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The cohomology of Kontesevich’s stack of stable maps to Pⁿ, the case of conics O’Halloran, Anne Fionnuala 2000

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T H E C O H O M O L O G Y OF KONTSEVICH'S STACK OF STABLE MAPS T O P*, T H E CASE OF CONICS. by A N N E FIONNUALA O ' H A L L O R A N B.A. (Mathematics and Legal Science) The National University of Ireland (Galway), 1992 M.A. (Mathematics) The National University of Ireland (Galway), 1993 A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T O F T H E REQUIREMENTS F O R T H E D E G R E E O F D O C T O R O F PHILOSOPHY in T H E F A C U L T Y OF G R A D U A T E STUDIES Department of Mathematics We accept this thesis as conforming to the required standard T H E UNIVERSITY O F BRITISH C OLU MBIA November 2000 © Anne Fionnuala O'Halloran, 2000 In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for refer-ence and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Mathematics The University of British Columbia Vancouver, Canada Date Abstract In this thesis we consider the singular cohomology of M 0 ,o(P n , 2), the coarse moduli space associated to Kontsevich's stack of degree two stable maps to P n , Mo,o{Pn, 2). We show that the cohomology ring is generated by a divisor d which corresponds to the locus of pairs (C,g) with C reducible, and the first and second Chern classes, C\ and C2, of the canonical rank three vector bundle E = 7r*/*0(l) on A^o,o(Pn, 2), where 7r is the canon-ical projection associated to the universal curve C and / is the universal map. We give the cohomology ring as a quotient of a polynomial ring in these generators. The relations are in degrees n, n + 1 and n + 2. We also give a representation of the cohomology ring in terms of the Chern roots of E. The results are conjectural for n » 0. s ii Table of Contents Abstract ii Table of Contents iii Acknowledgement v Chapter 1. Introduction 1 Chapter 2. Coordinate charts for M 0 ) 0 ( P n , 2) 14 2.1 What is AT0 )o(Pn,2)? 15 2.2 Gm and A1 actions on P n 20 2.2.1 Gm and A1 actions 20 2.2.2 Two special actions on P™ 21 2.2.3 A BB-minus decomposition of P n 26 2.3 Gm and A1 actions on_A*0lo(Pn> 2) 28 2.4 An open substack of_Alo,o(Pn,2) 31 2.5 An Open Cover for -M0,o'(Pn, 2) 72 2.6 Covering the Fixed Locus 74 2.6.1 Transition Functions 76 2.6.2 The Fixed Locus 81 2.7 A vector field on M>,o(Pn, 2) 82 Chapter 3. The Cohomology of M 0 ) 0 ( P n , 2) 91 3.1 The Hypercohomology of AT 0,o(P n,2) 91 3.1.1 Preliminary Ideas 91 3.1.2 The Hypercohomology Calculation 100 3.1.3 The Case n =_2_ 119 3.2 The Cohomology of M 0 ) 0 ( P n , 2) 128 Chapter 4. Geometric Interpretation 132 4.1 Chern Classes 133 4.2 Chern Classes and Hypercohomology 136 4.3 Some Divisors on A^o.o(P71,2) ^ 148 4.4 Chern Classes and the Cohomology of M 0 ,o(P n , 2) 150 4.5 Geometric Interpretation in the case n = 2 151 4.5.1 M 0,o(P 2,2) as the Blow-Up of the Veronese Surface in P 5 151 4.5.2 Cohomology of M 0 , 0 (P 2 , 2 ) via the Veronese 154 4.5.3 Geometric Interpretation for n — 2 158 4.6 Chern Roots and the cohomology of M 0,o(P", 2) 159 iii Table of Contents Bibliography Acknowledgement I would like to thank my research supervisor, Dr. K. Behrend, for all his help and sup-port. I also thank Dr. J. Carrell and Dr. W. Casselman for their careful reading of this thesis. Thanks to my friends and family for their patience, particularly Russell Joyce, Sonia Pawlak, Heather Ferguson, Christine Ferguson, David Burggraf and K. G. O'Halloran. C h a p t e r 1 Introduction Suppose we consider conies in projective n space, P n . If n = 2 a conic is given by a degree two homogeneous equation, if n > 2 a conic is given by a collection of homogeneous polynomials. Instead of focusing on the polynomial or polynomials giving the conic, one can represent the conic using a parameterization. Specifically, if we consider a fixed non-degenerate conic in P n (non-degenerate meaning that it is not given geometrically by a line or a pair of lines), one can find a degree two morphism / from P 1 to P n such that the image of P 1 under / is this fixed conic. So / is of the form f :< s,t > — K aos2 + bost + cot2, ais2 + bist + C i i 2 , . . . , ans2 + bnst + cnt2 > where the coordinate polynomials do not simultaneously vanish. Thus a non-degenerate conic in P n is given by a pair (C, /) where C = P 1 and / is a degree two morphism from P 1 to P n . Suppose we also consider pairs (C, /) where f(C) is a line in P n and / is as above, then / gives a two to one branched covering of this line with two branch points. We consider the coarse moduli space, M 0 i o(P n , 2), which parameterizes such pairs (C, / ) . The space M 0 ,o(P n , 2) has a natural compactification M 0 ,o(P n , 2), due to Kontsevich [12], where the points of M 0 ) o(P n , 2) also include pairs (C, /) where C is two copies of P 1 meet-ing transversally and / : C -> Pn is a morphism such that if we restrict / to either copy of P 1 the restriction is an isomorphism. The space M 0 ,o(P n ,2) is actually the coarse moduli space associated to the stack of stable maps A^o,o(Pn,2). For the properties of 1 Chapter 1. Introduction Mo,o(P",2) and ^f 0 >o(P n,2) see [8] or [3]. Suppose we consider pairs (C, /) where C is isomorphic to P 1 and / is a degree d mor-phism from C to P n , i.e. / is of the form f :<s,t >->< /0(s, t), fi(s, *), . . . , fn(s, t) > where the /j are degree d homogeneous polynomials which do not simultaneously vanish. The coarse moduli space M0,o(Pn,rf) parameterizes such pairs. Again we have a natural compactification M0,o(Pn,d) due to Kontsevich [12]. Kontsevich's stacks of stable maps are central to the ideas of quantum cohomology. The coefficients in the multiplication table of the quantum cohomology ring are the solutions to certain enumerative geometry problems. This is in general quite remarkable. Of course quantum cohomology is guided by ideas in physics, and in particular has applications in string theory. Therefore understanding the structure of these stacks of stable maps has become very important. In this thesis we calculate Hging(M0>o(Pn, 2)(C), C), the singular cohomology ring of Mo,o(Pn, 2). The method we use is one which we shall attempt to use in the future to calculate the singular cohomology of M0,o(Pn,d), for d > 2. We try to make the theory we develop in this thesis as general as possible, since the case of conies is really a test case. There is almost no information on this cohomology ring in the literature. The Betti numbers of Hging(MQfl(Pn, 2)(C),C) were calculated previously in [9] by Getzler and Pandharipande, but their methods give no information about the ring structure (al-though using the Betti numbers one might make some conjectures about the number of generators and relations). In [14] Pandharipande computes the Picard group, the Picard 2 Chapter 1. Introduction group has two generators, the first is a divisor corresponding to the pairs (C, /) with C reducible, and the second is a divisor corresponding to pairs (C, /) where f(C) meets a fixed codimension two linear subspace of P n . Again we cannot deduce any information on the ring structure. The coarse moduli space M 0 )o(P", 2) associated to the stack of stable maps Mo,o(Pn, 2) has a classical analogue, namely the space of complete conies in P n . This classical space specifies a plane, together with a complete conic contained in the plane. For n = 2, the plane is always just P 2 , so in particular the space of complete conies in P 2 is the same as M 0 ,o(P 2 ,2). For n > 2, Kontsevich's space has blown down all the planes containing a given line. For d > 2, the coarse moduli spaces M 0 ,o(P n ,d) do not have a classical analogue. The scheme M 0 ,o(P 2 ,2) has a second well known representation. Suppose we consider the Veronese surface in P 5 , then Mo,o(P2,2) is isomorphic to the blow-up of P 5 along the Veronese surface, P 5 . Although this is a well known fact, we could not find a proof in the literature. For this reason we provide an explicit isomorphism in Lemma 15, Sec-tion 4.5.1. One can easily calculate the Chow ring of P 5 , and thus the Chow ring of M 0 ) o(P 2 ,2) . For M 0,o(P 2,2) the Chow ring and the cohomology ring coincide, so we can calculate the cohomology ring of M 0 ) o(P 2 ,2) . We do this in Section 4.5.2, and verify that we get the same result as when we calculate using our method described below. We now state the main result of this thesis. Let n > 2. We have a canonical rank three vector bundle E = T T * / * 0(1), on A^ 0,o(Pn,2), where 7T is the canonical projection associated to the universal curve C for Mo,o(Pn, 2) and / is the universal map. The fiber for E over a point a is H°(Ca, f*0(l)), where Ca is the curve corresponding to a. Let 7i, 72 and 73 be the Chern roots of E, and let the group with two elements, Z 2 , act on 3 Chapter 1. Introduction C[7i, 72,73] as follows: Then where 71 -> 7i 72 -> 73 73 ->• 72-#WMo,o (P n , 2 ) (C) ,C) = (C[7l,l2,l3]/(R1,R2,R3)f R i = 72"+1 ~ 7? + 1 , 73"+1 ~ 7i"+1 72 - 7 i 73 ~ 7i «, = _ ( f e ^ ) ( 7 5 _ 7 ] ) _ ( f c ^ ) ( 7 2 _ 7 l ) V 72-Ti / \ 73-7i / +(72n+1 + 7r + 1 ) + (73 n + 1 + 7i + 1 ) Rs = 7r + 1((72-7i) + (73-7i)). We also give the cohomology ring in two other forms. The first involves the Chern classes, the second is not motivated by geometry but is the simplest representation. Let ci and C2 denote the first and second Chern classes of E and let d be the divi-sor of points (C, /) 6 M 0 ,o(P n , 2) such that C is reducible, where c\ and d have degree one and C2 has degree two. Then the cohomology ring #* i n f l(M 0 jo(P n , 2)(C), C) is C[d, ci, c 2]/{g n +i, fn+i, rn+i) where ^ 9n+l x fn+l ( d-2g 6 -d 2+4c 2 2c 1 2 2 6 V 0 d+4ci 3 , . d2+16c?+8dci -4c2 H j 4 Chapter 1. Introduction Finally, if we let u b 9 d- 2ci 3 d + ci d2 4c? -4c 2 1 1 3 3 then H*(M 0 ,o(P n ,2)(C),C) equals C[6,«, #] / ( s n+l , /n+i , r „ + i ) where / 1 2 0 \ fn+1 = is |« -2 \ rn+l J \ 0 0 n - l / « + 26 \ 5 + 462 \ 6 2 ( « - 2 6 ) y In terms of the Chern roots, 7i 72 72 2 These representations are conjectural for n » 0. The method we use to calculate the cohomology ring of M 0 ,o(P n ,2) is similar to one which has been used in the past to calculate the cohomology of a smooth complex pro-jective variety X, in the situation where we have a vector field on X with isolated non trivial zero set Z (we describe this method below). In our case X is an stack (which complicates matters) and we have a vector field on X with a one dimensional zero locus. As far as we know, this is the first time a method of this type as been used in the case 5 Chapter 1. Introduction where the vector field has a strictly positive dimensional zero locus (and the calculation is non-trivial). We now describe our method. Let X be a smooth and proper Deligne-Mumford stack, and X its associated coarse moduli space. The singular cohomology Hging(X(C), C) of X(C) with its C-topology is equal to the derived functor cohomology of the analytic space X(C) with the constant sheaf C, H*(X(C), C). This in turn equals the sheaf cohomology of the analytic stack Xanai, H*(Xanai,C) [2]. We consider the holomorphic deRham complex c -* oXhol -+ siXhol -+ tfXkol -» • • • which is exact. Thus the map of complexes 0 -+ °xhol -> nxhol t t t 0 -+ c -»• 0 -)• 0 is a quasi isomorphism, and the induced map on hypercohomology H* (xanal, q -> ET (xana,, o X h o i -+ nXkol ->•••) = (xanai, 04 j is an isomorphism. So in particular H*sing(x(c), c) = #*(xana,, q = ff (xanai, c) IT (xana,, n^), is an isomorphism of rings. We have a canonical spectral sequence 'E*!'q — H9(Xanai,DFhol) for p, q > 0, which abuts to the hypercohomology W+q(Xanai, £l*hol). Since X is a smooth and proper Deligne-Mumford stack, G A G A applies [15, 2], thus H<(Xmallffhal)XH<{X1W) 6 Chapter 1. Introduction where the latter sheaf is the pth exterior power of the algebraic cotangent sheaf. In the case where we have the added assumption that Hq(X,Qp) = 0 for p^q (this is true for the stack A^o,o(Pn, 2), see Lemma 36), the spectral sequence 'E^q degen-erates at E\ and M*(xanal,n*hol) = ®kHk(x,nk). Thus we have as rings. The idea at this stage is to choose a different complex of sheaves KP for p < 0, on X, with an O^-linear differential, such that we have a spectral sequence Er abutting to the hypercohomology H* (X, fC*) with the property that for q > 0 and p < 0, { Hg(X,tt-p) forp + q = 0, 0 otherwise, and whose hypercohomology is accessible. In this case H*(X,/C*) =0 for k^O. We explicitly calculate If(X, JC*). By standard spectral sequence arguments we have a filtration F9 of If{X, /C*), such that F^(W(X,JC*)) ~ H 1 ' j' Therefore the associated graded ring GrF(lf(X,K:*)) is isomorphic to @qHg(X, Qq). Thus GrF(W(X,lC*)) ^ H;ing(X(C),C). 7 Chapter 1. Introduction We do not have to compute this associated graded ring because if we choose our complex of sheaves /C* correctly, (using a result of Carrell and Akyildiz [1]) it turns out that W(X,K*) is already graded (from a Gm action) and the filtration FQ from the spectral sequence equals the filtration from this grading. Therefore H 0 pf./C*) = Gr F(H° (* ,£•) ) = H*sing(X(C),C) as graded rings. The complex we choose is the global Koszul complex associated to a vector field V. We first describe this notion for X a smooth complex projective variety and V a vector field with isolated but non trivial zero set Z. What we describe in this setting in due to Akyildiz and Carrell [1]. Given an A1 action on X we have an associated vector field V. Suppose we have a Gm action on X such that these actions constitute an A1, Gm pair, i.e. there exists a positive integer k such that for A e Gm, dX.V = XkV. We consider the contraction operator associated to V, which has the property that i(V)2 = 0. Thus we have a complex of sheaves on X, K? = Q,~p for p < 0, with differential i(V). Let Ui be an affine open cover for X, such that Q, restricted to Ui is trivial. The vec-tor field on Ui is given by a collection of regular functions, (v\,... ,vn), for Ui where n — d i m X The complex IC restricted to Ui is just the Koszul complex of the ring 17(17,) with respect to the sequence (vi,... ,vn). If V has an isolated zero in Ui then, since 8 Chapter 1. Introduction r([/j) is Cohen-Macauley, (vi,... , vn) is a regular sequence, so the complex restricted to Ui is exact except at position 0, and the homology at position zero is T(U)/(vi,... ,vn). If V does not vanish on U{ then the complex is automatically exact. Thus we have a quasi isomorphism of complexes ••• -> fix -> Ox 0 I I 0 -> ox/i(v)nx -»• o and the sheaf Ox/i(V)Q,x is zero away from the fixed locus. Now I P ( X , (...-• 0 Oxli{V)Slx)) = H°(X, Ox/i(V)Qx) = H°(Z, Oz), where Oz is the structure sheaf of the fixed locus viewed as a (possibly non-reduced) scheme, i.e. Oz = Ox/i(V)Qx, so H°(X,/C*) = H°(Z,Oz). We have a spectral sequence 'E{,q = Hq(X, Q, p) abutting to the hypercohomology W+q(X, fC). Since 'E{,q = 0 for p + q ^ 0, this spectral sequence degenerates at 'E\, so in particular M°(X,/C*) = H°(Z,Oz) has a filtration Fq such that Fq(H°(Z,Oz)) F^{H\Z,Oz))~ [ ' J' Carrell and Akilydiz show that this mysterious filtration is just the canonical filtration given where Fq is the qth weight space of the induced Gm action on H°(Z,Oz)- Thus H°(Z, Oz) is already graded and H*ing(X(C), C) is isomorphic to H°(Z, Oz) as a graded ring with the grading induced on the latter by the Gm action, i.e. H;ing(X(C),C)=H°(Z,Oz) as graded rings. In fact Carrell and Akilydiz show that even in the case where the dimension of Z is strictly positive we have such a result. In this case we have H;ing(x(c),c)=®P(x,)c*) Chapter 1. Introduction as graded rings. In our case X is the stack of stable maps Mofi(Pn,2). In order to work with this stack we construct for it an etale open cover. Specifically, we find a collection of afl&ne varieties, {Ui}, each isomorphic to AZn~l and each equipped with a morphism of stacks such that <f>i is etale, and the collection taken together map surjectively to Mo$(Pn, 2). Each Ui is equipped with an action of Z 2 (the group with two elements) such that the induced map from the quotient stack, [Ui/Z2] is an open immersion, It is well known that such covers exist for these stacks, however one has not been ex-plicitly constructed in the literature before now. Chapter 2 of this thesis focuses on the construction of this etale open cover. The next task is to find a vector field on A/(o,o(Pn, 2) and thus on the cover {Ui}. We first consider a vector field on P", which in turn induces a vector field on M-o,o(Pn, 2). Suppose we let A1 act on P n via the matrix (j>i:Ui^M0,o(Pn,2) cf>:[Ui/Z2}^Mo,oCPn,2). ( \ (n-l)! n! tn~2 tn~x (n-2)! (n-l)! 0 0 1 t m - 3 L T I - 2 M(t) = (n-3)! (n-2)! 0 0 0 0 1 \ 0 0 0 0 0 10 Chapter 1. Introduction then by differentiating we have a vector field V oh P n , which has a unique zero. We have an induced A1 action on the stack A /io,o(P7\2) which is described on geometric points (C,f) ofM),o(P n,2)(C) by t.(C,f) = (C,M(t)of). The fixed locus consists of closed points (C, /) such that f(C) is a conic in P n which is invariant under the action of A1, such conies are paramaterized by A1, plus one other closed point (C, f) where C is degenerate, / (C) is the unique A1 invariant line in Pn, and / maps the crossing point to the unique A1 fixed point in P n . We have an induced vector field on Afo,o(Pn, 2), and an associated global Koszul complex with differential i(V) as before. Since the maps fa : Ui -> A1o,o(Pn> 2) are etale, we have in particular and for the purposes of calculating hypercohomology we need only look at the Koszul complex associated to the induced vector field on the Ui (in each case this induced vector field is given by a collection of regular functions on Ui), and at the gluing data for these Ui. For each Ui which does not meet the zero locus of V, the Koszul complex on Ui is exact as in the scheme case. We need two neighborhoods to cover the fixed locus, we call them UQ and U\. Suppose the vector field is given by (v^i,... ,i>i,3 n-i) on Ui. For i = 0,1, we can find a subsequence of regular functions of length 3n — 2 such that this subsequence is a regular sequence (it is this part of the proof which remains conjectural for n » 0, since verifying that we have a regular subsequence is computational). So the Koszul complex is exact except in positions zero and one in each case. Thus we have a 11 Chapter 1. Introduction quasi isomorphism of complexes of sheaves % 0 , o (P n , 2 ) ~ * ^7CT0,o(Pn,2) °M0,o(Pn^) ~* 0 I I I • • ' 0 ^ o , o ( P « , 2 ) / ( ^ ( ^ ) % , o ( p n , 2 ) ) ^ o o ( p n ) 2 ) -> 0 and to calculate the hypercohomology H°(7Vt 0 ,o(P n ,2), /C*) we need only calculate the hypercohomology of the lower complex restricted to UQ and U\. Thus we are reduced to considering the following four term double complex: 0 0 t t . . . -+ 0 -> (^u0/(i(V)n2UO))(U01) -+ 0UO(UOL) -> 0 (*) t t • : - 0 © < = 0 l l ( ^ / ( * ( ^ ) ^ ) ) ( ^ ) ^ ©<=o,iO^(^) -> 0 Even when the double complex is reduced to four terms, calculating the hypercohomol-ogy is a difficult task. We introduce the idea of a companion pair of vector fields for V in Chapter 3 (Definition 11) as a simplification technique. One can define an action of the group Z 2 x Z 2 on each of the four terms of the double com-plex (*) such that with respect to this action all the maps are equivariant. Thus taking the Z 2 x Z 2 invariant part of the degree zero hypercohomology, denoted by H ° ( * ) Z 2 x Z 2 , is the same as taking the degree zero hypercohomology of the invariant subcomplex. This ring H ° ( * ) Z 2 x Z 2 maps injectively to the degree zero hypercohomology of the double com-plex (*), thus maps injectively to EP(A7 0,o(Pn, 2), JC*). We next find a ring R(n) which maps injectively to H ° ( * ) Z 2 > < Z 2 , thus is isomorphic to a subring of H° (AT 0 >o(P n, 2), JC*). The ring R(n) is actually a subring of T{U0)/(vi,... , v3n-i), where (vi,... , i>3n_i) is the collection of regular functions giving the vector field V on UQ. 12 Chapter 1. Introduction As in the case for schemes we can find a GM action on Mo,o(Pn,2). The GM action we choose comes from a GM action on P n given by a matrix M(A) for A G G M , such that this action together with the A1 action above is an (A1, G m )-pair (see Definition 7). We have an induced GM action on Moto(Pn, 2). In fact the cell UQ is the big BB-minus cell (see [4]) of the unique A1 fixed point corresponding to a pair (C, / ) with C re-ducible (this point is also GM fixed). We have an induced GM action on UQ and thus on R(n). This gives a grading of R(n), so R(n) is realized as a subring of the graded ring H° ( M ) , 0 ( P n , 2), JC*) thus is a subring of the graded ring H*sing(M0,0(Pn, 2), C). The last step is to show that R(n) is actually isomorphic to H*ing(M0fi(Pn, 2),C) as a graded ring. Getzler and Pandharipande calculate the Poincare Polynomial of Mo,o(Pn, 2) in [9], which equals the Hilbert polynomial ofthe ring H*ing(M0fi(Pn, 2), C). We calcu-late the Hilbert Polynomial of R(n) and show that these polynomials are equal, thus the rings are isomorphic. The cohomology ring R(n) is generated by three elements, two in degree 1 and one in degree 2. Chapter 4 of this thesis focuses on the geometric meaning of these generators. We show how one can calculate the Chern classes ofthe rank three bundle E = 7r*/*0(l) as elements of R(n). Thus we have the second representation for the cohomology ring given at the beginning of this introduction. 13 C h a p t e r 2 Coordinate charts for A4o,o(Pn?2) In this chapter we construct a collection of affine varieties, {Ui}, each isomorphic to A 3 " - 1 , each equipped with an action of Z 2 , the group with two elements, and each having an associated etale morphism & : c W M > , o ( P n , 2 ) . We shall show that the induced morphism of stacks Ji-. [Ui/Z2]^Mo!o(Pn,2) is an open immersion in each case, and that the collection maps surjectively to Moto(Pn, 2). We shall define an A1 action on Mofi(Pn,2), which will have a one dimensional fixed locus. Two members of our cover UQ and U\ will cover the fixed locus. These members of the cover shall be used to calculate the cohomology in Chapter 3. First the standard description of the moduli stacks .Mo,o(Pn, 2), and the associated coarse moduli spaces M0,o(P",2) will be given (Section 2.1). Then some general theory of A1 and Gm actions will be developed, on P n , and via P" on .Mo,o(P™,2) in Sections 2.2 and 2.3. Section 2.4 will focus on constructing one member of the cover {Ui}. We will show how to construct the whole collection in Section 2.5. In Section 2.6 we find a sub-collection which covers the fixed locus under the action of A1, this sub-collection consists of two cells we call UQ and U\. In Section 2.7 we find the vector field on UQ and U\ 14 Chapter 2. Coordinate charts for M 0 ,o(P n , 2) induced by the A1 action. Throughout this thesis we shall take the complex numbers C to be the ground field. 2.1 What is M0,o(Pn,2)? In this section we shall first describe the coarse moduli space M 0 > o(P n , 2). We shall then describe formally the stack Mo,o(Pn, 2), and explain the relationship between the two. We first consider the open subset M 0 )o(P n ,2) of the coarse moduli space Mo,o(P",2). A closed point in M 0 ) o(P n , 2) is an isomorphism class of pairs (C, /) where C is isomor-phic to P 1 and / is a morphism of degree two from C to P n . In other words if we identify C with P 1 with coordinates < s, t >, then / is of the form / (< S, t >) = < / 0 ( S , t), fi{S, t), . . . , fn{s, t) > where the / ; are homogeneous polynomials of degree two in s and t which do not simul-taneously vanish. The image of C in P n is either a non-singular conic or a line. We say that a pair (C, /) is isomorphic to a pair ( C , /') if there exists an isomor-phism r : C —> C such that f or = f. In other words if we identify both curves with P 1 we have a projective change of coordinates for P 1 , r : P 1 —>• P 1 , relating / and / ' . There is a compactification .due to Kontsevich [12], M 0 ! o ( P n , 2 ) c M 0 ) o ( P n , 2 ) where the points of M 0 ,o(P n ,2) consist of isomorphism classes (C , / ) where C is either isomorphic to P 1 or to a tree consisting of two copies of P 1 meeting transversally, in the first case / is as before, in the second case / , when restricted to each copy of P 1 , is an 15 Chapter 2. Coordinate charts for A4o,o(Pn, 2) isomorphism onto a line in P n . Thus f(C) is either two lines in P" meeting transversally or a single line in P n with a special point corresponding to the image of the 'crossing point' of C. The concept of isomorphism class is as before. There are certain points in M 0 ,o(P n ,2) which are special because they have automor-phisms associated to them. An automorphism of the pair (C, /) is an isomorphism r : C —» C such that for = f. We shall call a point which has non-trivial automor-phisms associated to it a stack point. We need to decide which points in M 0 ,o(P n , 2) are stack points and what automorphism groups are associated to them. Lemma 1 The closed points z /M 0 ,o(P n ,2) with non-trivial automorphism groups are follows: (i) Points (C , / ) where C is isomorphic to two copies of P1 meeting transversally, and f maps both copies isomorphically onto the same line in P41. (ii) Points (C, f) where C is isomorphic to P1 and f constitutes a double covering of a line in P71. In either case the automorphism group is Z 2 . Proof First suppose (C, f) is a class in M 0 ( o(P n ,2). So we identify C with P 1 as before and assume / is given by a series of n + 1 homogeneous polynomials. Suppose the image of C is not contained in a line in P n . Then (C, f) can not be a stack point. This is the case since the map / from C to / (C) is an isomorphism. Suppose (C , / ) is a class in M0fi(Pn,2) and / (C) is contained in a line in P n . Then 16 Chapter 2. Coordinate charts for Mo,o{Pn, 2) (C, /) constitutes a two sheeted branch covering of the line and we have two branch points. First we observe that any automorphism must fix the ramification points. Sup-pose pi and p2 are distinct points in P1 such that f(pi) = f(p2)- If T is an isomorphism such that f or = f then either r(pi) = pi or r(pi) = p2. Thus since / is a local iso-morphism r must either be trivial or be the automorphism which swaps the sheets of the branch covering. Therefore we have Z 2 as automorphism group at these points. If (C, f) is such that C is a tree of two copies of P 1 meeting transversally then the image of C in P 2 is either a line in P n or a pair of lines meeting transversally. In the second case we clearly have no automorphisms. In the first case, we can identify each component of C with P 1 in such a way that / restricted to either component is the same. With this choice of representative for the point (C, /) we have a unique automorphism, i.e. the one which interchanges the two lines. This automorphism again has order two. So again we have Z 2 as the automorphism group for these points. • We now shall describe the algebraic stack Afo,o(Pn 5 2). But first some definitions. Let pairs (C, f) be as before, i.e. C is a (arithmetic) genus zero, projective, reduced, connected, (at worst) nodal curve, and / is a morphism from C to P n of degree two (i.e. such that f*[C] = 2Hn~l where H G Ai(Pn) is the hyperplane class) such that no component of C is mapped to a point under / (this last condition is the so called stability condition and it ensures that the automorphism group of the pair (C, /) is finite), then (C, /) is called a degree two stable map to P n . Let X be a scheme over C. 17 Chapter 2. Coordinate charts for Mofi(Pn, 2) Definition 1 defn-family of curves A family of curves genus zero C over X is a flat, projective map TT-.C^X such that each geometric fiber CX is a (arithmetic) genus zero, projective, reduced, con-nected, (at worst nodal) curve. Definition 2 A family of stable maps from genus zero curves over X to P71 consists of the data (n : C ->• X, \x : C -> F) : (i) A family of genus zero curves ir: C —> X (ii) A morphism p, : C —> F such that \i restricted to any geometric fiber is a degree two stable map to F1. Definition 3 Two families of maps over X, ( T T ! : d -> : d -> F), ( T T 2 : C 2 -> X,p2 : C 2 -+ F) are isomorphic if there exists a scheme isomorphism r : C i —> C 2 such that iti — ~K2 O T and Hi = fj.2 o r. Definition 4 A4o,o(Pn,2) is the contravariant lax functor from schemes to groupoids, /vf 0 >o(P r\2) : {Schemes/C) (Groupoids) X -+ A7 0 ) o(P n ,2) (X) whereMofi(Pn,2)(X) is the groupoid with whose objects are families of stable maps from genus zero curves over X to F, and whose morphisms are isomorphisms between such families. 18 Chapter 2. Coordinate charts for Mo,o(Pn, 2) We note that the functor is contravariant since if we have a map of schemes g : X\ —> X2 then we get an induced map of groupoids g* : A V o ( P n , 2){X2) M ^ P " , 2)(XX) where g'(n :C^X2,fi:C^Pn) = (Pl: X1 xX2 C -+ Xl,p,op2 : Xx xX2 C -+ P B ) where and p2 are the coordinate projections. •Mofi(Pn, 2) is a lax functor since if we have a commutative of schemes X -> Y \ I Z the corresponding diagram of groupoids is not commutative, however there is a natural transformation between the composition of functors M , ,o(P n ,2)(Z) -+ M> l 0(Pn,2)(Y) M),o(P n ,2)(X) and the functor AT 0 ,o(P n ,2) (Z ) ->AT 0 ,o(P n ,2) (X) . Theorem 1 T/iere erciste a projective coarse moduli space M 0 , o (P n , 2) associated to Mo,o(P' i.e. M 0 , o (P n , 2 ) is a scheme and we have a natural transformation of functors 4> • A V o ( P n , 2 ) -> ^ o m 5 c / l ( * , M 0 , 0 ( P n , 2 ) ) with the following properties: (i) (f>(Spec(C)) : M0,o(Pn,2)(Spec{C)) -> Hom(Spec{C),M0)0(Pn,2)) is a set bijec-tion. 19 Chapter 2. Coordinate charts for Mo,o(Pn, 2) (ii) If Z is a scheme and I/J : M.o,o(Pn, 2) —> Z is a natural transformation of functors, then there exists a unique morphism of schemes 7 : M 0 ) 0 ( P n , 2 ) ^ Z such that ip^jcxf). Where j : Hom(*,Mofi(Pn,2)) -> Uom(*,Z). (iii) The dimension o /M 0 , o (P n , 2 ) is 3n — 1. (iv) JM 0,O(P",2) is smooth. Proof: [8, Theorem 1]. Theorem 2 in Fulton and Pandharipande's paper [8, Theorem 2] states that M 0 i 0 ( P " , 2) is locally a quotient of a non-singular variety by a finite group. We shall show in Section 2.4 that M 0 ) 0 ( P n , 2) can be represented locally as the quotient of A3n~l modulo an action of Z 2 . 2.2 G m and A 1 actions on P n 2.2.1 Gm and A 1 actions Definition 5 An Algebraic Group is a variety G, together with a morphism p:GxG^G such that (G, p) is a group. Gm and A1 are examples of algebraic groups. As a variety, Gm is the non-zero complex numbers C*. The morphism p is defined by p(Xi, A2) = AiA 2 20 Chapter 2. Coordinate charts for M0,o(Pn, 2) for Ai and \ 2 in C*. As a variety A1 is the complex numbers C. In this case the morphism \x is defined by p,{ti,t2) = h +t2 for ti and t2 in C. Let G be an arbitrary algebraic group. Definition 6 An action of G on a variety X is a morphism 4> '• G x X —> X such that each fixed g G G induces an isomorphism <j)(g) from X to X such that <f>(g){x) := <f)(g,x). These induced morphisms obey the group law, i.e. <t>{gi)4>{g2) = fiigm) ^(1G) = Ix where Ix is the identity morphism from X to X. 2.2.2 Two special actions on P n In our analysis we are concerned exclusively with A1 and Gm actions on P n and the in-duced actions on Mofi(Pn,2). The group P G L n + i ( P G L n + i = G L n + 1 / C * ) acts naturally on P n thus we can define an action of A1 or Gm on P n by constructing a morphism of algebraic groups from respectively A1 or Gm to P G L n + i . 21 Chapter 2. Coordinate charts for M 0,0 (P n, 2) Consider the following morphism of algebraic groups from A1 to P G L n + i : t -»• M(t) = 1 t § 0 1 t 0 0 1 3! £ 2! t -1 tn (n--1)! n\ * n -2 tn-l (n--2)! (n-l)! t n - 3 tn-2 (n--3)! (n-2)! 0 0 0 0 \ 0 0 0 0 i.e. if we let A — 0 1 0 0 0 0 1 0 1 0 0 0 \ \ 0 . . . 0 0 1 0 0 . . . 0 0 0 1 V 0 0 0 0 0 then M(t) = eAt. It is easy to see that M is a morphism of algebraic groups, since if t\ and t2 are in A1 then M't1)M't2) = M'tl+t2). We can also construct a morphism of algebraic groups from Gm to P G L n + i given by A -»• 7V(A) = A Q o 0 0 0 . . . 0 0 0 A Q 1 0 0 . . . 0 0 0 0 A°2 0 . . . 0 0 \ 0 0 0 0 0 0 0 0 A 0 " - 1 0 A Q " 22 Chapter 2. Coordinate charts for Mo,o(Pn, 2) where the on are integers. Without loss of generality we can assume that n = 0 i=0 in other words our matrix has determinant 1. We observe that for Ai and A 2 in G. Thus this is indeed a morphism of algebraic groups. The notion of an (A1, Gm)-pair shall be useful to us, [5, Chapter 1, Section 5]. Definition 7 Let Gm and A1 act on X via morphisms (j) and if) respectively. Then (tp, <f>) are called an (A1, Gm)-pair if there exists a positive integer k such that for any t £ A1 and A 6 Gm Suppose we require (M, /V) to be an (A1, Gm)-pair (associated to which is a fixed integer k) then we must have Lemma 2 Suppose (M,N) is a (A1, Gm)-pair. Let k be any fixed positive even integer, then the entries of N are determined by k and n, as follows: N(Xl)N(X2) 0(A)^(*)0(A-1) = ^(A**). N(X)M(t)N(\-1) = M{\H). nk a0 — 2 ^ = ctQ — ik, for i = 1,... , n. 23 Chapter 2. Coordinate charts for Moft(Pn, 2) Thus in the case where n is odd I N(X) , kn A 2 0 0 , k(n-2) 0 X^ir2 0 0 0 V 0 0 0 0 At 0 0 \ - \ 0 0 A 2 0 0 0 and in the case where n is even we have / A ^ 0 0 0 0 . k(n-2) 0 A ^ i H 0 0 0 N(X) V o o 0 0 0 A t 0 0 0 1 0 0 0 A " " k(n-2) 0 0 0 A 2 ^ 0 0 0 Proof If we consider the equation rV(A)M(t)JV(A _ 1) = M(Xkt), 24 Chapter 2. Coordinate charts for Mo,o{Pn, 2) the matrix on the left hand side is 0 1 *A Q 1 0 0 0 0 0 0 v ° 0 0 and the matrix on the right is I 1 t\k %\2k 0 \ Equating we see that j n - 2 tn~3 (£=3)T +n-2 \ - {n-2)\A (n-l)! ai—a„ 1 0 0 *"-2 \k(n-2) tn~3 \k(n-3) f\Ctn-2-<Xn-l 1 0 * i \ O n - 2 - a „ 2! 1 t"-1 \fc(n-l) (n^l)! A t"~2 \fc(n-2) t"-1 \fc(n-l) (rtT2)!A ( ^ T ) ! A 0 0 0 1 txk 0 0 0 0 1 txk 0 0 0 0 0 1 k — O>Q — ct\ 2k = OSQ — ai nk = ao — an. So in particular for each i — 1... n, on — CXQ — ki. Now n o = i=0 = aQ + (aQ - k) + (a0 - 2k) + ... + (aQ - nk) , ^ . fn(n + l)\ = ( n - H ) a o - f c ^ V 2 JJ-25 Chapter 2. Coordinate charts for Mofi(Pn, 2) Therefore nk *o = Y as required. • 2.2.3 A BB-minus decomposition of P n Suppose we consider the Gm action given by N on P". We will not specify a value for k (except for requiring that it must be even) since all our results are independent of k. We can decompose P n into locally closed subsets using this action. In general we have the following result due to Bialynicki-Birula [4]: Theorem 2 Let Gm act on a projective non-singular scheme X such that all the fixed points are isolated. Then X can be decomposed into locally closed subschemes each of which is isomorphic to a vector space and contains exactly one fixed point. One canonical decomposition satisfying the requirements above is the Bialynicki-Birula minus, or BB-minus decomposition. Definition 8 Suppose Gm acts on P71, with isolated fixed points Fo,... ,Fn. The Vth BB-minus cell C{ is d = {xe P*\ lim \.x = Fi\. A—>oo Before we can calculate the BB-minus decomposition of P n under the action N we must find the fixed points under this action. 26 Chapter 2. Coordinate charts for A4 0 lo(Pn, 2) Lemma 3 The fixed points under the action given by the matrix N are Fi =< 0,... , 0, 1 0,... , 0 > Ph position for % = 0,... , n. Proof Suppose x —< x0, XI, ... ,xn > is fixed, then at least one of the Xi is non-zero. Suppose addition xi ^ 0 also for I ^ j, then p = X!r~lk, which is impossible. Thus we can assume x\ = 0 for / ^ j, and Xj = 1. Clearly j can take on any value from zero to n. Thus for any even positive integer k, the Gm action has fixed points F{ for i = 0,... , n. • Lemma 4 Let Gm act on P71 via N then Ci = {< x0,Xi, ...xn>e F^lxj = 0 for j <i k i r = 1}, for i — 0,... ,n. Proof Let us first consider C 0 which is the set of points in P n which move to the fixed point F0 =< 1,0,... , 0 > as A oo. Now if x =< x0, xx,... , xn > is in C0 we clearly must have x0 ^ 0. If x0 = 1 we have Xj is non-zero. Then X"? ikXj = p,Xj for some p, ^  0. Thus p, = X^~jk. Suppose in lim (A. < l , x i , . . . ,xn >) = lim (< 1, X~kxu X~2kx2,... , X~nkxn >) =< 1,0,... , 0 > Thus Co = {<ar 0 ,xi , . . . , a ; n > G P n | a ; o ^ O } ^ ^ n . 27 Chapter 2. Coordinate charts for A^o,o(Pn, 2) The set C\ C C'0 — {< XQ,Xx, ... xn > \xQ = 0}. Using the same type of argument as above we see that Ci = {< x0,xlt...xn >e Pn\x0 = 0 & xx = 1} £ An~l. Continuing in this way we find that Ci = {< x0,Xi,.. .xn >e Pn\xi = 0 for i = 0,... ,i - 1 & xt = 1} = An~\ as required. • We call the cell Co the big cell (because it has highest dimension). It is special because it contains the unique A1 fixed point F0. 2.3 Gm and A1 actions on JW0,o(Pn, 2) Let G be an arbitrary algebraic group acting on P n . We can describe an induced action M 0 > o(P n ,2) as follows: Definition 9 Let (C, /) be a point in M 0 ,o(P n , 2) then for g G G, we define g.(C, f) to be g.(CJ) = (CM9)°f) where <p(g) is the isomorphism on P71 induced by g. This action is well defined since all representatives of the class (C, /) have the same image in P n . One can define a compatible action on the stack Mofl(Pn,2) as in [10]. Let A1 act on M0,o(P",2) via the action on P n given by M. We would like to find the fixed locus under this action. 28 Chapter 2. Coordinate charts for Mo,o(Pn, 2) Lemma 5 The fixed locus under the action given by M on M 0 ,o(P n , 2) is parameterized by P1 and is as follows: • Points (C , / ) in M 0 ,o(P n ,2) where f(C) is a conic in P71 given by the ideal Ic = (Xl — 2XQX2 + cX\, Xz, X t , . . . , Xn) for c G A1. We shall use F ^ 1 to denote these points. • The point (C, f) where C is isomorphic to two copies of P1 intersecting transversally and f maps each copy of P1 isomorphically onto the line given by the ideal loo — (X2, X3,... , Xn) and the crossing point maps to the unique A1 fixed point in P71 namely the point FQ =< 1,0,... , 0 >. We shall use F^ to denote this point. Proof If a point (C, f) is fixed then for each t G A1, f(C)=(M(t) o f)(C) as a sub-variety of P n , thus f[C) must be invariant under the action of A1. For (C, /) G M 0 ,o(P n , 2), / (C) is either a line or a conic in P n (the conic may be degenerate). Thus we are looking for conies or lines in P n which are invariant under the action of A1. Let Vi = {< x0,xi,... , xn > G P" \ Xj =0 for j > i}, for i = 0,... , n, then it is easy to see that the subvarieties Vi are invariant under the action of A1 and Vi = P \ If we let x eVi-Vi-i then the orbit of x under A1 is a curve of degree i. Thus all our invariant points (C, /) have the property that / (C) is contained in V2. The A1 action restricted to V2 is given by the following matrix (when we identify V2 29 Chapter 2. Coordinate charts for M 0 ] o(P n , 2) with P 2 ): f i t 2 0 1 One can easily verify that the equation of the conic parameterized by the points above is Xl - 2X0X2 + cX\ = 0, where c = 2xQ-x\. These conies are all nonsingular and we have a one-to-one correspondence between these conies and points c e A1. Thus if (C , / ) is such that / (C) lies in V2, and if when we identify V2 with P 2 , / (C) is one of the conies above, then (C, /) is a fixed point under the action of A1 on M 0 ,o(P n , 2). What about fixed points (C, /) in M 0 ,o(P n ,2) where / (C) is a line or a pair of dis-tinct lines in P n ? Clearly since we have only one A1 invariant line in P n , (namely the subvariety Vi) the latter is impossible. We observed above that any degree one invariant curve lies in Vi, so / (C) must lie in Vi and be the line X2 = 0 (identifying V2 with P 2 as before and viewing Vi as a subvariety of V2). We note that if (C, f)=(C, M(t) o /) then any 'special' points on / (C) must be A1 invariant, by special points we mean branch points or the image of the 'crossing point' if C happens to be reducible. Thus since A1 has only one fixed point / cannot constitute a branch covering. We conclude therefore that the only possibility is that C is reducible and / (C) is the line X2 = 0 and the crossing point maps to the point < 1,0,0 >. Translating this information to P n (and letting Xi be the coordinate functions in P r a 30 Chapter 2. Coordinate charts for Moto(Pn, 2) for i = 0 , . . . , n) the fixed locus is claimed. • 2.4 An open substack of A^o,o(Pn? 2) In this section we shall construct a morphism <j> from the affine space A3n~x viewed as a stack, to jMo,o(Pn) 2). We shall show that the image is an open substack U of Moto(Pn, 2), and that <f> is etale. In fact we can find an action of Z 2 on A371-1 such that the induced map from the quotient stack 0: [A 3 "-7Z 2 ] ^M, ,o(P n ,2 ) is an open immersion. A map of stacks from A371'1 to A^o,o(P", 2) is given by a pair ( T T : C A3n~\ f : C —> P n ) € M,,o(Pn, 2)(^ 3 "- 1 ) . Theorem 3 Let C = PL x A3n~l with projection n to A3n~l and let f be the rational map from C to P71 defined as follows: /(< s,t >, (m,6i,... A i .^n r2,2,-- - ,rn,i>r».2)) =< st, ms2 + bist + t2, mr2,is2 + b2st + r2j2t2,... , mr n j ls 2 + bnst + rn>2t2 > . Let C be the blowup of C along the subvariety where m = t = 0, and let f be the induced morphism from C to P 7 1 . Then the data c -UP71 7f J,, 31 Chapter 2. Coordinate charts for Mo,o(Pn, 2) gives a morphism of stacks 4>: A 3 - 1 -+ AT 0,o(P n,2), 4>{X): A*"-\X) -> M0<0(Pn,2)(X), where if a G Homsch(X, A 3 " - 1 ) then (f)(X)(a) is given by the following data: C x A 3 n - i X -> C i f 1 1 7T Proof We have to show that after blowing up we have a family of stable maps. The rational map / is clearly defined when m and t are not both zero since st and ms2 + b\st +12 do not simultaneously vanish. Let Ut and Us be respectively the subvarieties corresponding to the points where t ^ 0, and s ^ 0, so we have Us = Ut — A3n. Us has coordinates m,bu... , bm r 2 , i , r 2 ) 2 , • • • , rn>u rn>2^j , and Ut has coordinates (|, m,bu... , bn, r2,i, r 2 ) 2 , . . . , r n , i , r n , 2 ) • We construct the blow-up, Us, of Us along the subvariety V(t/s,m) where the map / is undefined. Let x £ Us and let Vs = | ( x ,<p ,9 >) G tf, x P 1 | mq = • Us can be covered by two affines Up and Uq both isomorphic to A 3 n where Up = {yeUs\p^0} with coordinates -, m, h,... , bn, r 2 ) 1 , r 2 , 2 , • • • , rnA, rn>2 ) where m ( - ) = - , vP / V P / S 32 Chapter 2. Coordinate charts for Ai 0 ,o (P n , 2) and with coordinates Uq = {yeUa\q^0] 0 , ^, 6i,... , bn, r2,i, r 2 , 2 , . . . , r„,i, rn>^ where Then / induces a morphism / as follows: / : Ut -)• P n / m, 6i,... , bn, r 2 , i , r 2 , 2 , • • • , r„,i, r „ ) 2 ) / : Up P " S / .<? \ 2 / -, m, bi, • • • , K, r 2 , i , r 2 > 2 , . . . , r n , i , r n > 2 - i 1 + &i ( - ) + m ( - j , r„,i + bn ( - ) + mr„, 2 P \Pj \pj \PJ f:Uq^ Pn ft1 P A = ( l , - + 6i + - , r 2 > 1 ( - ) + 62 + r 2 , 2 ( - ) , . . . ,r„,i ( - ) +6n + \ 9 s V ? / V s / W These glue to give a morphism from C = ^ l 3 " - 1 x P 1 to P n . We now need to show that (# : C ->• A3n~\ f]: C - ) • P n ) e Xo ,o (P n , 2)( J4 3 n" 1) It is easy to see that the morphism ?f : C ->• A 3n- l 33 Chapter 2. Coordinate charts for Ato,o(Pn, 2) is flat. The fiber Cx above x = (m, bi,... , r n ; 2 ) is isomorphic to P 1 in the case m ^ 0, and is isomorphic to {< s,t > x < p , g >G P 1 x P 1 | tp = 0} in the case m = 0, so in either case we have a projective, arithmetic genus zero, (at worst) nodal, reduced, connected curve. The induced morphism from the fiber is of degree two in either case. • The next lemma gives us more information about the images of geometric points un-der the map <j>. Specifically if a G AZn~l we would like to know what is the image of the object (a - * A3n~l) G AZn~l(Spec(C)) under <f>. Lemma 6 Let a — (m, 6X,... , bn, r 2 , i , r 2,2, • • • , rn>i, r n > 2 ) G AZn~l be fixed. Let Ca be the fiber of C over a. Then ifm^O, and r^i / T * J ) 2 for some i = 2,... , n, f(Ca) is a conic in P11 contained in the unique plane containing the points < 0,1, r 2 > i , . . . , r n > 1 >, < 0,1, r2,2,... , r„, 2 >, < l,bu . . . ,bn > . If m — 0, then Ca isomorphic to two copies of P1 meeting transversally and f(Ca) is a pair of (not necessarily distinct) lines, one through the pair of points < 1, &i, . . . , & „ > , < 0. U r2,i, • • • , rn,i > and one through < 1,6i,... , bn >, < 0,1, r 2 , 2 , . . . , r n , 2 > . In this case, the crossing point of Ca maps to < 1, b\,... ,bn > under f. 34 Chapter 2. Coordinate charts for Mo,o(Pn, 2) If m ^ 0 and r^i = r i ] 2 for i = 2 , . . . , n then f : Ca -4 P n is a double covering of the line through < 1,b\,... ,bn> and < 0 , 1 , r 2 > i , . . . , r n > i >. Proof Clear from the definition of / , and C. • L e m m a 7 Let Z 2 act on A371'1 as follows: (m, bi,... , bn, r 2 j i , r 2 ) 2 , . . . , r„,i, r n > 2 ) ->• (m, & i , . . . , bn, r 2 > 2 , r 2 , i , . . . , r „ | 2 > r n > i ) . TVien we /iaue an induced action on C suc/i r7ia£ 7r and f are equivariant, and in particular forpeZ2, f(pc) = /(c) force C. Proof Let a: be the non-identity element in Z 2 . Then a defines a rational map a : P 1 x A 3 " " 1 —» P 1 x A 3 n _ 1 a(< s, t >, (m, 6 i , . . . , bn, r 2 > i , r 2 | 2 , . . . , r n , i , r„ l 2 )) =< t, ms >, (m, 6 i , . . . , bn, r 2 j 2 , r 2 | i , . . . , r n , 2 , r„,i)), which is defined when m and t are not both zero. Clearly a2 = 0. One can readily verify that n (the projection TT : C -> A 3 n _ 1 ) is equivariant with re-spect to this action. We also have /(<*(< s, t >, (ra, 6 i , . . . , bn, r2,i, r 2 , 2 , . . . , r n , i , r„, 2))) = / ( < i , ms >, (m, &„, r 2 , 2 , r 2 , i , . . . , r n > 2 , r„,i)) 35 Chapter 2. Coordinate charts for Mo,o(Pn, 2) =< mst,mt2+bimst+m2s2,mr2,2t2+mb2st+r2)im2s2,... ,mrnj2t2+rnbnst+rnjim2s2 > =< st, ms2 + bist + t2, rar2,is2 + b2st + r2>2t2,... , mr n > 1 s 2 + bnst + rn>2t2 > = /((< s, t >, (m, 61,... , bn, r 2 , i , r2,2, ••• , r„,i, r„, 2 ))) . So / is equivariant with respect to this action. When we blow up along the subvariety m = t = 0 we have a morphism a from C to P 1 x A3n~\ C I a \ p i x A3n-1 p i x A3n-1 Blowing up again we have an isomorphism a from C to C, and the following diagram commutes: C a C 4- I p l x A3n-1 p i x A3n-1 Clearly / o a — f by our construction, so / is equivariant with respect to this action. • We now look at <j>(A3n~l) as a substack of A T 0 , 0 ( P n , 2). 36 Chapter 2. Coordinate charts for Mo,o(Pn, 2) Theorem 4 Let U be the stack defined as follows: U : (Schemes/C) —> (Groupoids) X -¥ U(X) Where U(X) C .Mo,o(Pn, 2)(X) is a subgroupoid of stable maps given by the data: (v.C^XJ-.C^P"), (where this data represents an element of Mo,o(Pn, 2)(X)) satisfying the following: Consider f*(Xi) 6 r ( C , / * 0 ( l ) ) where the Xi are the usual coordinate functions on P71, and let D{ = Z(f*(Xi)) for i = 0,1 then (i) The induced map 7 r : D0 —>• X is unramified, (ii) The divisors DQ and D\ do not intersect. Then U is an open substack and (j): A3™*1 —> A^o,o(Pn, 2) factors through U. Proof First we must verify that these conditions indeed define an open substack of A /fo,o(Pn5 2). Fix a scheme X and an element ( 7 r : C - > X , / : C - > P n ) , of M),o(P n,2)(.X). Then it suffices to show that if x e X is such that (Cx,f) satisfies the conditions above, then we can find an open set x 6 U C X such that the family C restricted to U satisfies the conditions also. Certainly the locus of ramification points on DQ, R, is closed in D0 and the locus where D0 and Di intersect is closed also. So U = X - ir((DQ n L>i) U R) is open and contains x. 37 Chapter 2. Coordinate charts for Ato,o(Pn> 2) To show that the image of A 3 n _ 1 is contained in the substack U it suffices to show that the fibers of the morphism / from C to P" satisfy the two conditions above. This is equivalent to showing that for x G A 3 n - 1 , the image of the fiber Cx under / is contained in Co U C i (where the Cj are the BB-minus cells of the action of N on G m , see Lemma 4) and exactly two points in Cx have their images in the hyperplane X0 — 0. We shall consider two cases, m = 0 and m ^  0: If m ^ 0 then Cx is identified with P 1 with homogeneous coordinates s and t, and / restricted to Cx is given by /(< 8,t>) = < st, ms2 + bist +12, mr2,is2 + b2st + r2>2t2,... , mr„ ; 1s 2 + bnst + rn,2t2 > . We notice also that the image has the property that the first two coordinates do not simultaneously vanish thus the image is contained in Co U C\, and exactly two points map to the hyperplane XQ = 0 namely < 1,0 > and < 0,1 >. When m = 0 the fiber Cx can be identified with C = {< s, t > x < p, q > \tp = 0} and / restricted to Cx is given by /(< s,t> x <p,q>) - (sq, sp + hsq + tq, r2Asp + b2sq + r2>2tq,... , rn>isp + bnsq + rnfitq). One sees easily that the points (< 0,1 >, < 0,1 >) and (< 1,0 >, < 1,0 >) map to the hyperplane X0 = 0, and the image is contained in Co U C\ as required. • Notation Convention 38 Chapter 2. Coordinate charts for A^o,o(Pn) 2) From now on we shall use / in place of / and C in place of C and a in place of a, and we let Uo = AZn~l i.e. the data C -4 P n Uo will represent our chosen map of stacks ^ : ^ 0 - ^ l M o , o ( P n , 2 ) , and we have an action of Z 2 on C given by a, and with respect to this action / and n are equivariant. Lemma 8 The induced map is surjective. To prove this we shall need the following lemma: Lemma 9 Let (C,g) € U(Spec(C)), then if (C,g) is a stack point there is exactly one point p e U0 such that (Cp,f) is isomorphic to (C,g), otherwise we have two such points, and these points lie in the same orbit under the action on Z 2 . Proof First let us assume that C = P 1 . In this case g is represented by g :< s,t>->< g0(s,t),gi(s,t),... ,gn(s,t) >, where the & are homogeneous of degree two and do not simultaneously vanish. <f>: A3n~l -)• U 39 Chapter 2. Coordinate charts for Ato,o(Pn> 2) Since (C, g) G U(Spec(C)) we have two distinct points of C mapping to the hyper-plane Xo = 0 and no points mapping to the intersection of this hyperplane with the hyperplane X\ = 0. Thus g0 has two distinct roots, so we may assume #0 = st. Since go and gx cannot vanish simultaneously we must have gx = a i^s 2 + a^st + a i ; 3 i 2 with fli.i) a i , 3 0- If w e compose G with the projective change of coordinates r for P 1 given by the matrix 1 0 0 — 11,3 we have that (C, J O T ) is isomorphic to (C,g) and gor is given by < 1 o CLi 2 ^1 3 9 st, ax is H -st + —j-t,.. fll,3 ' ai,3 < 3 = ( st,ahiait3s2 + ai)2st + t2,... , a n , iO i , 3 S 2 + an<2st + C^t2 \ Q 3 , l We shall use this representative for the isomorphism class (C, g). In this case the image is contained in the linear subspace of P n spanned by < 1, O i , 2 ) • • • , a n , 2 >, ( 0, 1, — , . . . , — > , ( 0, 1, — S . • • , — \ Ol . l O i , i / \ O i | 3 O i > 3 If this is a plane, then ^ ^ for some i = 2, . . . , n. In this case there are exactly 01,1 ' ai,3 two points pi and p2 in C/o such that (CPi, f) is isomorphic to (C, g) namely a2,l ^2,3 an,l °n ,3 ^ 1 , l a l , 3 , ^1,2; • • • ) Q.„,2, , , • • • j , ^1,3 a l , l °1 ,3 and G 2 ) 3 Q 2 , i o „ i 3 an,i\ fll,lGl,3> ^1,2, • • • , Qn,2) , , • • • > j I • Ol,3 0,1,1 Ol,3 0 1 , 1 / If the subspace spanned by the points above is a line then ^ — ^  iox i = 2,... , n, so we have exactly one point p in U0 such that (Cj,, / ) is isomorphic to (C, g) namely ^2,1 °2,1 an,l an,l p — I a i iax 3 , a i 2, • • • , an2, , , 1 a i . i 01,1 oi , i a i , i 40 Chapter 2. Coordinate charts for Mo,o(Pn, 2) In the case where C is reducible, we can assume that C = {< s,t > x < p,q >G P 1 x P 1 ^ = 0} and g : C —• P n is given by g{< s,t> x <p,q >) =< g0(< s,t> x < p, q >),..., gn{< s,t> x <p,q>) > =< a0,ips + aofiqs + a0t3tq,... , an>ips + anfiqs + an>3tq > . Since (C, g) G U(Spec(C)) we can assume that g0 vanishes at the two distinct points (< 1,0>,< 1,0>),(< 0,1 >,<0,1 >), therefore we can assume that go = qs, and since g0 and g\ do not vanish simultaneously, 01,1,01,3 0. Composing g with the isomorphism r' : C C given by T' :< s,t > x <p,q >-> (s,—t) x (—p,q) \ ai > 3 / \o i , i / g or' :< s,t > x <p,q> &21 2^ 3 1 ^ n 3 \ gs,ps + o,\ oqs + tq, —*-ps + a2,2gs H -tq,... , —-ps + a n, 2gs H ) , ai,i a i ) 3 ai,i ai, 3 / in this case the image is two (not necessarily distinct) lines, one through the points i /n i °2,1 an,l < 1, ai,2, • • • , an,2 >, ( 0,1, , • • • , \ Oi,i Oi,i and one through the points a2,3 ani3 < l,ai,2, • • • ,an,2 >, ( 0,1, ai,3 a i ) 3 41 Chapter 2. Coordinate charts for Mo,o(Pn, 2) If these lines are not distinct (C, g) is a stack point. If the lines are distinct, we have exactly two points p\ and p2 in UQ such that (CPi, f) is isomorphic to (C,g), namely Proof of Lemma 8: To show that a map of stacks is surjective it suffices to show surjectivity for geometric points (see [2]). Therefore we must show that if (C, g) G U(Spec(C)), we can find a point p e Uo such that (C xUop,fopx) = (Cp, f) is isomorphic to (C, g). This follows immediately from Lemma 9. • Theorem 5 We have an induced map of stacks and • <t>: [A3n-l/Z2] -»• U which is an open immersion. Proof We have shown at this stage that the image is U. Thus to prove this theorem we need to 42 Chapter 2. Coordinate charts for Mofi(Pn, 2) show that the map <f> is etale (Lemma 12), radiciel, and representable (Lemma 13), then, since the image is the open substack U of .Mo,o(Pn>2), the map is an open immersion (see [2]). • First we show that <f> is etale. From Deformation Theory we have the following lemma: Lemma 10 Let X be a scheme over C. A morphism tp iP:X^M0fi(Pn,2) given by the data C - 4 P n I X is etale at a 6 X if and only if the induced map H°(Ca,Tc)^H°(Ca,rT^) is bijective. Lemma 11 <j> is etale. Proof By Lemma 10 above it suffices to show that both H°(Ca,f*Tpn) and H°{Ca,Tc) are isomorphic to C 3 " 4 " 2 , and that the induced linear map between them is bijective. Let a = (m,6i, . . . , bn, r 2 , i , r 2 f 2 , . • • , r n , i , r n , 2 ) . We shall we shall deal with two cases, when m ^ O and when m — 0. 43 Chapter 2. Coordinate charts for A^o,o(Pn, 2) Let UXi = {< x0,xu... ,xn >E Pn\xi ^ 0}. Since /(C) C UXo U UXl we need only consider the tangent bundle for P n , Tpn, on these neighborhoods. Suppose we use affine coordinates ^ for i ^ j on UXj, and list them as follows: XQ Xj—^ Xj Xj^.\ > • • • ) Xj Xj Xj Xn X j We choose local trivializations for Q,pn in the standard way, i.e. & : f i P » | t f x 4 ^ (0UxiT d[Xj x, (0,... ,0, ^ ,0 , . . . ,0), jth position If we let <f)itj — <j>i o then in particular 0 o ,i is given by the matrix: / V - i f ^0X2 xf XQX$ xj x\ 0 XQ Xl 0 0 0 0 XQ Xl 0 0 0 XQ Xl 0 0 0 0 XQ Xl \ J Suppose we let $ be the locally trivializing maps for the dual sheaf 7p" then 4>\fi is the transpose of the matrix above. Suppose first that m ^ 0. Consider the following diagram: P 1 = Ca C - 4 P n 4 i 44 Chapter 2. Coordinate charts for A^o,o(Pn, 2) Suppose we give P 1 coordinates < s, t >, then the induced map from Ca to P n is given by < s,t >—K st, ms2 + bist +12, mr2,\S2 + b2st + r2^t2,..., rarniis2 + bnst + rn^t2 > • = < / o ( M ) , / l (*,*),••• Jn(s,t) > The trivial neighborhoods Ui pull back to the trivial neighborhoods £//., where U/{ — {< s,t ><= P\fi{s,t) ^ 0}. In particular Ufo = P 1 - ^ ) and Uh = P1-V{ms2+bxst+t2), and Ufo U Uh = P 1 . The gluing matrix <^0 pulls back to I t2 foh ft o Jl /o /n-1 ft h 0 0 fo fl 0 0 fo fl 0 0 0 h I An element of H°(Ca, / *Tp») is a global section of /*Tp", so is given by on Uf. for j = 0,1, where gitj is a homogeneous polynomial of degree four is s and t, and Mi j 0 (gi) = go-This forces —<7i,i — gift and + /oft,i = /i0«,o for % = 2, . . . , n, /o<7i,l = /l<7i,0 — /j<7l,0-45 Chapter 2. Coordinate charts for Mo,o(Pn, 2) Suppose we let gij = a^s 4 + bijsH + . . . + e^t4, then since git\ = —<7i,o we have in particular a±ti = —ai,o and e^i = —eii0, and the equation above becomes st(aiAsA + ... + eiAt4) = (ms2 + ftist + t2)(aifis* + ... + e i i 0t 4) - (mr^s 2 + M + rit2t2)(aifis4 + ... + eli0t4) Taking s = 0 we have taking t = 0 we have 0 = maifi - mri}iaifi =>• ritlalfi — aifi. Thus the polynomials g^\ are entirely determined by the polynomials p; )0. The polynomi-als gifi, for % = 2, . . . ,n , are not entirely free, their s 4 and r 4 coefficients are determined by (71,0. Thus for i = 2 , . . . , n, the polynomials (?;)0 have each exactly three free coefficients. Altogether we have 5 + 3(n - 1) = 3n + 2 free coefficients. Thus H°{Ca, / * T P » ) S C 3 n + 2 where W; = We shall now show that H°(Ca, Tc) = C 3 n + 2 also. The fiber above the point a is contained in Up U Ut since m ^  0, so we need only consider Tc restricted to these neighborhoods. Recall that Ut has coordinates 0 = & in the case m ^ O , and has a basis given on £// 0 by \ t2 dwi' t dwi' dwi' s dwi' s2 die i ' and Uv has coordinates - , m , 6 i , 46 Chapter 2. Coordinate charts for Mo,o(Pn, 2) where m P Since m ^ O w e can replace £ with (^)(| )• We have locally trivializing maps for fic|r/pnt/t given in the usual way: <j>t:Slc\Ut^(0Vtfn 4>p:Slc\Up^(0Ut)3n, and the gluing matrix 4>pt = <$>p o <^x is / - M (S) - (i) (!) l3n-l V If we allow Ua and Ut to denote the neighborhoods Up and Ut pulled back to C a (identified with P 1 ) , then the gluing data for 7c pulled back to P 1 is the transpose of the matrix above i.e. if we let Mst denote the gluing matrix which is the pullback of 0V 4, then Mst = M ( S ) -(i)(f) 0 ••• 0 \ I3n-1 V A global section of TQ is given by a 3n-tuple of rational functions on each neighborhood, such that if we multiple the 3n-tuple of functions for Us by Mat we get the 3n-tuple of functions for Ut. Clearly the functions in positions 2 to 3n must be constant, say the constant functions A; 0 , . . . , kn, 02,1,02,2, • • • , c n,i, cn>2- If we let £ and & denote the rational function in the first position for Us and Ut respectively, we see that i = j = 2 and -<»> (?) H 3 (?)*-*• 47 Chapter 2. Coordinate charts for Mo,o(Pn, 2) Therefore stk0 9 = mf. m Thus g is dependent on / , and / is a homogeneous polynomial of degree two. In particular if / = a 0 s 2 + aist + a2t2, then g = —ma0s2 + (—ma\ — — ^ st — ma 2t 2 . m Altogether we have 3n — 1 constant functions, and / has three free coefficients, so we have 3n + 2 in total, therefore H°(Ca, T c ) = C 3 n + 2 in the case m # 0, and a basis for T c on Ut is d s_d d d_ d_ d * 2 3 ( f ) ' t 9 ( f ) ' a ( f ) ' a m ' - 5 6 i ' - " ' c V n i 2 i / -We now need to show that the induced map H°(Ca,Tc)^H°(Ca,rTpr>) is bijective. The universal map / from C to P N induces a map from f*Qp* to tic- Restricting to f~l(UXQ) C Ut and using Ut coordinates, we have f ((?*)) = (? m (?)+b4+mr^ (?)+b4+r2-2' • • • ' m r ^ ( ? ) + b A + r - 2 ) and s o , = (™2,x-r2t2(^y^ ^ ( zoO = l ^ " ' 1 " r"'2 (s 2) ) ^ (?) + ^ (?) d m + d b n + m (?) d r ™ + ( d f n 48 2-Chapter 2. Coordinate charts for M0io(Pn, 2) Therefore the induced map from / * f i p n X 0 a c\f-iuxo is given by the 3 n x n matrix ( mr2,i - r 2 ) 2 (&) f r 2 i l (f) 0 V i o 0 0 0 0 0 0 0 0 0 0 m (!) s 0 mr 3 > i - r 3 ) 2 ( £ ) 3^,1 (f) 0 0 0 0 0 0 1 0 m (!) m (!) s 0 0 0 0 mr, \ m (!) (f) 0 0 0 1 0 0 0 0 0 0 0 0 The transpose of this matrix gives the induced map from T c | / - i r / z 0 - > / * 7 p » | / - i % 0 . We need to write down the matrix representing the induced map H0(Ca,Tc)->H0(CaJ*Tpn), with respect to our chosen bases. This matrix will be a (3n + 2) x (3n + 2) matrix. 49 Chapter 2. Coordinate charts for Mo,o(PN, 2) Let v = (a0,ai,a2,k0,... , kn, c 2,i, c 2 ) 2 , . •. ,0,1,1,(^,2) € C371"1"2 represent the section of To, given on Us by the 3n-tuple of functions / aos 2+ai st+atf2 \ «2 and Ut by A;0 \ Cn,2 J ( (-mao)s 2+(-J-fl-TOOi)^t+(-ma2)t 2 \ i 2 *1 \ Cn,2 / We multiply the Ut representative by the transpose of the matrix above to get the corre-sponding element of H°(Ca, / * T p » ) . Letting (-ma0)s2 + (-^ - mai)st + (-ma2)t2 9 = t2 we have / h (f) + k1 + (m-£)g ^ (fc0r2,i + mc 2 , i ) (f) + k2 + c 2 i 2 (f) + (mr 2 , i - r 2 , 2 # (fc0r3,i + mc 3 > 1) (f) +k3 + c 3, 2 (f) + (mr 3 , i - r 3 ) 2 ( j ) ) 5 \ (korn,i + mcn^) (f) +kn + c n > 2 (J) + (mr n , i - r n , 2 ( # / Writing the ith coordinate as a rational function with sH2 in the denominator, we have the following numerator: -m2a0r^is4 + (-rj )1m2a1 + mciti)s3t + (-m2a2rt,i + ma0rit2 + ki)s2t2+ 5 0 Chapter 2. Coordinate charts for Mo,o(Pn, 2) ci>2 + T ^ z r ^ + mai7-i,2 ) t3s + m a 2 r i t 2 t 4 . m When j = l we retrieve the numerator by taking r^i = r i j 2 = 1 and ci,i = Ci ) 2 = 0. We identified an element of H°(Ca, /*Tp«) with C 3 n + 2 by listing the coefficients of the coordinate functions in increasing powers of t and ignoring the r 4 and s 4 coefficients in positions 2 to n since they are not free. Thus the section above corresponds to the following element of C 3 n + 2 with respect to our chosen basis: —m2ao —m2ai \ \ —m2a2 + ma0 + ki ma2 —r2^mla\ + mc2)i —m2a2r2)1 + mao^ 2,2 + k2 C2,2 + ^ + ™air2,2 -r 3 iim 2ai 4- mc3ji -m 2 a 2 r 3 ! i + ma0r3 ) 2 + k3 c3,2 + r-^r + "W3,2 -rntimlai + mcn,i -m 2 a 2 r n ) i -I- ma0rn>2 + kn 51 Chapter 2. Coordinate charts for At o,o(Pn, 2) Therefore our map is given by the matrix: ( a0 ai a2 k0 kx .. . ki . s4 9 —mr 0 0 0 0 . . . 0 . . 0 s3t 0 —mr 0 0 0 . . . 0 . . 0 s2t2 m 0 —m2 0 1 . . . 0 . . 0 st3 0 m 0 m 0 . . . 0 . . 0 t4 0 0 m 0 0 . . . 0 . . 0 C i , l C i i 2  0 s3t 0 -m2riA 0 0 0 . . 0 . . m 0 s2t2 0 -m2riA 0 0 . . 1 . . . 0 0 st3 0 mn , 2 0 r.,2 m 0 . . . 0 . . . 0 1 52 Chapter 2. Coordinate charts for Mot0(Pn, 2) where the three rows indexed by i occur for % - 2,... , n. This matrix is invertible for all n > 2, with inverse Mm^Q: I 1_ m 2 0 0 0 0 0 0 0 .. . 0 0 0 .. . 0 0 0 0 1_ m 2 0 0 0 0 0 0 .. . 0 0 0 .. . 0 0 0 0 0 0 0 m 0 0 0 .. . 0 0 0 .. . 0 0 0 0 1 0 m 0 0 0 0 .. . 0 0 0 .. . 0 0 0 \_ m 0 1 0 m 0 0 0 .. . 0 0 0 .. . 0 0 0 T2,2 m 0 0 0 mr 2 , i 0 1 0 .. . 0 0 0 .. . 0 0 0 n,2 TO V o o o o 0 0 0 mr^i 0 0 0 .ILL m r n , i TO 0 0 0 0 0 0 o -rn,2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 £ 0 0 TO 0 0 1 0 0 0 0 0 0 \ 0 0 0 TO 0 0 0 mrnti 0 0 0 .. . 0 0 0 .. . 0 1 0 0 T2,l TO 0 0 0 m 0 0 .. . 0 0 0 .. . 0 0 0 0 0 0 -7*2,2 0 0 0 1 .. . 0 0 0 .. . 0 0 0 0 0 0 0 0 0 ^-0 0 0 0 1 Thus the induced map is an isomorphism. We shall now deal with the case m = 0. Let a = (0, bi,.. • , bn, r 2 , i , r 2 ) 2 , . . • , r B > i , r n > 2 ) be fixed. We identify C a with {<s,t> x <p,q>eP1 xP 1 |pt = 0}. 53 Chapter 2. Coordinate charts for Mo,o(Pn, 2) Then the universal map / restricted to Ca is given by f(<s,t> x <p,q>) = < sq,ps + bxsq + qt, r2Aps + b2sq + r2y2qt,... , rnAps + bnsq + rn,2qt > . Consider the following P 1 V Ca A, P 1 where the top copy of P 1 has coordinates < s,t> and fx:<s,t>^ (< s , t> ,<0 , l> ) , and the bottom copy of P 1 has coordinates < p,q > and f2:<p,q>^ (< 1,0 >,<p,q>). An element of H°(Ca, / * (Tp«) ) is a global section of each of the sheaves / * / * T p " for i = 1,2 such that these sections agree when p = t = 0. The universal map above restricted to the top copy of P 1 is given by / o fx :< s, t >->•< s, bxs +1, b2s + r2i2t,... , bns + rn>2t > and restricted to the bottom copy is given by / ° J 2 -<P,q >->< q,P + hq,pr2tl + b2q,... ,pr n > 1 + bnq > . The open neighborhoods UXo and UXl cover /(C). The open set UXo pulls back to Us via / o fx and Uq via / o / 2 , and UXl pulls back to Ui via / o / j where Ux = P 1 - < 1, -bx > and U2 = P 1 - < - 6 i , l >. 54 Chapter 2. Coordinate charts for M0fi(Pn, 2) The gluing data for Tpn (the matrix ^ 0 given in the m ^ 0 case) pulls back to the following gluing matrix, Mu on the top copy of P 1 : M i = V (hs+t)2 s2 (hs+t)(b2s+r2,2t) (bis+t)(b3s+r3i2t) (feis+f)(6n-is+r(n_1)t2t) s2 (bis+t)(bns+rn<2t) and to the matrix M2 where / iP+biq)2 g2 (p+6ig)(r2,ip+62g) M o (p+fcig)(r3,ip+63g) n2 V (P+fclg)(r(B-l),lP+ftn-lg) q2 (p+6ig)(rn,ip+6ng) 0 0 s 0 bis+t s 0 0 9 0 0 0 0 0 0 0 0 0 0 0 P+biq o 9 0 0 bis+t s 0 0 0 0 bis+t s 0 0 0 \ P+biq q P+hq q on the bottom copy of P 1 . We shall deal with the top copy of P 1 first. A global section is given by an n-tuple of rational functions of the form \ 3 ' t=l,...,n j=l,2 where hx = hs +1, and h2 = s, on Ui for j = 1 and Us for j = 2. Where M i g i = g 2-Thus gitl = - p i , 2 and = 0t,2v*i* + *) - 9i,iipiS + %2t) 55 Chapter 2. Coordinate charts for Mofi(Pn, 2) for i — 2,. . . , n. Hence the coordinate functions on Us entirely determine the coordinate functions on U\. The coordinate functions on Us are not entirely free, in fact, if we let 9i,j = Q'itjS b{jSt -\- Cijt then the above equation becomes s{aitls2-rbitlst+citit2) = (o j ,2S 2 +6 i ,2st+c i ) 2i 2 ) (6 iS+t ) - (a i ) 2S 2 +6 1 > 2 s i+c i > 2t 2 ) (6 iS*+r i i 2i 2 ) . Taking s = 0 we have thus 0 = Ci, 2 - C i ) 2 r i j 2 Ci,2 - c i i 2 r i ) 2 . So git2 has three free coefficients and the #;)2 have each two free coefficients (the s2 and st coefficients) for i = 2,... , n. So we have 3 + (n - 1)2 = 2n + 1 free coefficients in total, and the coordinate functions on Us are of the form / a i , 2 S 2 + 6 l , 2 S t + C l , 2 t 2 \ \ ? / Performing a similar analysis on the bottom copy of P 1 we see that the global section is completely determined by the section on Uq, and the section on Uq is of the form / a'i,2g2+6i,2Pg+ci,2P2 \ o'2 2<?2+6'2 2P9+r2,ic'1 2 p 2 V <,29 2+ 6n ,2P9+ rn.l c ' l ,2P 2 These sections must agree when t = p = 0 therefore = a'it2 for i = 1,... , n. Thus we have n + n + 1 + n + 1 = 3n + 2 free coefficients so F ° ( C a , / * (Tp») ) = C 3 n + 2 , with basis 56 Chapter 2. Coordinate charts for jVfo,o(Pn> 2) given on Uq by / d d t d p 8 t d p d t2 d p2 d t2 d p2 d \ \ dwx' dwn' 5 dwi q du>i" ' " ' s dwn q dwn' s2 dw\' q2 dw\ ' s2 dwn' q2 dwn J Thus the section above, with respect to this basis is given by the vector: ( ai,2> a2,2> • • • > Gn,2) &i,2> &1,2, 2^,2> &2,2, • • • , &n,2> &n,2, c ' 1 ) 2 , C i i 2 ) Now we will show that in the case m = 0 that # ° ( C a , T c ) = C 3 n + 2 also. We consider again the diagram P 1 V Ca ^ C A P 1 where ii : 1^  -+ m, &i,... , &„, r 2 ] 1, r2,2, • • • , r n,i, r f t ) 2) C £/ t 11 : 0»O>&i,--- A , r 2 l i , r 2 , 2 j . . . , r n ) i , r n > 2 ^ C C / , 12 : V ^ ' 1 ^ ~* A , r 2 > i , r 2 f 2 , . . . ,r n,i,r„, 2^ C f/9 z 2 : ^ ! > ^ - > • , &n, r 2,i, r 2 f 2 , . . . ,r n , i ,r n , 2 ^ C C/p. We cover the top copy of P 1 with Us and Ut and the bottom copy with Up and Uq. Abusing notation we shall also use Ut, Up and Uq as a cover for the universal curve C as before. We have that i2(P1)QUqUUp. The gluing data for T c , </>^ , where <frt : fic|r/t (Or/J 3" 57 Chapter 2. Coordinate charts for A^o,o(Pn, 2) <t>g : fic 0 , - > (0Ug) 3n is given by / - S 0 0 m f 0 0 0 0 0 0 0 I3n-2 and is given by V 0 -$ 0 ; i 0 0 0 0 0 Pulling back the gluing data back to the top copy of P 1 , via i\ we have V -5 0 0 0 f 0 0 0 0 0 \ I311-2 / and pulling back via to the bottom copy of P 1 via i2 we have 0 -** 0 f 0 0 0 0 0 0 \ l3n-2 58 Chapter 2. Coordinate charts for M0io(Pn, 2) Thus a section of i*(Tc) is given by / as2+bst+ct2 \ t2 0 h k2 \ hn-2 J on Ut and by / as2+bst+ct2 \ s2 0 ki k2 \ h n - 2 J on Us. Similarly a section of i2(Tc) is given by 0 a!q2+b'qp+dp2 Q2 k[ \ k2 \ Kn-2 J 59 Chapter 2. Coordinate charts for A^ 0 ] 0 (P n , 2) on Uq and by on Un ( a'q2+b'qp+c'p2 \ P2 0 k[ key \ ^3n-2 / Now these two sections must agree when p = t = 0 thus ' - * ] • ( o ^ 0 a' ki = k[ \ h n - 2 J y ^ 3n-2 / Therefore we have the following 3n + 2 free coefficients: (b,C,b',c',ki,. . . ,fc 3n-2)-Hence H°(Ca,Tc) = C 3 n + 2 in the case m = 0, and we have a basis t d t2 d p d p2 d d d d d d d « a ( j ) V a (J) ' q d ( E ) ' q2 d ( E ) ' dV • • • ' 36 n ' d r 2 i l ' d r 2 , 2 ' ' ' ' ' drny drn,2 We now need to show that the matrix, giving the induced map from H°(Ca, / * (Tp») ) to H°(Ca,Tc) with respect to our chosen bases, is injective. We shall calculate the induced map on each copy of P 1 and glue. We shall use the neighborhood Uq of C. Now Chapter 2. Coordinate charts for Mofl(Pn, 2) /-• P , t p t ( 1, - + 61 + - , r 2 , i - + b2 + r 2 , 2 - , • \ q s q s so d(f) is given on UXQ (with coordinates ^) by the matrix P L * r n , l - + 6 « + r n , 2 -5 S d '.5 d b i d b 2 dbj d b n d r 2 , i d r 2 , 2 d r i , i d r i j 2 d r n , i d r n , 2 1 72,2 1 7-2,1 1 0 0 1 0 0 0 0 0 f 0 i 0 0 0 0 0 0 \0 0 ^,2 0 0 0 0 0 0 0 rn,2 rn,i 0 0 1 0 0 0 0 where tp = 0. The transpose of this matrix gives the map from /* (Tp«) to T o In terms of the basis chosen the vector (b,c,b',c',ki,... ,fc 3n-2) 61 Chapter 2. Coordinate charts for .Mo,o(Pn, 2) corresponds to a section g g2 h k2 \ hn-2 J on Ca C Ug where pt = 0. Under d(f) this maps to f b'Z + c'ZL + h - b l - 4 N r 2,i {V* + d§) + kn+l (J) + fc2 + r 2 , 2 (-&{ - ) + * n + 2 (}) r M + c ' £ ) + ^-1+2(^-1) (f) + h + r < ) 2 ( - 6 f - c%) + A W ^ D (f) ^ r B l l (y; + d£) + A ; 3 N - 3 ( § ) + kn + r n , 2 ( - 6 J - c%) + (f) / which corresponds to fc2 + r 2 l 2 ( - 6 f - c £ ) + f c n + 2 (f) fci + r i i 2 - c ^ ) + A ^ + 2 ( < _ i ) (f) V fcn + r n , 2 ( - 6 f - c f ) + A ; 3 n - 2 ( f ) / 62 Chapter 2. Coordinate charts for Ma,o(Pn, 2) when we restrict to the top copy of P 1 , and look at the representative on Us, and corre-sponds to f * i + V\ + c'$ N k2 + r2,l(b^ + c'^)+kn+1^) ki + ritl (&'| + c ' £ ) + A:n-i +2 ( i-i) (f) V K + r B l l (tfj + c ' £ ) + fc3n-3 (j) / when we restrict to the bottom copy of P 1 , and look at the representative on Uq. In terms of our chosen basis for H°(Ca.Tc), this corresponds to the following element of (ki,... ,ki,... ,kn, b', -b, b'r2,i+kn+i, -br2t2+kn+2,... , b'riti+kn-i+2(i-i), -brit2+kn+2(i-i), ... , b'rn>i + A; 3 n _ 3 , - 6 r n > 2 + & 3 n _ 2 , d, -c) 63 Chapter 2. Coordinate charts for A^o,o(Pn, 2) Therefore the matrix for this map is: 0 0 r 2 > 1 0 - r 2 j 2 0 0 0 0 0 ? v 0 -r<,2 0 0 0 0 0 r„,i 0 ^ - r „ , 2 0 0 0 j 0 0 0 hn-2 0 0 0 0 \ 64 Chapter 2. Coordinate charts for Mo,o(Pn, 2) This matrix is also invertible with inverse Mm=0: • 0 ' o 0 0 1 0 0 0 I n 0 0 0 - r 2 l 2 -Ti,l 0 0 - r W l 2 - r n , i 0 0 - r n , 2 0 0 - 1 0 0 1 0 0 0 2(n-l) 0 Lemma 12 is eia/e Proof Consider the following diagram: [A 3 -VZ 2 ] A AT0,o(Pn,2) f A3"" 65 Chapter 2. Coordinate charts for Mofi(Pn, 2) The quotient map A3n~l -> [A3n~l/Z2] is etale and (j) is etale thus <j> is etale also. • Lemma 13 (p is radiciel and representable. Proof We first show that the map <f> is radiciel. Thus we must show that the fiber over a point is equivalent to a scheme consisting of one point, i.e. is a groupoid consisting of a collection of isomorphic objects such that none of these objects has any automorphisms. Let X be a scheme and suppose a map X —>• A^o,o(Pnj 2) is given by the data C x fA Pn X We consider F(Spec(C)), where F is the fiber product, F -» X I 4 [Uo/Zv] M),o(Pn,2) In this case an element of the set X(Spec(C)) is a morphism from Spec(C) to X, so is given by a point x in X, and the image of this element in A1o,o(Pn, 2) is the pair (Cx, fx), where Cx is the fiber over x. An object ofthe groupoid [U0/Z2] (Spec(C)) is as follows: B 4 Uo I Spec{C) 66 Chapter 2. Coordinate charts for .Mo,o(Pn, 2) where B is a principal Z 2 bundle over Spec(C) and g is Z2-equivariant. If we take the canonical section of B associated to the identity element Id of Z 2 , then g(Id) is a point in Uo, which we shall call p. We shall rename the bundle Bp and the map gp in this case. If we let a be the non-identity element of Z 2 then gp(a) = ap. If p is invariant under the action of Z 2 on Uo, then Bp has an automorphism, induced by the action of a on the fiber. If p is not invariant then its orbit contains a point p' distinct from p, and the bundle (Bp> —> p',gpi) (with the morphism gp< induced by the action of Z 2 on p') is isomorphic to (Bp —>• p,gp), via the isomorphism which sends (Id,p) to (a,p') and (a,p) to (Id,p'). Given an object (Bp-+p,gp) e [U0/Z2}(Spec(C)), this maps to the element of Afo,o(Pn, 2)(5pec(C)) given by ((Bp xUo C ) / Z 2 , / ) , which, if we consider the section S of Bp associated to the identity element, is isomorphic to (5 X r j 0 C , / ) , which is in turn isomorphic to (Cp, /) . If we assume that (Cx,fx) G U(Spec(C)), then if (Cx,fx) is a stack point there is one point p e UQ such that (Cp, /) is isomorphic to (Cx, fx) by Lemma 8, but we have exactly two choices for the isomorphism since (Cp, f) has exactly one non-trivial automorphism, induced by the action of a ouCp. Let us call these two isomorphisms tp : Cp -» Cx and ip o a : Cp -» Cx. In this case F(Spec(C)) consists of the two objects (x,Bp,ip), (x,Bp,ipoa). We shall show that these objects are isomorphic and neither has any automorphisms. 67 Chapter 2. Coordinate charts for Mofi(Pn, 2) We consider the automorphism of Bp induced by a. We need to find the induced mor-phism from Cp = p x Uo C to itself. Let (p, c) G Cp then (p, c) comes from the element ((Id,p),c) in B xUo C where p = 7r(c). Now (a(Id,p),c) = ((a,p),c) which lies in the same equivalence class as ((Id,p),ac) which corresponds to (p, ac) in Cp, thus the induced map from Cp to Cp is a, and we have the following commutative diagram: (Cp,p) A (Cx,fx) ai || (cp,P) *F (cx,fx) so the two objects are isomorphic. This argument tells us also that neither point has any automorphisms, since any au-tomorphism would have to be induced by the unique non-trivial automorphism of Bp, a. If (Cx, fx) is not a stack point then there are points p and p' in U0 such that (Cp,f), (Cp', f) and (Cx, fx) are isomorphic, and p and p' lie in the same equivalence class under the action of Z 2 on U0- Thus again F(Spec(C)) consists of two objects, in this case (x, Bp, f^1 o /), (x, Bpl, f^1 o f) where Bp and Bp> are as above, (since fx is an isomorphism when restricted to Cx, and / is an isomorphism when restricted to Cp or C'p). These objects are isomorphic, via the morphism (Id,p) -> (a,p'), (a,p) —> (Id,p'). This morphism induces a morphism given by a from (Cp, f) to (Cy, / ) , and we have the following commutative diagram: (CP,f) f ^ f (Cx,fx) ai || (Cp.,f) f ^ f (Cx,fx) since f(a(c)) = f(c). So the two objects above are isomorphic. 68 Chapter 2. Coordinate charts for Ato,o(Pn> 2) Neither of these objects has any automorphisms since neither Bp nor Bp> has any auto-morphisms. Therefore the map 4> is radiciel. To show that <f> is representable, we need to show that if X is a scheme then the fiber product F, F -> X i I [E/0/Z2] -> M),o(P n ,2) is a scheme. Let V be a scheme over C, we consider the diagram of groupoids F(V) ->• X(V) I I [t/0/Z 2](V) -> A4o,o(P",2)(V) An object of F(V) as a triple (h,B,*P) such that h : V —> X is a morphism, 5 is given by the data B U0 I V and ip is an isomorphism from (V xx Cx, fx) to (5 C, / ) , where S is the section of B associated to the identity element. An automorphism of this object is an automorphism of B, r , such that the following diagram commutes: (SxUoC,f) A (VxxCJx) r l II (SxUoC,f) A (VxxC,fx) 69 Chapter 2. Coordinate charts for Mo,o{Pn, 2) If r is not the identity it must be nontrivial when restricted to some fiber of B, and both elements of this fiber must map to the same (Z 2 invariant) point p in Uo, thus we can represent this fiber by (Bp -> v,gp), for v E V. Let x = h(v). When we restrict the diagram above to v, we have an automorphism of the object {x,Bp,4>) of F(v), induced by r. This is impossible, by the arguments above, thus r is the identity. Therefore t/> is representable as required. • We now consider the induced Gm action on Uo'-Lemma 14 The action of Gm on P71 induces an action on Uo which is given by the matrix ( X~2k 0 0 \ 0 Dx(A) 0 \ 0 0 D2(X) j where D\(X) is the nxn diagonal matrix with d^i — X~kl and D2(X) is the 2(n—1) x2(n—1) diagonal matrix with entries X~k, X~k, X~2k, X~2k, etc. to X~^n~l^k, A - ^ - 1 ^ along the main diagonal. This action is compatible with the Gm action on the Afo,o(Pn)2) induced by the action N on P71. It is also compatible with the Z 2 action on UQ. Proof Let x = (m, & i , . . . ,bn, r 2 , i , r 2 > 2 , . . . , r n i i , r n > 2 ) be a point in Uo- Then / restricted to Cx is given by the rational map < s, t >-*< st, ms2 + bist +12, mr2,\S2 + b2st + r2<2t2,... , mrn^s2 + bnst + rn,2t2 > 70 Chapter 2. Coordinate charts for A^o,o(Pn) 2) from P 1 to P n . We choose a different representative for (Cx, f) by composing / with the projective change of coordinates, T\, given by the matrix \ 0 0 At Thus letting fx = / o r> we have /A(< s,t >) = (st, ^s2 + bist + Xkt2, ^ r 2 A s 2 + b2st + Xkr2>2t2, ... , ^ r n A s 2 + bnst + \ k r n , 2 t 2 } . Now we recall that the action of Gm is given by the matrix N{\) = A a o 0 0 0 0 A a i 0 0 0 0 A Q 2 0 0 0 0 0 0 0 V 0 0 0 0 . . . A""-1 0 0 0 0 . . . 0 A a " where a 0 = ^ and on = a0-ki. Thus the point A.(C, /) = (C, N(\ofx) is parameterized by the map / kn % fc(n-2) / m o , , t , \ <s,t>-> (\TsttA~V- l^—s2 + bist + Xkt2J , A*i^i) ^ R 2 ) L 5 2 + b2St + xkr2at2^j ,X^ (jk-rntls2 + bnst + A f c r n , 2 t 2 ) ) . kn Dividing through by A ~r we have < s, t >-> {st, X~2kms2 + X'khst + t2, X~3kmr2<1s2 + X~~2kb2st + X~kr2>2t2,... A"**"*1 W „ , i s 2 + X~knbnst + A ~ f c ( n _ 1 ) r n ) 2 £ 2 ) . This corresponds to the map / restricted to the fiber Cx.x where X.x is the point (A" 2 f cm, X'%, X~2%,X~nkbn, A - f e r 2 ) 1 , A " f c r 2 , 2 , . . . , A f c ( n _ 1 V n j i , A * * " " 1 ^ ) 71 Chapter 2. Coordinate charts for A^o,o(Pn, 2) in U0. Since Gm acts with the same weight on and r^2 for i = 2,.. . , n, we see that this action is clearly compatible with the action of Z 2 above. • 2.5 An Open Cover for M 0,o(P n, 2) We have so far constructed one affine open cell UQ isomorphic to A3™'1 with an etale map <t> to M.ofl(J>N, 2), and a Z 2 action such that the induced map ^:[C/o/Z 2]-^ATo,o(P n ,2) is an open immersion. We have also calculated the induced Gm action on UQ. What we would like is a method, using which, one can generate a collection {Ui} which taken together map surjectively to Mofl(Pn, 2). This is very easy to do. For our purposes we only need two open cells, which cover the fixed locus under the A1 action. Lemma 15 Let P e PGLn+i be a projective change of coordinates for P71. Let C and f be as before. Let <f>P be the morphism of stacks from A3n+1 to Mo,o(Pn, 2) given by the following data: C - 4 F 1 4 F 1 7 T I Then fo-.A3"-1 ^AT 0 , o (P n , 2 ) is etale, and the induced map from the quotient stack 4P:[A3n-l/Z2}-^MQt0(Pn,2) 72 Chapter 2. Coordinate charts for Mo,o(Pn, 2) is an open immersion. The image is the open substack Up defined by the following data: Up : (Schemes/C) —• (Groupoids) X -> UP(X) Where an element ofUp(X) is an element of Mofi(Pn,2)(X) given by the data {7r:Cx-+X,fx:Cx->Pn), and such that this element satisfies the following: Consider fx (H{) G T(CX, fxO(l)) where the Hi is the image of the coordinate hyperplane Xi under P, and let Di = Z(fx(Hi)) for i = 0,1 then (i) The induced map TT : D0 —> X is unramified, (ii) The divisors Do and D\ do not intersect. Proof This is an immediate consequence of Theorem 4, Lemma 11, and Theorem 5. • Corollary 1 Let X be a scheme and (v.Cx^XJx-.Cx^n, an element of Mo,o(Pn, 2)(X). For each x € X we can find an open subscheme x e U C X and a projective change of coordinates for F1, P, such that if Cv is the restriction of Cx to U, then the data (n:Cu^XJx:Cu-+Pn), gives an element of Up. 73 Chapter 2. Coordinate charts for M 0 ] o ( P n , 2) Proof Let x e X. We can find hyperplanes H0 and Hx such that fx(Cx) D H0 D Hi is empty (where Cx is the fiber of Cx over x) and /x maps two distinct points of Cx into H0. Choose any projective change of coordinates P such that P maps the hyperplane XQ = 0 to # 0 and Xx = 0 to J?!. • 2.6 Covering the Fixed Locus In this section we shall find two cells UQ and U\ which cover the fixed locus under the A 1 action given by M . We shall take UQ to correspond to our original construction, i.e. UQ = A 3 N ~ L with coordinates (ra, & i , . . . , rnp), and the universal curve and map over UQ are C and / . We shall identify which points on the fixed locus lie outside the image of UQ in A^o,o(Pn, 2), and find a cell U\ which covers the missing points. To do this we need only find a suitable projective change of coordinates P (Corollary 1). Recall that the fixed locus consists of the following closed points: • Points (C, /) in M 0 ) o(P n , 2) where f(C) is a conies in P n given by the ideal Ic = (X2 — 2X0X2 + c X | , X3, X4,... , Xn) for ceA1, denoted by F C M . • The point (C, /) where C is isomorphic to two copies of P 1 meeting at a point and / maps each copy of P 1 to the line given by the ideal loo — (X2, X3,... , Xn), and the crossing point maps to the unique A 1 fixed point in P n , namely the point F0 =< 1,0,... , 0 >. This point corresponds to c = 00, and is denoted by F ^ . 74 Chapter 2. Coordinate charts for Mofl(Pn, 2) The conic given by the ideal Ic meets V(X0) at two distinct points, <0,7,1,0, . . . ,0>, where 7 2 = - c , if c ^ 0 and at the point < 0,0,1,0,... , 0 > if c = 0. Thus F C M is covered by UQ as long as c is non-zero. So we need to generate another cell, which contains the point F 0 M . There are many choices for this cell but we shall choose one which makes the gluing maps between the cells particularly nice. We notice that our conic does not intersect V(Xi,XQ — X2), and it intersects the hyper-plane Xi — 0 at two distinct points. We can take H0 = X\ and H\ = X0 — X2. Consider the projective change of coordinates for P n given by the matrix: I / n i i \ \ P = 0 1 1 1 0 0 \o o I) 0 0 I n - 2 V P maps the hyperplane XQ will suffice. 0 to HQ and the hyperplane Xx = 0 to H \ . Thus this P Lemma 16 Let 0 1 1 \ 1 0 0 Vo o 1 / \ 0 0 In-S 75 Chapter 2. Coordinate charts for !Mo,o(Pn> 2) Let U\ — A 3 " - 1 with coordinates (m ,bx,... ,b'n,r'2X,r22, • • • ,r'ni,r'n2). Consider the rational map from U\ x P1 given by fP((m'b[,... ,r'nt2),<s,t>) = < rn\r'2l+l)s2+(b[+b'2)st+(r'2>2+l)t2, st,m'r2ls2+b'2st+r'2>2t2,... ,m'r'nAs2+bnst+rn2t2 > Let C be the blowup of Ui x P1 along the subvariety where t = m' = 0, as before and let fp be the induced morphism from C to P"1. Then the induced map from Ui to jMo,o(Pn, 2) is etale and the image contains the closed point F0M. Proof Follows immediately. • 2.6.1 Transition Functions Since each conic in P n is contained in a plane, we calculate the transition functions by equating planes. First however we observe that the projection map p: P n -> P 2 < x0,xi,... ,xn > - » <a; 0 ,a; i ,X2> induces a map Ui,n -> Ui>2 (m, , bn, r 2 ) i , r 2 j 2 , . . . , rn,i, r n > 2 ) (m, bu b2, r 2 , i , r 2 > 2 ) . for i = 0,1, where Uitk is the is the ith member of the cover for M0to(Pk,2). If we let Uoitk be the intersection C/0,fe and t/i^, we have a commutative diagram: Ui,n Uoi>n —> Uo,n 4 4" 4' Ul,2 <r- UQI,2 —> Uo>2 76 Chapter 2. Coordinate charts for Mofi(Pn, 2) where all the horizontal maps are two to one. We shall initially calculate the gluing maps in the dimension two case and then calculate them in the general case. One can readily check that the map < s, t >—st, ms2 + bist +12, mr2,iS2 + b2st + r2}2t2 > in the case r2,\ i1 r 2 , 2 parameterizes a conic (which is degenerate when m = 0) with equation ((b2 - r 2 , i6i)X 0 + r 2 , i X i - X2) ((b2 - r2y2h)X0 + r 2 , 2 * i - X2) + m(r 2,i - r2,2)2X2 = 0 and the map < s,t > - K m'(r 2 > 1 + l)s2 + (b2 + b[)st + (r 2 > 2 + l)t2,st,m'r21s2 + b'2st + r'22t2 > in the case r'21 ^ r'22 parameterizes a conic (which is degenerate when m! = 0) with equation ((6'2 - r'^b'JX, + r'2A(X0 - X2) - X2) {(b'2 - rfvl«l)Xl + r'2>2(X0 - X2) - X2) + m'(r'2tl - r'2>2)2X2 = 0. Since conies corresponding to points in (7o,2 have the property that they do not pass through the point < 0,0,1 > (the intersection point of the hyper planes X0 — 0 and Xi = 0) and the line X0 = 0 is not tangent, if a conic in Uit2 lies in (7oi,2 then it has the following properties: « 2 = W,l + 1 ) ( ^ + 1)#0 and Ki - r'2<2)2 {(b[ + b'2)2 - 4m'(r2 > 1 + l)(r 2 , 2 + 1)) ^  0 => ux = (b[ + b'2)2 - 4ro'(r2>1 + l)(r 2 > 2 + 1) # 0. 77 Chapter 2. Coordinate charts for Mo,o(Pn, 2) Therefore on the intersection we can invert u\ and u2. Similarly in Uo,2 coordinates conies corresponding to points in the intersection do not pass through the point < 1,0,1 > (the intersection point of the hyperplanes X\ = 0 and XQ — X2 = 0), and the line X\ = 0 is not tangent so we can invert v2 = (62 ~ 2^,161 - 1)(&2 - r2,2h - 1) + m(r2,i - r 2 ) 2 ) 2 and =»• vi = b\ - Am ^  0. Equating the coefficients of the conies above we have the following relations: mv2 m = v\ , 1 _ 262 - (r 2,i + r 2 | 2)&i - 2(62 - r2,ifti)(fr2 - r 2 > 26i) - 2ro(r2,i - r 2 , 2 ) 2 ± (r 2,i - r 2 | 2 ) 0 ^ r2,l)r2,2 — -5i&2 + 61 + 2ro(r2>i + r 2 | 2 ) bib2) - 2m(r2,i + r 2 | 2 ) «i 2v2 Going in the opposite direction we have m = mu2 2^,1, ?"2,2 (62 ~ 6i)(r 2 | 1 + r 2 > 2 + 2% - 2 ^ ^ ) ± (r'2A - r'2>2)^ui b[ + b'2 2u2 Ui b2 = b'2(b'i + b'2) - 2m'(2r'2^2 + r 2 < 1 + r 2 > 2) « i These relations hold also in the case where r2,\ = T - 2 ) 2 and r'2l = r'22. Therefore we have the following maps of rings, in Uij2 coordinates: T(u0,2) -> r([/01>2) C[m,6i ,6 2 ,r 2 > 1 ,r 2 ) 2 ] -> C m', 6'1? 62, r2 v r2 2 , W 7 , — , — 1 ^ A i ^ Ui u2 I ({W'f - ui) 78 Chapter 2. Coordinate charts for M0,o(Pn, 2) where m'u2 m - » — 5 -}J 1 w 61 ftj + &2 , _ + ~ 2m'(2r 2 ] 1r 2 ) 2 + r 2 | 1 + r 2 > 2 ) 02 —r Ml (b'2 - t j ) ^ , ! + r'2,2 + 262 - 2 6 ^ , , ) + (r 2 > 1 - r ^ W (62 - b[)(r'2A + r 2 , 2 + 2b>2 - 2 6 ' ^ , ^ ) - ( r^ - ri^W' r 2 2 -> 2u2 In particular we see that the points (m'X,b'2,r'2tl,r'2>2,W') and (m'^b^r'^r'^-W') in t/01,2 both map to the same point in Uo,2 and {m\b\,b'2,r'2^2,W') and (m' ,6 ' 1 , 6 ' 2 , r 2 i l , r 2 ) 2 , -W) map to the same point in (71)2. Thus we have four points in C/01,2 giving us the same stable map. In order to calculate the transition functions in the general case we observe that if points (m, 61,... , bn, r 2 , i , r 2 ) 2 , • • • , r n , i , r n , 2) and ( m ' , . . . , b ' n , r 2 1 , r 2 2 , . . . , r n l , r n 2 ) , 79 Chapter 2. Coordinate charts for A4o,o(Pn, 2) with riti ^ rit2 and r'^ ^ r'^2 for some 2 < i, j < n, represent the same conic in P n then the plane spanned by the points < 1, bi, b2,... , bn >, < 0,1, r 2 ) i , . • • , r„,i >, < 0,1, r 2 , 2 , . . . , rn<2 > must be the same as the plane spanned by the points <b'2 + b[, 1, b'2,... , b'n >, < r'2jl + 1,0, r 2 j l ) . . . , r'njl >, < r 2 > 2 + 1,0, r 2 ) 2 , . . . , rj, > 2 > . Therefore the latter three points can be written as a linear combination of the former three, and vice versa. In this way one can solve for the rest of the coordinates and we get the following relations: r i , D ri,2 b' "t,l ± Vj,2y/Vl 2v2 -bxbj + 2m(riA + r< > 2) where Vi,l Vi,2 = -2m(r 2 ) 1 - r 2 ) 2 ) (r i ) 1 - rii2) - b\(ri>2r2A + r 2 > 2 r M ) +bibi(r2>l + r 2 ) 2 ) + bxb2(ritl + r i ) 2 ) - &i(r i fi + r i > 2) - 2626j + 2b{ -6 i (r i , 2 r 2 | 1 - r 2 , 2 r M ) + 6j(r2 ii - r 2 ) 2 ) - 6 2(r i j l - r i > 2) + ( r M - r i > 2 ) , and ^,1,^,2 2u2 -b\% + &;) + 2m'(r^ + rj, 2 + r[^2 + r^r a > 1 ) where = "(&2 + + T - - 2 + r^r^a + rj , 2 r 2 i l ) 4- (2r 2 i l r 2 ) 2 + r 2 ) 1 + r2 ) 2)&; +2^ + ^ ( r 2 i l + r 2 j 2 ) W*,2 = <1 - <2 + < 1 < 2 - < 2 < 1 for i = 3, . . . , n. 80 Chapter 2. Coordinate charts for Mo,o(Pn, 2) 2.6.2 The Fixed Locus We previously described the fixed locus in terms of geometric points of Mofi(Pn, 2). We would now like to express it in terms of the coordinates of the two neighborhoods. The fixed locus is contained in the subvariety V(X3, X±,... , Xn) of P n , thus we can project the fixed locus into P 2 . If a stable map corresponding to a point in Uo or Ui has its image contained in V(X$,X±,... , Xn) we must have, in each neighborhood respectively, bi = K = riti = ri%2 = r- ; 1 = r-> 2 = 0, for % = 3, . . . , n. Thus when dealing with fixed conies we identify each with a 5-tuple of complex numbers. In P 2 , each conic is given by a polynomial as above. Thus solving a series of equations, the fixed conic (for c ^ 0) 1 2 2 2 —X, XoX2 + X7 c c corresponds to the two points The stack point (corresponding to c = 00) corresponds to the origin. If F(XQ, Xi, X2) is a conic corresponding to a point in Ui then F(1,0,1) = 1 thus we express the fixed conies in the following way for c 7^  2. Solving a system of equations we see that this conic corresponds to the two points ( ^ T ' °' °' 7^2 ' °) ' °' °' °' rh) = (m''&;'625 r2'1'r2,2)-81 Chapter 2. Coordinate charts for Mo,o(Pn, 2) 2.7 A vector field on M 0 , 0 ( P n , 2) In the last section we gave a method using which on can construct a family of affine varieties {Ui}ieI such that each U{ is isomorphic to A 3 " - 1 , and in each case we have an etale map from Ui to Mo,o(Pn,2). In this section we shall find the vector fields on UQ and U\ which are pull-backs of the vector field on .Mo,o(Pn> 2). Lemma 17 Let UQ be as in Section 2.4. Then the vector field on UQ is ' m(-46i + r2,i + r 2 , 2) ^ - 4 m - b\ + b2 -2m(r 2 , i + r 2 j 2 ) - bxb2 + h -2m( r i j i + r i ) 2 ) - bxb{ + bi+i -2m(r„ , i + r B | 2 ) - hbn r 2 , i6i - r 2 , i - b2 + r 3 , i r 2 , 2 &i - r\2 -b2 + r 3 j 2 >"i,i&i - ri,ir2,i -bi + r ( i + i ) , i n,2^ i - rit2r2,2 -bi + r ( i + i ) i 2 r n , i&i - r„,ir 2 > i - 6„ \ fn,2h - r n , 2 r 2 , 2 - 6B / with respect to the basis ( d_ d_ d_ _d d_ _d d_\ Vdm"' db[''"" ' dbn' 9 r 2 , i ' d r 2 > 2 ' " ' ' 9 r n > 1 ' d r n , 2 ) Proof Let V be the section of Tp» given by the A1 action M . Let <f> be the map of stacks given 82 Chapter 2. Coordinate charts for M0to(Pn, 2) by the data C A P 7T 4-as before. Since the map of stacks given by this data is etale, we have an isomorphism H\Ca,Tc)^H* (Ca,f*Tp») induced by / , for each a e A3n~l. Suppose we restrict to the locus of points a e UQ such that Ca is irreducible (i.e. a = (m,bi,... ,rn%2) G UQ with m / 0 ) . Then the inverse of this isomorphism is given by the matrix Mm^0 of Lemma 11. We shall find the pull-back of the section V using M m ^o- We shall restrict V to the neighborhood U X O of P n . Recall that the A1 action given by M is as follows: t. < XQ, X \ , . . . , XN > — < XQ + tXi + . . . + —.Xn, Xi + tX2 + . . . + Thus the induced vector field on U X N is XQX2 — x\ XQXJ, — X2X\ XQXn Xn—\X\ XnX\ 2 ' 2 XQ XQ 83 Chapter 2. Coordinate charts for M0,o(Pn, 2) Pulling back via / to C, we have a section of / * 7 p « given on f~l{UXQ) c Ut by: ( st(mr2,is2+b2St+r2,2t2)-(ms2+bist+t2)2 \ s2t2 st(mr3,is2+b3st+r3,2t2)-(mr2,is2+b2St+r2,2t2)(ms2+bist+t2) s2t2 st(mri+i,1s2+bi+ist+ri+itlt2)-(mri,is2+bist+rii2t2)(ms2+bist+t2) I - (mr n , 1 s2+bn st+rn,212) (ms2+b3 t+t2) J \ sW / which, written in increasing powers of t, is / (-m2)s4+(m(-2fci+r2,i))a3(+(-2m-fe2+62)s2i2+(-2fc1+r2,2)^3-t4 \ s2t2 {-m2r2^)s4+(m(-b2-biT2yi+rzli))s3t+{-m{r2,\+r2a)-bib2+b3)s2t2+(-^^ ^ 2 ( - m 2 r ( i + 1 ) i l ) s 4 + ( m ( - 6 i - c n r i , 1 + r ( i + 1 ) , 1 ) ) s ^ ^2 •• ( -m 2 r n a )s 4 +(m ( - fe n -r n , i&i ) )^ 3 t - | - ( -m (T- T 1 , i+r n ,2 ) -6nfei )a 2 t 2 +(-&n-fcir n ,2)st 3 +(-rn ,2) t 4 / s2t2 ) 84 Chapter 2. Coordinate charts for Mo,o(Pn, 2) This corresponds to the following element of C 3 n + 2 with respect to our chosen basis: m(-26i +r2,i) -2m - b\ + b2 -2h + r 2 , 2 -1 m(-bi - 61^ ,1 + r(i+i),i) -m(r i ) i + r i ) 2) - M i + bi+i - h - 6ir i ) 2 + r(i+i),2 m(-6„ - rB,i&i) -»n(r n i i + r n > 2) - 6„6i -6„ - &ir„)2 \ 85 Chapter 2. Coordinate charts for Mo,o(Pn, 2) Multiplying by Mm^0 we have —m 2&i - r2,2 -1 m(-46i + r 2 , i + r 2, 2) - 4 m - b\ + b2 -2m(r 2 , i + r 2, 2) - M 2 + b3 -2m(r« ] 1 4- r i ) 2) - M i + - 2 m ( r „ , i + rn>2) - 6i6n r-2,i&i - r | p l - 62 + r 3 , i ?*2,2&i - r | i 2 - &2 + r 3 ) 2 _ r i , l r 2 , l - 6i + »"(»+i),i r^&i - ri>2r2y2 -bi + r ( i + i ) ) 2 J>,1&1 _ rn,1^2,l - &n ^ rB,2&i - r B | 2 r 2 ) 2 - bn y 86 Chapter 2. Coordinate charts for Mofi(Pn, 2) When we push forward to UQ we forget the first three coordinates. Thus our vector field on UQ — V(m) is given by ' m(-46i + r 2 , i + r 2 | 2 ) ^ -Am -b\ + b2 -2m(r 2 , i + r 2 i 2 ) - &i&2 + 63 -2m(ri,i + r i | 2 ) - bibt + bi+1 -2m(r n > i + r„ | 2 ) - 6x6n r2,i&i - r\\ -b2 + r 3 ) 1 r 2 ( 2 6i - r\2 - 62 + r 3 > 2 - n,ir 2 , i - 6» + ,^261 - riy2r%2 -bi + r ( i + 1 ) i 2 r n , ibi - r„, ir 2 ) i - bn \ rn>2bi - r n , 2 r 2 i 2 - bn J This extends uniquely to give a section of Tu0- • We now need to calculate the vector field on Ui. Recall that when tVi is identified with A 3 n _ 1 with coordinates (m',b'v... ,b'n,r'2A,r'2i2,...,r'n>1,r'nt2) 87 Chapter 2. Coordinate charts for A^0,o(Pn, 2) then the universal curve for Ux is the blowup U\ x P 1 and the universal map fP is P o f where / is the universal map for U0 and / / n i 1 \ \ P = 0 1  1 0 0 \0 0 1 ) 0 V L 0 * n - 2 7 Therefore in order to find the vector field induced on U\ we pull back the element of Tp"(J7Xl) given by the vector field V via P giving an element of 7 p » ( [ / X o ) which using the matrix M m ^ 0 (with all the entries replaced with the corresponding coordinates for Ui) above gives us a vector field on Ui. We choose affine coordinates for UXl which pull back via P to give the standard affine coordinates for UXo, i.e. we choose the affine coordinates Xp - X2 X2 £ 3 %n ) > ) • • • ) Xi Xi Xi Xi In these coordinates, the vector field V on UXl is given by the following: 'Xi(xi - X3) - X2(x0 - X2) XiX3 - x\ XiXn - Xn-XX2 ~XnX2 ™.2 ' 2 ' " * " ' 2 ' 2 Pulling back via P we have the following /x0(x0 - x3) - XXX2 XQXS - x\ X0Xn - Xn-iX2 -x2xn ~2 ' 2 ' ' ' ' ' 1 ' 2 XQ XQ XQ XQ 88 Chapter 2. Coordinate charts for Mo,o{Pn, 2) We pull back via / and list in increasing powers of t as before f -(m')2r'2>l —m'r'31 — m'b'2 — m'b\r'2l l-b'3-b[b'2-m'(r'2!l+ry ~ r3,2 ~b'2 — b\r22 b'3 - {b'2f - 2mr'2tlr'2<2 r3,2 — 2&2r2,2 ~ m'b2r'i,l ~ m'b'ir2,l K+i ~ b'ib2 - m'r'iAr'2t2 - m'r'^r'^ r'i+1,2 ~ b2<,2 ~ b'A,2 - b'n-lb2 ~ ™'r'n-lA,2 ~ m ' r n - l , 2 r 2 , l r'n,2 ~ b2r'n-l,2 ~~ b'n-Xr2,2 -m'b2r'n,l ~ m'b'nr2,l -b'nb'2 - m'r'n^'2t2 - m'r n > 2r 2 ) 1 ~b>2Tn,2 - b'nr2,2 89 Chapter 2. Coordinate charts for A^o,o(Pn, 2) We then multiply by M m # 0 and push-forward to Ui to get the following vector field: ( • -m'(r'3A + r 3 > 2 + (r2>1 + r'^V, + 2b'2) ^ -2m'(r 2 i l + r y + l - b'3 - b[b'2 -4m'r 2 ) 1 r 2 ) 2 - [b'2f + b'3 -2m'(rJ i 2^ | 1+rJ f l^ i 2)-6 a6J + 6{+1 -2m'(r n_ l i 2r 2 > 1 + r n _ M r 2 i 2 ) - b'^ + b'n -2m\r'nt2r'2!l + r'n>1r'2t2)-b'2b'n r'2,ir'3>l + (r'2!l)2b'l-b'2r'2il + r'3tl r2,2r3,2 + (r2,2)2^i — &2r2,2 + r3,2 r'i,A,i + r'iAA ~ b'Ax + * i + M ri,2r3,2 + ri,2r2,2^'l ~~ b'iT2,2 + V'i+l,2 r n - l , l r 3 , l + r n - l , l r 2 , l ^ i ~~ &n-l r2,l + r'n,l rn-l,2r3,2 + r'n-l,2r2,2b'\ ~ K-lr>2,2 + r'n,2 r'nA^+r'^r'^-by^ \ r'n,2r3,2 + r'n,2r2,2b'l — ^nr2,2 J in terms of the basis / d _d_ _d_ _d d_ _d d_\ 90 C h a p t e r 3 The Cohomology o f ¥ 0 , 0 ( P n , 2 ) 3.1 The Hypercohomology of M 0,o(P n, 2) Let {Ui} be the collection of affine varieties each isomorphic to A 3 n _ 1 constructed in Chapter 2. Our aim is to calculate the hypercohomology H°(.Mo,o(P n ,2) , /C*) where K? = Q,~p, for p < 0, is the complex of sheaves associated to the vector field, with differential i(V). We can achieve this by calculating the hypercohomology of the following double complex, using our etale cover {Ui}: t t t ••• -»• ®i<j<kto2{uijk) -»• ®i<j<kn(UiJk) -> ®i<j<kO(UiJk) -> o t t t ••• ©i<jfi2(yij) -»• 8i<jfi(^ij) ®i<jO(Uij) -> 0 (*) t t t • • • -> e ^ 2 ^ ) -+ ®MUi) -> ©<o(c/i) -> o 3.1.1 Preliminary Ideas In order to calculate the hypercohomology of the complex (*) above we shall use the concept of quasi-isomorphic complexes. 91 Chapter 3. The Cohomology of M 0 ) 0 (P n, 2) Definition 10 A map j : C* -)• IC between complexes of sheaves is a quasi-isomorphism if it induces an isomorphism on cohomology sheaves: j * : rlq(C) -> Hq{)C*) q>0. In our case KP = Q,~p for p < 0, and the differential i(V) is a map i(V) : KP '-»• KP~l and "Hp is the cohomology sheaf associated to the presheaf w ( n ) < t 7 ) " • Lemma 18 Suppose the map of complexes of sheaves on X, j : C ^ K* is a quasi-isomorphism. Then the induced map on hypercohomology j , : I F ( X , £ * ) ->W(X,1C) is an isomorphism. Proof One compares the spectral sequences associated to a each double complex. • Thus our idea shall be to replace the complex of sheaves K* with a quasi isomorphic complex whose hypercohomology is more accessible. We shall simplify in four main ways: 92 Chapter 3. The Cohomology of M0,o (P n, 2) (i) We shall show that one need only consider the members of the open cover U which meet the locus where the vector field vanishes. Thus since UQ and U\ cover the zero locus (in the notation of Chapter 2) we need only consider these members of the cover. (ii) The vector field is given, on each of U0 and U\, by a collection of 3n — 1 regular functions. Suppose that a we have subsequence of length (3n — 1) — 1 in each case which is a regular sequence (Definition 12). Then the cohomology sheaves rl~p vanish for p > 2 so we can truncate the complex of sheaves in question to 2 terms. (iii) We shall introduce the idea of a companion pair of vector fields for the vector field V (Definition 11) to simplify the remaining modules in the double complex. (iv) After simplifying as above, we reduce our problem to calculating the hypercoho-mology of a four term double complex. We observe at this stage that the group Z 2 x Z 2 acts on each of the four terms and all the maps are equivariant with respect to this action. Thus the complex has a Z 2 x Z 2 invariant subcomplex and taking the Z 2 x Z 2 invariant subring of the hypercohomology is the same as taking the hypercohomology of the Z 2 x Z 2 invariant subcomplex. Definition 11 A pair of vector fields (WQ,WI) on UQ and U\ respectively is called a companion pair for the vector field V it has the following properties: (i) Wi does not vanish on Ui. (ii) The kernel ofWi, K-i, is mapped injectively by i(V) to Ouif for i = 0,1. (Hi) W0 = fWx on UQI, where f e O*UQ{U0I). As a result of the three statements above we shall show that the double complex (*) is quasi-isomorphic to the four term double complex given in the statement of the following lemma: 93 Chapter 3. The Cohomology of M 0 ) 0 (P n, 2) Lemma 19 Let V and U be as above. Suppose that the zero locus of the vector field V is covered by U0 and Ux. Let V be given by the collection of rational functions Hi — (vi,i,--. ,Vitk) on Ui, and suppose that for i — 0,1 the sequence v{ has a regular subsequence of length k — 1. Suppose also that (W0, W\) is a companion pair for V. Then the double complex (*) is quasi-isomorphic to the following four term double complex: 0 0 t t o nUo/(i(v)n2Uo + Ko)(u0i) -»• oUo/(i'v)Ko){u01) o t t 0 (B^itouJiiW^+lQiUi) -4 ®i^,iOuJ{i{y)lCi){Ui) 0 where Ki is the kernel ofWi. Proof Lemma 20 below will show that we can throw out all the Ui except UQ and U\ since we can assume that the others do not meet the fixed locus (by replacing them with smaller affine open sets if necessary). Lemma 23 will show that the cohomology sheaves vanish for p > 2. Lemma 24 will give us the four terms of the double complex. The conclusion of Lemma 19 is then immediate. • Lemma 20 Let p>0 and suppose x lies in one of our affine open covering cells U for Mofl(Pn,2), such that the induced vector field V does not vanish on U. Then the stalk 'H~p is trivial. To prove Lemma 20 we shall use the Koszul complex. But first two definitions. Definition 12 Let R be a ring and M and R-module. A sequence v\,V2,.-- ,vr € R is called a regular sequence for M, if the following conditions hold: 94 Chapter 3. The Cohomology of M 0,o (P n, 2) (i) V\ is not a zero divisor in M, and for all i = 2,... ,r, V{ is not a zero divisor in M/(v1,v2,... ,Vi-i)M. (ii) M/(vlt...,vr)M^O. In particular we shall be interested in the case where M = R — C[xi,... , In this case if . . . , vr is a regular sequence for R , then it is a regular sequence for the localization RM for each maximal ideal M such that M ~D (i>i,... , vr). Def in i t ion 13 Given a ring R and a sequence vi,... , vr in R. We can define a complex Kp as follows: K0 = R, Kp = 0 for p ^ 1,.. . , r and ~ • ^ e * i . - . * p * i < i i < . . - < i p otherwise, where ij = 1,. . . , r and eiu,.^ip is a formal basis element. Thus Kp is a free R-module of rank (n). The differential d : Kp —> Kv_\ is defined by p deiu...,ip = ^-1)i~lvieiu...,ri,..,ip forp > 1, and for p = 1, dej = Vj for j = 1,.. . , r. This differential has the property that d2 = 0 and the resulting complex (Kp, d) is called the Koszul complex, and is written K.{vu... tvr) = K.{v). Suppose M is an .R-module then K.(v,M) is defined to be K.(v) ®R M. If C* is a complex of J?-modules then C.(v) is defined to be C* <B> K.(v). It is easy to see that K.(vu ... ,vr) = K.'vi) <8> K.{v2) ® • • • ® K.(vr). L e m m a 21 Let C* be a complex of R-modules, and v 6 R. Then we obtain an exact sequence of complexes o-»c\, -•£.(«) ^ c : ->0, 95 Chapter 3. The Cohomology of M0,o (P n, 2) where CI is the complex obtained by shifting the degrees in C* up by 1 (i.e. C'p = Cv-\). The associated long exact sequence in homology is • • • HP(C.) - + Hp(C.(v)) H^iC*) ( T 1 Hp-i(C*) —>•••. Proof: See [13, Theorem 16.4, p. 127]. Lemma 22 Suppose that ... , vr) is a regular sequence in R, then Hp(v, M) = 0 for p> 0 and H0(v, M) = M/vM. Proof: See [13, Theorem 16.5, p. 128.]. • Proof of Lemma 20: Since U is an open subset of Ak = Spec(C[xi,... ,Xk\), for some k, the vector field is given by a collection of regular functions vi,... ,vk G T(i7). Let x G U then since the vector field does not vanish at x we can assume that at least one of the Vi is a unit in Ou,x, let us assume that Vi is a unit in OutX. Let Ou,x = R, and suppose that Ak has coordinates xi,... ,xk then the complex associated to the vector field restricted to the stalk at x: can be identified with the Koszul complex by identifying e .^..,^  with dxiAdx2A.. .Adxk. Since v\ is a unit, HQ(K.(VI, . . . ,Vi)) = 0 for i = 1,... , r , because if we take / G . . . - • f i ^ n s - 1 - * . . . 96 Chapter 3. The Cohomology of M 0 , 0 (P n, 2) Thus Hi = 0. We shall show the other homology groups vanish using induction on i. Clearly Hp(K.(vi)) = 0 for all p. If we assume that Hp\K.(vu ... , vi)) = 0 for all p > 0 then the exact sequence • Hp(K.(vu ... , v^) -» Hp(K.(vu ... , vi+i)) -> Hp-iiK.ivu ... , vi)) ->• • • • shows that Hp(K.(vu... ,vi+1)) = 0 for p>l. Thus Hp(K.(vi,... , vi+i)) = 0 for i = 0,... , r—1 andp > 0, so in particular Hp(K.(v)) — Hp(K.(vi,... , vr)) = 0 for all p > 0. Now via our identification H~p = Hp(v, R) so ri-p = 0, for p > 0 , as required. • Lemma 23 Let U be an affine open subset of Ak and let the vector field V be given by (vi,... , vk) on U. Suppose (v\,... , vk-i) is a regular sequence. Then for all p > 2, the cohomology sheaf %~p = 0. Proof: Let x e U. We shall show that U~p = 0, when p > 2. We assume that the vector field vanishes on U since if it did not we would be done by Lemma 20. Let x £ U, and let R = 0Ux as before. By Lemma 21 above we have a long exact sequence ...-». Hp(K.(vi,... ,«*_!)) -> Hp(K.(vi,... ,vk)) # p _ i ( # > i > • • • > « * - i ) ) { ~ ^ 1 V k Since Hp^{K.{vu... , vk^)) = Hp{K.(vu vk^)) = 0 for p > 2, Hp(K.(vi,vk)) -0 for p > 2. Thus Upx = 0 for p > 2 as required. • 97 Chapter 3. The Cohomology of M 0 > 0 ( P n , 2) We would like to have a simple way of deciding when a pair (Wo, W\) of vector fields is a companion pair. The most difficult part ofthe definition to check is part (ii), i.e. that the kernel of a vector field W on Ui is mapped injectively by i(V) to Out- The following lemma gives a condition on the vector field V which ensures that this property holds, and gives us an explicit description of the four terms of the double complex in Lemma 19. Lemma 24 Suppose that Ak has coordinates yi,... ,yk and W is the vector field on U C Ak. Suppose i(V)dyi = V{ for i = 1,... , k — 1 where (v\,... , v k - i ) is a regular sequence in T(U). Then i(V) maps the subsheafK offlu generated by < dyi,... , dyk-i > injectively to OJJ (this subsheaf is the kernel ofW) and tlu/iUVn + K)^ (T(U)/(vu vk-!)) dyk, Ovl{i(V)K) ^r(U)/(vu... ,vk.1). Proof We shall show that i(V) maps the stalk Kv injectively to Ou,P for each p e U. First suppose p G V(vi,... , ffc-i), then (v\,... , vk-i) is a regular sequence in Ou,P- The stalk Kp for p € U is generated by dyi, for i = 1,... , k - 1 as an Ou,P module. Consider the submodule of Vlu,p generated by dyi. Then i(V) clearly maps this submodule to Ou,P injectively since Ou,v is an integral domain. Consider the induced map i(V) , UJ/)1 : ^p/(i(V)n2UtP + Ou,pdyi) Ou,p/viOUtP t t i(V): nvji(y)(^p) -> 0UtP The submodule i(V)Sl\j contains in elements of the form i(V)(dyiAdyj) = vxdyj-Vjdyi for j = 2 , . . . , k. Thus i(y)ti{jp + Ou,Pdy\ contains elements of the form vxdyj for 98 Chapter 3. The Cohomology of M 0 ,o(P n , 2) j = 2,... , k, so W ( » G 0 f # , + Ou,PdVl) - ( 0 (r(tOp/«i) ) /i(tOn* „. \j=2,...,A: / Suppose next we consider the submodule of 2 generated by dy2, and the induced map i(V) from this submodule to Ou,p/viOu,p, Wf • ^u,P/(i(V)^p + Ou,pdyi + Ou>pdy2) -> 0UtP/{vi,v2)Ou,p t t Wf • nUtP/(i{v)n2u>p + ou<pdyi) OujviOu,,, Then this induced map is injective if and only if fv2 = 0 in Ou,p/viOu,p implies that / = 0 in Ou,p/viOu,P, this is exactly the condition that vx,v2 is a regular sequence in Ou,p- Suppose that this is the case, then in the same way as before we observe that since i(y)Vtijp contains elements of the form i(V)(dy2 A dyj) — v2dyj — Vjdy2 for j = 3, . . . , k, i(V)Q2/p + Ou,pdyi + Ou,pdy2 contains elements of the form v2dyj for j as above. Thus Qu,p/(i(V)nlp + Ou,pdyi + Ou,pdy2) <* ( 0 (r(£Op/fa,t*))dVi) \j=3,... ,k J Thus continuing in this way we see that if the sequence vi,vs,... ,vk-i is a regular sequence in Ou,p for each p then Kp is mapped injectively by i(V) to Ou,P and in particular we have the following: SluJ(i(V)n2Ujt + Kp) * { r ( U ) p / { v u d y k , and O W W s W ( « i In the case where p V(vi,... , vk-i) we can assume that vx invertible in Ou,p thus the sub-sheaf of fir/,p/i(V)fi2/jP generated by ofr/i is all of Q,ujP/Hy)^l\j<p and Ou,P/v\Ou,P = 0 so our claim is trivially true. • 99 Chapter 3. The Cohomology of M 0 j 0 (P n, 2) 3.1.2 The Hypercohomology Calculation We shall first change coordinates on Uo and U\, the reason for this is to make the extraction of the invariant part of the hypercohomology more simple. We choose new coordinates as follows: di - n , 2 Si = n , i + n , 2 d'i = <i = < i + <2 We list coordinates on U0 as follows (m, &i, 6 2 , . . . , bn, s2, d2, s3, d3,... , sn, dn) and for t7i, the order is analogous. 100 Chapter 3. The Cohomology of M 0 > 0 (P n, 2) In these coordinates the vector field on UQ and U\ is m(-4&i + s2) - 4 m - b\ + b2 -2ms 2 - bib2 + b3 —2ms3 — bib3 + ft4 -2msi - bibi + bi+x -2msn - bibn s2h - | («1 + di) - 2b2 + s 3 d2bi — s2d2 + d3 S3W - \ (s 2s 3 + d2d3) - 2b3 + s 4 d3h - \ (s2d3 + d2s3) + tit «A ~ \ (s2Si + d2di) - 2bi + si+i dih - \ (s2di + d2Si) + di+i snh - \ {s2sn + d2dn) - 2bn ^ dnb\ - \ {s2dn + d2sn) J \ ( -m'(s'3 + s'2b\ + 2b'2) -2m's'2 + 1 - b'3 - b[b'2 -m'((s 2) 2 - (d2)2) + b'3 - (b'2f -m'{s'2s'3-d'2d'3)-b'2b'3 + b', -m'(s'2s'n - d'2d'n) - b'2b'n ± (s'2s'3 + d'2d'3) + I ((s2)2 + {d'2f) b[ - s'2b'2 + s'3 \ (s'2d'3 + d'2s'3) + s'2d'2b[ - d'2b'2 + d'3 \ « s 3 ) 2 + (d'3)2) + \ (s'2s'3 + d'2d>3) b[ - s'2b'3 + s4 s'3cf3 + i (s'2d'3 + d'2s'3) b[ - d'2b'3 + d'4 \ (s^ + d'^) + I (s'2s> + d'A) b[ - s'M + s'i+l \ Kd j + disj) + \ (s'A + d1^) b[ - djftj + d'i+l \ (s'3s'n + d'3d'n) + \ (s'2s'n + d'2d'n) b[ - s'2b'n V I (4< + d'3s'n) + \ (s'2d'n + d'2s'n) b[ - d'2b'n J We now need to find a companion pair for V. Suppose we replace Ui with (7i -^(m'd 2 ( s 2 + 2)), abusing notation we shall call this new neighborhood Ui also. We observe that V(m'd2(s'2+ 2)) meets the fixed locus in Ui at c = 1, since on the fixed locus, / c - 2 „ , 2 , n 2(c -1) and c ^  2. Thus UQ and our new neighborhood Ui still cover the fixed locus. 101 Chapter 3. The Cohomology of M0,o (P n, 2) Lemma 25 Let (W0,Wi)-Id d (s'2 + 2) d d2ddn ds'n d'2 dd'n Then (Wo, W\) is a companion vector field for V for n > 3. Suppose that we let vy be the image of dy under V. Then the proof of Lemma 25 is dependent on the following conjecture. Conjecture 1 The sequences (vm,vbl,vb2,... ,vbn,vS2,vd2,... ,vSn) = J 0 (*Wi tty, vy2, • • • , vyn, • • • , vs'n_1,'"d'n_l, (4 + 2)vs,n - d'2vd,n) = h are regular sequences for n > 3. This conjecture can be verified up to n = 10 on Maple for the first sequence, and to n = 6 for the second. In general proving that the first sequence is regular amounts to showing that the three term sequence (b\sn - ^ (s2sn + d2dn) -2bn,m (-4&i + s2), bxdn - i (s2dn + d2sn)j is a regular sequence in the ring C[m,s2,d2,b1] where sn, dn and bn are defined as follows: ( Sn \ ( - h + U2 U \bn) 2 "2 V'2 ( \d2 \ 2m -bi + \s2 0 0 6i ) S2 d2 \ \4m + bl J It is not hard to show that the first two form a regular sequence for all n, so we need to show that the last one is not a zero divisor when we mod out by the first two. 102 Chapter 3. The Cohomology of M 0,o (P n, 2) Showing that the second sequence is regular amounts to showing that in the ring c[m', 4 , 4 4 , 4 M^&'g] the following seven term sequence is regular for all n: -m'{s'3 + s'2b[+2b'2), -2m's'2 + 1 - b'3 - b[b'2, ( 4 + 2 ) where -m>((s'2f-(d'2)*) + b'3-(b'2f, \{s'2s'3 + d'2d'3) - s'2b'2 + i ( ( 4 2 + {d'2f)b'x + s'3 ^(s'2d'3 + 4 4 ) - d'2b2 + s'2d'2b[ 4- 4 - r o W „ - d'2d'n) - b'2b'n, ^ ( 4 4 + « ) + \{JA + dMK - 4 & ; ) - 4 Q(44 + 4 4 ) + ^ ( 4 4 + 4 4 W = (v, ^, vb>2, tvB, ^, vd>2, (s 2 + 2)vs>n - d2vd,n) dL If we replace Ui with ^-iw + ^ i ) - i t e + ^ i ) 4 ^ - § ( 4 + 4&'i ) - | ( 4 + 4&'i) 4 ^ m 's 2 —m'd'2 b2 y n - 2 Ui - v ( ( 4 ) 2 - ( 4 ) 2 + 3 4 + 2) - n ( 4 f - + 2 4 ) again abusing notation we shall call this new neighborhood U\. Then U\ and UQ still cover the fixed locus. We can solve the system above for m', b'2, b'3, s'3, d'3. First vm> = 0 => 4 - _ 4 & i _ 2 4 103 Chapter 3. The Cohomology of M 0 > 0 (P n, 2) since m! does not vanish on U\. vVi = 0^b'3 = -2m's'2 + 1 - b[b2. If we add and vy2 and solve for m' we have 1 - b[b>2 - {b'2f m 2s2 + ( * ' 2 )2 - (d 2 )2-vd'2 = 0 ==> d 3 Substituting in our expressions for s'3 we have •/ _ 2 S 2 S 3 + d2b'2 — s^d^b'i d , _ d'2(s'2b[ - Ab'2) 3 s'2 + 2 • Finally we can substitute our expressions for d'3 and s'3 into vs>2 = 0 and solve for b'2, so u_ -b'As'2y-(d'2f + 2s'2) 2 2((s'2y-(d'2y + ss'2 + 2y Thus we are reduced to showing that the pair K n , (s ' 2 + 2 K{. -give a regular sequence in C s2> b\i YT-1 1 (4) 2 - (4 ) 2 + 3s'2 + 2' {s'2f - {d'2Y + 2s 2' s'2 + 2' Proof of Lemma 25 One can clearly see that W{ does not vanish on U\. When n > 3 and Wo is written in Uy coordinates we have Wo = ^-Wu 2v\ so W0 and W\ differ by a scalar on UQ\. That the kernels /Q map injectively under i(V) is a consequence of Conjecture 1 using the criterion in Lemma 26. • . 104 Chapter 3. The Cohomology of M 0 ) 0 ( P n , 2) To show that one can dispense with the terms of the double complex involving for j > 2 , one must show that one has a regular subsequence of length 3n — 2 among the vy on each neighborhood. Thus is true on UQ if one can establish that the first sequence in Conjecture 1 is regular for all n and on U\ is a consequence of the following conjecture. Conjecture 2 The sequence (vm>, U f c i , . . . , vVn, vs>2 ,vd>2,... , vs'n^, , tvn) is regular for all n>2. Again this reduces to showing that (vyn,vs'n) is a regular sequence in r L h, i i L_ 1" [ 2 ' 2 ' ( 4 ) 2 - {d2f + 3s2 + 2 ' ( 4 ) 2 - (cf2)2 + 2 4 ' s 2 + 2 ' d'2_ • This has been verified using Maple up to n = 6. • Lemma 26 TTie double complex (*) is quasi-isomorphic to the following double complex when n > 3 (we shall deal with the case n = 2 at the end of this section): 0 0 t t o -> (Otfo(MM))<WK) -» oUo(uol)/i0 t t Jo = ( « m , Ufti, «6a, • • • , « fc „ W S 2 , ^ d 2 , • • • , VSn) 105 Chapter 3. The Cohomology of M 0 , 0 (P n, 2) Proof Immediate. • The group Z 2 x Z 2 acts on the double complex above. On 0r/o(^oi)/% the action is induced from the action of Z 2 x Z 2 on r(C/oi) (see Section 2.6.1). For the terms Ou0(Uo)/Io © OuiiU^/h the action is just induced from the Z 2 action on each compo-nent of the direct sum. One can check easily that the maps are equivariant. Thus our task at this stage is to calculate the degree zero hypercohomology of the Z 2 x Z 2 invariant part of this double complex (**), we shall use the notation H ° ( * * ) Z 2 > < Z 2 , we shall eventually show that in fact this is equal to the hypercohomology of the complex (*). di+l and in particular V " 1 / From this point on we shall use the notation, for % = 2,. . . , n ( -bx + \s2 \d2 2 ~h + \S2 0 \ 2m 0 bi J \d2 s2 d2 \ \ 4m + 62 J / Sn+l \ ( VSn ^ dn+l = Vdn \ bn+l J \ vbn ) ( -h + ls2 \d. 2 « 2 * \d2 -&x + \s2 0 \ 2m bj \ S2 d2 V 4m + b\ ) Theorem 6 The C-module homomorphism ij>: <C[g,s2M] -+ {0Uo{UQl)lh)d2d{dn)®{0Uo{UQ)lh@0Ul{Ul)/h) 106 Chapter 3. The Cohomology of M 0 ) 0 (P n, 2) where </>(!) = (0,1,1), 1>(g) = (o^i + iem^ft^ + ^  + ^ X ^ ) 2 - ^ ) 2 ) ) , ^(6i) = (0,6!, 6a), 1P(S2) = (0,S2,-*3 + 2&2-4&i) induces an isomorphism, ip, where • for ^ : C [ ^ , 5 2 , 6 1 ] / / ^ H 0 r ) Z 2 X Z 2 , I — ( s n+l , /n+1, 7"n+l) ^ S n + i \ n - l /n+1 V rn+l ) or equivalently / ± s 2 - h \ 0 Y f s2 \ lg l S 2 - b l _ 2 V o o h j 5 + 46? V &i(* 2 -4&i) ] I = (rB(&i), s n Q s 2 - 6^ + ^ / n , Q s 2 - 6^ / „ + ^ s n + 26^  (4&i -and Si and fi and r{ are defined recursively as follows: S2 h fi+l Si+1 Ti+l S2 g + 4b2 6 2 (s 2 -46!) \s2 - h\ fi +. \gsi + 2b[(4h - s2) fi bin for i = 3 , . . . , n. 107 Chapter 3. The Cohomology of MQfi (P n , 2) We shall first show that ip is injective, then surjectivity shall follow from a result of Get-zler and Pandharipande on the Betti numbers of Moto(Pn,2) [9]. We shall discuss this result in Section 3.2. First two preliminary lemmas. Lemma 27 Let sn € C[s 2, h, m, d2] be as above. Then sn is not divisible by s2 — 4&i. Proof We shall show that if we take m - and s2 = 4&i that sn ^ 0, thus sn is not divisible by s2 - 46x. >2 We show by induction that when s 2 = 4bi and m = - y | n-l sn = 2(n)b\l dn = ^ ^ « - 2 bn = so in particular sn ^ 0. Now 1 \ 1 and s 3 = -s2 \ bx - ^s2J + ^{d\ + 16m) + 2b\ = %b\ dz - -d2 (h - s2) = 3d2bi 3 b3 = 2ms2 + b1b2 = 12mbi + bl =--djbi + bl Si+i = Si (-bt + ^s2^j + ^d2di + 2bi ^ - 1 ) ^ 2 ^ - 2 , / i-Wj2,i-2 2(*)&i + -^-4-^^61 + ( 7^dz2b\-2 + 2b\ 2(i + l)b{, 108 Chapter 3. The Cohomology of M 0 ) 0 (P n, 2) similarly and di+i = di (-bi + ^s2^ + 7^d2Si 2 -d2b\~l + id2b\~ \ + l = 2msi + bibi 4m(i)b[-1 + ^d22b[-1 + b[+1^j i d 2 8 so we are done by induction. • Lemma 28 We have an isomorphism of rings C[s2 ,h,g]/ (pn+i, fn+1, r„+i) C[s2,bu d\ + 16m]/((sn+1, d2dn+1, bn+1,m(s2 - Abx)) n C[s 2, h, d\ + 16m]). for all n>2. Where /n+l \ rn+i / 0 \g \s2-bx -2 \ 0 0 bx J \ S2 g + Ab2 \ b2(s2 - Abx) ) Proof Consider ( s n + i , d2dn+x, bn+i, m(s2 — 46!)). We shall show first that these generators only 109 Chapter 3. The Cohomology of M 0 > 0 (P n, 2) contain even powers of d2, by induction. For % = 2 it is clear. If we assume that s*, d2di and bi only contain even powers of d2. Then since Si+i = Sih - ^ (s2Si + d2di) + 2bi d2di+i = d2bi - i (s2d2di + d\sij bi+x = 2msi + bibi we see that S j + i , d2di+i and 6j+i only contain even powers of d2 also. Therefore we can view ( s n + i , d2dn+i, bn+i, m(s2 — Abi)) as an ideal in C[m, s2,bi, d2,] and intersect this ideal with C[s2,bi,dl + 16m]. We observe that (a„+i, d2dn+i, bn+i, m(s2 - Abi)) = (sn+u d2dn+x + 46 B + 1 , bn+i, m{s2 - Abi)). We shall show that in C[m,s2,bud22]/(m{s2-Abl)) / n + i = d2dn+i + Abn+i Pn+l = Sn+1-We use induction on i. For i = 1 we have f2 = g+ Abl = d2 + 16m + Ab2, and d2 + Ab2 = d2 + 4(6f + 4m), and p2 = s2 so equality holds. If we assume the result holds for i then d2di+i + Abi+i — -d2dibi + ]-d2 (s2di + d2Si) + A(2mSi + bik) = -bi(d2di + Abi) + ^d2 (s2di + d2Si) + 8(mSj + bxbi) = -bi{d2di + Abi) + \si{d\ + 16m) + 861 bt - 2s2h + \s2{d2di + Abi) = -bi(d2di + Abi) + \si{d\ + 16m) + ^s2(d2di + 46;) +(bi - b\)(8bi - 2s2) + 6* (861 - 2s2) 110 Chapter 3. The Cohomology of M 0 , 0 ( P " , 2) Now (bi - b{)(8bi - 2s2) = fm(s2 — Abx) for some / e C[m, s 2, bi, df,] since bi - b\ is divisible by m. Therefore in C[m, s2,bx,d2]/m(s2 — Abx) we have d 2 d i + i + Abi+1 = Q s 2 - 61^  d2di + Abi + + 16m) + b i ( 8 b i - 2s 2) = Q s 2 - 6 ^ fi + ^ Pi(dl + 16m) + b[(8h - 2s2) = fi+1-Now in C[m, s 2,b x, d|]/m(s 2 — 4&i) we have Si+i = Si Q s 2 -b^j +^ (d2di + Abi) as required. = Pi+i Thus in C[m, s 2, bx, d^] (Pn+i, /n+ i , ra(s2 - 46i), 6 n + i ) = (sn+i, d2dn+x + Abn+1, b n + u m(s 2 - Abi)), and if we intersect these ideals with the subring C[s2, bx, d\ + 16m] equality still holds. So for this proof it suffices to show that (Pn+i, /n+i,m(s 2 - Abx), 6 n + i ) n C[s2,61,d2 + 16m] = (pn+i, / B + 1 , r B + i ) . Since p n + i and / B + i are elements of C[s2,61, d2, + 16m] it suffices to show that (m(s 2 - 4&i), 6 B +i) n C[s2,61, d\ + 16m] = ( r B + 1 ) . We now replace C[m,s2,61,d2] with the isomorphic ring C[m,<?,s2,&i]. Suppose / £ (m(s2—46x), 6B + 1)nC[s2,61, rf2+16m], then for some polynomials^ and#2 in C[s2,bx,g, m] we have f(h,s2,g) = gi(m, bx, s2,g)m(s2 - Abx) + g2(m, bx, s2,g)(2msn + bxbn). I l l Chapter 3. The Cohomology of M 0 ) 0 (P n , 2) Taking m — 0 we have f(bi,s2,g)=g2(0,b1,s2,g)(bn1+1) thus divides / . Taking and taking a common denominator we have snf{bi, s2, g) = - (sngi ( - ^ > h, s2, g^j 7^0*2 - 46i)^ , where this is an equation of polynomials. But by Lemma 27, s2 — Abi does not divide sn so s2 — Ab\ divides / and / G (bn+1(s2 — Abi)) as claimed. • For simplicity we shall write sn+i in place o f 'p n + i from now on, since these polyno-mials are equal in C[m,s2,bi,dl\/m(s2 — 4&i) although their recursive definitions are slightly different. Lemma 29 ip is injective. Proof We shall first show that the vertical Cech maps in (**) S1 : (OU0(UQ)/I0) (d2)d(dn) © {OvMVh) (d"2)d(d'n) (OU0(U01)/I0) d2d(dn) and $2 : OUo(U0)/I0 © OuMyii -+ OU0(U01)/I0 have isomorphic kernels. We need to find an / G O^JJI^/h such that fd'2d(d'n)%d2d(dn). 112 Chapter 3. The Cohomology of M 0 ) 0 ( P n , 2) Now in Ui coordinates, _{d'n + \ (s'2d'n -d'2s'n)) y/u[ U2 therefore in (Ou^U^/I^did'J, d(dn) -^dn ./ i\ ddn . ,. dd„ -di, ddn ddl 7 d(d'n) d2d(dn) = ds'n \s'2 + 2 d'2m (W2 + 2f-{d'2Y) 4 + 2 8 d(d'n) ( 4 + 2 ) ( 2 4 + ( 4 ) 2 - ( 4 ) 2 ) , the last equality results when we write all of the variables in terms of s2, d!2 and b[. Thus in particular / = 8 is invertible and (s'2 + 2)(2s'2 + (s'2y-(d'2r) fd'2d(d'n) % d2d{dn). We observe that C Z 2 x Z 2 m,bi,s2,dl, m (OU0(U0i)/h) / {snbi - - (s2sn + d2dn), -2msn - bxbn, m(-46i + s2) C m,bi,dl, 1 m / (snbi - ^ (46is„ + d2dn), -2msn -where, for i = 3 , . . . , n, S2 b2 Si+l di+i = 46i = 4m + b\ = - S i h + ^ (4feiSj + d2di) + 2bi = - d i h + i (4Mi + d2Si) = 2msi + bibi. 113 Chapter 3. The Cohomology of M0,0 (P n, 2) For simplicity we shall let m, bi, d2,, — m R = C Thus we need to show that / (snh - ^ (4&!S n + d2dn), -2msn - bxbn^j 52(-s,3 + 2b,2-s'2b,l) 62(dl + 16m) = -(61) (6W2 + 4 ) 2 + 4m ' ( (4 ) 2 - (d 2 ) 2 ) in R. The first is easy since b' -2ms2 + 6i62 _ —8mbi + b\ + 4m&i 61. b\ — Am b2 — Am For the second, using the transition functions -s'3 + 2b'2-S'2b[=Abl in R, so it has the same image as s2. In the last case one writes d2 + 16m and (b'^ + d'3)2 + 4m'((s2)2 — (d2)2) in terms of b[, s2 and d'2 using in the transition functions and the coordinate conversions given in Conjecture 1, and observes that they are equal. We omit the explicit expressions since they are very cumbersome. Let A — ker#2. Since kertfi = kev82 where the isomorphism is given by the map (gid2d(dn),g2fd'2d(d'n)) -> (gi,g2) we can identify ker^i with A also. Therefore we have homomorphisms A-> (Oei) ->H°(**) Z 2 X Z 2 114 Chapter 3. The Cohomology of M0,o (Pn, 2) where the kernel of the induced map A - > B ° ( * * ) Z 2 X Z 2 is the image of A under i(V), thus we have an injective map i / j ( y ) A ^ t f ( * * ) Z 2 X Z 2 . We shall now calculate the kernel of the induced map ^:C[g,bus2}^A/(i(V)A) We observe that the image of (gi,g2) £ A under i(V) is (gid2vdn,fg2d'2vd.n) since (gi,g2) € A corresponds to (gid2d(dn),g2fd'2d(d'n) in ker^ which maps to {g\d2vdn,gifd'2vd>n) = (gid2dn+i, gifd'2d'n+1) under i{V). Thus i/> factors through C[d22 + 16m, 3 2 , h]/ ((d2dn+i, sn+i, bn+u m(s2 - 4&i)) n C[d22 + 16m, s2,&i]) which maps injectively to A/(i(V)A). Therefore it suffices to find the kernel of the map C[#,s2,h] A C[d22 + 16m, s 2 , h ] l ( (d 2 d n + l , s n + 1 , b n + l ,m( s 2 - 4&i)) D C[d22 + 16m,s2,h]). But by Lemma 28 we know that this kernel is ( s n + i , fn+i,rn+\)- So the induced map C[82,b1,g]/(Sn+1,fn+1,rn+1) -> H° (**)Z2*Z2 is injective as required. • 115 Chapter 3. The Cohomology of M 0,o (P n, 2) We shall now choose new generators d, cx and c2 for the ring C[s2,bi, g]/I. We shall explain the geometric significance of these new generators in the next chapter. We will show that the generators C\ and c2 are the first and second Chern classes of the bundle E = 7r*/*Op"(l) where / is the universal map, and 7r is the projection from the universal curve C to A^o,o(Pn,2), and the generator d corresponds to the divisor of points (C, /) such that C is reducible. Lemma 30 Consider the map of rings d? 4c? A 9 - j + - 3 ^ - 4 * s2 61 d + 4ci 3 d + ci Then the induced map C[g, bt, s2]/I C[d, ci, c2]/J = R(n) is an isomorphism for each n > 3, where J = (#n+l) /n+l) rn+l) where g2 and f2 and r2 are defined by gi = d + Aci -4c2 + d2 + 16c2 + 8dci 9 r2 = d d + ci and / 9n+l ^ /n+l V rn+1 ) ( <*-2ci 6 -d2+4c? ! - 2 c 2 4=f^  d+ci 1 3 / -4c 2 + d+4ci 3 d2+16c2+8dci 9 V d(^tf 116 Chapter 3. The Cohomology of M0,o (P n, 2) Proof Immediate, by taking d = s2 - 46i, ci = h - s2 and c 2 = -\g + \s22 - b\. • We would like to calculate the Hilbert series for R(n). The following lemma shall be useful when we attempt this. Lemma 31 When d = c2 — 0, / „ + i ( 0 , C l , 0 ) = a 0 c ? + 1 Pn+i(0,ci,0) = c*ie?. for on i=- 0 /or z = 0,1. For Ci = d = 0 and n even, / n + 1 (0 ,0 ,c 2 ) = 0 and for n is odd o n + 1(0,0,c 2) = 2 ( - l ) f cf /n + i(0,0,c 2 ) = 4(- l ) a * 1 c 2 2 * 1 # n +i(0,0,c 2) = 0. Proof Clear if one substitutes the appropriate values into the equation t 9n+l ^ /n+1 \ ^n+1 / d-2ci 6 n - l ( z ^ ± i _ 2 c 2 -2 0 6 0 3 d+4ci 3 . . d2+16c?+8dci —4C 2 H ^ • 117 Chapter 3. The Cohomology of M0,o ( P n , 2) Theorem 7 Let d, C\ and c2 have weights 1,1, and 2 respectively. Then the Hilbert series of the graded C[d, cx, c2]-module R{n), oo equals the polynomial (l-gn)(l-gn+1)(l-g"+2) (l-q)2(l-q2) Proof First we notice that r n + 1 , fn+i, and gn+i have (weighted) homogeneous degree n + 2, n + l and n respectively. For all n, dn+2 appears with non-zero coefficient in the polynomial r n + i . By Lemma 31 n above, for n even c | appears with non-zero coefficient in gn+x and c n + appears with non-zero coefficient in fn+i- Thus for n even the dimension of the C-vector space of monomials of total weighted degree k in R(n) is the coefficient of qk in the polynomial (1 + q + q2 + ... + qn+1){l + q + q2 + ... + qn)(l + q2 + q* + . . . + (a2)*'1) _ (l-g"+2)(l-g"+1)(l-(g2)t) (1 - q){l - q)(l - q*) ' n+l For n odd, c2 2 appears with non-zero coefficient in / n + i and c" appears with non-zero coefficient in gn+\. So the dimension of the C-vector space of monomials of total weighted degree k in R{n) is the coefficient of qk in (l + q + q 2 + ... + q^)(l + q + ... + q^)(l + q2 + qi + ... + (q2)^-1) _ (i - <r+2xi - <?")(! - (g2)^ ) (1 - q)(l - q)(l - q2) ' also. • 118 Chapter 3. The Cohomology of M0,o (P n, 2) 3.1.3 The Case n = 2 In the case n = 2 the vector fields on U0 and U\ are m(s2 - 46i) - 4 m - b\ + b2 —2ms2 — bib2 hs2 - I (s| + 4) - 2b2 b\d2 — s2d2 \ -m'(s'2b[-r2b'2) -2m's'2 + 1 - b[b'2 -m'((s'2f - (d'2f) - (6'2)2 § «4 ) 2 +Km - 4&2 y 0 i d 2 - s2d2 J y s'2d!2b\ - d'2b'2 J We replace Ui with Ui - V(m'd'2s2) and as before UQ and ^ still cover the fixed locus. Lemma 32 (Wn W)~( 1 5 m ' 9 + f,'+2^ 3 I 5 ( ( g 2 ) 2 + ( d 2 ) 2 ) + 2 5 2 + 2 9 ^ o, Wl) - - m — + (s2 + 2)— + — J is a companion pair for V. Proof The vector field W0 written in U\ coordinates is ^ W\, so in particular they differ by a scalar on J7oi. We notice that the vector field W\ does not vanish on Ui and in particular its kernel is generated by Thus in order for (Wo, W\) to be a companion pair, we must verify that the sequences (vb[,vb>2,m'vS2 + (s2 + l)vm>,m'd'2vdl2 + Q ((s'2)2 + (d'2f) + 2s'2 + 2^ vd>2 and (vm,vbl,vb2,vS2) 119 Chapter 3. The Cohomology of M0,o (P n, 2) are regular. This is easily verifiable using Maple. • We now would like to write down a simple four term double complex quasi-isomorphic to the double complex (*), in the case n = 2. The following lemma is useful in this regard: Lemma 33 Suppose U C Ak is open, and we have two vector fields on U, V and W, such that W does not vanish on U, and the equation V-(iW = 0 has no solutions except p, = 0 in U x A1. If we assume that where the vertical maps are isomorphisms. Proof For each open set V c U the tangent bundle on V is a free Ou(V) module, generated by the for i — 1,... ,k. Since W\ does not vanish on U, we can chose a new basis for the tangent bundle TJJ. Let where w\ does not vanish on U, and then if K is the kernel of W we have a commutative diagram (il„/(i(V)m) + K)(U) l±> (Ou/i(V)K)(U) {Ou(U)/(v1,...ivk))dz1 A OuW/iV! ,v2,--- ,vn) _d_ dzi d_ dzi _d_ dyi d_ k 120 Chapter 3. The Cohomology of M 0,o (P n, 2) In these coordinates W\ = and V = vi— + > I «i Wi —. The subsheaf of corresponding to the kernel of Wi, JC, is generated by dz2, dzz,. as a sheaf of 0\j modules. Now Ov/i(V)K = Oul < i(V)dz2,i(V)dzk > Ouf [v2- — w2,... ,vn- —wk = Ov[p]/{vi - fiWi, v2 - pw2,... ,vk- pwk). But since V — pW = 0 does not vanish in U x A1 except when p = 0, if we take /=(«!- ftwi, v2 - pw2,... ,vn- pwn) and V(I) = X, then the natural projection p: U x A1 -> U x {0} -i> 1/ a; x ;u —>• i x O —>• £ induces an isomorphism p: X - » p ( X ) . Therefore the induced map Ou/{vi,v2,... ,vn) -> Oir[A*]/(vi - A*«;i,U2-/itw 2 , ••• ,vn-pwn) is an isomorphism also. Therefore cwi(v)/c = Ouj{vu v2,..., Vn) s <Vi(v)0V). 121 Chapter 3. The Cohomology of M0,o(Pn, 2) So the map induces a map Ou/iiV^iy^Ov toul{i{V)($ll)+K) %] Ou/i{V)K ll ; nu/(i(Y)(%)+K) A Ov/i(V)(Qu) where the vertical map on the right is an isomorphism. Thus i(V) is the zero map also. We also have, using Lemma 26, that (Qu/iiiVWl) + IC) (U) * (ov(U)/ (v2 - ^-w2, ...,vn- ^™*) ) dzx *(Ou(U)/(vu... ,vk))dzv • Lemma 34 Let U\, V, and W\ be as above. Then V - pWi has no solutions inU x A1 except p = 0, where U = m- V(4(s>2)2 - 2(4)2(d'2)2 + (S'2)4 + 4(s 2) 3 + 2(d'2)2 + (4)4 - 4s'2(d2)2)-Proof We shall show that the system of equations -m'{s'2b\+2b'2) = -fim' (3.1) - 2 m V 2 + 1 - b\b'2 = 0 (3.2) -m'((s'2)2-(d'2)2)-(b'2)2 = 0 (3.3) \ ((4)2 + (4)2) K - s'2b'2 = p(s'2 + 2) (3.4) , + + (,5 ) 122 Chapter 3. The Cohomology of M 0 , 0 (P n, 2) has no solution except p — 0 in U, which incidentally contains the points ( - | , 0,0, - 1 , -1), (—|,0,0, -1,1), which correspond to c = 0 on the fixed locus. Adding jr times (3.5) and times (3.1) and solving for b[ we have, M(4 + 2(d2)2 + ( g 2 ) 2 + 4s2) Subtracting jr times (3.5) from — ^ times (3.1) and solving for fr2 we have, / i ( 4 - K 2 ) 2 + ( 5 2 ) 2 + 4 g 2) 6 2 - # ' ( } Substituting (3.6) and (3.7) into (3.4) we have Thus ju = 0 on (7 as required. • Suppose we replace U\ with U from Lemma 34, abusing notation we shall still call this neighborhood Ui, then we have the following: L e m m a 35 Let {vm,vbl,vb2,vS2) = J 0 ( v , ^ , ^ , ^ , ^ 2 ) = J l . The double complex (*) is quasi-isomorphic to the following double complex (**) in the case n = 2; 0 0 t t 0 ->• (0Uo(Uoi)/Io)d2d(d2) -+ 0Uo(Uoi)/I0 t t 0 -4 (o t / o(c/o)/Jo)(rf2)rf(rf2)e(o f / l(c/ 1)/J 1)(i(m ,) -> o r / 0 ( [ / 0 ) / J 0 e 0^(^)7A -> 123 Chapter 3. The Cohomology of M 0 , 0 ( P n , 2) Proof Follows from Lemma 31. • We shall restrict to the Z 2 x Z 2 invariant part of the double complex above. In this case vbl = 0 =>• 62 = b\ + Am, and vS2 = 0 => d\ = 2s 26i - s2, - Abi ~ lfim, and Ouo(U0)/Io = C[m,s2,h]/(vm,vb2). For Ui, vmi = 0 and rf2^d'2 = 0 =>• b[ = b'2 = 0 since (d'2)2, s 2 and m' are invertible. Also vyx = 0 =>- m! = ^ r , and = 0 (d'2)2 = (s'2)2. Thus 0Ux{Ux)/I2 = C s> 1 Theorem 8 The map if> : C[s2, h] -> (0Uo(U0i)/Io)d2d(d2) © (0Uo{Uo)/I0 © Ov^/h) where </>(l) = (0,1,1) (^61) = (0,6i,0), ^ 2 ) = (0,s 2,0), induces an isomorphism, ip, from C[s 2 ,6i]/7 to u ° ( * * ) z 2 x Z 2 ) where I = (6? (s2 - Ah), (s 2 - 261)(s2 - s2h + 462)) We shall again show injectivity and surjectivity shall follow from Getzler and Pandhari-pande's result in [9] on the Betti numbers of A f 0 ) 0 ( P n , 2 ) , we discuss this in Section 3.2. Theorem 9 ip is injective. 124 Chapter 3. The Cohomology of M 0 , 0 ( P n , 2) Proof As in the proof of the general case, we first show that the vertical Cech maps in (**), 1 <*i : (C[m, s2, bx]/{vm, vb2)) (d2)d(d2) © C S 2 ' j dm' ^{0Uo(U0i)/Io)d2d(d2) and S2:0Uo(U0)/Io®C s< 1 ^OU0(U01)/I0 have isomorphic kernels. This will follow once we show that / Si d2 Now dm' A- —d(d2), Si(dm') = ^d(d2) dd2 2d2m 4(6? - 4m) m'u2 d(d2) Ui d2d(d2) where the last equality comes from writing the previous expression in Ui coordinates. In the ring Ou^U^/h, m' = u2 = s'2 + 1 and ux = so 2 2 m'u2 _ 1 Ml 4 Thus S^drri) = %d(d2). Let A denote the Z 2 x Z 2 invariant part of the kernel of S. Then there is an injective map 0®A/{i(V)A) - * . H ° ( * * ) Z 2 induced from the inclusion 0 © A - » (OU0(U01)/I0) d2d(d2) © (0Uo(U0)/Io © 0Vl{Ux)/h). 125 Chapter 3. The Cohomology of M 0 ) 0 (P n, 2) Therefore we need only show that ip(C[bi,s2]) C A and the kernel of the induced map To show that ip(C[bi,s2]) C A we must show that <5i(&i,0) = Si(s2,0) = 0, i.e. that s 2 = O l = 0 in OU0(U01)/I0. First we note that m and 12m + b\ do not vanish on the fixed locus in Uo except at the point F ^ , thus (replacing UQ with a smaller neighborhood if necessary) on U0i we can invert these functions. C[bus2]^A/(i(V)A) is I. Since h = \ m(s2 - 46i), - 4 m - b\ in 0Uo{Uoi)/Io we have -I- o2, -2ms2 - bib2, bxs2 - - (s2. + d\ s2 = 46i 4m + b\ and -2ms 2 - 6i&2 = 0 =• -6i(12m + b\) = 0 =• &i = 0. Thus S2 = 6i = 0 as required. We shall now calculate the kernel of the induced map iP:C[bus2]^A/{i{V)A). First we observe that the image of (gid2d(d2), g2dm') G A under i(V) is {gid2vd2,0) 126 Chapter 3. The Cohomology of M 0 , 0 (P n, 2) Thus i\> factors through C[m, s2,61]/ ((d2vd2, vb2, vm) n C[m, s2, h]) which maps injectively to A/(i(V)A). Thus it suffices to find the kernel of the map C[s 2,6i] 4 C[m ,S2 ,6 i]/(d2««fe ,U6a , t ;m) n (C[m,s2,b1]). In terms of m, s2 and 61, d2?jrf2 = (2s26i - s\ - 46? - 16m) (61 - s2) , thus we need to find the kernel of the map C[s 2,&i] i C[m,s 2,6 1]/((2s 26i -s\- 46? - 16m) (6X - s 2), -6X(6? + 12m),m(s2 - 460). Now ((2s2&! - 4 - 46? - 16m)(6i - s 2), -6X(6? + 12m), m(s 2 - Abi)) = ((2a26i - s| - 46?)(6X - s 2) + 48m6i, -61(6? + 12m), m(s 2 - 4&1)) = ((2s26x - s \ - AbDih - s2) - 46?, -bx{b\ + 12m), m(s 2 - 46x)). Thus we need to find the kernel of C[s2,61] -> C[m, s 2 ,6 i ] / ( -6i(6 2 + 12m), m(s 2 - 46i)). Suppose / is in the kernel then f(s2, bx) = -gi(m, 82,61)61(6? + 12m) + g2{m, s2, bl)m(s2 - 46i) for polynomials gx and #2. Taking m = 0 we see that 6? divides / , and taking m = —^2 we see that (s2 — 46i) divides / , thus / € (6?(s2 — 4&i)), so the kernel contains the ideal (6?(s2 — 46i)). Now 6?(s2 — 46i) lies in the kernel since bl(s2 - 4&i) = - ( s 2 - 46i)6i(6? + 12m) + 12m(s2 - 46i). Thus the kernel equals the ideal 6? (s2 — 46i). Therefore the kernel of the map is the ideal {b\(s2 - 4&i), (2s26i - s2, - 46?)(6i - s2) + 46?). Now the polynomial (2s26i - s 2 -127 Chapter 3. The Cohomology of M0,o (P n, 2) 46?)(61 - s2) + 46? factors and equals (s2 - s2bx + 46?) (s2 - 26i) as required. • In this situation we also choose new generators d and C\ for the ring C[6i,s2]/7 where d = s2 — 46i, ci = 61 — s2. Their geometric significance is explained in the next chapter. In terms of these generators, I = ((d + cxfd, (Ad2 + l ldci + 16c2)(d - 2ci)). If we let R(2) = C[d, ci]/((d + ci) 3d, (Ad2 + l ldci + 16c2)(d - 2ci)), where d and C i each have weight 1, then R(2) has Hilbert polynomial P(R(2), q) = (l + q + q2 + q3)(l + q + q2) = ( 1 " ( f^~ ( f l ( ^ g 4 ) -3.2 The Cohomology of M 0 , 0 (P n , 2) First we mention an important result. Lemma 36 Hq(M0fi(Pn, 2), W) = 0 when p^q. Proof One can deduce this fact using the methods of stratification of A^o,o(P") 2) as in [9]. • As explained in the introduction, the result above is used to show that H*sing(MQfi(?n,2)X) = GrF(1$(MQfi(V\2),1C)) as graded rings where the filtration F comes from the spectral sequence. But the result of Akyildiz and Carrell [1] mentioned in the introduction tells us that we do not have to 128 Chapter 3. The Cohomology of M0,o (P n, 2) calculate the graded ring associated to this filtration since H° [Mo,o{Pn, 2), K.*) is already graded (this grading is given by the action of Gm), so if*n9(M0,o(P", 2), C) S B°(M0fi{Pn, 2), KT) as graded rings with respect to this grading. In particular the Hilbert polynomial of the graded ring H°(AT 0 ,o (P n , 2), £* ) equals the Hilbert polynomial of H*sing(M0,o(pn> 2)> C)-Getzler and Pandharipande [9] show that the Poincare polynomial of Moto(Pn, 2), e(M),o(Pn,2)) = ^(-t)Mimfr(Mo,o(P n,2),q i (1 - qn)(l - qn+l)(l - qn+2) (1 - qf{l - q*) {™} where t2 = q. We shall show that the induced Gm action on the subring R(n) of H°(Afo,o(P n ,2) , f2*) has weight respectively 1,1, and 2 on the generators d, C\ and c2. We calculated the Hilbert polynomial of R(n) with respect to these weights and found that it was equal to the polynomial in Equation 3.9 above. Thus R(n) = H ° ( ^ t 0 ! o ( P n , 2 ) , / C * ) = H:ing(M0fi(Pn,2),C) as graded rings. Therefore we have the following theorem. Theorem 10 Let d, cx and c 2 have weights 1, 1, and 2 respectively, then R(n) = C[d, C l , c2]/J = #* n s (M 0 ) o(P n , 2)C, C) as graded rings, where J — (^n+lj /n+lj rn+l) 129 Chapter 3. The Cohomology of M 0 > 0 (P n, 2) and / 9n+l ^ /n+l V r„+i J d-2ci 6 ^ n - 1 V 0 0 d+ci 3 d+4ci V , . d2+16c?+8dci - 4 C 2 H ^ — Proof We need only show that the induced Gm action on R(n) gives the generators d, C\ and c 2 weights 1,1, and 2 respectively. The result of Carrell and Akyildiz in [1] mentioned in the introduction gives us that H° (.Mo )o(PnJ 2), Q*) is graded via the action of Gm and the filtration F9 is the canonical filtration associated to this grading. In particular the induced Gm action on R(n) (which can be viewed as a subring of H°(.Mo,o(P 7 \ 2), Q,*)) gives a filtration of R(n) as follows: FQ(R(n)) = {/ € R(n)\X*f = X~kqf. Since R(n) = C[d,c1,c2}/J it suffices to calculate the weights of the Gm action on the generators. Now from Lemma 14 we have X*m X*h X*s2 X*d2 X~2km X'% X~ks2 x-kd?. Thus since d Cl c2 = 8 2 - 46i = b i - 8 2 ~(d2 + l6m) + ±s2-b2, 130 Chapter 3. The Cohomology of M 0 ,o(P n Gm acts with the required weights. • C h a p t e r 4 Geometric Interpretation Let U — {Ui} be the etale cover for Mofi(Pn, 2) we constructed in Chapter 2. Each Ui equipped with universal curve Ci and universal map fi. Consider the following diagram: f*0(l) -> 0(1) d 4 p n Ui We can push forward the bundle /j*0(l) to get a rank three vector bundle onUi,ir*f*0(l). Thus gives a vector bundle E on Afo,o(P", 2). In this section we shall calculate the Chern classes of this bundle, in terms of the generators of the cohomology ring of M 0 ,o(P n , 2). We shall show that the cohomology ring can be generated by the first and second Chern classes of the bundle E and one other degree one generator, which is a divisor consisting of the points (C , / ) 6 M 0 ) o(P n ,2) such that C is reducible. We shall give the relations in terms of these generators. We can also compute the Chern roots of E, 71, 72 and 73 of E. In Section 4.6 of this chapter we give the cohomology ring as the invariant subring of Cpyi, 72,73]/(i?i, R2, R3) under an action of Z 2 , where Ri, R2 and R3 are invariant polynomials under this action. 132 Chapter 4. Geometric Interpretation For arbitrary n we shall show how one can describe some other divisors with geometric significance in terms of the generators s2 and bx (Section 4.3), in particular we shall con-sider the divisor corresponding to pairs (C, /) where f(C) intersects a fixed codimension two linear subspace of P n . This divisor will correspond to — s2 in our cohomology ring. In the case n = 2, M 0 ) o(P 2 ,2) is the space of complete conies, which is isomorphic to the blow-up of P 5 along the Veronese Surface, P 5 . We shall construct an isomorphism from the scheme XJ§jTL2 to an open subscheme of P 5 . We shall also calculate the coho-mology of P 5 in terms of the hyperplane class in P 5 , H, and the exceptional divisor T , and give the relationship between these generators and the generators s2 and b\. 4.1 Chern Classes In this section we shall define the Chern Classes of a vector bundle E. But first some preliminary definitions. Let X be a smooth variety and E a vector bundle on X. Definition 14 A connection is a homomorphism of sheaves of C vector spaces V : E -¥ fix ® E satisfying Leibnitz' rule, i.e. if U is an open subscheme of X, and f G 0(U) and e e E(U), then V(U)(fe) = df®e + fVe. Lemma 37 Let V and V be connections, then V - V : E -> Qx ® E 133 Chapter 4. Geometric Interpretation is Ox-linear. Thus is a homomorphism of vector bundles. Proof Let / G 0(U) and e G E(U), then (V - V')/e = (df ® e + /Ve) - (d/ ® e + / V e ) = / ( V - V')e. • Definition 15 Suppose thatU = {Ui} is a collection of locally trivial open neighborhoods for the rank h vector bundle E, i. e. E\ut = (QVt)h then we can define a family of local connections {Vj} as follows: Vi:OhVi A QUt ® OhVi = QhVi (/i, ••-,//>) ->• {dfi,... ,dfh). Definition 16 Let (Ui, Vj) 6e /oca/ connections. Then c(E) = (Uij,Vi-Vj)eH\X,Uom(E,nx®E)) = H\X,Uom{E,E)®Slx) is called the Atiyah class of the vector bundle E, and is independent of the local connec-tions chosen. Let M(k, C) be the variety of k x k matrices over C. We introduce now the notion of invariant polynomial due to Carrell and Lieberman [6]: Definition 17 Suppose p is a polynomial map p:M(k,C) ->C 134 Chapter 4. Geometric Interpretation such that p(N~1AN) = p(A). Then p is called an invariant polynomial of degree d if there exists a C-linear map 4>:M(k,C) ® . . . ® M ( f c , q - > C v v ' d times such that the following diagram commutes: M(k, C) C A \ ^<f> M(k, C) <g> . . . <g> M(k, C) d times tu/iere A is the diagonal map. For example, suppose Xk+p1Xk-1 + ...pk.1X+pk = 0 is the characteristic polynomial of the k x k matrix A = {ai3,}. Then each of the coef-ficients Pi is a degree i polynomial in the and is an invariant polynomial of degree i (see [6]). Suppose E has rank k. Let A G End(E), and let {Ui} be a collection of affine open neighborhoods such that E is trivial on each C/j. Then A is given on Ui by a matrix Ai G M(k, 0{Ui)) and if b : E\Ut-4 (0Ut)k then thus Ai and A,- have the same characteristic polynomial. We define the characteristic polynomial of the endomorphism A to be the characteristic polynomial of any A^ 135 Chapter 4. Geometric Interpretation Let pd(A) be the degree d coefficient of the characteristic polynomial of A 6 End(E). Then as above we have a commutative diagram of sheaves: End(E) Ox End(E) ® ... ® End{E\ d times where pd is Ox-linear. This gives us a commutative diagram End(E) <g> nx ( f i* )® d A \ / * p d (8> irfn®d (End(E) ® fix) ® • • • ® (End(E) ® fix) " v ' d times Definition 18 Ie£ 0(75) G Hl(X,End(E) ® fix) &e i/ie Atiyah class, then the image of pd{c{E) U • • • U c(E)) in Hd(X, fi^rf) is called the d t / l C/iern Class, where pd is the degree d coefficient of the characteristic polynomial. 4.2 Chern Classes and Hypercohomology In this section we shall show how to find elements in H ° ( A t 0 , o ( P n , 2), JC*) corresponding to the Chern classes of the rank three vector bundle E on M0,o(Pn, 2), given on Ut by TT./;0(1). Let VM be a vector field on MQfi(Pn,2). Let JC* be the complex associated to the 136 Chapter 4. Geometric Interpretation vector field VM on A40,o(Pn, 2). Consider the double complex End(E) <g> K*: t t ••• -> ®i<j<kEnd(E) ® Q(Uijk) @i<j<kEnd(E) ® 0(Uijk) -+ 0 ••• ->• ®i<jEnd(E)^Cl(Uij) ®i<jEnd(E)®0{Uij) -> 0 ••• ®iEnd(E) <g> Q(Ui) -> ®{End(E) ® 0(Ui) ->• 0 with differential id <g> Z(VM)-Consider the Atiyah class c(E') G i / ^ M ^ P " , 2), End(E) <g> fi), then c(£?) can be rep-resented by an element of ®i<jEnd(E) ® Sl{U%j) which we shall call c(E) also, with the property that 5(c(E)) = 0 where S is the Cech differential. Now 5{i(VM)c(E))=i(VM)(6c{E)) = 0 thus there exists an element a G ®iEnd(E)®Ou{(Ui) such that I(VM)C(E) = 5(a). Then the pair /3 = (c(E), a) is an element of H°(iMo,o(P B , 2),End(E) ® /C*). Thus the Atiyah class can be viewed as an element of H° (A^o,o(Pn, 2), End(E) <g> £* ) once we find for it a suitable partner a. Suppose we can find such a partner a. Then we can compute the image of the pair P = (c(E), a) under the map pd p~d:T$(X,End(E)®1C) -4 H°(X,/C*) (3 ^ pd(pu---Ul3) d times and the image will be the dth Chern class of E as an element of H°(A / fo ,o(P n ) 2), JC*). 137 Chapter 4. Geometric Interpretation Definition 19 Let V:Ox^ Ox be a derivation. Let E be a vector bundle over X. Then a lift of V to E is a homomor-phism of sheaves of C-vector spaces V :E^E such that V(fe) = V(f)e + fV(e), for all f e 0(U) and e e E(U). Consider our original vector field on P n induced by the A1 action M on P n (given in Chapter 2). We have an induced a vector field on .Mo,o(Pn,2) and thus a differential VM on -M0,o(P™, 2). Let E be as before, i.e. E is given by 7r*/*0(l) on Ui. We shall show that the vector bundle E is trivial on Ui. We shall show also that we can find a lift VM '• E —> E. Once we have established these facts the following Lemma gives us a partner a for the Atiyah class c(E): Lemma 38 Let a e ®iEnd(E) <g> 0Vi (Ui) a = {Ui,{id®VM)(Vi) - VM) then c(E) = (UijtVi-Vj) -> (U&iidQiiVMNVi-Vj) {uh{id®vM)iyi)-vM) so (c{E), a) € H° (M0 > 0(Pn, 2), End{E) <g> tC). Proof We just have to show that (id® VM)(Vi) - VM € End(E)(Ui). Let e € E(Ui) and 138 Chapter 4. Geometric Interpretation feOutiUi). Then {(id®VM)(Vi)-VM)fe = id®VM(Vi)(fe)-VMfe = id® VM(e ®df + / V i e ) - (e <g> VM(f) + fVu(e)) = (id®VM)(fVie)-fVM{e) = f((id®VM)(Vie)-V^(e)) = f(id®VM(Vi)-VM)e as required. • Lemma 39 The vector bundle E is trivial on Ui and has basis ms2, st and t2 on UQ. Proof We shall show that the vector bundle is trivial on Uo and since the universal maps /o and fi differ only by a projective change of coordinates for P n it follows that the bundle is trivial on each Ui. First we shall show that if we restrict E to the subvariety m = 0 in Uo, E has a basis | , 1, | , which extends to give a basis for E on UQ-Consider f*0(l) on C. Since /(C) C UXo U UXl we need only consider 0(l)\UxQuuxl-On UXi, 0(1) has local trivializations & : 0 ( l ) k 0Vi i - 1 Thus for the relevant neighborhoods, UXo and UXl, we have gluing data, ^ I ^ Q 1 = f^ -139 Chapter 4. Geometric Interpretation Using the notation from Chapter 1, let a = (m, 6X,... , rn^) € Uo with m = 0. Then the fiber C a is isomorphic to {< s, £ > x < p, q >£ P 1 x P 1 | pt = 0}. and the fiber of TT*/*0(1) at a is # ° ( C « , / * 0 ( 1 ) ) . Consider the following diagram: < s,t> \ Ca A P n <p,q> The induced map from each copy of P 1 to P n is an isomorphism. The map from the top copy of P 1 to P n is < 8,t >^< 8,t + biS,... > and from the bottom copy is <p,q>^<q,p + blq,... > . Thus a section of f*0(l) on Ca is given by a section of i)f*0{\) on P 1 for j = 1,2 such that these sections agree when p = t — 0. The vector bundle i*/*0(l) has trivial neighborhoods U0 = P 1 - < 0,1 >= i71f~1UXo and C/x = P 1 - < 1, -bi >= i7lf~lUXl, and gluing data G + M ) ' 140 Chapter 4. Geometric Interpretation so a global section of i\f*0(l) is given on UQ by a0s + ait a o , a i 6 L s and on Ui by OJOS + ait t + bis Therefore H0(P1,ilf*O(l)) has a basis which is given on UQ by s The vector bundle i*2f*0(l) has the same trivial neighborhoods, and gluing data Q ,P + biqJ ' so a global section is given on UQ by Q and on Ui by Ax? + PIP p + hs Therefore ^ ( P 1 , i\f*0(l)) has a basis which is given on UQ by Q Now these sections must agree when p = t = 0, thus a0 = (30. Since ij(Uo) C Uq, j = 1,2, the vector bundle /*0(1)) has basis on Ca fl i7g, which extends to a basis for H°(Ca, f*0(l)) as claimed. 141 Chapter 4. Geometric Interpretation We must now show this extends to a basis for E on UQ. Let a = (ra,b\,... ,rnt2) G UQ, with m / 0 . Then C a is isomorphic to P 1 and is contained in UpUUt. Consider < s , £ > A Ca -4< si,ms 2 + hst + t2, ...> . Let t/ / o = r 1 / " 1 ^ = P 1 - V(st) and C77l = r 1 / " 1 ^ = P 1 - V(ms 2 + 6ist + r 2). Thus when we pull 0(1) back to P 1 we have two trivial neighborhoods, Uf0 and Ufx and gluing data ( 5* ) \ms2 + b1st + t2 J and a global section of i*f*0(l) is given on Uf0 by / aQs2 + ctist + a2t2\ V 7t J and on Ujx by / OJOS2 + a.\st + a2t2\ V ms 2 + 6ist + i 2 ) ' Thus # ° ( C a , /*0(1)) has a basis given by s_t_ t s on Uf0. However since m ^ O w e can choose as a basis l , m - , - . t s We have five trivial neighborhoods on C, namely the intersections of f~l{UXi) and each of Ut, Up and Uq, for i = 0,1. We have five instead of six because f(Uq) C Uxo. We call them Uto, Utl, UPo, UP1 and Uq, where t/ to = / _ 1 (^x 0 ) n etc. The neighborhood C/ t 0 = { ( | , m , . . . , r n , 2 ) G I7t| s # o} 142 Chapter 4. Geometric Interpretation contains the image of U/0 under i. Thus we need to find the gluing data for UQ and Uto. Since both of these neighborhoods are contained in f~l(UXo), which is a locally trivial neighborhood for /*0(1), the gluing data is trivial so we just need to write the basis q s in terms of the basis l , r a - , - . *' t We use the coordinate conversion given by the equation t p s mq = -p =>• - — m-. s q t Thus these bases are the same, so in particular the basis 1, | , | extends to give a global basis for E on UQ as required. • We shall now deal with finding a lift of the derivation VM on .Mo,o(Pn,2). First we shall find a lift of the derivation V on P n , V , to 0(1), and this will induce a derivation VM on 7r*/*0(l) = E, which is a lift of VM-Lemma 40 Let V be the derivation V : Op» -4 Op» associated to the A1 action M on F1. Consider the C-linear map from 0(1) to 0(1) given by d V : f(x0,... ,xn)->— (f{t.(xQ,... , xn))) t = 0 where t.(x0,... , xn) = M(t)(x0,... ,xn). Then V is a lift of V. Proof It is easy to see that V is a derivation, so we must just show that it is a lift for V. Let 143 Chapter 4. Geometric Interpretation f G Op»(U), and e G 0(1)(U). Then V{fe) = ^f(t.(x0,... ,xn))e(t.(x0,... ,xn)) t=o d = ^/ ( t . (x 0 > . . . ,xn)) = V{f)e + fVe d t=oe + —e(t.(x0,... ,xn)) -of since by definition, if / G Op"(£7), then d V(f) = -(f(t.(x0,...,xn))) t=0-We note that in particular Vfa) = xi+i for i = 0,... , n - 1 and V(xn) = 0. Since 0(l)(U) is generated by the global sections 0(C/)-module, the lift is completely determined by its action on these global sections. • This derivation on 0(1) induces a derivation VM on 7 r » / * 0 ( l ) = E, which is a lift of VM since VM is induced by the derivation V on P n . We need to find this induced deriva-tion V M - We shall calculate VM only on C/0 since this is the important neighborhood when we calculate the hypercohomology. Since 0(1) is generated the global sections X o 5 • ' . j Xfi cLS £111 sheaf of Op«-modules , E is generated locally by f*{x0),... , f*(xn). Thus the lift VM is determined by its action on the f*X{. Lemma 41 The endomorphism id® V M ( V O ) — VM with respect to the basis ms2, st, t2 of E on UQ is given by the matrix C where ( C = r2,i - h 1 0 \ 2m b\ 2m \ 0 1 r 2 ) 2 -bx J 144 Chapter 4. Geometric Interpretation Proof Since and the universal map is given by blowing up the rational map (m, b ••• , rnfl) X < S,t < st, ms2 + hst +12, mr2,is2 + b2st + r2,2t2,... , mrntls2 + bnst + rn,2t2 > we have VM{st) = ms2 + hst +12 VM(ms2 + hst + t2) = mr2>1s2 + hst + r2>2t2 VM(mr2tis2 + hst + r2>2t2) = mr 3 ] 1s 2 + hst + r3>2t2 (4.1) (4.2) (4.3) Thus 4.2 gives us VM(ms2) + VM(t2) = mr2,is2 + b2st + r2>2t2 - V (^M*) = rnr2As2 + b2st + r2<2t2 - VM{h)st - h(ms2 + hst +12) = ( r 2 , i - h)ms2 + (62 - VM(h) - b\)st + ( r 2 > 2 - h)t2 and 4.3 gives VM(rnr2tls2 + r2j2t2) = mr3tls2 + b3st + r 3 ) 2 t 2 - VM(b2st) = mr 3 ) 1s 2 + b3st + r3i2t2 - VM(h)st - b2(ms2 + hst + t2) =>- r2,iVM(ms2) + r2>2VM{t2) = mr3tls2 + b3st + r3>2t2 - VM{b2)st -b2(ms2 + hst +12) - VM(r2ti)ms2 - VM(r2>2)t2 = (fs,i ~ h - VM(r2,i))ms2 + (b3 - VM(b2) - hb2)st +(r-3,2 - h - VM(r2,2))t2 145 Chapter 4. Geometric Interpretation Now VM(h) = - 4 m - 6 2 + 62 VM{b2) = - 2 m ( r 2 , i + r 2 , 2 ) - hb2 + 63 ( r 2 > i ) = 6 i r 2 ] i - r 2 j l - 62 + r 3 , i V M ( r 2 , 2 ) = 6ir 2, 2 - r\2 -b2 + r 3 > 2 , so VW(ms2) + VW(t2) = ( r 2 ) i - h)ms2 + {4m)st + (r 2 ) 2 - 6i)t2 (4 r 2 , i i ^ ( m s 2 ) + r2>2VM(t2) = ( r 2 , i ( r 2 ) 1 - 6i))ms2 + (2m(r 2 ] 1 + r 2 | 2 ))st +(r2)2(r2,2 - h))t2. (4 Subtracting (4.5) from r 2 ) 1 times (4.4) we have (r 2 , i - r 2 , 2 ) V M ( i 2 ) = 2ra(r2,i - r 2 > 2)s* + ( r 2 , i - r 2 , 2 )(r 2 i 2 - o^r 2 Thus VM(t2) = {2m)st-r(r2t2-b1)t2. Similarly, subtracting (4.5) from r 2 ) 2 times (4.4) we have VM(ms2) = (r 2 ) i - bi)ms2 + 2m(st). Thus the matrix is as required. • To find the Chern classes we must calculate the characteristic polynomial of the domorphism given by the matrix C. 146 Chapter 4. Geometric Interpretation Theorem 11 The Chern classes of the vector bundle E = 7r»/*0(l) are ci = -s2 + h 1 1 2 ,2 C2 = -4^+4*2 -C 3 = 61 (-c2 - 61 ci - 6?) Proof We calculate the determinant of a;/ — C. Now ^ ar — r 2 , i + 61 -1 0 N xI-C= -2m x-bi -2m \ 0 -1 z-r2>2 + bi / So the characteristic polynomial of C is z3+(-Ki+r2,2)+&i):z2+(r2!l^  and ci = -(r2,i+ r2)2)+ 61 c2 = r 2 > 1 r 2 > 2 - 4m - 6? c 3 = - r 2 ! l r 2 ) 2 6 i + ( r 2 1 + r 2 ) 2 ) 6 2 - ^ + 2m(r 2 ii + r 2 ) 2 ) - 4 m 6 i . 147 Chapter 4. Geometric Interpretation In terms of the coordinates of the cohomology ring we have Ci - - s 2 + 61 c 2 = \{s\-dl)-Am-b\ 1 1 2 ,2 c 3 = ~(sl - dl)bi + s2b\ -b\ + 2ms2 - 4 m 6 i = 61 - + s 2 0i - b2 = 61 ( -C2 - 262 + s26i) = h ( -c 2 - 61(61 - 82) - b\) = h ( -C2 - 61C1 - 6 2 ) as required. • 4.3 Some Divisors on A40,o(Pn, 2) Suppose a divisor D on A/to,o(P",2) is given by the local equation fi = 0 on Ui, where {Ui} is our etale open cover for Mo,o(Pn, 2). Let L(D) be the associated rank one vector bundle. Let V* be local connections as before. Then in this case on Uij, v - v * ' " f " / * fi' Thus the Atiyah class c(L(D)) e H^MoA?", 2)> End(L(D)) ® £ * ) is Now 148 Chapter 4. Geometric Interpretation Therefore, as long as G 0Vi(Ui), a = (Ui, - ™ ) G ®iEnd(L(D))®0(Ui) is a partner for the Atiyah class c(L(D)) in the sense that (c(L(D)), a)etf (M),o(F\ 2), End(L(D)) ® JC). If the condition ^jp- G Ou^Ui) is satisfied, then to calculate the first Chern class of L(D) one merely calculates the trace of the endomorphism given by a on UQ, this is exactly V(/o) fo • Theorem 12 In the degree one graded piece of the cohomology ring o /M 0 ,o (P n , 2) we have an element d, which corresponds to the set of all points (C, f) such that C is re-ducible. In the cohomology ring d = s2 — 4b\. Proof If we consider the divisor corresponding to the points (C, f) such that C is degenerate, on UQ it is given by the equation /o = m = 0, on U\ it is given by the equation / i = m! = 0. Since V(m) = m(s2-Abl), and V(m') = -m'(s'3+s'2b'1+2b>2), we see that ^ G 0Vi(Ui). The corresponding element in the cohomology ring is s2 — Abi. • Theorem 13 The divisor of points (C,f) such that f(C) meets the linear subspace of P71 where a:n_i = xn = 0 is given by —s2 in the cohomology ring of Mo,o(Pn, 2). Proof We consider first UQ. If x is such that Cx is irreducible and f(Cx) passes through the required linear subspace, we have that ww„_i , i s 2 + 6n_ist + r n _ i ) 2 t 2 = 0 mrn>is2 + bnst + rn,2t2 = 0 149 Chapter 4. Geometric Interpretation have a common solution. Thus fo = (&n»n- l , l - &n-l»n I l ) (&n'n- l ) 2 ~ K^n?) + ^(^n,l rn-l,2 ~ r n > 2 r n _ i , i ) 2 = 0. In the case Cx reducible the same equation holds. We calculate On Ui we require that V(h) . ^ , —}— = - ( r 2 i + r 2 2 ) = - s 2 . Jo m'r'n_hls2 + b'n_lSt + r'n_lt2t2 = 0 Jr'nAs2 + b'nst + r'nt2t2 = 0 m : h = (Kr'n-1,1 - d < i ) ( & n < - i , 2 - C i < , 2 ) + m(r'njlr'n_lt2 - r n > 2 r n - i , i ) 2 = 0. Again we calculate ^ = r 3 ! l + r 3 , 2 + 6 ' 1 ( r 2 i l +r 2 ) 2 ) -26 2 . The associated element in the cohomology ring is —s2 as required. • 4.4 Chern Classes and the Cohomology of M0,o(Pn? 2) We have d = s 2 — Ab\ ci = - s 2 + 61 1 1 2 ,2 c 3 = 61 (-c 2 - 6xci - bf) . 150 Chapter 4. Geometric Interpretation thus we can write the generators 61, g and s2 in terms of d, C\, and c2 as follows: d + ci 61 = s2 = -g • 3 d + 4ci d 4c2 _ 4 C 2 - y + _ l . Theorem 14 Let C\ and c 2 be the first and second Chern classes of the vector bundle E = 7r*/*0(l) on A^o,o(Pn) 2). Let d be the divisor of points (C , / ) such that C is reducible. Let n>2, then the cohomology ring o/M 0,o(P",2) is C[d,Cl,c2]/I where I — (<7n+l, /n+15 1'n+l) and ( gn+i ^ /n+1 \ n^+1 / V d-2ci 6 d+ci , 3 / V d+4ci \ 3 . , d2+16c?+8dci -4c 2 + g d ( ^ ) 2 / 4.5 Geometric Interpretation in the case n = 2 4.5.1 M 0,o(P 2, 2) as the Blow-Up of the Veronese Surface in P 5 The image of the projective embedding i:P2 P 5 i:<a,b,c> < a2,b2,c2,2ab,2ac,2bc> is called the Veronese Surface. One can realize the space of complete conies (which is the classical equivalent of the coarse moduli space M 0 ]o(P 2 ,2)) as the blow-up of P 5 151 Chapter 4. Geometric Interpretation along the Veronese Surface. We shall construct a mapping from our neighborhood UQ to this blow-up. We identify a point in P 5 with a conic in P 2 in the normal way, i.e. if a =< x0, •.. , x5 >G P 5 then as long as x g" i(P 2), x corresponds to the (possibly degenerate) conic with equa-tion x0X2 + xxXl + x2X\ + XZXQXX + x4X0X2 + x5XxX2 = 0. This conic has the property that the variety associated to it is not a line (although it may be a pair of distinct lines crossing transversally). Thus there is a one to one correspondence between such points in P 5 and points (C, /) in Mo,o(P2,2) where / ( C ) is a non-degenerate conic or a pair of distinct lines (none of these points are stack points). Theorem 15 The coarse moduli space M 0 ,o(P 2 ,2) is isomorphic to the blowup of P 5 along the Veronese, P5. Proof We first analyze the blow-up in the neighborhood where x2 ^ 0, U2, with coordinates jj£ = uii for % — 0, . . . , 5, i ^ 2. This neighborhood meets the Veronese at the images of points of the form < a, 6,1 > in P 2 . Thus in this neighborhood the Veronese is described by the following ideal: h = (w\ - 4w0, w\ - 4twi, w4w5 - 2w3) = (/ 2 ) 0, / 2 , i , fo). When we blow-up along the variety given by this ideal we have the following Ui = {(x,y) G U2 x P 2 |y =< y0, yu y2 >, and yjf2,i(x) = VihM for % < j , i,j = 0,1,2}. 152 Chapter 4. Geometric Interpretation The neighborhood Lr2 is covered by three affines B2>i for i = 0,1,2, where B2>i = {(x,y)eU2\yi^0}. In particular B2,i can be described as follows: Since yi ^ 0, we have Vof -%(w25-4wi) + wl 72,0 = —/2,i ™o = — -A 72,2 = —/2,1 W3 = . Vi 2 so £?2,i can be identified with A5 with coordinates 2/o 2/2 Vi 2/i Since the point (m, 61, b2, r 2 , i , r 2 ) 2) corresponds to the conic ((62 - r 2 , i&i)X 0 + 2^,1^ 1 - X2) {(b2 - r2,2h)X0 + r2tlXx - X2) + m(r 2 > 1 - r 2 > 2 ) 2 X 0 2 = 0 we have the following map from C7 0 to B2y. (m, by, bi, r 2 , i , r 2 ) 2 ) -»• ( r 2 > 1 r 2 i 2 , -26 2 + (r 2 ) 1 + r 2 ) 2)6i, - ( r 2 , i + r 2 > 2 ) , 6? - 4m, -61), since 2/o = ^0 = w\ - 4w 0 = (r2,i -r 2, 2) 2(6? - 4 m ) 2/1 /2,1 «>5 _ 4 w l (r2,l - r2,2)2 and m = j%2= ^4^5 ~ 2^3 = -fel(?~2,l - r 2 | 2 ) 2 2/i /2,i w2-4wx ( r 2 , i - r 2 ) 2 ) 2 If we take the subring of C[m, 6X, b2, r 2 ) i , r2 ) 2] which is invariant under the action of Z 2 , it is generated by m, bx,b2,s2, and (r 2,i ~ r 2 | 2 ) 2 2 2 = 4 • So in "invariant" coordinates the map is as follows (m, 61, b2, s2, z2) -> (z2 + - 26 2 + s2bx, - s 2 , b\ - 4m, -b^j , 153 Chapter 4. Geometric Interpretation which is an isomorphism with inverse 2/o m\ v [ Ti + {m) 2 / 2 -Wi + w^ w l 2 / 1 2 / 1 / 4 yx 2 4 Let Ui be any other member of our etale open cover U for Mo,o(P 2,2). Then since the universal map for Ui differs from the universal map for UQ only by a projective change of coordinates for P 2 , by choosing new coordinates for P 2 we can construct an isomorphism from Uil%i to an open neighborhood of P 5 in the same way as above. Therefore since M 0 ,o(P 2 ,2) is covered by the family {£/i /Z 2 } , we have M 0 i o(P 2 , 2 ) = P^ as required. • 4.5.2 Cohomology of M 0,o(P 2 5 2) via the Veronese The cohomology of the blow-up of the Veronese is easy to calculate, since the Chow Ring coincides with the cohomology in this case and the Chow Ring of a blow-up is readily calculable. Theorem 16 let X be a regularly embedded subscheme of a scheme Y, and let Y be the blowup of Y along X, X A Y X A Y if the restriction map i* : A*Y -> A*X is surjective then A*(Y) is generated by the class of the exceptional divisor as an A*(Y) - algebra. The relations are as follows: If we let T = j*[X], then aT = 0 */ aeA*(Y) and i*(a) = 0, r - l Tr + J2jkTr-k + (-lY(i*[X}) = 0, 154 Chapter 4. Geometric Interpretation where r is the codimension of X in Y and^k are any classes of A*(Y) such that i*(7fe) = (-i)k[X]r)ck(Nx/Y). Proof see [7]. • In order to write down the cohomology ring we need to calculate the Chern classes of the normal bundle of the Veronese Surface in P 5 . To calculate these Chern classes we shall use standard properties of Chern classes (for example see Hartshorne, [11, p 429-431]). Lemma 42 The Chern classes of the normal bundle of the Veronese Surface in P5 are ci(N) = 9h c2(N) = 30h2 where h is the hyperplane class in P2. Proof Since the Veronese is a non-singular irreducible closed subscheme in P 5 , we have an exact sequence 0 ->• T p 2 - > z * T p 5 iV -> 0 where N is the normal bundle of the Veronese in P 5 and i is the Veronese embedding. Therefore ct(Tp2)ct(N) = ct(i*Tp5). (4.6) where ct(F) is the Chern polynomial of the locally free sheaf T. The exact sequence 0 O p 2 -> O p 2 ( l ) 3 ->> T p 2 0, 155 Chapter 4. Geometric Interpretation where the superscript 3, means the direct sum three times, gives c t (O p 2 ) C ( (Tp 2 ) = C t(Op2(l) 3). Now c4(Op2) = 1 and Ct(0p 2 (l) 3 ) = (c t(0p 2(l))) 3 = (1 + ht)3 where h is the hyperplane class in P 2 (and h3 = 0). Thus Q(T P 2) = 1 + Sht + ShH2. (4.7) To calculate Ct(i*Tps) we observe that Ci(i*TpS) = i*Ci{Tps) = + Htf where H is the hyperplane class in P 5 . Now i*H = 2h since i i s a degree 2 map. Thus Ct(i* Tp 5 ) = (1 + 2htf = 1 + \2ht + 60h2t2 (4.8) since h3 = 0. We substitute 4.7 and 4.8 into 4.6, and letting Ci(N) be the ith Chern class of the normal bundle JV, we have (1 + Ci(N)t + c2(N)t2) (1 + 3ht + 3h2t2) = 1 + 12/rt + 60^2t2 thus 3h + Cl(N) = 12h =>ci{N) = 9h 3h2 + 27h2 + c2(N) = 60h2 ^c2(N) = 30h2. • 156 Chapter 4. Geometric Interpretation Lemma 43 The cohomology ring of the blow-up of the Veronese Surface in P6 is C[T, H]/(TH3,T3 - (9/2)T2H + (15/2)T# 2 - AH3), where T is the class of the exceptional divisor, and H is the hyperplane class in P5. Proof Let h be the hyperplane class in P 2 . Certainly the restriction map i* : A*P5 -> A*P2 H ->• 2h is surjective so A*(P5) is generated by T as an A*(P5)—algebra. Thus is generated by T and H as a polynomial ring over C. For the relations i* 7i = (-l)1^ -Yi = ~ H r 7 2 = (-i)2soh2 = * ^  = ^  = 1±H. Thus the Chow ring has two generators H and T and the second relation is T 3 _ lT2R + y * T H 2 _ 4 i f 3 = Q 2 2 The last coefficient comes from the fact that U P 2 ] = [K(P2) : K(i(P2))]J^) where [if(P 2) : K(i(P)2)] is the degree of the field extension (finite since i is an isomor-phism) induced by the map i. The degree of this field extension is 2, and i(P2) = 2H3. For the first relation we look for elements a of A*(P5) such that i*(a) — 0, now the 157 Chapter 4. Geometric Interpretation generators of A*(P5) are of the form 7T and i*(7T) = (2/i)* which is zero exactly when i > 3. Thus in particular i*(H3) = 0. Hence the first relation is TH3 = 0. The Chow ring is C[T, H}/(TH\T3 - (9/2)T 2# + (15/2)T# 2 - AH3). • 4.5.3 Geometric Interpretation for n = 2 Theorem 17 In terms of s2 and b\, T = —2(s2 — h) o,nd H — —s2. Geometrically H corresponds to the divisor of points (C,f) such that f(C) contains the point < 1,0,0 > in P2, and T corresponds to the stack locus. Proof The result for H is just a special case of Theorem 13 The stack locus (which corresponds to the exceptional divisor in the blowup) is given by Z2 = 0 on UQ- Thus corresponds to the element in the cohomology ring as long as z 2 divides V(z 2). Now V(z2) = -\d2V(d2) = - s2) = 2z2(h - s2) So the corresponding element in the cohomology ring is —2(s2 — &i). • Suppose we take T = -2(s 2 - bi) and H = -s2 we get the same relations as in the blow-up case. In our original calculation we had relations rx = b\ (s2 - 46i) 158 Chapter 4. Geometric Interpretation r2 = (s2 - 26i)(s| - s 26i + 462). Repeated division yields r 2 = - ( -86? + 6s 26 2 - 36is 2 + sij) = 9 15 (-232 + 260 3 - -(-23 2 + 26 1 ) 2 (-s 2 ) + — (-232 + 2 6 1 ) ( - 3 2 ) 2 - 4 ( - s 2 ) 3 = r 3 _ 9 r 2 / / + }$TH2_4H3 and - 8 r i + (2s2 + 46i)r 2 = -86?(s 2 - 46x) + (2s2 + 46i)(-86 3 + 6s 26 2 - 3&i32 + s 3) = ( - s 2 ) 3 ( - 2 s 2 + 261) = H3T Therefore our cohomology calculation is consistent with the blow-up calculation. 4.6 Chern Roots and the cohomology of Mo,o(Pn, 2) In this section we give the cohomology ring of M 0 ) o(P n , 2) in terms of the Chern roots of the vector bundle E. Theorem 18 Let 7 l ; 7 2 and 73 6e the roots of the Chern polynomial C(t) of E, where C{t) = r 3 + cxt2 + c2t + c 3 for Ci the Chern classes of E, and 71 = 61 in the notation of before. Let Z 2 act on the ring C [ 7 i , 72,73] as follows: 71 7i 72 ->• 73 73 ->• 72 Chapter 4. Geometric Interpretation Then the cohomology ring o/M 0 )o(P",2) is the invariant subring ofthe quotient ring C[ 7i,72,73]/(#i,#2,#3) where the Ri are invariant polynomials and are given by Ri 7 2 "+ 1 ~ 7 ? + 1 + 7 ? + 1 ~ 7 i " + 1 72 - 7i 73 - 7i R 2 = _ ( 2 £ ^ ) ( 7 3 _ 7 l ) _ ( 2 3 ^ r , ( 7 2 _ 7 l ) \ 7 2 - 7 i / V 73 - 7i + ( 7 2 n + l + 7 r i ) + ( 7 3n + l + 7 r i ) Rs = 7 r + 1 ( ( 7 2 - 7 i ) + ( 7 3 - 7 i ) ) Proof Let u = s 2 - 26i and b = bx. Then the cohomology relations for M 0 ,o(P n , 2) are given by the matrix ( -u 1 2" 2  s 0 \ \g \u -2 V 0 0 b J and vector V -u + 2b \ p + 462 \ b2{u - 26) J i.e. the cohomology ring of M 0,o(P", 2) is C[b,u,g]/I where / = (sn+1, fn+l, rn+1) and ( Sn+1 ^ fn+l An~lv. 160 Chapter 4. Geometric Interpretation We shall show that the eigenvalues of A are exactly the Chern roots, and we shall write A in terms of the Chern roots and diagonalize. We then calculate the An~l using the diagonalization and the relations will follow. The eigenvalues of A are the roots of (b-\)\[\u-\) - \ g \ = V thus We write the Chern polynomial in terms of b, u and g. Recall that Ci = bi — s 2 = —u — b °2 = -\g + \sl-b2 = ~g + ^(u + 2b)2-b2 = ~g + ^u2 + ub c3 = h - ^s2 + s2h -b2^=b - ^ Thus C(t) = t3 + (-« - b)t2 + {~\g + \u2 + ub)t + h(jg - ^u2) ( * - * ) ( ( | « - t ) a - | / so the Chern roots are the same as the eigenvalues. Suppose we take 7i = b u + y/g 72 73 2 u-^/g 2 ' u = 72 + 73 9 = ( 7 2 - 7 s ) 2 -Thus C[b,u,g] is isomorphic to the invariant subring of €[71,72,73] under the Z 2 action which sends 72 to 73, 73 to 72 and fixes 71. 161 Chapter 4. Geometric Interpretation In terms of the Chern roots, the matrix A is (72 + 7s) \ 0 \ |(72 - 7 s ) 2 5(72 + 73) -2 V 0 0 7i J We now need to find eigenvectors of A corresponding to the eigenvalues, 71, 72 and Let / 1 ^ ( 1 ) ( 1 ^ - ( 7 2 - 7 1 ) - (73-71) ,v2 = (72 - 73) ,v3 = -(72 - 73) - ( 7 2 - 7 1 X 7 3 - 7 1 ) / \ 0 J I 0 / then Vi, V2 and V3 are eigenvectors and if E \ vi v2 v3 ; ; ; / Then An = EDn lE , where D is the diagonal matrix of eigenvalues. Now E -1 0 1 2 2(72-73 1 0 1 (73-71) (72-71) 1 (72-73X72-71) \ and V I i ± / \ 2 2(72-73) (72-73X73-71) / An~lV = EDn-lE~lV R2 as required. • 162 Bibliography [1] E . Akildiz and J.B. Carrell. Cohomology of projective varieties with reglular SL2 actions. Manuscripta Math., pages 473-486, 1987. [2] K. Behrend, D. Edidin, B. Fantechi, W. Fulton, L. Gottsche, and A. Kresh. An Introduction to Stacks, in preparation. [3] K. Behrend and Yu. Manin. Stacks of stable maps and Gromov-Witten invariants. alg-geom/9506203. [4] A. Bialnicki-Birula. Some theroems on actions of algebraic groups. Ann. Math., 98:480-497, 1973. [5] J.B. Carrell. Vector fields, flag varieties and the Schubert calculus. Proceedings of the Hyderabad Conference on Algebraic Groups, pages 23-57, 1989. [6] J.B. Carrell and D.I. Lieberman. Vector fields, Chern classes, and cohomology. Proc. Sympos. Pure Math., Vol. XXX:251-254, 1975. [7] B. Fantechi and L. Gottsche. The cohomology ring of the Hilbert scheme of 3 points on a smooth projective variety. J. Reine Angew. Math., 439:147-158, 1993. [8] W. Fulton and R. Pandharipande. Notes on stable maps and quantum cohomology. alg-geom/9608011. [9] E . Getzler and R. Pandharipande. The Poincare Polynomial of Kontsevich's space of stable maps M.o>n(pn, d). in preparation. [10] T. Graber and R. Pandharipande. Localization of virtual classes, alg-geom/9708011. [11] R. Hartshorne. Algebraic Geometry. Springer-Verlag, 1997. [12] M . Kontsevich. Enumeration of rational curves via torus actions, hep-th/9405035. [13] H. Matsumura. Commutative Ring Theory. Cambridge University Press, 1992. [14] R. Pandharipande. Intersections of Q-divisors on Kontsevich's moduli space Mo,n(pr,d) and enumerative geometry, alg-geom/9504004-[15] J.P. Serre. Geometrie algebrique et geometrie analytique. Ann. Inst. Fourier, pages 1-42, 1955-1956. 163 

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