•, a∗G• ⊗ ωa[n]). (I.14)Hence one obtains the following natural morphism by looking at the pre-image of theidentity:Ra∗(a∗G• ⊗ ωa)[n] −→ G•. (I.15)Now, consider the following sub-diagram of (I.13) coming from the topmost horizontalsquare and the diagonal square:M[1/r]νC[1/r]ρooν̂f // TotLτˇM[r] C[r]pioo f′// TotLrThere are two natural maps arising from this diagram:Rρ∗(ρ∗Lν ⊗ ωρ)[1] −→ Lν and Lf∗Lτˇ −→ Lν̂ ∼= ρ∗Lν .Combining these two we can we define the following morphism:φν : Rρ∗(Lf∗Lτˇ ⊗ ωρ)[1] −→ Lν .We will show in this subsection that this morphism is a perfect relative obstructiontheory.We will begin by examining a related morphism constructed in the same way. Specif-ically, we consider the the following sub-diagram of (I.13) coming from the middlehorizontal squares:Totpi∗LτCTotpi∗Lψooτ̂e // TotLτˇTotpi∗Lr CTotpi∗Lrϕoo e′// TotLr(I.16)As before we have two natural mapsRψ∗(ψ∗Lτ ⊗ ωψ)[1] −→ Lτ and Le∗Lτˇ −→ Lτ̂ ∼= ψ∗Lτ .which combine to obtain the morphism:φτ : Rψ∗(Le∗Lτˇ ⊗ ωψ)[1] −→ Lτ .The following lemma shows that φτ is a relative obstruction theory.Lemma 4.1.1. There is a commuting diagram where the rows are distinguished triangles:Lτ∗Rϕ∗(α̂∗L-r ⊗ ωϕ)[1] //Lτ∗φαRψ∗(β̂∗L-1 ⊗ ωψ)[1] //φβRψ∗(Le∗Lτˇ ⊗ ωψ)[1]φτ// Lτ∗Rϕ∗(α̂∗L-r ⊗ ωϕ)[2]φτ [1]Lτ∗Lα // Lβ // Lτ // Lτ∗Lα[1]such that φβ and φα are relative obstruction theories. Moreover, φτ is also a relativeobstruction theory.4. COTANGENT COMPLEX OFM1/rg (X, d) 27Proof. Consider the leftmost square of (I.16) and note that it is cartesian. The dis-tinguished triangle arising from the cotangent complex gives the following diagramwhere the rows are distinguished triangles:Rψ∗(ψ∗Lτ∗Lα ⊗ ωψ)[1] //Rψ∗(ψ∗Lβ ⊗ ωψ)[1] //Rψ∗(ψ∗Lτ ⊗ ωψ)[1]// Rψ∗(ψ∗Lτ∗Lα ⊗ ωψ)[2]Lτ∗Lα // Lβ // Lτ // Lτ∗Lα[1](I.17)We also have isomorphisms:Rψ∗(ψ∗Lτ ∗Lα ⊗ ωψ) ∼= Rψ∗Lτ̂ ∗(ϕ∗Lα ⊗ ωϕ) ∼= Lτ ∗Rϕ∗(ϕ∗Lα ⊗ ωϕ)making the first column into the derived pullback of the canonical morphism fromequation (I.15):Rϕ∗(ϕ∗Lα ⊗ ωϕ)[1] −→ Lα.Now consider the rightmost square of (I.16) and note that it has all morphisms overC. This gives the following commutative diagram with distinguished triangles as rows,noting that Lτ̂ ∗Le′∗Lαˇ ∼= Le∗Lτˇ ∗Lαˇ:Lτ̂ ∗Le′∗Lαˇ //Le∗Lβˇ //Le∗Lτˇ// Lτ̂ ∗Le′∗Lαˇ[1]Lτ̂ ∗Lα̂ // Lβ̂ // Lτ̂ // Lτ̂∗Lα̂[1]Lτ̂ ∗ϕ∗Lα //∼=OOψ∗Lβ //∼=OOψ∗Lτ∼=OO// Lτ̂ ∗ϕ∗Lα[1]∼=OO(I.18)Also, note that Le∗Lβˇ ∼= Le∗(βˇ∗L∨) ∼= L(βˇ◦e)∗L∨ ∼= β̂∗L∨ and similarly, Le′∗Lαˇ ∼=α̂∗(Lr)∨. We now obtain the desired diagram by combining (I.18) with (I.17). We de-note the appropriate morphisms by φτ , φβ and φα. It is shown in [CL, Prop. 2.5] thatφβ and φα are perfect relative obstruction theories.To show that φτ is an obstruction theory it will suffice to show that H-1(cone(φτ )) =H0(cone(φτ )) = 0. We have that β : Totpi∗Lr →M is representable, so H1(Lβ) =0 andHi(φβ) = 0 for all i ≥ −1. Also, cone(φα) is quasi-isomorphic to a flat complexF• which is zero in all degrees greater than −2. Now by definition Lτ ∗cone(φα) =τ ∗F• also vanished in degrees greater than −2, making Hi(Lτ ∗φα) = 0 for alli ≥ −1. The result now follows from taking the cohomology exact sequence of thedistinguished triangle of the cones:H-1(cone(φβ)) // H-1(cone(φτ )) // H0(cone(Lτ ∗φα))// H0(cone(φβ)) // H0(cone(φτ )) // H1(cone(Lτ ∗φα)).Lemma 4.1.2. φν is the composition of Li∗φτ and the natural differential morphismLi∗Lτ → Lν . In particular, φν is a relative obstruction theory.28 CHAPTER I. STABLE MAPS WITH DIVISIBLE RAMIFICATIONProof. There is a commuting diagram, noting that there is an isomorphism Lj∗ψ∗Lτ ∼=ρ∗Lj∗Lτ :Lj∗Le∗Lτˇ // Lj∗Lτ̂Lj∗ψ∗Lτ∼=ooLf∗Lτˇ // Lν̂ ρ∗Lν∼=oowhich gives the left square of the following diagram after applying the functor Rρ∗( _⊗ωρ)[1] and using the isomorphism of functors Rρ∗(Lj∗ _ ⊗ ωρ)[1] ∼= Li∗Rψ∗( _ ⊗ωψ)[1] :Li∗Rψ∗(Le∗Lτˇ ⊗ ωρ)[1] //∼=Li∗Rψ∗(ψ∗Lτ ⊗ ωψ)[1]// Li∗LτRρ∗(Lf∗Lτˇ ⊗ ωρ)[1] // Rρ∗(ρ∗Lν ⊗ ωρ)[1] // LτˇNow, Li∗φτ is the composition of the top row and φν is the composition of the bottomrow. Hence, φν is the composition of the desired morphisms.The maps i and i′ are immersions and τ and ν are representable, so Li∗Lτ → Lνis a relative obstruction theory (see for example [BF, §7]). We now consider the dis-tinguished triangle of cones coming from composition of lemma 3.2.5. The reasoningthat φν is a relative obstruction theory is now the same as for φτ in the previouslemma, lemma 4.1.1.Lemma 4.1.3. The left derived pullback by f of the map τˇ ∗Lαˇ → Lβˇ is the map:r j∗σr−1 : j∗β̂∗L-r −→ j∗β̂∗L-1where σ is the universal rth root.Proof. It will suffice to show this locally. The local situation is described by the diagramA1 × U //t[A1 × U/G] //TotLτˇA1 × U //a[A1 × U/G] //TotLrαˇU // [U/G] // C[r]where U = SpecB, G is a finite group and t is defined by the morphism of B-algebrasB[z]→ B[w] with z 7→ wr . On U we have that L is given by an equivariant line bun-dle E defined by a local generator φ. Then Lr corresponds to Er with φr .We have that Lβˇ ∼= Ωβˇ ∼= βˇ∗L-1 and the last morphism is defined locally by dw 7→ 1φ .Similarly, Lαˇ ∼= Ωαˇ ∼= αˇ∗L-r is defined by dz 7→ 1φr and the map dτˇ is defined by:dt : B[w]⊗A[z]dz −→ A[w]dw1⊗ dz 7−→ rwr−1dw.Let σB ∈ B be the pullback of σ to U . Locally on U the map j is defined by thesection σB via the following map B[w]→ B : w 7→ σB . Hence, pulling back via j wehave that the map j∗dτˇ is locally defined by:j∗dt : B 1φr −→ B 1φ1φr 7−→ r(σB)r−1 1φ .4. COTANGENT COMPLEX OFM1/rg (X, d) 29Theorem 4.1.4. The map φν : Rρ∗(Lf∗Lτˇ ⊗ ωρ)[1] −→ Lν is a perfect relative ob-struction theory with relative virtual dimension (1 − r)m. (Note that rm is the virtualdimension ofM[r]. This follows from the discussion after this theorem.)Proof. Let ρ : C → S be a family in M[ 1r ]. In the derived category we have thefollowing isomorphismsLf∗Lτˇ ⊗ ωρ ∼= Lf∗([τˇ ∗Lαˇ −→ Lβˇ]⊗ ωρ) ∼= [ β̂∗L-r ⊗ ωρ −→ β̂∗L-1 ⊗ ωρ ].Denote the restriction to S of this complex byE• = [E−1θ−→ E0] = [f∗ωX ⊗OC id⊗σr−1−→ f∗ωX ⊗ Lr−1] (I.19)with the last equality following from lemma 4.1.3. Let M be a line bundle on C whichis ample on the fibres of ρ. Then for sufficiently large n we have for each Ei thefollowing properties:1. ρ∗ρ∗(Ei ⊗Mn)⊗M -n −→ Ei is surjective.2. R1ρ∗Ei ⊗Mn = 0.3. For all z ∈ S we have H0(Cz, ρ∗ρ∗(Ei ⊗Mn)⊗M -n) = 0.Denote the locally free sheaf ρ∗ρ∗(E0⊗Mn)⊗M−n by AE0 and the associated mapfrom property 1 above by a. Then using the fibre product for modules we have thefollowing commuting diagram with exact rows0 // ker(a) // G //θ˜E-1 //θ00 // ker(a) // AE0a // E0 // 0(I.20)where G also fits into the exact sequence:0 // G // E-1 ⊕AE0(-θa) // E0 // 0. (I.21)The diagram in (I.20) shows that there is an isomorphism [Gθ˜−→ AE0 ] ∼= [E-1 θ−→ E0]in the derived category.The exact sequence in (I.21) shows that G is locally free and hence the diagram in(I.20) contains only flat modules. Hence for z ∈ S we may restrict to the fibre Cz andmaintain exactness. Then using the snake lemma we have an isomorphismker θz ∼= ker θ˜z.We claim that H0(Cz, ker θz) = 0. To see this take s ∈ H0(Cz, ker θz) and note thats ∈ H0(Cz, E-1) and s is in the kernel of θz . From (I.19) we know that (E-1)z = f∗zωXand θz = σr−1z , so θz only vanishes where fz is constant. Hence, we let B ⊂ Czbe the union of components contracted by fz . Then we have Supp s ⊂ B and(E-1)z|B = (f∗zωX)|B ∼= OB so we must have s = 0. Hence, H0(Cz, ker θz) =H0(Cz, ker θ˜z) = 0.30 CHAPTER I. STABLE MAPS WITH DIVISIBLE RAMIFICATIONFrom property 3 of the definition of AE0 we have H0(Cz, (AE0)z)∼= 0. So thefollowing exact sequence shows that H0(Cz, Gz) = 0:0 −→ H0(Cz, ker θ˜z) −→ H0(Cz, Gz) −→ H0(Cz, (AL˜)z)Hence, we have that R1ρ∗G is locally free and Rρ∗G ∼= [R1ρ∗G][−1]. MoreoverRρ∗[Gθ˜−→ AL˜] ∼= [R1ρ∗GR1ρ∗θ˜−→ R1ρ∗AL˜][−1] is a complex of locally free sheavesconcentrated in degree [0, 1].The virtual dimension follows immediately from Riemann-Roch for twisted curves (see[AGV, §7.2]) applied to E•.The spaceM[r] has a natural perfect obstruction theory following from the construc-tion of [B2, L2]. It will suffice to show that the space of r-stable maps Mr has aperfect obstruction theory since by lemma 2.1.2 the mapM[r] →Mr is étale. Recallthe construction of the perfect obstruction theory for the moduli space of stable mapsM is defined via the relative to the forgetful morphismMg(X, d) −→Mg.It is pointed out in [GV, §2.8] that in the case of relative stable maps the perfectobstruction theory can be constructed relative to the morphismMg(X,µ) −→Mg,l(µ) × TXwhere TX is the moduli space parameterising the degenerated targets. We have thefollowing two cartesian squares where the bottom arrows are flat:Mrg(X, d) //pabsMg(X, d)Mrg //MgMrg(X,µ) //prelMg(X,µ)Mrg,l(µ) × TX //Mg,l(µ) × TXWe let p : Mr → X be one of pabs or prel maps depending on the choice ofMr . Then we have a natural perfect relative obstruction for p by pulling back viaMr →M:φp : E•p −→ Lp.Then a perfect obstruction theory forMr is given by the following cone construction:E•p[−1] //φp[−1]p∗LX // F •Mm //φE•pφpLp[−1] // p∗LX // LMr // LνCorollary 4.1.5. (Theorem C) If g = 0 there is a perfect obstruction theory for M[1/r]giving virtual dimension m. Moreover, sinceM[1/r] →M1/r is étale in genus 0, there isa perfect obstruction theory forM1/r .Proof. Let E• = Rρ∗(Lf∗Lτˇ ⊗ ωρ). Then there the following is a commutativediagram with distinguished triangles for rows:E•[−1] //φν [−1]ν∗LM[r] // F • //φE•φνLν [−1] // ν∗LM[r] // LM[1/r] // Lν4. COTANGENT COMPLEX OFM1/rg (X, d) 31Here F • is defined via the cone construction. As before we have an exact sequence ofcohomology of the cones:H-1(cone(id)) // H-1(cone(φ)) // H-1(cone(φν))// H0(cone(id)) // H0(cone(φ)) // H0(cone(φν)).Which shows that H-1(cone(φ)) = H0(cone(φ)) = 0.32 CHAPTER I. STABLE MAPS WITH DIVISIBLE RAMIFICATIONChapter IIThe Donaldson-ThomasTheory of the BananaThreefold with Section Classes1 Introduction1.1 Donaldson-Thomas Partition FunctionsDonaldson-Thomas theory provides a virtual count of curves on a threefold. It givesus valuable information about the structure of the threefold and has strong links tohigh-energy physics.For a non-singular Calabi-Yau threefold Y over C we letHilbβ,n(Y ) ={Z ⊂ Y∣∣∣ [Z] = β ∈ H2(Y ), n = χ(OZ)}be the Hilbert scheme of one dimensional proper subschemes with fixed homologyclass and holomorphic Euler characteristic. We can define the (β, n) Donaldson-Thomas invariant of Y by:DTβ,n(Y ) = 1 ∩ [Hilbβ,n(Y )]vir.Behrend proved the surprising result in [B1] that the Donaldson-Thomas invariantsinvariants are actually weighted Euler characteristics of the Hilbert scheme:DTβ,n(Y ) = e(Hilbβ,n(Y ), ν) :=∑k∈Zk · e(ν−1(k)).Here ν : Hilbβ,n(Y ) → Z is a constructible function called the Behrend function andits values depend formally locally on the scheme structure of Hilbβ,n(Y ). We alsodefine the unweighted Donaldson-Thomas invariants to be:D̂Tβ,n(Y ) = e(Hilbβ,n(Y )).These are often closely related to Donaldson-Thomas invariants and their calculationprovides insight to the structure of the threefold. Moreover, many important prop-erties Donaldson-Thomas invariants such as the PT/DT correspondence and the flopformula also hold for the unweighted case [T1, T2].3334 CHAPTER II. DT THEORY OF THE BANANA 3-FOLD WITH SECTION CLASSESFigure II.1: A visual representation of the banana threefold. On the left the diagonalS∆ and the anti-diagonal Sop are highlighted. On the right the two rational ellipticsurfaces S1 and S2 are highlighted.The depth of Donaldson-Thomas theory is often not clear until one assembles theinvariants into a partition function. Let {C1, . . . , CN} be a basis for H2(Y,Z), chosenso that if β ∈ H2(Y,Z) is effective then β = d1C1 + · · · + dNCN with each di ≥ 0.The Donaldson-Thomas partition function of Y is:Z(Y ) :=∑β∈H2(Y,Z)∑n∈ZDTβ,n(Y )Qβpn=∑d1,...,dN≥0∑n∈ZDT(∑i diCi),n(Y )Qdii pn.We also define the analogous partition function Ẑ for the unweighted Donaldson-Thomas invariants.Remark 1.1.1. This choice of variable is not necessarily the most canonical as shownin [Br] where the variable p is substituted for −p. However, in this chapter we willbe focusing on the unweighted Donaldson-Thomas invariants where this choice makesthe most sense.This partition function is very hard to compute and for proper Calabi-Yau threefolds,the only known complete examples are in computationally trivial cases. However,when we restrict our attention to subsets of H2(Y,Z) there are many remarkable re-sults. Two case which we will be related to computations are the Schoen (Calabi-Yau)threefold of [S] and the banana (Calabi-Yau) threefold of [Br].We will employ computational techniques developed in [BK] for studying Donaldson-Thomas theory of local elliptic surfaces.1.2 Donaldson-Thomas Theory of Banana ThreefoldsThe banana threefold is of primary interest to us and is defined as follows. Letpi : S → P1 be a generic rational elliptic surface with a section ζ : P1 → S. We willtake S to be P2 blown-up at 9 points which gives rise to 9 natural choices for ζ . Theassociated banana threefold is the blow-upX := Bl∆(S ×P1 S) (1)1. INTRODUCTION 35Figure II.2: On the left is a visual representation of the rational elliptic surfaces S1, S2and Sop. On the right is the diagonal surface S∆. Note that the exceptional curves inthe fibres of the pencil have order 2.where ∆ is the diagonal divisor in S×P1 S. The surface S is smooth but the morphismpi : S → P1 is not. It is singular at 12 points of S and this gives rise to 12 conifoldsingularities of S ×P1 S that all lie on the divisor ∆. This makes X a conifold resolu-tion of S ×P1 S. It is a non-singular simply connected proper Calabi-Yau threefold asshown in [Br, Prop. 28].The section ζ : P1 → S gives a section σ : P1 → X of the natural map pr : X → P1.It also gives natural sections of the projections pri : X → S which we denote by S1and S2. These are both divisors of X that are copies of the rational elliptic surface.The diagonal ∆ and anti-diagonal ∆op of S ×P1 S are also divisors which are copiesof S. The anti-diagonal intersects the diagonal in a curve on ∆op, so it is unaffectedby the blow-up. We denote the anti-diagonal divisor in X by Sop and the propertransform of the diagonal by S∆. The latter is a rational elliptic surface blown-up atthe 12 nodal points of the fibres.The generic fibres of the map pr : X → P1 are Abelian surfaces of the form E × Ewhere E = pi−1(x) is an elliptic curve that is the fibre of a point x ∈ P1. The projec-tion map pr also has 12 singular fibres which are non-normal toric surfaces. They areeach compactifications of C∗ ×C∗ by a banana configuration and their normalisationsare isomorphic to P1 × P1 blown up at 2 points [Br, Prop. 24].Definition 1.2.1. A banana configuration is a union of three curves C1∪C2∪C3 whereCi ∼= P1 with NCi/X ∼= O(−1)⊕O(−1) and C1∩C2 = C1∩C3 = C2∩C3 = {z1, z2}where z1, z2 ∈ X are distinct points. Also, there exist formal neighbourhoods of z1and z2 such that the curves Ci become the coordinate axes in those coordinates. Welabel these curves by their intersection with the natural surfaces in X . That is C1 isthe unique banana curve that intersects S1 at one point. Similarly, C2 intersects S2and C3 intersects Sop.The banana threefold contains 12 copies of the banana configuration. We label theindividual banana curves by C(j)i (and simply Ci when there is no confusion). Thebanana curves C1, C2, C3 generate a sub-lattice Γ0 ⊂ H2(X,Z) and we can considerthe partition function restricted to these classes:ZΓ0 :=∑β∈Γ0∑n∈ZDTβ,n(X)Qβpn.In [Br, Thm. 4], this rank three partition function is computed to be:ZΓ0 =∏d1,d2,d3≥0∏k(1−Qd11 Qd22 Qd33 (−p)k)−12c(‖d‖,k) (2)36 CHAPTER II. DT THEORY OF THE BANANA 3-FOLD WITH SECTION CLASSESFigure II.3: On the left is a depiction of the banana configuration. On the right is thenormalisation of the singular fibre Fban = pr−1(x) with the restrictions of the surfacesS1, S2, Sop.where d = (d1, d2, d3) and the second product is over k ∈ Z unless d = (0, 0, 0) inwhich case k > 0. (Note the change in variables from [Br].) The powers c(‖d‖, k) aredefined by∞∑a=−1∑k∈Zc(a, k)Qayk :=∑k∈ZQk2(−y)k(∑k∈Z+ 12 Q2k2(−y)k)2 = ϑ4(2τ, z)ϑ1(4τ, z)2and ‖d‖ := 2d1d2 + 2d2d3 + 2d3d1 − d21 − d22 − d23.Remark 1.2.2. We can pass to the unweighted ẐΓ0 from the weighted partition func-tion ZΓ0 by the change of variables Qi 7→ −Qi and p 7→ −p.We can include the class of the section σ to generate a larger sub-lattice Γ ⊂ H2(X,Z).The partition function of this sub-lattice is currently unknown. The purpose of thischapter is to make progress towards understanding this partition function. We will becalculating the unweighted Donaldson-Thomas theory in the classes:β = σ + (0, d2, d3) := σ + 0C1 + d2 C2 + d3 C3,by computing the following the partition functionẐσ+(0,•,•) :=∑d2,d3≥0∑k∈ZD̂Tβ,n(Y )Qd22 Qd33 pn,which we give in terms of the MacMahon functions M(p,Q) =∏m>0(1 − pmQ)−mand their simpler version M(p) = M(p, 1).Theorem A The above unweighted Donaldson-Thomas functions are:Ẑσ+(0,•,•) is:Ẑ(0,•,•)(1− p)2∏m>01(1−Qm2 Qm3 )8(1− pQm2 Qm3 )2(1− p−1Qm2 Qm3 )2where Ẑ(0,•,•) is the Q01 part of the unweighted version of the Γ0 partition function(2) and is given by:M(p)24∏d>0M(Qd2Qd3, p)24(1−Qd2Qd3)12M(−Qd−12 Qd3, p)12M(−Qd2Qd−13 , p)12.1. INTRODUCTION 37The connected unweighted Pandharipande-Thomas version of the above formula isidentified as the connected version of the Pandharipande-Thomas theory for a rationalelliptic surface [BK, Cor. 2] in the following corollary.Corollary B The connected unweighted Pandharipande-Thomas partition function is:ẐPT,Conσ+(0,•,•) := log(Ẑσ+(0,•,•)Ẑ(0,•,•)|Qi=0)=−1(1− p)2∏m>0−1(1−Qm2 Qm3 )8(1− pQm2 Qm3 )2(1− p−1Qm2 Qm3 )2.We will also be computing the unweighted Donaldson-Thomas theory in the classes:β = b σ + (0, 0, d3), β = b σ + (0, 1, d3) and β = b σ + (1, 1, d3)and the permutations involving C1, C2. So for i, j ∈ {0, 1} we defineẐ•σ+(i,j,•) :=∑b,d3≥0∑k∈ZD̂Tβ,n(Y )QbσQd33 pn.The formulas will be given in terms of the functions which are defined for g ∈ Z:ψg = ψg(p) :=(p12 − p− 12)2g−2=(p(1− p)2)1−g.Theorem C The above unweighted Donaldson Thomas functions are:1. Ẑ•σ+(0,0,•) is:M(p)24∏m>0(1 + pmQσ)m(1 + pmQ3)12m.2. Ẑ•σ+(0,1,•) = Ẑ•σ+(1,0,•) is:Ẑ•σ+(0,0,•) ·((12ψ0 +Q3(24ψ0 + 12ψ1) +Q23(12ψ0))+QσQ3(12ψ0 + 2ψ1))3. Ẑ•σ+(1,1,•) is:Ẑ•σ+(0,0,•)·( ((144ψ-1 + 24ψ0 + 12ψ1) +Q3(576ψ-1 + 384ψ0 + 72ψ1 + 12ψ2)+Q23(864ψ-1 + 720ψ0 + 264ψ1 + 24ψ2)+Q33(576ψ-1 + 384ψ0 + 72ψ1 + 12ψ2) +Q43(144ψ-1 + 24ψ0 + 12ψ1))+Qσ((12ψ0 + 2ψ1)+Q3(288ψ-1 + 96ψ0 + 44ψ1)+Q23(576ψ-1 + 600ψ0 + 156ψ1 + 24ψ2)+Q33(288ψ-1 + 96ψ0 + 44ψ1)+Q43(12ψ0 + 2ψ1))+Q2σQ23(144ψ-1 + 48ψ0 + 4)).38 CHAPTER II. DT THEORY OF THE BANANA 3-FOLD WITH SECTION CLASSESThe connected unweighted Pandharipande-Thomas versions of the above formula con-tain the same information but are given in the much more compact form. In fact wecan present the same information in an even more compact form using the unweightedGopakumar-Vafa invariants n̂gβ via the expansionẐPT,ConΓ (X)=∑β∈Γ\{0}∑g≥0∑m>0n̂gβ ψg(pm) (−Q)mβ=∑b, d1, d2, d3 ≥ 0(b, d1, d2, d3) 6= 0∑g≥0∑m>0n̂g(b,d1,d2,d2) ψg(pm) (-Qσ)mb(-Q1)md1(-Q2)md2(-Q3)md3 .As noted before, these express the same information as the above generating functions.For β = (d1, d2, d3), these invariants are given in [Br, §A.5]. We present the newinvariants for β = bσ + (i, j, d3) where b > 0.Corollary D Let i, j ∈ {0, 1}, b > 0 and β = bσ + (i, j, d3). The unweightedGopakumar-Vafa invariants n̂gβ are given by:1. If b > 1 we have n̂gβ = 0.2. If b = 1 then the non-zero invariants are given in the following table:Table 1: The non-zero n̂gβ for β = σ + (i, j, d3) where i, j ∈ {0, 1} and d3 ≥ 0.(d1, d2, d3) (0, 0, 0) (0, 1, 1) (1, 0, 1) (1, 1, 0) (1, 1, 1) (1, 1, 2) (1, 1, 3) (1, 1, 4)g = 0 1 12 12 12 48 216 48 12g = 1 0 2 2 2 44 108 44 2g = 2 0 0 0 0 0 24 0 0Remark 1.2.3. We note that the values given only depend on the quadratic form‖d‖ := 2d1d2 + 2d1d3 + 2d2d3 − d21 − d22 − d33 appearing in the rank 3 Donaldson-Thomas partition function of [Br, Thm. 4]. However, there is no immediate geometricexplanation for this fact.Corollaries B and D will be proved in section 6.1.1.3 NotationThe main notations for this chapter have been defined above in section 1.2. In partic-ular X will always denote the banana threefold as defined in equation (1).1.4 FutureThe calculation here is for the unweighted Donaldson-Thomas partition function.However, the method of [BK] also provides a route (up to a conjecture) of comput-ing the Donaldson-Thomas partition function. The following are needed in order toconvert the given calculation:1. A proof showing the invariance of the Behrend function under the (C∗)2-actionused on the strata.2. A computation of the dimensions of the Zariski tangent spaces for the variousstrata.2. OVERVIEW OF THE COMPUTATION 39A comparison of the unweighted and weighted partition functions of the rank 3 latticeof [Br] reveals the likely differences:In the variables chosen in this chapter one can pass from the unweighted to theweighted partition functions by the change of variables Qi 7→ −Qi and p 7→ −p.Moreover, the conifold transition formula reveals further insight by comparing to theDonaldson-Thomas partition function of the Schoen variety with a single section andall fibre classes, which was shown in [ObPi] (via the reduced theory of the product ofa K3 surface with an elliptic curve) to be given by the weight 10 Igusa cusp form.As we mentioned previously the Donaldson-Thomas partition function is very hard tocompute. So much so that for proper Calabi-Yau threefolds, the only known completeexamples are in computationally trivial cases. This is even true conjecturally and evena conjecture for the rank 4 partition function is highly desirable. The work here showsunderlying structures that a conjectured partition function must have.2 Overview of the Computation2.1 Overview of the Method of CalculationWe will closely follow the method of [BK] developed for studying the Donaldson-Thomas theory of local elliptic surfaces. However, due to some differences in geome-try a more subtle approach is required in some areas. In particular, the local ellipticsurfaces have a global action which reduces the calculation to considering only theso-called partition thickened curves.Our method is based around the following continuous map:Cyc : Hilb1(X)→ Chow1(X).which takes a one dimensional subscheme to its 1-cycle. The fibres of this map are ofparticular importance and we denote them by Hilb•Cyc(X, q)where q ∈ Chow1(X).The bullet notation will be elaborated on further in this section.Remark 2.1.1. No such morphism exists in the algebraic category. In fact we notefrom [K, Thm. 6.3] that there is only a morphism from the semi-normalisationHilb1(X)SN → Chow1(X). However, Hilb1(X)SN is homeomorphic to Hilb1(X),which gives rise to the above continuous map.Broadly, we will be calculating the Euler characteristics e(Hilbβ,n(X))using the fol-lowing method:1. Push forward the calculation to an Euler characteristic on Chow1(X), weightedby the constructible function (Cyc∗1)(q) := e(Hilb•Cyc(X, q)). This is furtherdescribed in sections 2.2 and 2.3.2. Analyse the image of Cyc and decompose it into combinations of symmetricproducts where the strata are based on the types of subscheme in the fibresHilb•Cyc(X, q). This is done in section 3.3. Compute the Euler characteristic of the fibres e(Hilb•Cyc(X, q))and show thatthey form a constructible function on the combinations of symmetric products.This is done in section 5.4. Use the following lemma to give the Euler characteristic partition function.40 CHAPTER II. DT THEORY OF THE BANANA 3-FOLD WITH SECTION CLASSESFigure II.4: A depiction of the process for reducing to partition thickened curves.Clockwise from the top-left we have: a) Consider a 1-cycle in the Chow scheme; b)Consider the fibre of the given 1-cycle; c) Reduce to a computation on the open subsetof Cohen-Macaulay subschemes; d) Reduce to a computation on partition thickenedschemes.Lemma 2.1.2. [BK, Lemma 32] Let Y be finite type over C and let g : Z≥0 → Z((p)) beany function with g(0) = 1. Let G : Symd(Y ) → Z((p)) be the constructible functiondefined byG(ax) =∏ig(ai)where ax =∑i aixi ∈ Symd(Y ) and xi ∈ Y are distinct points. Then∞∑d=0e(Symd(Y ), G)qd =( ∞∑a=0g(a)qa)e(Y )To compute the Euler characteristics of the fibres (Cyc∗1)(q) := e(Hilb•Cyc(X, q))weuse the following method made rigorous in section 4:1. Consider the image of the fibre under the constructible morphism denoted κ :Hilb1(X) → Hilb1(X) which takes a subscheme Z to the maximal Cohen-Macaulay subscheme ZCM ⊂ Z .2. Denote the open subset contain Cohen-Macaulay subschemes by Hilb•CM(X, q) ⊂Hilb•Cyc(X, q).2. OVERVIEW OF THE COMPUTATION 41Figure II.5: A depiction of how the topological vertex is applied to calculate Eulercharacteristic of a given strata.3. Note the equality of the Euler characteristic e(Hilb•Cyc(X, q))and that of theweighted Euler characteristic e(Hilb•CM(X, q), κ∗1)where κ∗1 is the constructiblefunction (κ∗1)(p) = e(κ−1(p)).4. Define a (C∗)2-action on Hilb•CM(X, q) and show that κ∗1(p) = κ∗1(α · p)meaning e(Hilb•Cyc(X, q))= e(Hilb•CM(X, q)(C∗)2 , κ∗1). This technique is dis-cussed in section 4.2.5. Identify the (C∗)2-fixed points Hilb•CM(X, q)(C∗)2 as a discrete subset containingpartition thickened curves. These neighbourhoods and this action are givenexplicitly in section 4.4.6. Calculate the Euler characteristics e(Hilb•CM(X, q)(C∗)2 , κ∗1)using the Quotscheme decomposition and topological vertex method of [BK]. The concept ofthis is depicted in figure 2.1 and described below. Further technical details aregiven in section 4.5.The Euler characteristic calculation of e(Hilb•CM(X, q)(C∗)2 , κ∗1)for theorems A andC follow similar methods but have different decompositions. The calculations are com-pleted by considering the different types of topological vertex that occur for each fixedpoint in Hilb•CM(X, q)(C∗)2 .Since the fixed locus Hilb•CM(X, q)(C∗)2 will be disjoint we can consider individualsubschemes C ∈ Hilb•CM(X, q)(C∗)2 and their contribution to the Euler characteristic42 CHAPTER II. DT THEORY OF THE BANANA 3-FOLD WITH SECTION CLASSESe(Hilb•CM(X, q)(C∗)2 , κ∗1). To compute the contribution from C we must decomposeX as follows:1. Take the complement W = X \ C .2. Consider set of singularities of the underlying reduced curve. Denote this setC.3. Define C◦ = Cred \ C to be its complement.The curve C will be partition thickened. So each formal neighbourhood of a pointx ∈ C will give rise to a 3D partition. Similarly points on U ∈ C◦ will also giverise to 3D partitions and points on W will give rise to the empty partition. Usingtechniques from section 4.5 the Euler characteristics can then be determined.This calculation for theorem A is finalised in section 5.1. Generalities for the proof oftheorem C are given in section 5.2 and the individual calculations are given in sections5.3, 5.4 and 5.5.2.2 Review of Euler characteristicWe begin by recalling some facts about the (topological) Euler characteristic. Fora scheme Y over C we denote by e(Y ) the topological Euler characteristic in thecomplex analytic topology on Y . This is independent of any non-reduced structure ofY , is additive under decompositions of Y into open sets and their complements, andis multiplicative on Cartesian products. In this way we see that the Euler characteristicdefines a ring homomorphism from the Grothendieck ring of varieties to the integers:e : K0(VarC) −→ Z.If Y has a C∗-action with fixed locus Y C∗ ⊂ Y the Euler characteristic also has theproperty e(Y C∗) = e(Y ).The interaction of Euler characteristic with constructible functions and morphismsalso plays a key role in this chapter. Recall that a function µ : T → Z is con-structible if µ(T ) is finite and µ−1(c) is the union of finitely many locally closedsets for all non-zero c ∈ µ(T ). The µ-weight Euler characteristic is defined to bee(Y, µ) =∑k∈Z k · e(µ−1(k)). Note that we have e(Y ) = e(Y, 1) where 1 is theconstant function.For a scheme Z over C, a constructible morphism f : Y → Z is a finite collection ofmorphisms fi : Yi → Zi where Y =∐i Yi and Z =∐i Z are decompositions intolocally closed subschemes. We can defined a constructible function f∗µ : Z → Z by(f∗µ)(x) := e(f−1(x), µ).This has the important property e(Z, f∗µ) = e(Y, µ). If ν : Z → Z is a constructiblefunction, then µ · ν is a constructible function on Y × Z and e(Y × Z, µ · ν) =e(Y, µ) · e(Z, ν).It will also be helpful to extend these definitions to the rings of formal power series inQi and formal Laurent series in p. This will allow us to make use of lemma 2.1.2.2. OVERVIEW OF THE COMPUTATION 432.3 Pushing Forward to the Chow VarietyRecall that the Chow scheme Chow1(X) is a space parametrising the one dimensionalcycles of X . We will consider the subspace of this Chowβ(X) parametrising 1-cyclesin the class β ∈ H2(X,Z). We will then define a constructible morphismρβ :∑npnHilbβ,n(X)→ Chowβ(X).The strategy for calculating the partition functions is to analyse Chowβ(X) and thefibres of the map ρβ . These will often involve the symmetric product, and where pos-sible we will apply lemma 2.1.2.It will be convenient to employ the following • notations for the Hilbert schemes:Hilbσ+(0,•,•),n(X) :=∑d2,d3≥0∑n∈ZQσQd22 Qd33 pn Hilbσ+(0,d2,d3),n(X)Hilb•σ+(i,j,•),n(X) :=∑b,d3≥0∑n∈ZQbσQi1Qj2Qd33 pn Hilbbσ+(i,j,d3),n(X)and for the Chow schemes:Chowσ+(0,•,•)(X) :=∑d2,d3≥0QσQd22 Qd33 Chowσ+(0,d2,d3)(X)Chow•σ+(i,j,•)(X) :=∑b,d3≥0QbσQi1Qj2Qd33 Chowbσ+(i,j,d3)(X)where i, j ∈ {0, 1}. Note, that here he have viewed the Hilbert and Chow schemesin the Grothendieck ring of varieties. We also extend the • notation to symmetricproducts in the following way:Sym•(Y ) :=∑n∈Z≥0QnSymn(Y ),an we use the following notation for elements of the symmetric productay :=∑iaiyi ∈ Symn(Y )where yi are distinct points on Y and ai ∈ Z≥0. We also denote a tuple of partitionsα of a tuple of non-negative integers a by α ` a.Using the •-notation for the maps ρβ we create the following constructible morphisms:ρ• : Hilbσ+(0,•,•),n(X) −→ Chowσ+(0,•,•)(X)ηij• : Hilb•σ+(i,j,•),n(X) −→ Chow•σ+(i,j,•)(X)and we also use the notation η• = η00• + η01• + η11• . The fibres of these morphisms willbe subspaces of the Hilbert scheme parametrising one dimensional subschemes witha fixed 1-cycle. Specifically, let C ⊂ X be a one dimensional subscheme in the classβ ∈ H2(X) with 1-cycle Cyc(C). Define Hilbn(X,Cyc(C)) ⊂ Hilbβ,n(X) to be thesubscheme.Hilbn(X,Cyc(C)) ={Z ∈ Hilbβ,n(X) ∣∣ Cyc(Z) = Cyc(C)}.The maps ρ• and η• are explicitly described in lemmas 3.5.3 and 3.5.1 respectively.44 CHAPTER II. DT THEORY OF THE BANANA 3-FOLD WITH SECTION CLASSES3 Parametrising Underlying 1-cycles3.1 Related Linear Systems in Rational Elliptic SurfacesIn this section we consider some basic results about linear systems on a rational ellip-tic surface. Some of these result can be found in [BK, §A.1].Recall our notation that pi : S → P1 is a generic rational elliptic surface with acanonical section ζ : P1 → S. Consider the following classical results for rationalelliptic surfaces from [Mi, II.3]:pi∗OS ∼= pi∗OS(ζ) ∼= OP1 , R1pi∗OS ∼= OP1(−1) and R1pi∗OS(ζ) ∼= 0.After applying the projection formula we have the following:pi∗OS(dF ) ∼= pi∗OS(ζ + dF ) ∼= OP1(d) (II.5)as well asR1pi∗OS(dF ) ∼= OP1(d− 1) and R1pi∗OS(ζ + dF ) ∼= 0. (II.6)Lemma 3.1.1. We have the following isomorphisms:H1(S,OS(dF )) ∼= H0(P1,OP1(d− 1)) and H1(S,OS(ζ + dF )) ∼= 0.Proof. The second isomorphism is immediate from the vanishing of Ripi∗OS(ζ + dF )for i > 0 (see for example [H, III Ex. 8.1]) and H0(P1,OP1(d)) ∼= 0.To show the first isomorphism we consider the following exact sequence arising fromthe Leray spectral sequence:H1(P1, pi∗OS(dF ))→ H1(S,OS(dF ))→ H0(P1, R1pi∗OS(dF ))→ 0We have from (II.5) that H1(P1, pi∗OS(dF )) ∼= 0 and we have the desired isomorphismafter considering (II.6).Lemma 3.1.2. Consider a fibre F of a point z ∈ P1 by the map S → P1 and the image ofa section ζ : P1 → S. Then there are isomorphisms of the linear systems|dF |S ∼= |ζ + dF |S ∼= |d z|P1 and |b ζ + F |S ∼= |z|P1 .Proof. The isomorphism |ζ + dF |S ∼= |d z|P1 is immediate from the vanishing ofRipi∗OS(ζ + dF ) for i > 0 and (II.5) (see for example [H, III Ex. 8.1]).We continue by showing |dF |S ∼= |ζ+dF |S . Consider the long exact sequence arisingfrom the divisor sequence for ζ twisted by OS(ζ + dF ):0→ H0(S,OS(dF )) f→ H0(S,OS(ζ + dF ))→ H0(S, ζ∗OP1(ζ + dF ))g→ H0(P1,OP1(d− 1))→ 0where we have applied the results from lemma 3.1.1. from intersection theory we havethat ζ∗OP1(ζ + dF ) ∼= ζ∗OP1(d − 1). Hence, g is an isomorphism making f an iso-morphism also.The isomorphism |b ζ + F |S ∼= |z|P1 will follow inductively from the divisor sequencefor ζ on S:0 −→ OS(kζ + F ) −→ OS((k + 1)ζ + F) −→ Oζ((k + 1)ζ + F ) −→ 0.3. PARAMETRISING UNDERLYING 1-CYCLES 45Intersection theory shows us that Oζ((k+ 1)ζ +F)is a degree −k line bundle on P1which shows that its 0th cohomology vanishes. Hence, we have isomorphisms:H0(S,OS(F )) ∼= · · · ∼= H0(S,OS(bζ + F )).3.2 Curve Classes and 1-cycles in the ThreefoldRecall from definition 1.2.1 that the banana curves Ci are labelled by their uniqueintersections with the rational elliptic surfacesS1, S2 and Sop.These are smooth effective divisors on X . Hence a curve C in the class (d1, d2, d3)will have the following intersections with these divisors:C · S1 = d2, C · S2 = d1 and C · Sop = d3.The full lattice H2(X,Z) is generated byC1, C2, C3, σ11, σ12, . . . , σ19, σ21, . . . , σ99where the σij are the 81 canonical sections of pr : X → P1 arising from the 9canonical sections of pi : S → P1. However, there are 64 relations between the σij ’sgiving the lattice rank of 20 (see [Br, Prop. 28 and Prop. 29]).Lemma 3.2.1. There are no relations in H2(X,Z) of the form:n · σi,j + d1C1 + d2C2 + d3C3 =∑(k,l) 6=(i,j)ak,l · σk,l + d′1C1 + d′2C2 + d′3C3where n, ak,l, dt, d′t ∈ Z≥0 for all k, l ∈ {1, . . . , 9} and t ∈ {1, 2, 3}.Proof. Any such relation must push forward to relations on S via the projectionspri : X → Si. However, S is isomorphic to P1 blown up at 9 points. The exceptionaldivisors of these blow-ups correspond to the sections ζi : P1 → S. HencePicS ∼= PicP2 × ζ1 × · · · × ζ9 ∼= Z10and there are are no relations of this form.The next lemma allows us to consider the curves in our desired classes by decomposingthem.Lemma 3.2.2. Let d1, d2, d3, b ∈ Z≥0 and i, j ∈ {0, 1}.1. Let C be a Cohen-Macaulay curve in the class (d1, d2, d3). Then the support of C iscontained in fibres of the projection map pr : X → P1.2. A curve C in the class σ + (d1, d2, d3) is of the formC = σ ∪ C0where C0 is a curve in the class (d1, d2, d3).3. A curve in the class b σ + (i, j, d3) is of the formC = Cσ ∪ C0where Cσ is a curve in the class b σ and C0 is a curve in the class (i, j, d3). Thesame result holds for permutations of b σ + (i, j, d3).Proof. Consider a curve in one of the given classes and it’s image under the twoprojections pri : X → Si. For (1) these must be in the classes |d1f1| and |d1f1|, for (2)the classes |ζ + d1F1| and |ζ + d2F2|, and for (3) the classes |if1| and |jf1|. Lemma3.1.2 now shows that the curve must have the given form.46 CHAPTER II. DT THEORY OF THE BANANA 3-FOLD WITH SECTION CLASSES3.3 Analysis of 1-cycles in Smooth Fibres of prConsider a fibre Fx = pr−1(x) which is smooth. Then there is an elliptic curve Esuch that Fx ∼= E × E. Consider a curve C with underlying 1-cycle contained inE × E, then this gives rise to a divisor D in E × E. Hence we must analyse divisorsin E ×E and their classes in X . The class of such a curve is determined uniquely byits intersection with the surfaces S1, S2 and Sop.Lemma 3.3.1. Let C ⊂ X correspond to a divisor D in E × E.1. If C is in the class (0, d2, d3) then d2 = d3 and D is the pullback of a degree d2divisor on E via the projection to the second factor.2. The result in (2) is true for (d1, 0, d3) and projection to the first factor.Proof. If C is in the class (0, d2, d3) then it doesn’t intersect with the surface S2. Whenwe restrict to E × E this is the same condition as not intersecting with a fibre of theprojection to the second factor. The only divisors that this is true for are those pulledback from E via this projection. It is clear that that the intersection with S2 is d2, andthat the intersection with Sop is d2 as well. Hence we have that d2 = d3. The prooffor part (2) is completely analogous.Lemma 3.3.2. Let C ⊂ X be in the class (1, 1, d) and correspond to a divisor D inE × E. Then d ∈ {0, . . . , 4} and occurs in the following situations:1. If E has j(E) 6= 0, 1728 then:(a) d = 0 occurs when D is a translation of the graph {(x,−x)}.(b) d = 4 occurs when D is a translation of the graph {(x, x)}.(c) d = 2 occurs when D is the union of a fibre from the projection to the firstfactor and a fibre from the projection to the second factor.2. If j(E) = 1728 and E ∼= C/i we have the cases (a) to (c) as well as:(d) d = 2 occurs when D is a translation of the graph {(x,±ix)}.3. If j(E) = 0 and E ∼= C/τ with τ = 12 (1 + i√3) we have the cases (a) to (c) aswell as:(e) d = 1 occurs when D is a translation of the graph {(x,−τx)} or the graph{(x, (τ − 1)x)}.(f ) d = 3 occurs when D is a translation of the graph {(x, τx)} or the graph{(x, (−τ + 1)x)}.Proof. Denote the projection maps by pi : E × E → E and let C ⊂ X be in the class(1, 1, d) and correspond to a divisor D in E ×E. Suppose D is reducible. Then fromlemma 3.3.1 we see that D must be the union p−11 (x1) ∪ p−12 (x2) where x1, x2 ∈ Eare generic points. We also have that D is in the class (1, 1, 2).Suppose D is irreducible. The surfaces S1 and S2 intersect D exactly once and theirrestrictions correspond the fibres of the projection maps pi : E ×E → E. So the pro-jection maps must be isomorphisms when restricted to D. Hence D is the translationof the graph of an automorphism of E.All elliptic curves have the automorphisms x 7→ ±x. Also• if E ∼= C/i (j-invariant j(E) = 1728) the E also has the automorphisms x 7→±ix, and3. PARAMETRISING UNDERLYING 1-CYCLES 47• if E = C/τ with τ = 12 (1 + i√3) (j-invariant j(E) = 0) then E also has theautomorphisms x 7→ ±τx and x 7→ ±(τ − 1)x.So to complete the proof we have to calculate the intersections #(Γξ ∩ Sop) whereΓξ is the graph of an automorphism ξ. Also, Sop|Fx ∼= Γ−1 hence we calculate#(Γξ ∩ Γ−1) = #{(x, ξ(x)) = (x,−x)} in the surface Fx. For all the elliptic curveswe have:(a) #(Γ1 ∩ Γ−1) is given by the four 2-torsion points {0, 12 , 12τ, 12 (1 + τ)}.(b) #(Γ−1 ∩ Γ−1) = 0 since one copy can be translated away from the other.For E ∼= C/i (j-invariant j(E) = 1728) we have:(d) #(Γ±i ∩ Γ−1) is given by the two points {0, 12 (1 + τ)}.For E = C/τ with τ = 12 (1 + i√3) (j-invariant j(E) = 0) we have:(e) #(Γτ ∩ Γ−1) and #(Γ(1−τ)i ∩ Γ−1) are both determined by the three points{0, 13 (1 + τ), 23 (1 + τ)}.(f) #(Γ−τ ∩ Γ−1) and #(Γ(τ−1)i ∩ Γ−1) are both given by the single point {0}.3.4 Analysis of 1-cycles in Singular Fibres of prWe denote the fibres of the projection pr by Fx := pr−1(x). The singular fibresare all isomorphic so we denote a singular fibre by Fban and its normalisation byν : F˜ban → Fban. From [Br, Prop. 24] we have that F˜ban ∼= Bl2 pt(P1 × P1) and if wechoose the coordinates on the P1’s so that the 0 and ∞ map to a nodal singularity,then the two points blown-up are z1 = (0,∞) and z2 = (∞, 0).Blz1,z2(P1 × P1)blν // FbanP1 × P1Also, we recall the decomposition of Si intoS◦i = Smi q Niwhere Ni are the 12 nodal fibres with their nodes removed and Smi = S◦i \ Ni. LetNi ={N(1)i , . . . , N(12)i}be the 12 nodal fibres with the nodes.3.4.1. Denote the divisors in F˜ban corresponding to the banana curve Ci by C˜i andC˜ ′i . They are identified in Fban byν(C˜i) = ν(C˜′i) = Ci.For i = 1, 2 we also denote Ĉi = bl(C˜i) and Ĉ ′i = bl(C˜′i) inside P1 × P1. The curveclasses in F˜ban are generated by the collection of C˜i and C˜ ′i’s with the relations:C˜1 + C˜3 ∼ C˜ ′1 + C˜ ′3 and C˜2 + C˜3 ∼ C˜ ′2 + C˜ ′3.48 CHAPTER II. DT THEORY OF THE BANANA 3-FOLD WITH SECTION CLASSESFigure II.6: On the left is a depiction of the normalisation F˜ban and on the right is adepiction of P1 × P1. Here bl is the map blowing up (0,∞) and (∞, 0). On the rightf1 and f2 are generic fibres of the projection maps P1 × P1 → P1 and on the left f˜1and f˜2 are their proper transforms.3.4.2. Let f1 and f2 be fibres of the projections P1 × P1 → P1 not equal to any Ĉi orĈ ′i and let f˜1 and f˜2 be their proper transforms. Then we also have the relations:f˜1 ∼ C˜1 + C˜3 and f˜2 ∼ C˜2 + C˜3.Moreover, if D˜ is a divisor in F˜ban such that ν(D˜) is in the class (d1, d2, d3) then Dis in a classa1C˜1 + a′1C˜′1 + a2C˜2 + a′2C˜′2 + a3C˜3 + a′3C˜′3where ai + a′i = di.Lemma 3.4.3. Let C ⊂ X correspond to a divisor D in Fban.1. C is in the class (0, 0, d3) if and only if D has 1-cycle d3C3.2. C is in the class (0, d2, d3) if and only if D has 1-cycle D˜ + a2C(j)2 + a3C(j)3where D˜ is the pullback of a degree af divisor from the smooth part of N(j)2 via theprojection Fban → N(j)2 such that af + a2 = D2 and af + a3 = D3. Moreover, D˜is in the class (0, af , af ).Proof. Let C ⊂ X be a curve in the class (0, d2, d3) and correspond to a divisor D inFban. There exists a divisor D˜ in F˜ban ∼= Blz1,z2(P1 × P1) with ν(D˜) = D.From the discussion in 3.4.2 we have that bl(D˜) is in the class of d2f2 and is hencein its corresponding linear system. So, D˜ is the union of the the proper transform ofbl(D˜) and curves supported at C˜3 and C˜ ′3. The result now follows.Lemma 3.4.4. Let C ⊂ X be an irreducible curve in the class (1, 1, d) and correspondto a divisor D in Fban. Then D is the image under ν of the proper transform under bl ofa smooth divisor in |f1 + f2| on |P1 × P1|. Moreover, the value of d is determined theintersection of D with points in P = {(0, 0), (0,∞), (∞, 0), (∞,∞)}. That is, if Dintersects1. (0, 0) and (∞,∞) only, then d = 2.2. (0,∞) and (∞, 0) only, then d = 0.3. (0, 0) only or (∞,∞) only, then d = 2.4. (0,∞) only or (∞, 0) only, then d = 1.3. PARAMETRISING UNDERLYING 1-CYCLES 495. no points of P, then d = 2.Moreover, there are no smooth divisors in |f1 + f2| on |P1 × P1| that intersect other combi-nations of these points.Proof. Let C ⊂ X be an irreducible curve in the class (1, 1, d) and correspond to adivisor D in Fban. There exists an irreducible divisor D˜ in F˜ban ∼= Blz1,z2(P1 × P1)with ν(D˜) = D. D˜ does not contain either of the exceptional divisor C˜3 and C˜ ′3.Hence, it must be the proper transform of a curve in P1 × P1.From the discussion in 3.4.2 we have that bl(D˜) is in the class of f1+f2 and is hence inits corresponding linear system. The only irreducible divisors in |f1 + f2| are smoothand can only pass through the combinations of points in P that are given. We refer tothe appendix 6.2.3 for the proof of this. The total transform in any divisor in |f1 + f2|will correspond to a curve in the class C1 +C2 + 2C2. Hence the classes of the propertransforms depend the number of intersections with the set {(0,∞), (∞, 0)}. Thevalues are immediately calculated to be those given.3.5 Parametrising 1-cyclesWe use the notation:1. Bi = {b1i , . . . , b12i } is the set of the 12 points in Si that correspond to nodes inthe fibres of the projection pi : Si → P1.2. S◦i = Si \Bi is the complement of Bi in SiLemma 3.5.1. In the case β = σ + (0, d2, d3) there is a constructible morphism ρ• whereChowσ+(0,•,•)(X) has the decomposition:Chowσ+(0,•,•)(X) = Sym•(S◦2 )× Sym•(B2)× Sym•(Bop).Moreover, if x = (ay,mb2,nbop) ∈ Chowσ+(0,•,•)(X) then the fibre is given byρ−1• (x) = Hilb•Cyc(X, q) whereq = σ +∑iaipr−12 (yi) +∑imiC(i)2 +∑iniC(i)3 .Proof. From lemma 3.2.2 part 2 it is enough to consider curves in the class (0, d2, d3).Also from 3.2.2 part 1 we know that the curves are supported on fibres of the mappr : X → P1. From lemma 3.3.1 part 1 we know that the curves supported on smoothfibres of pr must be thicken fibres of the projection pr2 : X → S. Similarly we knowfrom lemma 3.4.3 part 2 that the curves supported on singular fibres of pr must be theunion of thicken fibres of pr2 and curves supported on the C2 and C3 banana curves.The result now follows.We also use the notation:1. Ni ⊂ Si are the 12 nodal fibres of pi : Si → P1 with the nodes removed and:Ni = Nσi q N∅i where Nσi := Ni ∩ σ and N∅i := Ni \ σ.2. Smi = S◦i \ Ni is the complement of Ni in S◦i and:Smi = Smσi q Sm∅i where Smσi := Smi ∩ σ and Sm∅i := Smi \ σ.3. J0 and J1728 to be the subsets of points x ∈ P1 such that pi−1(x) has j-invariant0 or 1728 respectively and J = J0 q J1728.50 CHAPTER II. DT THEORY OF THE BANANA 3-FOLD WITH SECTION CLASSES4. L to be the linear system |f1 +f2| on P1×P1 with the singular divisors removedwhere f1 and f2 are fibres of the two projection maps.5. A˜ut(E) := Aut(E) \ {±1}.Remark 3.5.2. The following lemma should be parsed in the following way. Fori, j ∈ {0, 1} and b, d3 ∈ Z≥0, a subscheme in the class β = dσ + (i, j, d3) will have1-cycle of the following form:q = bσ +D +∑iniC(i)3where D is reduced and does not contain σ or and C(i)3 . Then D is in the class (i, j, n)for some n ∈ Z≥0.The Chow groups parameterise the different possible D and these possibilities dependon i and j:• If i = j = 0 then D is the empty curve. If• If i = 0 and j = 1 then D can be either a fibre of the projection pr2 or C(i)2 .• If i = j = i then and D then it can be combinations of fibres and bananacurves. It can also be neither of these in the cases we call diagonals.Lemma 3.5.3. In the cases β = dσ + (i, j, d3) we haveChow•σ+(i,j,•)(X) ∼= Z≥0 × Chow(i,j,•)(X)which agrees with constructible morphisms ηij• and the following decompositions of Chow(i,j,•)(X):1. For i = j = 0 we have the decomposition of Chow(0,0,•)(X) with parts:(a) Sym•(Bop).The corresponding fibres are then (η00• )−1(x) = Hilb•Cyc(X, q) where:(a) If x = nbop then q =∑i niC(i)3 .2. For i = 0 and j = 1 we have a decomposition of Chow(0,1,•)(X) with parts:(a) S◦2 × Sym•(Bop)(b)12qk=1Sym•({bkop})× Sym•(Bop \ {bkop}).The corresponding fibres are then (η01• )−1(x) = Hilb•Cyc(X, q) where:(a) If x = (y,nbop) then q = pr−12 +∑i niC(i)3 .(b) If x = (akbkop,nbop) then q = akC(k)3 +∑i niC(i)3 .3. For i = j = 1 we have a decomposition of Chow(1,1,•)(X) with parts:(a) S◦1 × S◦2 × Sym•(Bop)3. PARAMETRISING UNDERLYING 1-CYCLES 51(b)12qk=1S◦1 × Sym•({bkop})× Sym•(Bop \ {bkop})(c)12qk=1S◦2 × Sym•({bkop})× Sym•(Bop \ {bkop})(d)12qk, l = 1k 6= lSym•({bkop})× Sym•({blop})× Sym•(Bop \ {bkop, blop})(e)12qk=1Sym•({bkop})× Sym•(Bop \ {bkop})(f ) q Diag•where Diag• will be defined by a further decomposition. The corresponding fibres of(a)-(e) are (η11• )−1(x) = Hilb•Cyc(X, q) where:(a) If x = (y1, y2,nbop) then q = pr−11 (y1) + pr−11 (y2) +∑i niC(i)3 .(b) If x = (y1, akbkop,nbop) then q = pr−11 (y1) + C(k)2 + akC(k)3 +∑i niC(i)3 .(c) If x = (y2, akbkop,nbop) then q = pr−12 (y2) + C(k)1 + akC(k)3 +∑i niC(i)3 .(d) If x = (akbkop, alblop,nbop) then q = C(k)1 +C(l)2 + akC(k)3 + alC(l)3 +∑i niC(i)3 .(e) If x = (akbkop,nbop) then q = C(k)1 + C(k)2 + akC(k)3 +∑i niC(i)3 .For part (f ), Diag• is defined by the further decomposition:(g) Sm1 × Sym•(Bop)(h) q Sm2 × Sym•(Bop)(i) qy∈J Epi(y) × A˜ut(Epi(y))× Sym•(Bop)(j)12qk=1L× Sym•({bkop})× Sym•(Bop \ {bkop}).The corresponding fibres of (g)-(j) are (η11• )−1(x) = Hilbn(X, q) where:(g) If x = (y,nbop) then q = Dy +∑i niC(i)3 where Dy is the graph of the mapf(z) = z + x|Epi(y) in the fibre Fpi(y) = Epi(y) × Epi(y).(h) If x = (y,nbop) then q = Dy +∑i niC(i)3 where Dy is the graph of the mapf(z) = −z + x|Epi(y) in the fibre Fpi(y) = Epi(y) × Epi(y).(i) If x = (y,nbop) then q = Dy +∑i niC(i)3 where Dy is the graph of the mapf(z) = A(z) + x for some A ∈ Aut(Epi(y)) \ {±1}.(j) If x = (z, akbkop,nbop) then q = ν(L˜z) + akC(k)3 +∑i niC(i)3 where L˜z isthe proper transform of the divisor Lz in P1 × P1 and ν is the normalisationof the kth singular fibre.Proof. The decomposition Chow•σ+(i,j,•)(X) ∼= Z≥0 × Chow(i,j,•)(X) is immediatefrom lemma 3.2.2 part 3. Hence it is enough to parametrise the curves in the classβ = (i, j, •). Also from 3.2.2 part 1 we know that the curves are supported on fibresof the map pr : X → P1. We must have thatCyc(C) = aσ +D +12∑i=1miC(i)3 .for some minimal reduces curve D in the class (1, 1, n) for n ≥ 0 minimal. Thepossible D curves are described in lemmas 3.3.1, 3.3.2, 3.4.3 and 3.4.4. The result nowfollows.52 CHAPTER II. DT THEORY OF THE BANANA 3-FOLD WITH SECTION CLASSES4 Techniques for Calculating Euler Characteristic4.1 Quot Schemes and their DecompositionThis section is a summary of required results from [BK]. First we consider the followingsubscheme of the Hilbert scheme.Definition 4.1.1. Let C ⊂ X be a Cohen-Macaulay subscheme of dimension 1. Con-sider the Hilbert scheme of subschemes Z ⊂ X of class [Z] = [C] ∈ H2(X) andχ(OZ) = χ(OC) + n for some n ∈ Z≥0. This contains the following closed sub-scheme:Hilbn(X,C) :={Z ⊂ X such that C ⊂ Z and IC/IZ has finite length n}.It is convenient to replace the Hilbert scheme here with a Quot scheme. Recall theQuot scheme QuotnX(F) parametrising quotients F  Q on X , where Q is zero-dimensional of length n. It is related to the above Hilbert scheme in the followingway.Lemma 4.1.2. [BK, Lemma 5]. The following equality holds in K0(VarC)((p)):Hilb•(X,C) = Quot•X(IC).We also consider the following subscheme of these Quot schemes.Definition 4.1.3. [BK, Def. 12] Let F be a coherent sheaf on X , and S ⊂ X a locallyclosed subset. We define the locally closed subset of QuotnX(F)QuotnX(F , S) :={[F  Q] ∈ QuotnX(F)∣∣ Supp(Qred) ⊂ S}.This allows us to decompose the Quot schemes in the following way.Lemma 4.1.4. [BK, Prop. 13] Let F be a coherent sheaf on X , S ⊂ X a locally closedsubset and Z ⊂ X a closed subset. Then if Z ⊂ S and and n ∈ Z≥0 there is a geometricallybijective constructible morphism:QuotnX(F , S) −→∐n1+n2=nQuotn1X (F , S \ Z)×Quotn2X (F , Z).4.2 An Action on the Formal NeighbourhoodsLet C ⊂ X be a one dimensional subscheme in the class β ∈ H2(X) with 1-cycleq = Cyc(C). We recall the our notation that HilbnCyc(X, q) ⊂ Hilbβ,n(X) is thefollowing subschemeHilbnCyc(X, q) :={[Z] ∈ Hilbβ,n(X) | Cyc(C) = q}.Furthermore, we defineHilbnCM(X, q) ⊂ HilbnCyc(X, q)to be the open subscheme containing Cohen-Macaulay subschemes of Z .Lemma 4.2.1. Suppose Z ⊂ X is a one dimensional Cohen-Macaulay subscheme such that:1. Z has the decomposition Z = C∪iZi where C is reduced and Zi∩Zj = ∅ for i 6= j.4. TECHNIQUES FOR CALCULATING EULER CHARACTERISTIC 532. There are formal neighbourhoods Vi of Zi in X such that (C∗)2 acts on each andfixes Zredi .3. If C˜ := C \ (∪Vi) then C˜ ∩ (∪Vi) is invariant under the (C∗)2-action on Vi.Then there is a (C∗)2-action on HilbnCM(X,Cyc(Z)) such that if α ∈ (C∗)2 and Y ∈HilbnCM(X,Cyc(Z)) then:α · Y = C˜ ∪ α · (Y |∪Vi).Proof. We show the action is well defined on a flat family in HilbnCM(X,Cyc(Z)). Letsuch a family be given by the diagram:Z   //''X × SSThe reduced curves C,Zredi and the neighbourhoods Vi must all be constant on thefamily and we have a decompositionZ = (C × S) ∪iZiwhere Zi ⊂ Vi × S. Hence, the action is given byα · Z = (C˜ × S) ∪ α · (Z|∪(Vi×S)).Consider the constructible mapκ : Hilb•Cyc(X, q) −→ Hilb•CM(X, q)where Z ⊂ X is mapped to the maximal Cohen-Macaulay subscheme ZCM ⊂ Z . Thenwe havee(Hilb•Cyc(X, q))= e(Hilb•CM(X, q), κ∗1)= e(Hilb•CM(X, q)(C∗)2 , κ∗1)(7)where (κ∗1)(z) := e(κ−1(z)) and the last line comes from the following lemma.Lemma 4.2.2. The constructible function κ∗1 is invariant under the (C∗)2-action. Thatis if α ∈ (C∗)2 and z ∈ HilbnCM(X, q) then (κ∗1)(z) = (κ∗1)(α · x).Proof. Let α ∈ (C∗)2 and z ∈ HilbnCM(X, q) correspond to Z ⊂ X . Also let Z˜ = ∪Ziand V˜ = Vi be as in lemma 4.2.1. Then the fibre κ−1(x) is theκ−1(x) = Hilb•(X,Z) = Quot•X(IZ)where the last equality is in K0(VarC)((p)) from lemma 4.1.2. Also from lemma 4.1.4we have a geometrically bijective constructible morphism:QuotnX(IZ) −→∐n1+n2=nQuotn1X (IZ , X \ V˜ )×Quotn2X (IZ , V˜ ).We have Iα·Z |X\V˜ = IZ |X\V˜ so Quotn1X (IZ , X \ V˜ ) ∼= Quotn1X (Iα·Z , X \ V˜ ). More-over, we have isomorphismsQuotn2X (IZ , V˜ )∼= Quotn2V˜(IZ |V˜ )54 CHAPTER II. DT THEORY OF THE BANANA 3-FOLD WITH SECTION CLASSESand ZV˜∼= α · ZV˜ so we have an isomorphismQuotn2X (IZ , V˜ )∼= Quotn2X (Iα·Z , V˜ )Taking Euler characteristic now shows that e(κ−1(x))= e(κ−1(α · x)).4.2.3. We will now consider a useful tool in calculating Euler characteristics of theform given in (7). First let z ∈ HilbnCM(X, q) correspond to Z ⊂ X such that Z islocally monomial. Then the fibre κ−1(x) isκ−1(x) = Hilb•(X,Z) = Quot•X(IZ)where the last equality is in K0(VarC)((p)) from lemma 4.1.2. To compute this fibre weemploy the following method:1. Decompose X by X = Z q W where W := X \ Z2. Let Z be set of singularities of Zred.3. Let∐i Zi = Z \ Z be a decomposition into irreducible components.Then applying Euler characteristic to lemma 4.1.4 we have:e(Quot•X(IZ))= e(Quot•X(IZ ,W )) ∏z∈Ze(Quot•X(IZ , {z}))∏ie(Quot•X(IZ , Zi)).4.3 Partitions and the topological vertexWe recall the terminology of 2D partitions, 3D partitions and the topological vertexfrom [ORV, BCY]. A 2D partition λ is an infinite sequence of decreasing integers thatis zero except for a finite number of terms. The size of a 2D partition |λ| is the sum ofthe elements in the sequence and the length l(λ) is the number of non-zero elements.We will also think of a 2D partition as a subset of (Z≥0)2 in the following way:λ ! {(i, j) ∈ (Z≥0)2 | λi ≥ j ≥ 0 or i = 0}A 3D partition is a subset η ⊂ (Z≥0)3 satisfying the following condition:1. (i, j, k) ∈ η if and only if one of i, j or k is zero or one of (i−1, j, k), (i, j−1, k)or (i, j, k − 1) is also in η.Given a triple of 2D partitions (λ, µ, ν) we also define a 3D partition asymptotic to(λ, µ, ν) is a 3D partition η that also satisfies the conditions:1. (j, k) ∈ λ if and only if (i, j, k) ∈ η for all i 0.2. (k, i) ∈ µ if and only if (i, j, k) ∈ η for all j  0.3. (i, j) ∈ ν if and only if (i, j, k) ∈ η for all k  0.The leg of η in the ith direction is the subset {(i, j, k) ∈ η | (j, k) ∈ λ}. We analogouslydefine the legs of η in the j and k directions. The weight of a point in η is defined tobeξη(i, j, k) := 1−# {legs of η containing (i, j, k)}.Using this we define the renormalised volume of η by:|η| :=∑(i,j,k)∈ηξη(i, j, k). (II.9)4. TECHNIQUES FOR CALCULATING EULER CHARACTERISTIC 55Figure II.7: A 3D partition asymptotic to((2, 1), (3, 2, 2), (1, 1, 1)). The partitioncontaining only the white boxes has renormalised volume −14. The partition includingthe green boxes has renormalised volume −11.The topological vertex is the formal Laurent series:Vλµν :=∑ηp|η|where the sum is over all 3D partitions asymptotic to (λ, µ, ν). An explicit formula forVλµν is derived in [ORV, Eq. 3.18] to be:Vλµν = M(p)p− 12 (‖λ‖2+‖µt‖2+‖ν‖2)Sνt(p−ρ)∑ηSλt/η(p−ν−ρ)Sµ/η(p−νt−ρ)4.4 Partition Thickened Section, Fibre and Banana CurvesIn this subsection we consider non-reduced structure for curves in our desired classes.The partition thickened structure will be the fixed points of a (C∗)2-action.4.4.1. Recall that the section ζ ∈ S is the blow-up of a point in z ∈ P2. Choose onceand for all a formal neighbourhood SpecC[[s, t]] of z ∈ P2. The blow-up gives theformal neighbourhood of ζ ∈ S with 2 coordinate charts:C[[s, t]][u]/(t− su) ∼= C[[s]][u] and C[[s, t]][v]/(s− tv) ∼= C[[t]][v]with change of coordinates s 7→ tv and u 7→ v−1. This gives the formal neighbourhoodof σ ∈ X with 2 coordinate charts:C[[s1, s2]][u] and C[[t1, t2]][v]with change of coordinates si 7→ tiv and u 7→ v−1. We call these coordinates thecanonical formal coordinates around σ ∈ X .4.4.2. Now consider a reduced curve D in X that intersects σ transversely with length1. When D is restricted to the formal neighbourhood of σ it is given byC[[s1, s2]][u]/(a0u− a1, b0s1 − b1s2) and C[[t1, t2]][v]/(a0 − a1v, b0t1 − b1t2)for some [a0 : a1], [b0 : b1] ∈ P1. We use this to define the change of coordinates:s˜1 7→ b0s1 − b1s2 and s˜2 7→ b1s1 + b0s2t˜1 7→ b0t1 − b1t2 and t˜2 7→ b1t1 + b0t2We call these coordinates the canonical formal coordinates relative to D.56 CHAPTER II. DT THEORY OF THE BANANA 3-FOLD WITH SECTION CLASSESyy xyxx3224Figure II.8: Depiction of the subscheme in C2 given by the monomial ideal(y3, y2x, y1x2, y1x3, x4) associated to the partition (3, 2, 1, 1, 0 . . .).Definition 4.4.3. Let C[[s1, s2]][u] and C[[t1, t2]][v] be either of the above canonicalcoordinates. Then we define1. The canonical (C∗)2-action on these coordinates by (s1, s2) 7→ (λ1s1, λ2s2)and (t1, t2) 7→ (λ1t1, λ2t2).2. Let λ = (λ1, . . . , λl, 0, . . .) be a 2D partition. The λ-thickened section denotedby λσ is the subscheme of X defined by the ideal given in the coordinates by(sλ12 , . . . , sl−11 sλl2 , sl) and (tλ12 , . . . , tl−11 tλl2 , tl).We can now consider fibres of the projection map pr2 : X → S.Definition 4.4.4. Let x ∈ S◦ and fx = pr−12 (x) the fibre. Then we define1. Canonical coordinates on a formal neighbourhood Vx of fx are given by formalcoordinates C[[s, t]] of x in S under where Vx = fx × SpecC[[s, t]].2. The canonical (C∗)2-action on these coordinates by (s, t) 7→ (λ1s, λ2t).3. Let λ = (λ1, . . . , λl, 0, . . .) be a 2D partition. The λ-thickened fibre at xdenoted by λfx is the subscheme of X given by the ideal:(tλ1 , . . . , sl−1tλl , sl)4.4.5. We now consider a canonical formal neighbourhood of the banana curve C3.We follow much of the reasoning from [Br, §5.2]. Let x ∈ S correspond to a pointwhere pi : S → P1 is singular. Let formal neighbourhoods in the two isomorphiccopies of S be given bySpecC[[s1, t1]] and SpecC[[s2, t2]]and the map S → P1 be given by r 7→ siti. Then the formal neighbourhood of aconifold singularity in X is given bySpecC[[s1, t1, s2, t2]]/(s1t1 − s2t2),and the restriction to a fibre of the projection S ×P1 S → P1 isSpecC[[s1, t1, s2, t2]]/(s1t1, s2t2).Now, blowing up along {s1 = t2 = 0} (which is canonically equivalent to blowing upalong {s1 − t1 = s2 − t2 = 0}), we have the two coordinate charts:C[[s1, t2, s2, t2]][u]/(s1 − ut2, s2 − ut1) ∼= C[[t1, t2]][u], andC[[s1, t2, s2, t2]][v]/(t1 − vs2, t2 − vs1) ∼= C[[s1, s2]][v],4. TECHNIQUES FOR CALCULATING EULER CHARACTERISTIC 57where the change of coordinates is given by t1 7→ vs2 , t2 7→ vs1 and u 7→ v−1. Wecall these coordinates the canonical formal coordinates around the banana curveC3.4.4.6. With these coordinates we have:1. Then the restriction to the fibre of pr : X → P1 isC[[t1, t2]][u]/(t1t2u) and C[[s1, s2]][v]/(s1s2v).2. The banana curve C3 is given byC[[t1, t2]][u]/(t1, t2) and C[[s1, s2]][v]/(s1, s2).4.4.7. Similar to 4.4.2 we also consider canonical relative coordinates for a C3 bananacurve. Recall 3.4.4 and let D is the image under ν : F˜ban → Fban of the proper trans-form under bl : Bl(0,∞),(∞,0)(P1 × P1)→ P1 × P1 of a smooth divisor in |f1 + f2| on|P1 × P1|.If D intersects (0, 0) then the restriction of D to the formal neighbourhood of C3 isgiven by:C[[s1, s2]][v]/(s1 − as2, v)for some a ∈ C∗. In this case we define canonical formal coordinates relative to Daround a C3 banana by the following change of coordinates.s˜1 7→ s1 − as2 and s˜2 7→ s1 + as2t˜1 7→ at1 + t2 and t˜2 7→ −at1 + t2We similarly define the same relative coordinates if for D intersects (∞,∞) in theideal (−at1 + t2, u). Note that these coordinates are compatible if D intersects both(0, 0 and (∞,∞).Definition 4.4.8. Let C[[s1, s2]][u] and C[[t1, t2]][v] be either the canonical coordinatesor relative coordinates.1. The canonical (C∗)2-action on these coordinates is defined by(s1, s2, v) 7→ (λ1s1, λ2s2, v) and (t1, t2, u) 7→ (λ2t1, λ1t2, u).2. Let λ = (λ1, . . . , λl, 0, . . .) be a 2D partition. The λ-thickened banana curveC3 denoted by λC3 is the subscheme of X defined by the ideal given in thecoordinates by(sλ12 , . . . , sl−11 sλl2 , sl1) and (tλ11 , . . . , tl−12 tλl1 , tl2).(Note the change in coordinates compared to definition 4.4.3.)Remark 4.4.9. If D intersects both (0, 0) and (∞,∞) and λC3 is partition thickenedin the coordinates relative to D. Then ideals for D ∪ λC3 at the points (0, 0) and(∞,∞) are(sλ12 , . . . , sl−11 sλl2 , sl1) ∩ (s1, v) and (tλ11 , . . . , tl−12 tλl1 , tl2) ∩ (t2, u)respectively. These both give 3D partitions asymptotic to (λ, ∅,).58 CHAPTER II. DT THEORY OF THE BANANA 3-FOLD WITH SECTION CLASSESLemma 4.4.10. Let D be as described in the first paragraph of 4.4.7. If let V be theformal neighbourhood of C3 in X . If D intersects (0, 0) and/or (∞,∞) then use therelative coordinates of 4.4.7, otherwise use the canonical coordinates of 4.4.5. Then D ∩ Vis invariant under the (C∗)2-action.Proof. We have D ∩ V 6= 0 if and only if it intersects at least one of (0, 0), (0,∞),(∞, 0), (∞,∞). The possible combinations are:1. (0, 0) and/or (∞,∞): This is by construction of the relative coordinates.2. Exactly one of (0,∞) or (∞, 0): Then D is given by the ideal (v − a, s1) or(v − a, s2) for some a ∈ C∗, which are (C∗)2-invariant.3. (0,∞) and (∞, 0): Then D is given by the ideal (v− a, s1s2) for some a ∈ C∗which is (C∗)2-invariant.4.4.11. It is also shown in [Br, §5.2] that there are the following formal coordinates onC2 compatible with the canonical formal coordinates around C3:C[[s1, v]][s2] and C[[t1, u]][t2]where the change on coordinates is given by s2 7→ t2, s1 7→ t1t2 and v 7→ t2u. Wecan define partition thickenings and a compatible (C∗)2-action in these coordinates.Definition 4.4.12. Let C[[s1, v]][s2] and C[[t1, u]][t2] be the above canonical coordi-nates.1. The canonical (C∗)2-action on these coordinates is defined by:(s1, v, s2) 7→ (λ1s1, v, λ2s2) and (t1, u, t2) 7→ (λ2t1, u, λ1t2).2. Let µ = (µ1, . . . , µk, 0, . . .) be a 2D partition. The µ-thickened banana curveC2 denoted by µC2 is the subscheme of X defined by the ideal given in thecoordinates by(sµ11 , . . . , vk−1sµk1 , vk) and (tµ11 , . . . , uk−1tµk1 , uk).(Note the change in coordinates compared to definition 4.4.8.)3. Let λ = (λ1, . . . , λl, 0, . . .) be another 2D partition. The (µ, λ)-thickenedbanana curve denoted is the union µC2 + λC3.Remark 4.4.13. The C2 and C3 banana curves meet in exactly 2 points. At these twopoints a (µ, λ)-thickened banana curve will define define two 3D partitions. One willbe asymptotic to (µ, λ, ∅) the other will be asymptotic to (µt, λt, ∅) (or equivalently(λ, µ, ∅)).Remark 4.4.14. The partition thickened curves described in this section are easilyshown to be the only Cohen-Macaulay subschemes supported in these neighbour-hoods that are invariant under the (C∗)2-action. This is because the invariant Cohen-Macaulay subschemes must be generated by monomial ideals.Lemma 4.4.15. Let λ = (λ1, . . . , λl, 0, . . .) and µ = (µ1, . . . , µk, 0, . . .) be a 2D parti-tions. Then we have the holomorphic Euler characteristics :1. χ(Oλσ) = 12(‖λ‖2 + ‖λt‖2),2. χ(Oλfx) = 0,3. χ(OµC2 ∪ λC3) = |η1|+ |η2|+ 12(‖µ‖2 + ‖µt‖2 + ‖λ‖2 + ‖λt‖2) where |ηi| arethe renormalised volumes of the minimal 3D partitions associated to (µ, λ, ∅) and(µt, λt, ∅).Proof. (2) is straightforward and the rest are from [Br, Prop. 23].4. TECHNIQUES FOR CALCULATING EULER CHARACTERISTIC 594.5 Relation between Quot Schemes on C3 and the Topological VertexThis section is predominately a summary of required results from [BK]. For 2D parti-tions λ, µ and ν we define the following subscheme of C3:Cλ,µ,ν = Cλ,∅,∅ ∪ C∅,µ,∅ ∪ C∅,∅,ν ⊂ SpecC[r, s, t]where Cλ,∅,∅ is defined by the idea Iλ,∅,∅ := (tλ1 , . . . , tl−1sλl , sl), with C∅,µ,∅ andC∅,∅,ν being cyclic permutations of this. Also define the ideal by Iλµν = Iλ∅∅ ∩ I∅µ∅ ∩I∅∅νNow we consider the Quot scheme of length n quotients that are supported at theorigin and we employ the following simplifying notation:Quotn(λ, µ, ν) := QuotnC3(Iλµν , {0})The quotients parametrised here have kernels that are the ideal sheaf of a one-dimensional scheme Z with underlying Cohen-Macaulay curve Cλ,µ,ν . The embeddedpoints of this scheme are all supported at the origin, but Z doesn’t have to be locallymonomial. We use the following variation of the notation for the topological vertex:V˜λµν := e(Quot•(λ, µ, ν)) ∈ Z[[p]].Lemma 4.5.1. Let C be a partition thickened section, fibre or C3-banana curve thickenedby λ. Then1. If x ∈ C is a smooth point then e(QuotnX(IC , {x})) = V˜λ∅∅.2. If C is a thickened nodal fibre then e(QuotnX(IC , {x}))= V˜λλt∅.Let C ′ be a reduced curve intersecting C at y ∈ C such that IC′ ∩ IC is locally monomialand there are formal local coordinates C[[r, s, t]] at y such that:1. IC′ ∩ IC = (tλ1 , . . . sl−1tλl , sl) ∩ (r, s) then e(QuotnX(IC , {x}))= V˜λ∅.2. IC′∩IC = (tλ1 , . . . sl−1tλl , sl)∩(r, s)∩(r, t) then e(QuotnX(IC , {x}))= V˜λ.Proof. The proof is the same as [BK] Lemma 15.Lemma 4.5.2. Let D be a one dimensional Cohen-Macaulay subscheme of X .1. We have:e(QuotnX(ID, X \D))=(V˜∅∅∅)e(X)−e(C).2. Let λ be a 2D partition and λC ⊂ D be either a partition thickened section, fibre orC3 banana and let T be finite set of points on C such that C \ T is smooth. Thene(QuotnX(ID, C \ T ))=(V˜λ∅∅)e(C)−e(T ).Proof. The argument is the same as that given for equation (9) in [BK].The standard (C∗)3-action on C3 induces an action on the Quot schemes. The in-variant ideals I ⊂ C[r, s, t] are precisely those generated by monomials. Also, sincethere is a bijection between locally monomial ideals and 3D partitions we see thatV˜λµν = e(Quot•(λ, µ, ν)(C∗)3)=∑ηpn(η)60 CHAPTER II. DT THEORY OF THE BANANA 3-FOLD WITH SECTION CLASSESwhere we are summing over 3D partitions asymptotic to (λ, µ, ν) and n(η) is thenumber of boxes not contained in any legs. Note that that the lowest order term inV˜λµν is one, which is not true about Vλµν in general. In fact we have the relationship:Vλµν = p|ηmin|V˜λµνwhere ηmin is the 3D partition associated to Cλµν , and | · | is the renormalised volumedefined in eqn (II.9).Lemma 4.5.3. If λ is a 2D partition then we have the following equalities:1. Vλ∅∅ = V˜λ∅∅2. Vλ∅ = p−λ1 V˜λ∅3. Vλ = p−λ1−λt1 V˜λ4. Vλλt∅ = p−‖λ‖2V˜λλt∅Proof. Parts (1), (2) and (4) are directly from [BK] lemma 17. For part 3, there are λ1boxes that are in the λ-leg and one of the -legs. There are λt1 boxes that are inthe λ-leg and the other -leg. There is one box that is contained in all three so therenormalised volume is calculated to be(λi − 1)(1− 2) + (λti − 1)(1− 2) + (1)(1− 3) = −λi − λtiLetΨ•,•(a,m) :=∑α`a∑µ`mp12 (‖α‖2+‖αt‖2+‖µ‖2+‖µt‖2)(V∅µαV∅µtαt)5 Calculating the Euler Characteristic from the Fibresof the Chow Map5.1 Calculation for the class σ + (0, •, •)We now recall some previously introduced notation:1. Bi = {b1i , . . . , b12i } is the set of the 12 points in Si that correspond to nodes inthe fibres of the projection pi : Si → P1.2. S◦i = Si \Bi is the complement of Bi in Si3. Ni ⊂ Si are the 12 nodal fibres of pi : Si → P1 with the nodes removed and:Ni = Nσi q N∅i where Nσi := Ni ∩ σ and N∅i := Ni \ σ.4. Smi = S◦i \ Ni is the complement of Ni in S◦i and:Smi = Smσi q Sm∅i where Smσi := Smi ∩ σ and Sm∅i := Smi \ σ.Now from lemma 3.5.1 we can further decompose Chowσ+(0,•,•)(X) as:Sym•(Smσ2 )× Sym•(Nσ2 )× Sym•(Sm∅2)× Sym•(N∅2)× Sym•(B2)× Sym•(Bop).5. EULER CHARACTERISTIC OF THE FIBRES OF THE CHOW MAP 61Figure II.9: Depiction of the decomposition of the Chow sub-scheme that parametrisesthe vertical fibres of pr2. The red dots indicate when the fibres don’t intersect thesection i.e. Sm∅2 and N∅2. The white dots indicate when the fibres do intersect thesection i.e. Smσ2 and Nσ2 .Moreover, if q = (ax, cy,dz, lw,mb2,nbop) ∈ Chowσ+(0,•,•)(X) then the fibre isgiven by ρ−1• (q) ∼= Hilb•Cyc(X, q) whereq = σ +∑iaipr−12 (xi) +∑icipr−12 (yi)+∑idipr−12 (zi) +∑ilipr−12 (wi) +∑imiC(i)2 +∑iniC(i)3 .5.1.1. Suppose C is Cohen-Macaulay with the cycle given above. Note that C can bedecomposed into a part supported on C2 and C3 and a part supported away fromthe banana configuration. This gives the following formal neighbourhoods and (C∗)2-actions:1. Let Ui be the formal neighbourhood of C(i)2 ∪C(i)3 in X . These have a canonical(C∗)2-action described in 4.4.8 and 4.4.12.2. Let Vi be the formal neighbourhood of pr−12 (yi) in X . These have a canonical(C∗)2-action described in definition 4.4.4 and σ∩Vi is either empty of invariantunder this action.Hence the conditions of lemma 4.2.1 are satisfied and there is a (C∗)2-action definedon HilbnCM(X, q). Using the partition thickened notation introduced in section 4.4 weintroduce the subschemes:Cα,γ,δ,λ,µ,ν := σ∪i(α(i)fxi)∪i(γ(i)fyi)∪i(δ(i)fzi)∪i(λ(i)fwi)∪i(µ(i)C(i)2)∪i(ν(i)C(i)3)and their ideals Iα,γ,δ,λ,µ,ν in X where α, γ, δ, λ, µ and ν are tuples of partitionsof a, c, d, l,m and n respectively. Then using this notation we can identify the fixedpoints of the action as the following discrete set:Hilb•CM(X, q)(C∗)2=∐α ` a, γ ` c, δ ` d,λ ` l, µ `m, ν ` n{Cα,γ,δ,λ,µ,ν}.62 CHAPTER II. DT THEORY OF THE BANANA 3-FOLD WITH SECTION CLASSESFigure II.10: Visual depictions of Cohen-Macaulay subschemes in the fibres of ρ•that are supported away from the banana configurations. On the left is a generalsubscheme and on the right is a partition thickened curve which is a fixed point of the(C∗)2-action.Using the result of 4.2.2 we havee(Hilb•Cyc(X, q))= e(Hilb•CM(X, q)(C∗)2 , κ∗1)=∑α ` a, γ ` c, δ ` d,λ ` l, µ `m, ν ` ne((Hilb•(X,Cα,γ,δ,λ,µ,ν))=∑α ` a, γ ` c, δ ` d,λ ` l, µ `m, ν ` ne(Quot•X(Iα,γ,δ,λ,µ,ν)).5.1.2. Using the decomposition method of 4.2.3 following method:1. Decompose X by X = W q Cα,γ,δ,λ,µ,ν where W := X \ Cα,γ,δ,λ,µ,ν .2. Let Cα,γ,δ,λ,µ,ν be set points given by the following disjoint sets:(a) σα := σ ∩ Credα(b) σγ := σ ∩ Credγ(c) Cγ the set of nodes of Cγ(d) Cλ the set of nodes of Cλ(e) B = ∪i(C(i)2 ∩ C(i)3 ).3. Denote the components supported on smooth reduced sub-curves by:(a) σ◦ := σ \ σ(b) C◦α := Cα \ σα(c) C◦γ := Cγ \ (σγ ∪ Cλ)(d) C◦λ := Cλ \ Cλ(e) C◦µ := Cµ \B(f) C◦ν := Cν \B5. EULER CHARACTERISTIC OF THE FIBRES OF THE CHOW MAP 635.1.3. Then applying Euler characteristic to lemma 4.1.4 we have:e(Quot•X(Iα,γ,δ,λ,µ,ν))= e(Quot•X(Iα,γ,δ,λ,µ,ν ,W ))e(Quot•X(Iα,γ,δ,λ,µ,ν , σ◦))e(Quot•X(Iα,γ,δ,λ,µ,ν , σα))e(Quot•X(Iα,γ,δ,λ,µ,ν , C◦α))e(Quot•X(Iα,γ,δ,λ,µ,ν , σγ))e(Quot•X(Iα,γ,δ,λ,µ,ν , Cγ))e(Quot•X(Iα,γ,δ,λ,µ,ν , C◦γ))e(Quot•X(Iα,γ,δ,λ,µ,ν , Cδ))e(Quot•X(Iα,γ,δ,λ,µ,ν , Cλ))e(Quot•X(Iα,γ,δ,λ,µ,ν , C◦λ))e(Quot•X(Iα,γ,δ,λ,µ,ν , B))e(Quot•X(Iα,γ,δ,λ,µ,ν , C◦µ))e(Quot•X(Iα,γ,δ,λ,µ,ν , C◦ν))Applying lemmas 4.5.1 and 4.5.2 we have:e(Quot•X(Iα,γ,δ,λ,µ,ν))=(V˜∅∅∅)e(W )(V˜∅∅)e(σ◦)∏i(p−αiV˜α(i)∅)∏i(V˜α(i)∅∅)−1∏i(p−γiV˜γ(i)∅)∏i(V˜γ(i)(γ(i))t∅)∏i(V˜γ(i)∅∅)−1∏i(V˜δ(i)∅∅)0∏i(V˜λ(i)(λ(i))t∅)∏i(V˜λ(i)∅∅)0∏i(pχ(Oµ(i)C(i)2 ∪ ν(i)C(i)3)V˜µ(i)ν(i)∅V˜(µ(i))t(ν(i))t∅)∏i(V˜µ(i)∅∅)0∏i(V˜ν(i)∅∅)0.We note that e(X) = 24 and e(σ) = 2 and:pχ(Oµ(i)C(i)2 ∪ ν(i)C(i)3)V˜µ(i)ν(i)∅V˜(µ(i))t(ν(i))t∅= p12 (‖µ(i)‖2+‖(µ(i))t‖2+‖ν(i)‖2+‖(ν(i))t‖2)Vµ(i)ν(i)∅V(µ(i))t(ν(i))t∅.So from lemma 4.5.3 we now have we havee(Quot•X(Iα,γ,δ,λ,µ,ν))=(V∅∅∅)24(V∅∅V∅∅∅)2∏i( V∅∅∅V∅∅Vα(i)∅Vα(i)∅∅)∏i( V∅∅∅V∅∅p‖γ(i)‖2 Vγ(i)∅Vγ(i)(γ(i))t∅V∅∅∅Vγ(i)∅∅)∏i(p‖λ(i)‖2 Vλ(i)(λ(i))t∅V∅∅∅)∏i(p12 (‖µ(i)‖2+‖(µ(i))t‖2+‖ν(i)‖2+‖(ν(i))t‖2)Vµ(i)ν(i)∅V(µ(i))t(ν(i))t∅V∅∅∅V∅∅∅)64 CHAPTER II. DT THEORY OF THE BANANA 3-FOLD WITH SECTION CLASSESWe now define the functions:1. gSmσ : Sym•(Smσ2 ) −→ Z((p)) is defined by gSmσ (a) = V∅∅∅V∅∅∑α`aVα∅Vα(i)∅∅,2. gNσ : Sym•(Nσ2 ) −→ Z((p)) is defined by gNσ (c) = V∅∅∅V∅∅∑γ`cp‖γ(i)‖2 Vγ∅Vγγt∅V∅∅∅Vγ∅∅,3. gSm∅ : Sym•(Sm∅2) −→ Z((p)) is defined by gSm∅(d) =∑δ`d1,4. gN∅ : Sym•(N∅2) −→ Z((p)) is defined by gN∅(l) =∑λ`lp‖λ(i)‖2 Vλλt∅V∅∅∅,5. gB : Sym•(B2)× Sym•(Bop) −→ Z((p)) is defined by the equationgB(m,n) =∑µ(i) ` miν(i) ` nip12 (‖µ(i)‖2+‖(µ(i))t‖2+‖ν(i)‖2+‖(ν(i))t‖2)Vµ(i)ν(i)∅V(µ(i))t(ν(i))t∅V∅∅∅V∅∅∅.So the constructible function (ρ•)∗1 : Chowσ+(0,•,•)(X) → Z((p)) is calculated forq = (ax, cy,dz, lw,mb2,nbop) by:((ρ•)∗1)(q)= e(ρ−1• (q))=∑α ` a, γ ` c, δ ` d,λ ` l, µ `m, ν ` ne(Quot•X(Iα,γ,δ,λ,µ,ν))=(V∅∅∅)24(V∅∅V∅∅∅)2∏igSmσ (ai)∏igNσ (ci)∏igSm∅(di)∏igN∅(li)∏igB(mi, ni).So we can now apply lemma 2.1.2 to obtain:e(Chowσ+(0,•,•)(X), (ρ•)∗1))=(V∅∅∅)24(V∅∅V∅∅∅)2( V∅∅∅V∅∅∑α(Q2Q3)|α|Vα∅Vα∅∅)e(Smσ)( V∅∅∅V∅∅∑γ(Q2Q3)|γ|p‖γ‖2 Vγ∅Vγγt∅V∅∅∅Vγ∅∅)e(Nσ)(∑δ(Q2Q3)|δ|)e(Sm∅)(∑λ(Q2Q3)|λ|p‖λ‖2 Vλλt∅V∅∅∅)e(N∅)e(Sym•(B2)× Sym•(Bop), GB)where GB is the constructible functionGB : Sym•(B2)× Sym•(Bop)→ Z((p))defined by GB(mb2, bop) :=12∏i=1gB(mi, ni). However, since B2 = {b12, . . . , b122 } andBop = {b1op, . . . , b12op} we have:Sym•(B2)× Sym•(Bop) ∼=12∏i=1Sym•({b(i)2 })× Sym•({b(i)op }).5. EULER CHARACTERISTIC OF THE FIBRES OF THE CHOW MAP 65Which gives us:e(Sym•(B2)× Sym•(Bop), GB)=12∏i=1e(Sym•({b(i)2 })× Sym•({b(i)op }), gB)=(∑µ,νQµ2Qν3p12 (‖µ‖2+‖µt‖2+‖ν‖2+‖νt‖2)Vµν∅Vµtνt∅V∅∅∅V∅∅∅)125.1.4. Applying the vertex formulas of lemmas 6.3.6, 6.3.2 and corollary 6.3.4 we havee(Chowσ+(0,•,•)(X), (ρ•)∗1))= M(p)24(11− p)2(∏d>0(1−Qd2Qd3)(1− pQd2Qd3)(1− p−1Qd2Qd3))−10(∏d>0M(p,Qd2Qd3)(1− pQd2Qd3)(1− p−1Qd2Qd3))12(∏d>01(1−Qd2Qd3))10(∏d>0M(p,Qd2Qd3)(1−Qd2Qd3))−12(∏d>0M(Qd2Qd3, p)2(1−Qd2Qd3)M(−Qd−12 Qd3, p)M(−Qd2Qd−13 , p))12=M(p)24(1− p)2∏d>01(1−Qd2Qd3)8(1− pQd2Qd3)2(1− p−1Qd2Qd3)2(∏d>0M(Qd2Qd3, p)2(1−Qd2Qd3)M(−Qd−12 Qd3, p)M(−Qd2Qd−13 , p))12Which completes the proof of theorem A.5.2 Preliminaries for classes of the form •σ + (i, j, •)We recall from lemma 3.5.3 that there is a decomposition of Chow•σ+(i,j,•)(X) suchthat for any point q ∈ Chow•σ+(i,j,•)(X) the fibre is(η•)−1(q) ∼= Hilb•Cyc(X,Cyc(C))for some one dimensional subscheme C of X withCyc(C) = q = aσ +D +12∑i=1miC(i)3 (II.10)where D is a one dimensional reduced subscheme of X . We see from lemma 3.5.3 thatthe intersection of D with σ has length 0, 1 or 2. We consider the following formalneighbourhoods around components of C :1. Let Ui be the formal neighbourhood of C(i)3 in X . These have a canonical(C∗)2-action described in 4.4.8 and the (C∗)2-invariance of D ∩ Ui is shown inlemma 4.4.10.66 CHAPTER II. DT THEORY OF THE BANANA 3-FOLD WITH SECTION CLASSESFigure II.11: Depiction of two typical curves (away from C3) in the class bσ + (1, 1, d).2. Let V be the formal neighbourhood of σ in X with the coordinates:(a) If #(D ∩ σ) = 0, 2 the let V have the canonical coordinates of 4.4.1 of and(C∗)2-action described in 4.4.3.(b) If #(D ∩ σ) = 1 the let V have the canonical coordinates of 4.4.2 of and(C∗)2-action described in 4.4.3.By construction the restrictions of D to these neighbourhoods are invariant underthese actions. Hence the conditions of lemma 4.2.1 are satisfied and there is a (C∗)2-action defined on HilbnCM(X,Cyc(C)). We introduce the notation for subschemes ofX :Cα,µ = Cα,µ(1),...,µ(12) = ασ ∪ D12∪i=1µiC(i)3and their ideals Iα,µ. Then using this notation we can identify the fixed points of theaction as the following discrete set:Hilb•CM(X, q)(C∗)2=∐α`a, µ`m{Cα,µ}.Using the result of 4.2.2 we havee(Hilb•Cyc(X, q))= e(Hilb•CM(X, q)(C∗)2 , κ∗1)=∑α`a, µ`mpχ(OCα,µ )e((Hilb•(X,Cα,µ)=∑α`a, µ`mpχ(OCα,µ )e(Quot•X(Iα,µ).Where the holomorphic Euler characteristic χ(OCα,µ) is given by the following lemma.Lemma 5.2.1. The holomorphic Euler characteristic of Cα,µ is:χ(OCα,µ) = χ(OD) +(χ(Oασ)− |D ∩ ασ|)+( 12∑i=1χ(Oµ(i)C(i)3)−12∑i=1|D ∩ µ(i)C(i)3 |).5. EULER CHARACTERISTIC OF THE FIBRES OF THE CHOW MAP 67Proof. This is immediate from the exact sequence decomposing Cα,µ into irreduciblecomponents:0→ OCα,µ → OD ⊕ χ(Oασ)⊕iOµ(i)C(i)3→ χ(OD∩ασ)⊕iOD∩µ(i)C(i)3→ 05.2.2. Using the decomposition method of 4.2.3 we take the following steps:1. Decompose X by X = W q Cα,µ where W := X \ Cα,µ.2. Let Cα,µ be set points given by the following disjoint sets:(a) D is the set of nodes of D \ (σ ∪i C(i)3 ).(b) D∗ is the set singularities of D \ (σ ∪i C(i)3 ) that are locally isomorphic itthe coordinate axes in C3.(c) σ := σ ∩D,(d) Bi = (C(i)3 ∩D) for i ∈ {1, . . . , 12},Note that D ∪D∗ is the set of singularities of D \ (σ ∪i C(i)3 ).3. Denote the components supported on smooth reduced sub-curves by:(a) D◦ = D \ (D ∪D∗),(b) σ◦ := σ \ σ,(c) B◦i = C(i)3 \Bi for i ∈ {1, . . . , 12}.5.2.3. Then applying Euler characteristic to lemma 4.1.4 we have:pχ(OCα,µ )e(Quot•X(Iα,µ)= e(Quot•X(Iα,µ,W)pχ(OD)e(Quot•X(Iα,µ, D◦)e(Quot•X(Iα,µ, D)e(Quot•X(Iα,µ, D∗)pχ(Oασ)−|D∩ασ|e(Quot•X(Iα,µ, σ◦)e(Quot•X(Iα,µ, σ)12∏i=1pχ(Oµ(i)C(i)3)−|D∩µ(i)C(i)3 |e(Quot•X(Iα,µ, B◦i)e(Quot•X(Iα,µ, Bi)5.2.4. We have that e(X) = 24 and e(σ) = e(C(i)3 ) = 2. So the Euler characteristicof W is:e(W ) = e(X)− e(σ)−12∑i=1e(C(i)3 )− e(D◦)− e(D)− e(D∗)= −2− e(D◦)− e(D)− e(D∗)Hence now have from lemma 4.5.2 that the first two lines from above will be:Ψ(D) := pχ(OD)(V˜∅∅∅)e(W )(V˜∅∅)e(D◦)(V˜∅)e(D)(V˜)e(D∗)= pχ(OD)(V∅∅∅)−2(V∅∅V∅∅∅)e(D◦)(pV∅V∅∅∅)e(D)(p2VV∅∅∅)e(D∗)The intersection of D and ασ will determine the third line. From lemma 4.5.2 andlemma 4.5.3 it will be one of:68 CHAPTER II. DT THEORY OF THE BANANA 3-FOLD WITH SECTION CLASSES1. p12 (‖α‖2+‖αt‖2)(V˜∅∅αV˜∅∅α)= p12 (‖α‖2+‖αt‖2)(V∅∅αV∅∅αt)2. p12 (‖α‖2+‖αt‖2)−l(αt)(V˜∅αV˜∅∅α)= p12 (‖α‖2+‖αt‖2)(V∅αV∅∅αt)3. p12 (‖α‖2+‖αt‖2)−l(αt)−l(α)(V˜∅αV˜∅α)= p12 (‖α‖2+‖αt‖2)(V∅αV∅αt)4. p12 (‖α‖2+‖αt‖2)−(l(α)+l(αt)−1)(V˜αV˜∅∅α)= p12 (‖α‖2+‖αt‖2)+1(VαV∅∅αt)Similarly the factors of the fourth line will determined by the intersections D∩C(i)3 tobe (the fourth comes from 4.4.9):1. p12 (‖α‖2+‖αt‖2)(V˜∅∅αV˜∅∅αt)= p12 (‖α‖2+‖αt‖2)(V∅∅αV∅∅αt)2. p12 (‖α‖2+‖αt‖2)−(l(αt)+l(α))(V˜∅αV˜∅αt)= p12 (‖α‖2+‖αt‖2)(V∅αV∅αt)3. p12 (‖α‖2+‖αt‖2)−(l(α)+l(αt))(V˜∅αV˜∅αt)= p12 (‖α‖2+‖αt‖2)(V∅αV∅αt)4. p12 (‖α‖2+‖αt‖2)−2l(αt)(V˜∅α)2= p12 (‖α‖2+‖αt‖2)(V∅α)25. p12 (‖α‖2+‖αt‖2)−2(l(α)+l(αt)−1)(V˜αV˜αt)= p12 (‖α‖2+‖αt‖2)+2(VαVαt)5.2.5. We can calculate e(Hilb•Cyc(X, q))using the above results and notation from5.2.4:e(Hilb•Cyc(X, q))=∑α`a, µ`mpχ(OCα,µ )e(Quot•X(Iα,µ)= Ψ(D)Φσ(a)12∏i=1Φi(mi).where Φσ and Φi are determined by the intersections of σ and C(i)3 respectively to beone of the following functions:1. Φ∅,∅(a) :=∑α`ap12 (‖α‖2+‖αt‖2)(V∅∅αV∅∅αt)2. Φ−,∅(a) :=∑α`ap12 (‖α‖2+‖αt‖2)(V∅αV∅∅αt)3. Φ−,−(a) :=∑α`ap12 (‖α‖2+‖αt‖2)(V∅αV∅αt)4. Φ−, | (a) :=∑α`ap12 (‖α‖2+‖αt‖2)(V∅α)25. Φ+,∅(a) :=∑α`ap12 (‖α‖2+‖αt‖2)+1(VαV∅∅αt)6. Φ+,+(a) :=∑α`ap12 (‖α‖2+‖αt‖2)+2(VαVαt)5. EULER CHARACTERISTIC OF THE FIBRES OF THE CHOW MAP 695.3 Calculation for the class •σ + (0, 0, •)From lemma 3.5.3 have the decomposition of Chow•σ+(0,0,•)(X) into:Z≥0 × Sym•(Bop)Recall equation (II.10) from section 5.2 and the notation:Cyc(C) = aσ +D +12∑i=1miC(i)3 .In this class we have D = ∅. Hence we have the following summary of results from5.2.4 and 5.2.5.χ(OD) = 0e(η-1• (a,m)) =1(V∅∅∅)2·QaσΦ∅,∅(a) ·12∏i=1Qmi3 Φ∅,∅(mi)Now we have:e(Z≥0 × Sym•(Bop), (η•)∗1)=1(V∅∅∅)2(∑aQaσΦ∅,∅(a))(∑mQm3 Φ∅,∅(m))12= M(p)24∏m>0(1 + pmQσ)m(1 + pmQ3)12m.Where the last equality is from 6.3.4 part 2 and 6.3.2 part 1.5.4 Calculation for the class •σ + (0, 1, •)Recall the previously introduced notation:1. Bi = {b1i , . . . , b12i } is the set of the 12 points in Si that correspond to nodes inthe fibres of the projection pi : Si → P1.2. S◦i = Si \Bi is the complement of Bi in Si3. Ni ⊂ Si are the 12 nodal fibres of pi : Si → P1 with the nodes removed and:Ni = Nσi q N∅i where Nσi := Ni ∩ σ and N∅i := Ni \ σ.4. Smi = S◦i \ Ni is the complement of Ni in S◦i and:Smi = Smσi q Sm∅i where Smσi := Smi ∩ σ and Sm∅i := Smi \ σ.Now from lemma 3.5.3 we can further decompose Chow•σ+(0,1,•)(X) into the fourparts:1. Z≥0 × Smσ2 × Sym•(Bop)2. Z≥0 × Sm∅2 × Sym•(Bop)3. Z≥0 × Nσ2 × Sym•(Bop)4. Z≥0 × N∅2 × Sym•(Bop)5.12qk=1Z≥0 × Sym•({bkop})× Sym•(Bop \ {bkop})70 CHAPTER II. DT THEORY OF THE BANANA 3-FOLD WITH SECTION CLASSESRecall equation (II.10) from section 5.2 and the notation:Cyc(C) = aσ +D +12∑i=1miC(i)3 .Each part will be characterised by the type of D. We consider parts (1)-(4) separatelyto part (5).5.4.1. Parts (1)-(4): In parts (1)-(4) the curve D is a fibre of the projection pr2 : X → S.The following table is the summary of results from 5.2.4 and 5.2.5 when applied to theparticular D’s arising in each strata:Z≥0 × U × Sym•(Bop).U = Nσ1 e(U) = 12 χ(OD) = 0e(η-1• (a, x,m)) = Q1Q3p(V∅)(V∅∅)(V∅∅∅)2·QaσΦ−,∅(a) ·12∏i=1Qmi3 Φ∅,∅(mi)U = N∅1 e(U) = −12 χ(OD) = 0e(η-1• (a, x,m)) = Q1Q3p(V∅)(V∅∅∅)3·QaσΦ∅,∅(a) ·12∏i=1Qmi3 Φ∅,∅(mi)U = Smσ1 e(U) = −10 χ(OD) = 0e(η-1• (a, x,m)) = Q1Q31(V∅∅)(V∅∅∅)·QaσΦ−,∅(a) ·12∏i=1Qmi3 Φ∅,∅(mi)U = Sm∅1 e(U) = 10 χ(OD) = 0e(η-1• (a, x,m)) = Q1Q31(V∅∅∅)2·QaσΦ∅,∅(a) ·12∏i=1Qmi3 Φ∅,∅(mi)The union of parts (1)-(4) is Z≥0 × S◦2 × Sym•(Bop) so we have:e(Z≥0 × S◦2 × Sym•(Bop), (η•)∗1)= e(Z≥0 × Smσ2 × Sym•(Bop), (η•)∗1)q e(Z≥0 × Sm∅2 × Sym•(Bop), (η•)∗1)q e(Z≥0 × Nσ2 × Sym•(Bop), (η•)∗1)q e(Z≥0 × N∅2 × Sym•(Bop), (η•)∗1)5. EULER CHARACTERISTIC OF THE FIBRES OF THE CHOW MAP 71Which becomes:e(Z≥0 × S◦2 × Sym•(Bop), (η•)∗1)= e(Smσ2 )Q1Q3p(V∅)(V∅∅)(V∅∅∅)2(∑aQaσΦ−,∅(a))(∑mQm3 Φ∅,∅(m))12q e(Sm∅2)Q1Q3p(V∅)(V∅∅∅)3∑a≥0QaσΦ∅,∅(a)∑m≥0Qm3 Φ∅,∅(m)12q e(Nσ2 )Q1Q31(V∅∅)(V∅∅∅)∑a≥0QaσΦ−,∅(a)∑m≥0Qm3 Φ∅,∅(m)12q e(N∅2)Q1Q31(V∅∅∅)2∑a≥0QaσΦ∅,∅(a)∑m≥0Qm3 Φ∅,∅(m)12From lemmas 6.3.2, 6.3.4 and 6.3.5 we have:1. V∅∅∅ = M(p)2. V∅∅ = M(p) 11−p3. V∅ = M(p)p2−p+1p(1−p)24.∑m≥0QmΦ∅,∅(m) = M(p)2∏m>0(1 + pmQ)m5.∑m≥0QmΦ−,∅(m) = M(p)21 +Q1− p∏m>0(1 + pmQ)mSo we have:e(Z≥0 × S◦2 × Sym•(Bop), (η•)∗1)= QσQ1Q3M(p)24(∏m>0(1 + pmQσ)m(1 + pmQ3)12m)(2 + 12p(1− p)2)= QσQ1Q3M(p)24(∏m>0(1 + pmQσ)m(1 + pmQ3)12m)(2ψ1 + 12ψ0)5.4.2. Part (5): We have 12 separate isomorphic strata:Z≥0 × Sym•({bkop})× Sym•(Bop \ {bkop}).These parameterise when D = C(k)2 . The following is the summary of results from5.2.4 and 5.2.5.U = {k} e(U) = 1 χ(OD) = 1e(η-1• (a,mk,m)) =Q1p1(V∅∅∅)2·QaσΦ∅,∅(a) ·Qmk3 Φ−,−(mk) ·12∏i = 1i 6= kQmi3 Φ∅,∅(mi)From lemmas 6.3.2 and 6.3.4 we have:1. V∅∅∅ = M(p)72 CHAPTER II. DT THEORY OF THE BANANA 3-FOLD WITH SECTION CLASSES2.∑m≥0QmΦ∅,∅(m) = M(p)2∏m>0(1 + pmQ)m3.∑m≥0QmΦ−,−(m) = M(p)2(ψ0 + (ψ1 + 2ψ0)Q+ ψ0Q2)∏m>0(1 + pmQ)mSince the strata are isomorphic we have:e( 12qk=1Z≥0 × Sym•({bkop})× Sym•(Bop \ {bkop}), (η•)∗1)= 12 e(Z≥0 × Sym•({bkop})× Sym•(Bop \ {bkop}), (η•)∗1)= 12 Q11(V∅∅∅)2∑a≥0QaσΦ∅,∅(a)∑m≥0Qm3 Φ−,−(m)∑m≥0Qm3 Φ∅,∅(m)11= 12 Q1M(p)24(∏m>0(1 + pmQσ)m(1 + pmQ3)12m)(ψ0 + (ψ1 + 2ψ0)Q3 + ψ0Q23)5.4.3. Thus combining parts (1)-(5) we have that the overall formula is:e(Chow•σ+(0,1,•)(X), (η•)∗1)= Q1M(p)24(∏m>0(1 + pmQσ)m(1 + pmQ3)12m)·(12(ψ0 + (2ψ0 + ψ1)Q3 + ψ0Q23)+QσQ3(12ψ0 + 2ψ1))5.5 Calculation for the class •σ + (1, 1, •)We have a decomposition from lemma 3.5.3 of Chow(1,1,•)(X) into the parts:(a) S◦1 × S◦2 × Sym•(Bop)(b)12qk=1S◦1 × Sym•({bkop})× Sym•(Bop \ {bkop})(c)12qk=1S◦2 × Sym•({bkop})× Sym•(Bop \ {bkop})(d)12qk, l = 1k 6= lSym•({bkop})× Sym•({blop})× Sym•(Bop \ {bkop, blop})(e)12qk=1Sym•({bkop})× Sym•(Bop \ {bkop})(f) q Diag•We also recall the notation from equation (II.10) from section 5.2 and the notation:Cyc(C) = aσ +D +12∑i=1miC(i)3 .Each part will be characterised by the type of D. We will consider each case (a)-(f)separately and will use the following the previously introduced notation throughout:5. EULER CHARACTERISTIC OF THE FIBRES OF THE CHOW MAP 731. Bi = {b1i , . . . , b12i } is the set of the 12 points in Si that correspond to nodes inthe fibres of the projection pi : Si → P1.2. S◦i = Si \Bi is the complement of Bi in Si3. Ni ⊂ Si are the 12 nodal fibres of pi : Si → P1 with the nodes removed and:Ni = Nσi q N∅i where Nσi := Ni ∩ σ and N∅i := Ni \ σ.4. Smi = S◦i \ Ni is the complement of Ni in S◦i and:Smi = Smσi q Sm∅i where Smσi := Smi ∩ σ and Sm∅i := Smi \ σ.We will also use the new notation:D :={(x, x) ∈ S◦1 × S}.5.5.1. Part (a): We have the following stratification of S◦1 × S◦1 :1.((Nσ1 × Nσ2 ) ∩D q (Smσ1 × Smσ2 ) ∩D)2. q(Nσ1 × Nσ2 \D q Nσ1 × Smσ2 q Smσ1 × Nσ2 q Smσ1 × Smσ2 \D)3. q(Nσ1 × N∅2 \D q (Nσ1 × N∅2) ∩D q Nσ1 × Sm∅2 q Smσ1 × N∅2q Smσ1 × Sm∅2 \D q (Sm∅1 × Smσ2 ) ∩D)4. q(N∅1 × Nσ2 \D q (N∅1 × Nσ2 ) ∩D q N∅1 × Smσ2 q Sm∅1 × Nσ2q Sm∅1 × Smσ2 \D q (Smσ1 × Sm∅2) ∩D)5. q(N∅1 × N∅2 \D q (N∅1 × N∅2) ∩D q Sm∅1 × N∅2 q N∅1 × Sm∅2q Sm∅1 × Sm∅2 \D q (Sm∅1 × Sm∅2) ∩D)Here we have grouped by the number and type of intersection with σ.Grouping (1): The following table is the summary of results from 5.2.4 and 5.2.5 for thestrata in grouping (1):Z≥0 × U × Sym•(Bop).U = (Nσ1 × Nσ2 ) ∩ D e(U) = 12 χ(OD) = −1e(η-1• (a, x,m)) =Q1Q2Q23p(V∅)2(V∅∅)2(V∅∅∅)2·QaσΦ+,∅(a) ·12∏i=1Qmi3 Φ∅,∅(mi)U=(Smσ1 × Smσ2 ) ∩ D e(U)=−10 χ(OD) = −1e(η-1• (a, x,m)) = Q1Q2Q23p−1 1(V∅∅)2·QaσΦ+,∅(a) ·12∏i=1Qmi3 Φ∅,∅(mi)From lemmas 6.3.2 and 6.3.4 we have:1. V∅∅∅ = M(p)2. V∅∅ = M(p) 11−p3. V∅ = M(p)p2−p+1p(1−p)274 CHAPTER II. DT THEORY OF THE BANANA 3-FOLD WITH SECTION CLASSES4.∑m≥0QmΦ∅,∅(m) = M(p)2∏m>0(1 + pmQ)m5.∑m≥0QmΦ+,∅(m) = M(p)2(1+ψ0 +(ψ1 +2ψ0)Q+ψ0Q2)∏m>0(1+pmQ)m.So the contribution is:Q1Q2Q23M(p)24(∏m>0(1 + pmQσ)m(1 + pmQ3)12m)(ψ0 + (ψ1 + 2ψ0)Qσ + ψ0Q2σ)·(2(p4 + 8p3 − 12p2 + 8p+ 1)(p− 1)2p)Grouping (2): The following table is the summary of results from 5.2.4 and 5.2.5 forthe strata in grouping (2):Z≥0 × U × Sym•(Bop).U = Nσ1 × Nσ2 \ D e(U) = 132 χ(OD) = 0e(η-1• (a, x,m)) =Q1Q2Q23p2 (V∅)2(V∅∅)2(V∅∅∅)2·QaσΦ−,−(a) ·12∏i=1Qmi3 Φ∅,∅(mi)U = Nσ1 × Smσ2 e(U) = −120 χ(OD) = 0e(η-1• (a, x,m)) =Q1Q2Q23p(V∅)(V∅∅)2(V∅∅∅)·QaσΦ−,−(a) ·12∏i=1Qmi3 Φ∅,∅(mi)U = Smσ1 × Nσ2 e(U) = −120 χ(OD) = 0e(η-1• (a, x,m)) =Q1Q2Q23p(V∅)(V∅∅)2(V∅∅∅)·QaσΦ−,−(a) ·12∏i=1Qmi3 Φ∅,∅(mi)U=Smσ1 × Smσ2 \ D e(U) = 110 χ(OD) = 0e(η-1• (a, x,m)) = Q1Q2Q231(V∅∅)2·QaσΦ−,−(a) ·12∏i=1Qmi3 Φ∅,∅(mi)From lemmas 6.3.2 and 6.3.4 we have:1. V∅∅∅ = M(p)2. V∅∅ = M(p) 11−p3. V∅ = M(p)p2−p+1p(1−p)24.∑m≥0QmΦ∅,∅(m) = M(p)2∏m>0(1 + pmQ)m5.∑m≥0QmΦ−,−(m) = M(p)21p(ψ0 + (ψ1 + 2ψ0)Q+ ψ0Q2)∏m>0(1 + pmQ)m.5. EULER CHARACTERISTIC OF THE FIBRES OF THE CHOW MAP 75So the contribution is:Q1Q2Q23M(p)24(∏m>0(1 + pmQσ)m(1 + pmQ3)12m)1p(ψ0 + (ψ1 + 2ψ0)Q+ ψ0Q2)·(2(p4 + 8p3 + 48p2 + 8p+ 1)(p− 1)2)Grouping (3): The following table is the summary of results from 5.2.4 and 5.2.5 forthe strata in grouping (3):Z≥0 × U × Sym•(Bop).U = Nσ1 × N∅2 \ D e(U) = −132 χ(OD) = 0e(η-1• (a, x,m)) =Q1Q2Q23p2 (V∅)2(V∅∅)(V∅∅∅)3·QaσΦ−,∅(a) ·12∏i=1Qmi3 Φ∅,∅(mi)U = (Nσ1 × N∅2) ∩ D e(U)=−12 χ(OD) = −1e(η-1• (a, x,m)) =Q1Q2Q23p2 (V∅)3(V∅∅)3(V∅∅∅)2·QaσΦ−,∅(a) ·12∏i=1Qmi3 Φ∅,∅(mi)U = Nσ1 × Sm∅2 e(U) = 120 χ(OD) = 0e(η-1• (a, x,m)) =Q1Q2Q23p(V∅)(V∅∅)(V∅∅∅)2·QaσΦ−,∅(a) ·12∏i=1Qmi3 Φ∅,∅(mi)U = Smσ1 × N∅2 e(U) = 120 χ(OD) = 0e(η-1• (a, x,m)) =Q1Q2Q23p(V∅)(V∅∅)(V∅∅∅)2·QaσΦ−,∅(a) ·12∏i=1Qmi3 Φ∅,∅(mi)U=Smσ1 × Sm∅2 \ D e(U) = −110 χ(OD) = 0e(η-1• (a, x,m)) = Q1Q2Q231(V∅∅)(V∅∅∅)·QaσΦ−,∅(a) ·12∏i=1Qmi3 Φ∅,∅(mi)U=(Sm∅1 × Smσ2 ) ∩ D e(U) = 10 χ(OD) = −1e(η-1• (a, x,m)) = Q1Q2Q23(V∅)(V∅∅)3·QaσΦ−,∅(a) ·12∏i=1Qmi3 Φ∅,∅(mi)From lemmas 6.3.2, 6.3.4 and 6.3.5 we have:1. V∅∅∅ = M(p)2. V∅∅ = M(p) 11−p76 CHAPTER II. DT THEORY OF THE BANANA 3-FOLD WITH SECTION CLASSES3. V∅ = M(p)p2−p+1p(1−p)24.∑m≥0QmΦ∅,∅(m) = M(p)2∏m>0(1 + pmQ)m5.∑m≥0QmΦ−,∅(m) = M(p)21 +Q1− p∏m>0(1 + pmQ)mThe contribution from grouping (3) is:Q1Q2Q23M(p)24(∏m>0(1 + pmQσ)m(1 + pmQ3)12m)(1 +Qσ1− p)·(2(p2 + 10p+ 1) (p4 − 2p3 + 8p2 − 2p+ 1)(p− 1)3p)Grouping (4): The results for grouping (4) are identical to those of grouping (3) underthe symmetry of the banana threefold.The contribution from grouping (4) is:Q1Q2Q23M(p)24(∏m>0(1 + pmQσ)m(1 + pmQ3)12m)(1 +Qσ1− p)·(2(p2 + 10p+ 1) (p4 − 2p3 + 8p2 − 2p+ 1)(p− 1)3p)Grouping (5): The following table is the summary of results from 5.2.4 and 5.2.5 forthe strata in grouping (5):Z≥0 × U × Sym•(Bop).U = N∅1 × N∅2 \ D e(U) = 132 χ(OD) = 0e(η-1• (a, x,m)) = Q1Q2Q23p2 V∅)2(V∅∅∅)4·QaσΨ∅,∅(a) ·12∏i=1Qmi3 Φ∅,∅(mi)U = (N∅1 × N∅2) ∩ D e(U) = 12 χ(OD) = −1e(η-1• (a, x,m)) =Q1Q2Q23p2 (V∅)3(V∅∅)2(V∅∅∅)3·QaσΨ∅,∅(a) ·12∏i=1Qmi3 Φ∅,∅(mi)U = Sm∅1 × N∅2 e(U) = −120 χ(OD) = 0e(η-1• (a, x,m)) = Q1Q2Q23p(V∅)(V∅∅∅)3·QaσΨ∅,∅(a) ·12∏i=1Qmi3 Φ∅,∅(mi)5. EULER CHARACTERISTIC OF THE FIBRES OF THE CHOW MAP 77U = N∅1 × Sm∅2 e(U) = −120 χ(OD) = 0e(η-1• (a, x,m)) = Q1Q2Q23p(V∅)(V∅∅∅)3·QaσΨ∅,∅(a) ·12∏i=1Qmi3 Φ∅,∅(mi)U=Sm∅1 × Sm∅2 \ D e(U) = 110 χ(OD) = 0e(η-1• (a, x,m)) = Q1Q2Q231(V∅∅∅)2·QaσΨ∅,∅(a) ·12∏i=1Qmi3 Φ∅,∅(mi)U=(Sm∅1 × Sm∅2) ∩ D e(U)=−10 χ(OD) = −1e(η-1• (a, x,m)) = Q1Q2Q23(V∅)(V∅∅)2(V∅∅∅)·QaσΨ∅,∅(a) ·12∏i=1Qmi3 Φ∅,∅(mi)From lemmas 6.3.2 and 6.3.4 we have:1. V∅∅∅ = M(p)2. V∅∅ = M(p) 11−p3. V∅ = M(p)p2−p+1p(1−p)24.∑m≥0QmΦ∅,∅(m) = M(p)2∏m>0(1 + pmQ)mQ1Q2Q23M(p)24(∏m>0(1 + pmQσ)m(1 + pmQ3)12m)·(2(p2 + 10p+ 1) (p4 − 2p3 + 8p2 − 2p+ 1)(p− 1)4p)Summing the contributions from the above groupings we arrive at the overall contri-bution from part (a):e(Z≥0 × S◦1 × S◦2 × Sym•(Bop), (η•)∗1)= Q1Q2Q23QσM(p)24(∏m>0(1 + pmQσ)m(1 + pmQ3)12m)· (120ψ0 +Qσ (144ψ20 + 48ψ0 + 4))5.5.2. Part (b)-(c): By the symmetry of X we only need to consider part (b), with part(c) being completely analogous. For each k ∈ {1, . . . , 12} we begin by decomposingS◦1 into the following six parts:Smσ1 q Sm∅1 q Nσ,(k)1 q Nσ,c1 q N∅,(k)1 q N∅,c1where Nσ,(k)1 is the connected component of Nσ1 corresponding the the kth bananaconfiguration and Nσ,c1 is its complement in Nσ1 . The same definition is true for N∅1.78 CHAPTER II. DT THEORY OF THE BANANA 3-FOLD WITH SECTION CLASSESWe use the above size part decomposition forZ≥0 × S◦1 × Sym•({bkop})× Sym•(Bop \ {bkop}).The following table is the summary of results from 5.2.4 and 5.2.5 for this stratification.U = Nσ,c1 e(U) = 11 χ(OD) = 1e(η-1• (a, x, n,m)) =Q1Q2Q3p2 (V∅)(V∅∅)(V∅∅∅)2·QaσΦ−,∅(a) ·Qn3 Φ-,-(n)12∏i = 1i 6= jQmi3 Φ∅,∅(mi)U = N∅,c1 e(U) = −11 χ(OD) = 1e(η-1• (a, x, n,m)) =Q1Q2Q3p2 (V∅)(V∅∅∅)3·QaσΦ∅,∅(a) ·Qn3 Φ-,-(n)12∏i = 1i 6= jQmi3 Φ∅,∅(mi)U = Nσ,(k)1 e(U) = 1 χ(OD) = 0e(η-1• (a, x, n,m)) =Q1Q2Q3p2 (V)(V∅∅)2(V∅∅∅)·QaσΦ−,∅(a) ·Qn3 Φ-,-(n)12∏i = 1i 6= jQmi3 Φ∅,∅(mi)U = N∅,(k)1 e(U) = −1 χ(OD) = 0e(η-1• (a, x, n,m)) =Q1Q2Q3p2 (V)(V∅∅)(V∅∅∅)2·QaσΦ∅,∅(a) ·Qn3 Φ-,-(n)12∏i = 1i 6= jQmi3 Φ∅,∅(mi)U = Smσ1 e(U) = −10 χ(OD) = 1e(η-1• (a, x, n,m)) =Q1Q2Q3p1(V∅∅)(V∅∅∅)·QaσΦ−,∅(a) ·Qn3 Φ-,-(n)12∏i = 1i 6= jQmi3 Φ∅,∅(mi)U = Sm∅1 e(U) = 10 χ(OD) = 1e(η-1• (a, x, n,m)) =Q1Q2Q3p1(V∅∅∅)2·QaσΦ∅,∅(a) ·Qn3 Φ-,-(n)12∏i = 1i 6= jQmi3 Φ∅,∅(mi)From lemmas 6.3.2, 6.3.4 and 6.3.5 we have:1. V∅∅∅ = M(p)2. V∅∅ = M(p) 11−p3. V∅ = M(p)p2−p+1p(1−p)24. V = M(p)p4−p3+p2−p+1p2(1−p)35.∑m≥0QmΦ∅,∅(m) = M(p)2∏m>0(1 + pmQ)m5. EULER CHARACTERISTIC OF THE FIBRES OF THE CHOW MAP 796.∑m≥0QmΦ−,∅(m) = M(p)21 +Q1− p∏m>0(1 + pmQ)m7.∑m≥0QmΦ−,−(m) = M(p)21p(ψ0 + (ψ1 + 2ψ0)Q+ ψ0Q2)∏m>0(1 + pmQ)m.There are 12 singular fibres of pr. So, we have that the combined contribution fromparts (c) and (d) is:e( 12qk=1S◦1 × Sym•({bkop})× Sym•(Bop \ {bkop}), (η•)∗1)+e( 12qk=1S◦2 × Sym•({bkop})× Sym•(Bop \ {bkop}), (η•)∗1)= 24Q1Q2Q3QσM(p)24(∏m>0(1 + pmQσ)m(1 + pmQ3)12m)· (ψ0 + (ψ1 + 2ψ0)Q3 + ψ0Q23) (12ψ0 + 4ψ1 + ψ2)5.5.3. Part (d)-(e): Parts (d) and (e) parametrise the cases when D is the union of C(k)2and C(l)2 . We have the spaces:1.12qk, l = 1k 6= lSym•({bkop})× Sym•({blop})× Sym•(Bop \ {bkop, blop}),2.12qk=1Sym•({bkop})× Sym•(Bop \ {bkop}).The following table is the summary of results from 5.2.4 and 5.2.5 for this stratification.U = {(k, l)}, k 6= l e(U) = 1 χ(OD) = 2e(η-1• (a, c, d,m)) =Q1Q2p2 1(V∅∅∅)2·QaσΦ∅,∅(a) ·Qc3Φ-,-(c) ·Qd3Φ-,-(d)12∏i = 1i 6= j, kQmi3 Φ∅,∅(mi)U = {(k, k)} e(U) = 1 χ(OD) = 0e(η-1• (a,mk,m)) =Q1Q21(V∅∅∅)2·QaσΨ∅,∅(a) ·Qmk3 Φ+,+(mk)12∏i = 1i 6= jQmi3 Φ∅,∅(mi)From lemmas 6.3.2, 6.3.4 and 6.3.5 we have:1. V∅∅∅ = M(p)2.∑m≥0QmΦ∅,∅(m) = M(p)2∏m>0(1 + pmQ)m3.∑m≥0QmΦ−,−(m) = M(p)21p(ψ0 + (ψ1 + 2ψ0)Q+ ψ0Q2)∏m>0(1 + pmQ)m.4.∑m≥0QmΦ+,+(m)= M(p)2∏m>0(1+pmQ)m(Q4(2ψ0+ψ1)+Q3(8ψ0+6ψ1+ψ2)+Q2(12ψ0+10ψ1 +2ψ2)+Q(8ψ0 +6ψ1 +ψ2)+(2ψ0 +ψ1))80 CHAPTER II. DT THEORY OF THE BANANA 3-FOLD WITH SECTION CLASSESThere are 136 choices for two distinct fibres. Hence the contribution from part (d) is:e( 12qk, l = 1k 6= lSym•({bkop})× Sym•({blop})× Sym•(Bop \ {bkop, blop}), (η•)∗1)= 132Q1Q2M(p)24(∏m>0(1 + pmQσ)m(1 + pmQ3)12m)(ψ0 + (ψ1 + 2ψ0)Q3 + ψ0Q23)2.The 12 singular fibres give the contribution of (e) as:e( 12qk=1Sym•({bkop})× Sym•(Bop \ {bkop}), (η•)∗1)= 12Q1Q2M(p)24(∏m>0(1 + pmQσ)m(1 + pmQ3)12m)·((Q23ψ0 +Q3(2ψ0 + ψ1) + ψ0)2+(Q43(2ψ0 + ψ1) +Q33(8ψ0 + 6ψ1 + ψ2)+Q23(12ψ0 + 10ψ1 + 2ψ2)+Q3(8ψ0 + 6ψ1 + ψ2) + (2ψ0 + ψ1))).Summing the contributions of (d) and (e) we have:Q1Q2M(p)24(∏m>0(1 + pmQσ)m(1 + pmQ3)12m)·(144(Q23ψ0 +Q3(2ψ0 + ψ1) + ψ0)2+ 12(Q43(2ψ0 + ψ1) +Q33(8ψ0 + 6ψ1 + ψ2)+Q23(12ψ0 + 10ψ1 + 2ψ2)+Q3(8ψ0 + 6ψ1 + ψ2) + (2ψ0 + ψ1))).5.5.4. Part (f): Recall from lemma 3.5.3 that part (f), Diag• has the further decompo-sition:(g) Sm1 × Sym•(Bop)(h) q Sm2 × Sym•(Bop)(i) qy∈J Epi(y) × A˜ut(Epi(y))× Sym•(Bop)(j)12qk=1L× Sym•({bkop})× Sym•(Bop \ {bkop}).Where we have used the notation:1. J0 and J1728 to be the subsets of points x ∈ P1 such that pi−1(x) has j-invariant0 or 1728 respectively and J = J0 q J1728.2. L to be the linear system |f1 +f2| on P1×P1 with the singular divisors removedwhere f1 and f2 are fibres of the two projection maps.3. A˜ut(E) := Aut(E) \ {±1}.5.5.5. Parts (g)-(i):The results for parts (g)-(i) will all be very similar. The key differences are:1. The overall factor of Q3 may be different.5. EULER CHARACTERISTIC OF THE FIBRES OF THE CHOW MAP 812. The Euler characteristics of the space parametrising the D’s may be different.We define U to be one of(g) Sm1 noting that e(Sm1 ∩ {σ}) = −10 and e(Sm1 \ {σ}) = 10.(h) Sm2 noting that e(Sm2 ∩ {σ}) = −10 and e(Sm2 \ {σ}) = 10.(i) Epi(y) for y ∈ J noting that e(Epi(y) ∩ {σ}) = 1 and e(Epi(y) \ {σ}) = −1.U ∩ {σ} χ(OD) = 0e(η-1• (a, x,mk,m)) =Q1Q2Qn31(V∅∅)(V∅∅∅)·QaσΦ−,∅(a) ·12∏i=1Qmi3 Φ∅,∅(mi)U \ {σ} χ(OD) = 0e(η-1• (a, x,mk,m)) = Q1Q2Qn31(V∅∅∅)2·QaσΦ∅,∅(a) ·12∏i=1Qmi3 Φ∅,∅(mi)From lemmas 6.3.2, 6.3.4 and 6.3.5 we have:1. V∅∅∅ = M(p)2. V∅∅ = M(p) 11−p3.∑m≥0QmΦ∅,∅(m) = M(p)2∏m>0(1 + pmQ)m4.∑m≥0QmΦ−,∅(m) = M(p)21 +Q1− p∏m>0(1 + pmQ)mThe overall factors of Qn3 are calculated in 3.3.2 to be:1. n = 4 for (g) and n = 0 for (h).2. If j(E) = 1728 and E ∼= C/i then- n = 2 occurs when D is a translation of the graph {(x,±ix)}.3. If j(E) = 0 and E ∼= C/τ with τ = 12 (1 + i√3) then- n = 1 occurs when D is a translation of the graph {(x,−τx)} or the graph{(x, (τ − 1)x)}.- n = 3 occurs when D is a translation of the graph {(x, τx)} or the graph{(x, (−τ + 1)x)}.Lastly, in a generic pencil we have e(J0) = 4 and e(J1728) = 6.Hence the contribution for parts (g)− (i) is:Q1Q2QσM(p)24(∏m>0(1 + pmQσ)m(1 + pmQ3)12m)(−10 + 8Q3 + 12Q23 + 8Q33 − 10Q43)5.5.6. Part (j):In the appendix 6.2.2 we give the following decomposition for L into parts:82 CHAPTER II. DT THEORY OF THE BANANA 3-FOLD WITH SECTION CLASSES1. Lσ(0,0),(∞,∞) q L∅(0,0),(∞,∞) q Lσ(0,∞),(∞,0) q L∅(0,∞),(∞,0)2. q Lσ(0,0) q L∅(0,0) q Lσ(∞,∞) q L∅(∞,∞)3. q Lσ(0,∞) q L∅(0,∞) q Lσ(∞,0) q L∅(∞,0)4. q Lσ∅ q L∅∅.The Euler characteristics of the parts of this decomposition are computed in 6.2.3 andthe overall factors of Q1, Q2 and Q3 are calculated in lemma 3.4.4.Grouping (1): The following table is the summary of results from 5.2.4 and 5.2.5 for thestrata in grouping (1):Z≥0 × U × Sym•({bkop})× Sym•(Bop \ {bkop}).Note that the vertex is different for L(0,0),(∞,∞) as described in 4.4.9.U = Lσ(0,0),(∞,∞) e(U)=1 χ(OD) = 1e(η-1• (a, x,mk,m)) =Q1Q2Q23p1(V∅∅)(V∅∅∅)QaσΦ−,∅(a)Qmk3 Φ−, | (mk)12∏i = 1i 6= jQmi3 Φ∅,∅(mi)U = L∅(0,0),(∞,∞) e(U)=−1 χ(OD) = 1e(η-1• (a, x,mk,m)) =Q1Q2Q23p1(V∅∅∅)2QaσΦ∅,∅(a)Qmk3 Φ−, | (mk)12∏i = 1i 6= jQmi3 Φ∅,∅(mi)U =Lσ(0,∞),(∞,0) e(U) = 1 χ(OD) = 0e(η-1• (a, x,mk,m)) =Q1Q21(V∅∅)(V∅∅∅)·QaσΦ−,∅(a) ·Qmk3 Φ+,∅(mk)12∏i = 1i 6= jQmi3 Φ∅,∅(mi)U =L∅(0,∞),(∞,0) e(U) = −1 χ(OD) = 0e(η-1• (a, x,mk,m)) =Q1Q21(V∅∅∅)2·QbσΦ∅,∅(b) ·Qm3 Φ+,∅(m)12∏i = 1i 6= jQdi3 Φ∅,∅(di)From lemmas 6.3.2, 6.3.4 and 6.3.5 we have:1. V∅∅∅ = M(p)2. V∅∅ = M(p) 11−p3.∑m≥0QmΦ∅,∅(m) = M(p)2∏m>0(1 + pmQ)m4.∑m≥0QmΦ−,∅(m) = M(p)21 +Q1− p∏m>0(1 + pmQ)m5.∑m≥0QmΦ−,−(m) = M(p)2 1p (ψ0 + (ψ1 + 2ψ0)Q+ ψ0Q2)∏m>0(1 + pmQ)m.5. EULER CHARACTERISTIC OF THE FIBRES OF THE CHOW MAP 836.∑m≥0QmΦ−, | (m) = M(p)2(ψ0 + (2ψ0 + ψ1)Q+ (ψ0 + ψ1)Q2)∏m>0(1 + pmQ)m.7.∑m≥0QmΦ+,∅(m) = M(p)2(ψ1 +ψ0 +(ψ1 +2ψ0)Q+ψ0Q2)∏m>0(1+pmQ)mSo the after accounting for the 12 singular fibres we have the contribution from group-ing (1) as:Q1Q2M(p)24(∏m>0(1− pmQσ)m(1− pmQ3)12m)· 12QσQ23((ψ0 + ψ1) +Q3(2ψ0 + ψ1) + 2Q23ψ0 +Q33(2ψ0 + ψ1) +Q43(ψ0 + ψ1))Grouping (2): We compute the results for L(0,0) with L(∞,∞) being completely analo-gous. The following table is the summary of results from 5.2.4 and 5.2.5 for the stratain grouping (2):Z≥0 × U × Sym•({bkop})× Sym•(Bop \ {bkop}).U = Lσ(0,0) e(U)=−1 χ(OD) = 1e(η-1• (a, x,mk,m)) =Q1Q2Q23p1(V∅∅∅)2QaσΦ−,∅(a)Qmk3 Φ−,∅(mk)12∏i = 1i 6= jQmi3 Φ∅,∅(mi)U = L∅(0,0) e(U)=1 χ(OD) = 1e(η-1• (a, x,mk,m)) =Q1Q2Q23p(V∅∅)(V∅∅∅)3QaσΦ∅,∅(a)Qmk3 Φ−,∅(mk)12∏i = 1i 6= jQmi3 Φ∅,∅(mi)From lemmas 6.3.2, 6.3.4 and 6.3.5 we have:1. V∅∅∅ = M(p)2. V∅∅ = M(p) 11−p3.∑m≥0QmΦ∅,∅(m) = M(p)2∏m>0(1 + pmQ)m4.∑m≥0QmΦ−,∅(m) = M(p)21 +Q1− p∏m>0(1 + pmQ)mAccounting for both L(0,0) and L(∞,∞), the contribution for grouping (2) is:Q1Q2M(p)24(∏m>0(1 + pmQσ)m(1 + pmQ3)12m)· (−24)QσQ23 (ψ0 +Q3ψ0)Grouping (3): We compute the results for L(0,∞) with L(∞,0) being completely analo-gous. The following table is the summary of results from 5.2.4 and 5.2.5 for the stratain grouping (3):Z≥0 × U × Sym•({bkop})× Sym•(Bop \ {bkop}).84 CHAPTER II. DT THEORY OF THE BANANA 3-FOLD WITH SECTION CLASSESU = Lσ(0,∞) e(U)=−1 χ(OD) = 1e(η-1• (a, x,mk,m)) =Q1Q2Q3p1(V∅∅∅)2QaσΦ−,∅(a)Qmk3 Φ−,∅(mk)12∏i = 1i 6= jQmi3 Φ∅,∅(mi)U = L∅(0,∞) e(U)=1 χ(OD) = 1e(η-1• (a, x,mk,m)) =Q1Q2Q3p(V∅∅)(V∅∅∅)3QaσΦ∅,∅(a)Qmk3 Φ−,∅(mk)12∏i = 1i 6= jQmi3 Φ∅,∅(mi)From lemmas 6.3.2, 6.3.4 and 6.3.5 we have:1. V∅∅∅ = M(p)2. V∅∅ = M(p) 11−p3.∑m≥0QmΦ∅,∅(m) = M(p)2∏m>0(1 + pmQ)m4.∑m≥0QmΦ−,∅(m) = M(p)21 +Q1− p∏m>0(1 + pmQ)mAccounting for both L(0,∞) and L(∞,0), the contribution for grouping (3) is:Q1Q2M(p)24(∏m>0(1 + pmQσ)m(1 + pmQ3)12m)· (−24)QσQ23 (ψ0 +Q3ψ0)Grouping (4): The following table is the summary of results from 5.2.4 and 5.2.5 forthe strata in grouping (4):Z≥0 × U × Sym•({bkop})× Sym•(Bop \ {bkop}).U = Lσ∅ e(U)=2 χ(OD) = 1e(η-1• (a, x,mk,m)) = Q1Q2Q23p(V∅∅)(V∅∅∅)3QaσΦ−,∅(a)12∏i=1Qmi3 Φ∅,∅(mi)U = L∅∅ e(U)=−2 χ(OD) = 1e(η-1• (a, x,mk,m)) = Q1Q2Q23p(V∅∅)2(V∅∅∅)4QaσΦ∅,∅(a)12∏i=1Qmi3 Φ∅,∅(mi)From lemmas 6.3.2, 6.3.4 and 6.3.5 we have:1. V∅∅∅ = M(p)2. V∅∅ = M(p) 11−p3.∑m≥0QmΦ∅,∅(m) = M(p)2∏m>0(1 + pmQ)m6. APPENDIX 854.∑m≥0QmΦ−,∅(m) = M(p)21 +Q1− p∏m>0(1 + pmQ)mSo the contribution for grouping (4) is:Q1Q2M(p)24(∏m>0(1 + pmQσ)m(1 + pmQ3)12m)· 24QσQ23ψ0Combining groupings (1)-(4) we have the overall contribution for part (j) is:Q1Q2M(p)24(∏m>0(1 + pmQσ)m(1 + pmQ3)12m)· 12Qσ((ψ0 + ψ1) +Q3ψ1 +Q33ψ1 +Q43(ψ0 + ψ1))6 Appendix6.1 Connected Invariants and their Partition FunctionsFor the rank four sub-lattice Γ ⊂ H2(X,Z) generated by a section and banana curves,we can consider the connected unweighted Pandharipande-Thomas invariants. Theyare defined formally via the following partition functionẐPT,ConΓ (X) := log(ẐΓ(X)Ẑ(0,•,•)|Qi=0).For the partition function in theorem A we consider the first terms of the expansionin Qσ and Q1:ẐΓ(X)Ẑ(0,•,•)|Qi=0=Ẑ(0,•,•)Ẑ(0,•,•)|Qi=0+QσẐσ+(0,•,•)Ẑ(0,•,•)|Qi=0+ · · ·=Ẑ(0,•,•)Ẑ(0,•,•)|Qi=0(1 +QσẐσ+(0,•,•)Ẑ(0,•,•)+ · · ·).So the first terms of the expansion in Qσ and Q1 of the connected partition functionare:ẐPT,ConΓ (X) =Ẑ(0,•,•)Ẑ(0,•,•)|Qi=0−QσẐσ+(0,•,•)Ẑ(0,•,•)+ · · · .In particular we have the connected version of Ẑσ+(0,•,•) as:ẐPT,Conσ+(0,•,•) =−1(1− p)2∏m>01(1−Qm2 Qm3 )8(1− pQm2 Qm3 )2(1− p−1Qm2 Qm3 )2,proving corollary B. For the partition function in theorem C we consider the firstterms of the expansion in Q1 and Q2:ẐΓ(X)Ẑ(0,•,•)|Qi=0=Ẑ(0,•,•)Ẑ(0,•,•)|Qi=0(1 +Q1Ẑ•σ+(1,0,•)Ẑ•σ+(0,0,•)+Q2Ẑ•σ+(0,1,•)Ẑ•σ+(0,0,•)+Q1Q2Ẑ•σ+(1,1,•)Ẑ•σ+(0,0,•)+ · · ·).86 CHAPTER II. DT THEORY OF THE BANANA 3-FOLD WITH SECTION CLASSESSo the first terms of the expansion in Q1 and Q2 of the connected partition functionare:ẐPT,ConΓ (X) = log(Ẑ(0,•,•)Ẑ(0,•,•)|Qi=0)−Q1Ẑ•σ+(1,0,•)Ẑ•σ+(0,0,•)−Q2Ẑ•σ+(0,1,•)Ẑ•σ+(0,0,•)+Q1Q2(Ẑ•σ+(1,0,•)Ẑ•σ+(0,1,•)(Ẑ•σ+(0,0,•))2 − Ẑ•σ+(1,1,•)Ẑ•σ+(0,0,•))+ · · ·In particular we have the connected version of Ẑ•σ+(0,0,•) asẐPT,Con•σ+(0,0,•) = log(Ẑ•σ+(0,0,•)Ẑ(0,•,•)|Qi=0)= log(∏m>0(1 + pmQσ)m(1 + pmQ3)12m)=∑n>0pn(1− pn)2(−Qσ)nn+∑n>012pn(1− pn)2(−Q3)nn=∑n>0ψ0(pn)(−Qσ)nn+∑n>012ψ0(pn)(−Q3)nnand the connected version of Ẑ•σ+(0,1,•) (and also of Ẑ•σ+(1,0,•)) given by:ẐPT,Con•σ+(0,1,•) = −((12ψ0 +Q3(24ψ0 + 12ψ1) +Q23(12ψ0))+QσQ3(ψ0 + 2ψ1))and the connected version of Ẑ•σ+(1,1,•) given by:ẐPT,Con•σ+(1,1,•)=(12(Q43(2ψ0 + ψ1) +Q33(8ψ0 + 6ψ1 + ψ2) +Q23(12ψ0 + 10ψ1 + 2ψ2)+Q3(8ψ0 + 6ψ1 + ψ2) + (2ψ0 + ψ1)))+Qσ((12ψ0 + 2ψ1)+Q3(48ψ0 + 44ψ1)+Q23(216ψ0 + 108ψ1 + 24ψ2)+Q33(48ψ0 + 44ψ1)+Q43(12ψ0 + 2ψ1)).Corollary D now follows immediately.6.2 Linear System in P1 × P1In this section we consider a stratification of the following linear system in P1 × P1with strata determined by the intersections of the associated divisors with a collectionof points.Consider the fibres of the projection maps pri : P1 × P1 → P1 and a fibre from eachfi. The linear system in P1 × P1 defined by the sum of a fibre from each map is|f1 + f2| = P3. This is the collection of bi-homogeneous polynomials of degree (1, 1):{ax0y0 + bx0y1 + cx1y0 + dx1y1 = 0∣∣∣ [a : b : c : d] ∈ P3 }6.2.1. There are five points in P1 × P1 that are of interest to us:σ =([1 : 1], [1 : 1])and P :={(0, 0), (0,∞), (∞, 0), (∞,∞)}.6. APPENDIX 87Lσ(0,0),(∞,∞) : Lσ(0,0) : Lσ∅ :L∅(0,0),(∞,∞) : L∅(0,0) : L∅∅ :Figure II.12: Depictions of the curves in the decomposition of the linear system |f1+f2|on P1 × P1.where we have used the standard notation 0 = [0 : 1] and ∞ = [1 : 0]. We willdecompose |f1 +f2| into strata based on which points the divisor intersects. Considera divisor D ∈ |f1 + f2|. Then D passes through:1.(0, 0)if and only if d = 0;2.(0,∞) if and only if c = 0;3.(∞, 0) if and only if b = 0;4.(∞,∞) if and only if a = 0.6.2.2. Define the following convenient notation for y, x ∈ P:1. Sing ⊂ |f1 + f2| is the subset of singular divisors.2. L∅ ⊂(|f1 + f2| \ Sing) is the subset of smooth curves not passing through anypoints of P .3. Lx ⊂(|f1 + f2| \ Sing) is the subset of smooth curve passing through x but noother points of P .4. Lx,y ⊂(|f1 + f2| \ Sing) is the subset of smooth curve passing through x and ybut no other points of P.5. Also let Lσ∅ , Lσx and Lσx,y be subsets of L∅, Lx and Lx,y respectively with thefurther condition that the curves pass through σ.6. Let L∅∅, L∅x and L∅x,y be the complements of Lσ∅ , Lσx and Lσx,y in L∅, Lx and Lx,yrespectively.With this notation we have the following decomposition of |F1 + F2|:|F1 + F2| = Sing q L(0,0),(∞,∞) q L(0,∞),(∞,0)q L(0,0) q L(0,∞) q L(∞,0) q L(∞,∞)q L∅.6.2.3. We now consider the strata of this collection and their Euler characteristics:88 CHAPTER II. DT THEORY OF THE BANANA 3-FOLD WITH SECTION CLASSESban: A curve in |f1 +f2| is singular if and only if the equation for the curve factorises:ax0y0 + bx0y1 + cx1y0 + dx1y1 = (αx0 + βx1)(γy0 + δy1) = 0where [α : β], [γ : δ] ∈ P1. Hence Sing ∼= P1 × P1 and the Euler characteristicis e(ban) = e(|f1 + f2|) = 4.Lx,y : We consider for x = (0, 0) and y = (∞,∞) with the case (0,∞) and (∞, 0)being completely analogous. The points [a : b : d : c] ∈ |f1 + f2| correspondto a curve passing through x and y if and only if a = d = 0. Moreover, this issingular when either b = 0 or c = 0. Hence Lx,y ∼= P1 \{0,∞} and e(Lx,y) = 0.The set Lσx,y is when b+ c = 0, which is a point in P1. So we have e(Lσx,y) = 1and e(L∅x,y) = 1.Lx: We consider the case x = (0, 0) with the other cases being completely analogous.So the subspace of all divisors passing through x is [a : b : d : c] ∈ |f1 + f2|where d = 0. This is a P2 ⊂ P3. The subspace where the curve doesn’tpass through one of the other points is where a, b, c 6= 0 which is given byC∗×C∗ ∼= P2 \ ({a = 0}∪{b = 0}∪{c = 0}). None of the equations for thesecurves factorise since such a factorisation would require either b = 0 or c = 0.Hence, Lx ∼= C∗ × C∗ and e(Lx) = 0.The subset Lσx is defined by the further condition a+ b+ c = 0 which givesLσx ={[a : b : c] ∈ P2∣∣∣ a, b, c 6= 0 and a+ b = 1 } ∼= C∗ \ ptHence we have the Euler characteristics e(Lσx) = −1 and e(L∅x) = 1.L∅: The set of curves not passing through any points of P is given by{[a : b : c : d] ∈ |f1 + f2|∣∣∣ a, b, c, d 6= 0 } ∼= { b, c, d ∈ (C∗)3 }The singular curves are given by the factorisation condition:x0y0 + bx0y1 + cx1y0 + dx1y1 = (x0 + βx1)(y0 + δy1)which is the condition that d = bc. So the subspace of curves which are singularis (C∗)2 ⊂ (C∗)3. Hence L∅ ∼= {(b, c, d) ∈ (C∗)3|b 6= dc} and e(L∅) = 0.Lσ∅ is given by the further condition that b+ c+ d = 0, so we have:Lσ∅ ∼={(b, c, d) ∈ (C∗)3∣∣∣ d 6= bc and 1 + b+ c+ d = 0 }∼={(b, c) ∈ (C∗)2 ∣∣∣ (b+ 1)(c+ 1) 6= 0 and b+ c 6= −1 }∼={(b, c) ∈ (C∗ \ {−1})2 ∣∣∣ b+ c 6= −1 }∼= (C∗ \ {−1})2 − (C \ {2pt}).Hence we have the Euler characteristics e(Lσ∅ ) = 2 and e(L∅∅) = −2.6. APPENDIX 896.3 Topological Vertex FormulasIn this section of the appendix we collect some useful formulas for partition functionsinvolving the topological vertex.Define the “MacMahon” notation:M(p,Q) =∏m>0(1− pmQ)−mand the simpler version M(p) = M(p, 1).Lemma 6.3.1. We have the equality:VλVλ∅∅ =1pVλ∅∅Vλ∅∅ + Vλ∅Vλ∅Proof. We prove the equivalent equation:VνV∅∅ν=1p+V∅νV∅ν(V∅∅ν)2From the definition we have:VνV∅∅ν=1p∑η⊂S/η(p−ν−ρ)S/η(p−νt−ρ)=1p(S/∅(p−ν−ρ)S/∅(p−νt−ρ) + S/(p−ν−ρ)S/(p−νt−ρ))=V∅νV∅ν(V∅∅ν)2+1pLemma 6.3.2. We have1. V∅∅∅ = M(p)2. V∅∅ = M(p) 11−p3. V∅ = M(p)p2−p+1p(1−p)24. V = M(p)p4−p3+p2−p+1p2(1−p)3Proof. Part (1) is immediate from the definition. For part (2) we have:V∅∅ = M(p)p−12S∅(p−ρ)∑ηS/η(p−ρ)S∅/η(p−ρ)= M(p)11− pFor part (3) we have:V∅ = M(p)p−1S∅(p−ρ)∑ηS/η(p−ρ)S/η(p−ρ)= M(p)p−1(S/∅(p−ρ)S/∅(p−ρ) + S/(p−ρ)S/(p−ρ))= M(p)p−1( p(1− p)2 + 1)= M(p)p2 − p+ 1p(1− p)290 CHAPTER II. DT THEORY OF THE BANANA 3-FOLD WITH SECTION CLASSESPart (4) follows from parts (2) and (3) and lemma 6.3.1:V =1pV∅∅ +V∅V∅V∅∅6.3.3. It is shown in [Br, §4.3] the Donaldson-Thomas partition function of this iscomputed to be:∑ν,α,µQ|ν|1 Q|α|2 Q|µ|3 p12 (‖ν‖2+‖νt‖2+‖α‖2+‖αt‖2+‖µ‖2+‖µt‖2)(VνµαVνtµtαt).=∏d1,d2,d3≥0∏k(1− (−Q1)d1(−Q2)d2(−Q3)d3pk)−c(‖d‖,k)where d = (d1, d2, d3) and the second product is over k ∈ Z unless d = (0, 0, 0) inwhich case k > 0. The powers c(‖d‖, k) are defined by∞∑a=−1∑k∈Zc(a, k)Qayk :=∑k∈ZQk2(−y)k(∑k∈Z+ 12 Q2k2(−y)k)2 = ϑ4(2τ, z)ϑ1(4τ, z)2and ‖d‖ := 2d1d2 + 2d2d3 + 2d3d1− d21− d22− d23. Also, if we recall the notation thatψg :=(p(1− p)2)1−gthen we have the following corollary.Corollary 6.3.4. We have:1.∑α,µQ|α|2 Q|µ|3 p12 (‖α‖2+‖αt‖2+‖µ‖2+‖µt‖2)(V∅µαV∅µtαt)= M(p)2∏m>0M(Qm2 Qm3 , p)2(1−Qm2 Qm3 )M(−Qm−12 Qm3 , p)M(−Qm2 Qm−13 , p)2.∑αQ|α|p12 (‖α‖2+‖αt‖2)(V∅∅αV∅∅αt) = M(p)2∏m>0(1 + pmQ)m3.∑αQ|α|p12 (‖α‖2+‖αt‖2)+1(V∅αV∅αt)= M(p)2(ψ0 + (ψ1 + 2ψ0)Q+ ψ0Q2)∏m>0(1 + pmQ)m4.∑αQ|α|p12 (‖α‖2+‖αt‖2)+2(VαVαt)= M(p)2∏m>0(1+pmQ)m(Q4(2ψ0+ψ1)+Q3(8ψ0+6ψ1+ψ2)+Q2(12ψ0+10ψ1 +2ψ2)+Q(8ψ0 +6ψ1 +ψ2)+(2ψ0 +ψ1))Proof. These are all coefficients of the partition function in 6.3.3. For example part (3)is the coefficient of Q11Q02.Lemma 6.3.5. We have the following equalities:1.∑αQ|α|p12 (‖α‖2+‖αt‖2)(V∅αV∅∅αt) = M(p)21 +Q1− p∏m>0(1 + pmQ)m6. APPENDIX 912.∑αQ|α|p12 (‖α‖2+‖αt‖2)+1(VαV∅∅αt)= M(p)2((ψ0 + ψ1) + (2ψ0 + ψ1)Q+ ψ0Q2)∏m>0(1 + pmQ)m3.∑αQ|α|p12 (‖α‖2+‖αt‖2)+1(V∅α)2= M(p)2(ψ0 + (2ψ0 + ψ1)Q+ (ψ0 + ψ1)Q2)∏m>0(1 + pmQ)mProof. Part (1) is given by:∑αQ|α|p12 (‖α‖2+‖αt‖2)(Vα∅Vαt∅∅)= p−12M(p)2∑αQ|α|∑ηSαt/η(p−ρ)S/η(p−ρ)∑δSα/δ(p−ρ)S∅/δ(p−ρ)= p−12M(p)2∑αQ|α|(Sαt(p−ρ)S(p−ρ) + Sαt/(p−ρ))Sα(p−ρ)= p−12M(p)2(S(p−ρ)∑α⊃∅Sαt/∅(p−ρ)Sα/∅(Qp−ρ) +∑α⊃Sαt/(p−ρ)Sα/∅(Qp−ρ))After applying [Ma, Eqn. 2, pg. 96] the equation becomesp−12M(p)2∏i,j>0(1 + pi+jQ)(S(p−ρ)∑τ⊂∅S∅/τ (p−ρ)S∅/τ (Qp−ρ) +∑τ⊂∅S∅/τt(p−ρ)S/τ (Qp−ρ))= p−12M(p)2∏m>0(1 + pmQ)m(1 +Q)p121− p .Part (2) follows from lemma 6.3.1 and corollary 6.3.4:∑αQ|α|p12 (‖α‖2+‖αt‖2)+1(VαV∅∅αt)=∑αQ|α|p12 (‖α‖2+‖αt‖2)(V∅∅αV∅∅αt) +∑αQ|α|p12 (‖α‖2+‖αt‖2)+1(V∅αV∅αt).Part (3) is given by:∑αQ|α|p12 (‖α‖2+‖αt‖2)+1(V∅α)2=∑αQ|α|p12 (‖α‖2+1)(Vα∅) p12 (‖αt‖2+1)(V∅α)= M(p)2∑αQ|α|S(p−ρ)∑δSα/δ(p−−ρ)S∅/δ(p−−ρ)S∅(p−ρ)∑ηSαt/η(p−ρ)S/η(p−ρ)= M(p)2S(p−ρ)∑αSα(Qp−−ρ)(Sαt(p−ρ)S(p−ρ) + Sαt/(p−ρ))After applying [Ma, Eqn. 2, pg. 96] the equation becomesM(p)2S(p−ρ)(1 +Q)∏m>0(1 +Qpm)m(S(p−ρ) + S(p−−ρ)).92 CHAPTER II. DT THEORY OF THE BANANA 3-FOLD WITH SECTION CLASSESThe result follows from a quick computation involving V∅ = V∅ showing thatS(p−ρ)S(p−−ρ) = S(p−ρ)2 + 1.Lemma 6.3.6. The following are true1.∑αQ|α| =∏d>01(1−Qd)2.∑αQ|α|(Vα∅)(Vα∅∅)=11− p∏d>0(1−Qd)(1− pQd)(1− p−1Qd)3.∑αp‖α‖2Q|α|(Vααt∅)(V∅∅∅)=∏d>0M(p,Qd)(1−Qd)4.∑αp‖α‖2Q|α|(Vααt∅)(Vα∅)(Vα∅∅)(V∅∅∅)=11− p∏d>0M(p,Qd)(1− pQd)(1− p−1Qd)Proof. The first is a classical result and the other three are the content of [BKY, Thm.3].Bibliography[ACV] D. Abramovich, A. Corti and A. Vistoli, Twisted bundles and admissible covers,Special issue in honor of Steven L. Kleiman. Comm. Algebra 31 (2003), no. 8,3547-3618. MR 2007376[AGV] D. Abramovich, T. Graber and A. Vistoli, Gromov-Witten theory of Deligne-Mumford stacks Amer. J. Math. 130 (2008), no. 5, 1337-1398. MR 2450211[AJ] D. Abramovich and T. Jarvis, Moduli of twisted spin curves, Proc. Amer. Math.Soc. 131 3, (2003), 685-699. MR 1937405[AV] D. Abramovich and A.Vistoli, Compactifying the space of stable maps, J. Amer.Math. Soc. 15 (2002), no. 1, 27-75. MR 1862797[BBH] C. Bartocci, U. Bruzzo and D. Hernández-Ruipérez, Fourier-Mukai and Nahmtransforms in geometry and mathematical physics, Progress in Mathematics, 276.Birkhäuser Boston, Inc., Boston, MA, 2009. xvi+423 pp. ISBN: 978-0-8176-3246-5. MR 2511017[B1] K. Behrend, Donaldson-Thomas type invariants via microlocal geometry. Ann. ofMath. (2) 170 (2009), no. 3, 1307-1338. MR2600874[B2] K. Behrend, Gromov-Witten invariants in algebraic geometry, Invent. Math. 127(1997), no. 3, 601-617. MR 1431140[BF] K. Behrend and B. Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997),no. 1, 45-88. MR 1437495[BKLPS] G. Borot, R. Kramer, D. Lewanski, A. Popolitov and S. Shadrin Special casesof the orbifold version of Zvonkine’s r-ELSV formula, arXiv:1705.10811[Br] J. Bryan, The Donaldson-Thomas partition function of the banana manifold,arXiv:1902.08695[BCY] J. Bryan, C. Cadman, B. Young, The orbifold topological vertex, Adv. Math. 229(2012), no. 1, 531-595. MR2854183[BK] J. Bryan, M. Kool Donaldson-Thomas invariants of local elliptic surfaces via thetopological vertex, Forum Math. Sigma 7 (2019), e7, 45 pp. MR3925498[BKY] J. Bryan, M. Kool, B. Young, Trace identities for the topological vertex, SelectaMath. (N.S.) 24 (2018), no. 2, 1527-1548. MR3782428[CCC] L. Caporaso, C. Casagrande and M. Cornalba Moduli of roots of line bundles oncurves, Trans. Amer. Math. Soc. 359 (2007), no. 8, 3733-3768. MR 2302513[CL] H.-L. Chang and J. Li, Gromov-Witten Invariants of Stable Maps with Fields, Int.Math. Res. Not. IMRN 2012, no. 18, 4163-4217. MR 29753799394 BIBLIOGRAPHY[CLL] H.-L. Chang, J. Li and W.P. Li, Witten’s top Chern class via cosection localization,Invent. Math. 200 (2015), no. 3, 1015-1063. MR 3348143[Ch1] A. Chiodo Stable twisted curves and their r-spin structures, Ann. Inst. Fourier,Grenoble, 58, 5, (2008), 1635-1689. MR 2445829[Ch2] A. Chiodo Towards an enumerative geometry of the moduli space of twisted curvesand rth roots, Compositio Math. 144 (2008) 1461-1496. MR 2474317[Co] M. Cornalba, Moduli of curves and theta-characteristics, Lectures on Riemannsurfaces (Trieste, 1987), 560-589, World Sci. Publ., Teaneck, NJ, 1989. MR1082361[FJR1] H. Fan, T. Jarvis and Y. Ruan Quantum singularity theory for Ar−1 and r-spintheory, Ann. Inst. Fourier (Grenoble) 61 (2011), no. 7, 2781-2802. MR 3112508[FJR2] H. Fan, T. Jarvis and Y. Ruan The Witten equation, mirror symmetry, and quan-tum singularity theory, Ann. of Math. (2) 178 (2013), no. 1, 1-106. MR3043578[FP] B. Fantechi and R. Pandharipande, Stable maps and branch divisors, CompositioMath. 130 (2002), no. 3, 345-364. MR 1887119[GP] T. Graber, R. Pandharipande, Localization of virtual classes, Invent. Math. 135(1999), no. 2, 487-518. MR 1666787[GV] T. Graber and R. Vakil, Hodge integrals and Hurwitz numbers via virtual local-ization, Compositio Math. 135 (2003), no. 1, 25-36. MR 1955162[H] R. Hartshorne Algebraic Geometry, Graduate Texts in Mathematics, No. 52.Springer-Verlag, New York-Heidelberg, 1977. xvi+496 pp. ISBN: 0-387-90244-9. MR0463157[J1] T. Jarvis, Torsion-free sheaves and moduli of generalized spin curves, CompositioMath. 110 (1998), no. 3, 291-333. MR 1602060[J2] T. Jarvis Geometry of the moduli of higher spin curves, Internat. J. Math. 11 (2000),no. 5, 637-663. MR 1780734[K] J. Kollár, Rational curves on algebraic varieties, Ergebnisse der Mathematik undihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Re-sults in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveysin Mathematics], 32. Springer-Verlag, Berlin, 1996. viii+320 pp. ISBN: 3-540-60168-6. MR1440180[L1] J. Li, Stable morphisms to singular schemes and relative stable morphisms, J. Differ-ential Geom. 57 (2001), no. 3, 509-578. MR 1882667[L2] J. Li, A degeneration formula of GW-invariants, J. Differential Geom. 60 (2002),no. 2, 199-293. MR 1938113[Mi] R. Miranda Basic Theory of Rational Elliptic Surfaces, Dottorato di Ricerca inMatematica, Dipartimento di Matematica dell’ Universita di Pisa, ETS EditricePisa (1989).[Ma] I. G. Macdonald, Symmetric functions and Hall polynomials, Second edition.With contributions by A. Zelevinsky. Oxford Mathematical Monographs. Ox-ford Science Publications. The Clarendon Press, Oxford University Press, NewYork, 1995. x+475 pp. ISBN: 0-19-853489-2. MR1354144BIBLIOGRAPHY 95[MFK] D. Mumford, J. Fogarty, F. Kirwan, Geometric invariant theory. Third edition.Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematicsand Related Areas (2)], 34. Springer-Verlag, Berlin, 1994. xiv+292 pp. ISBN:3-540-56963-4. MR 1304906[ObPi] G. Oberdieck, A. Pixton, Holomorphic anomaly equations and the Igusa cusp formconjecture, Invent. Math. 213 (2018), no. 2, 507-587. MR3827207[OP] A. Okounkov and R. Pandharipande, Gromov-Witten theory, Hurwitz theory, andCompleted cycles, Ann. of Math. (2), 163 (2006), no. 2, 517-560. MR 2199225[ORV] A. Okounkov, N. Reshetikhin, C. Vafa, Quantum Calabi-Yau and classical crys-tals, The unity of mathematics, 597-618, Progr. Math., 244, Birkhäuser Boston,Boston, MA, 2006. MR2181817[O] M. Olsson, (Log) twisted curves, Compos. Math. 143 (2007), no. 2, 476-494. MR2309994[S] C. Schoen, On fiber products of rational elliptic surfaces with section, Math. Z. 197(1988), no. 2, 177-199. MR0923487[SSZ] S. Shadrin, L. Spitz, D. Zvonkine, Equivalence of ELSV and Bouchard-Mariñoconjectures for r-spin Hurwitz numbers Math. Ann. 361 (2015), no. 3-4, 611-645.MR 3319543[Stacks] The Stacks Project Authors, The Stacks Project,http://stacks.math.columbia.edu/, 2019[T1] Y. Toda, Curve counting theories via stable objects I. DT/PT correspondence, J.Amer. Math. Soc. 23 (2010), no. 4, 1119-1157. MR2669709[T2] Y. Toda, Curve counting theories via stable objects II: DT/ncDT flop formula, J.Reine Angew. Math. 675 (2013), 1-51. MR3021446[V] R.Vakil, The enumerative geometry of rational and elliptic curves in projective space,J. reine angew. Math. 529 (2000), 101-153. MR 1799935[W] E. Witten, Algebraic geometry associated with matrix models of two-dimensionalgravity, Topological methods in modern mathematics (Stony Brook, NY, 1991),235-269, Publish or Perish, Houston, TX, 1993. MR1215968

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