On The Existence of Jet Schemes Logarithmic Along Families of Divisors by Andrew Philippe Staal B.Sc., University of Ottawa, 2006 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in The Faculty of Graduate Studies (Mathematics) The University of British Columbia (Vancouver) December 2008 c© Andrew Philippe Staal, 2008 Abstract A section of the total tangent space of a scheme X of finite type over a field k, i.e. a vector field on X, corresponds to an X-valued 1-jet on X. In the language of jets the notion of a vector field becomes functorial, and the total tangent space constitutes one of an infinite family of jet schemes Jm(X) for m ≥ 0. We prove that there exist families of “logarithmic” jet schemes JDm(X) for m ≥ 0, in the category of k-schemes of finite type, associated to any given X and its family of divisors D = (D1, . . . , Dr). The sections of JD1 (X) correspond to so-called vector fields on X with logarithmic poles along the family of divisors D = (D1, . . . , Dr). To prove this, we first introduce the categories of pairs (X,D) where D is as mentioned, an r-tuple of (effective Cartier) divisors on the scheme X. The categories of pairs provide a convenient framework for working with only those jets that pull back families of divisors. ii Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1 Functors of Points and Yoneda’s Lemma . . . . . . . . . . . . . . . . . . . . 5 2.2 Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2.1 Definitions of Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2.2 Pullbacks of Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Categories of Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.1 Defining the Categories of Pairs . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Definitions and Examples in the Categories of Pairs . . . . . . . . . . . . . . 12 3.2.1 Open Subpairs and a Gluing Construction for Pairs . . . . . . . . . . 12 3.2.2 m-jets in Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4 Jet Pairs and Logarithmic Jet Schemes . . . . . . . . . . . . . . . . . . . . . 16 4.1 The Main Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5.2 Discussion and Further Research . . . . . . . . . . . . . . . . . . . . . . . . 24 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 iii Acknowledgments I would like to thank all those people whose support, encouragement, and assistance has lead and inspired me through the successful completion of this thesis. First, I wish to thank the various friends and colleagues of mine who have enriched my personal pursuit of mathematical knowledge. Especially among these, I thank Jesse Collingwood, Chris Dionne, Samantha Marion, Matthew Mazowita, Slava Pestov, and Eric Serré at the University of Ottawa; Alex Ustian during our visit at the Independent University of Moscow; and Adam Clay, Alexander Duncan, Benjamin Purcell, Simon Rose and the rest of the motivated group of algebra and algebraic geometry graduate students at the University of British Columbia. Further I would like to express sincere gratitude to the professional mathematicians who have nurtured my interests, especially Professors Thierry Giordano, Barry Jessup, Monica Nevins, and Vladimir Pestov at the University of Ottawa, and Professor Julia Gordon at the University of British Columbia. I would like to extend special thanks to Professor Kai Behrend at the University of British Columbia for taking the time to read and evaluate my thesis. Finally, I wish to express my deepest appreciation to Professor Kalle Karu, my research supervisor at the University of British Columbia; without the constant guidance, instruction, ideas, suggestions, and encouragement of Professor Karu this work simply could not have been possible. On a personal note, I would like to thank my family; it is impossible to sum up all the reasons for which they deserve my utmost gratitude with regards to this thesis, so I simply thank Mom, Dad, and Nico with all my heart. Thanks also to Gladys for her company and encouragement (and food!) after my move to Vancouver. iv To Ralph and Isabel, en honor de Salomón y AuraMaŕıa v 1 Introduction Let X and Z denote schemes of finite type over C. A Z-valued m-jet in X is a mor- phism γ : Z × Spec C[t]/(tm+1) → X. Let LXm denote the contravariant functor Hom(− × SpecC[t]/(tm+1), X). The functor LXm is representable; that is, the functor−×SpecC[t]/(tm+1) has a right adjoint, which we denote Jm(−). Thus, any Z-valued m-jet γ in X corresponds uniquely, and functorially, to a Z-valued point γ̃ : Z → Jm(X) of the scheme Jm(X). The scheme Jm(X) is referred to as the m th jet scheme of X, and is of finite type over C. Let us sketch a constructive proof that Jm(X) exists for such a scheme X; for more thorough treatments on jet schemes see for example the articles [Mus01], [EM08], [Ish07]. First, we may assume that Z and X are affine; in the first case this follows after refining Yoneda’s lemma applied to the category C-Schemes of schemes of finite type over C, and in the second case from the gluing construction on schemes. Thus, let Z = Spec A, where A is a finitely generated C-algebra, and let X = SpecC[X1, . . . , Xn]/(f1, . . . , fs) be a closed immersion of X into complex affine n-space. Then a Z-valued m-jet γ is determined by a homomorphism γ∗ : C[X1, . . . , Xn] → A[t]/(tm+1) such that γ∗(fj) = 0 for each 1 ≤ j ≤ s. Let γ∗(fj) = fj0+ fj1t+ · · ·+ fjmtm; the condition γ∗(fj) = 0 translates to fjl′ = 0 for every 0 ≤ l′ ≤ m. Note that each fjl′ is a polynomial in the coefficients (ail)i,l of the elements γ∗(Xi) = ai0 + ai1t+ · · ·+ aimtm. Thus, consider the homomorphism C[X(0)1 , . . . , X(0)n , X (1) 1 , . . . , X (1) n , . . . , X (m) 1 , . . . , X (m) n ]→ A mappingX (l) i 7→ ail; this Z-valued point of affine n(m+1)-space determines and is determined by γ∗ if and only if the condition fjl′ 7→ 0 holds for every 1 ≤ j ≤ s, 0 ≤ l′ ≤ m when we consider fjl′ as a polynomial in the variables X (l) i . Moreover, this correspondence is functorial; hence Jm(X) is the closed immersion Jm(X) = Spec C[X(0)1 , . . . , X(0)n , X (1) 1 , . . . , X (1) n , . . . , X (m) 1 , . . . , X (m) n ]/(fjl′)j,l′ in complex affine n(m+ 1)-space. We can describe the equations fjl′ explicitly. First, let m ′ and m be integers, m′ > m. The projection morphism pim′,m : Jm′(X) → Jm(X) is the morphism of schemes induced by the truncation homomorphism C[t]/(tm′+1)→ C[t]/(tm+1). The jet schemes of X with their projection morphisms form a projective system · · · → Jm(X) pim,m−1−−−−→ Jm−1(X)→ · · · → X, whose projective limit J∞(X) is called the arc space of X. Further, there are projec- tion morphisms ρm : J∞(X) → Jm(X) (it will always be clear to which “projection” we refer). Similarly to the jet schemes, over C the arc space of an affine scheme X = Spec C[X1, . . . , Xn]/(f1, . . . , fs) immerses into J∞(AnC) = Spec C[X (0) 1 , . . . , X (0) n , . . . , X (l) n , X (l+1) 1 . . .]. 1 Notice that J∞(X) is not generally a scheme of finite type over C. One obtains explicit equations for J∞(X) as follows (borrowing notation from [EM08]): let S∞ = C[X(0)1 , . . . , X(0)n , . . . , X(l)n , X (l+1) 1 , . . .] denote the polynomial ring in infinitely and denumerably many variables. There is a deriva- tion d on S∞, mapping X (l) i 7→ X(l+1)i . For any f ∈ C[X1, . . . , Xn], consider f as an element of S∞ by substituting X (0) i for Xi. Denote f (0) def= f and let f (l+1) def = df (l) recursively. Writing I∞ for the ideal (f (l) j |1 ≤ j ≤ s, l ∈ N ∪ {0}), it is straight-forward to prove that Spec S∞/I∞ = J∞(X); that is, the equations fj, dfj, d2fj, . . . over all 1 ≤ j ≤ s cut out the scheme J∞(X) from (infinite-dimensional) affine space. Under the truncation homo- morphism S∞ → Sm = C[X(0)1 , . . . , X(m)n ] we obtain the equations for Jm(X). Namely, Jm(X) = Spec Sm/Im, where Im = (fj, dfj, . . . , d mfj : 1 ≤ j ≤ s). Let us shift focus momentarily to the complex-analytic setting. Let X now be a smooth complex-analytic variety. Recall that a divisor Y in X is a formal, locally finite linear combination Y = ∑ aiYi of irreducible analytic hypersurfaces Yi in X. So, each Yi is locally the zero-locus of a single irreducible holomorphic function fi on X such that for any open U ⊂ X, Yi ∩ U 6= ∅ for only finitely many i. Such a divisor is said to have normal crossings if Y = ∑ Yi and the components Yi meet transversally; that is, when k of the components, say Yi1 , . . . , Yik , pass through x ∈ X, one can always choose local coordinates x1, . . . , xn on some U containing x such that x = (0, . . . , 0) and fij = xj for 1 ≤ j ≤ k. In particular Y ∩U is geometrically the zero-locus of x1 · · ·xk. Let Y be a divisor with normal crossings on X, let X∗ = X − Y , and let ϑ : X∗ → X be the inclusion morphism. Recall that Ω1X(logY ) denotes the locally free sub-OX-module of the direct image sheaf ϑ∗Ω1X∗ , locally generated by elements dx1 x1 , . . . , dxr xr , dxr+1, . . . , dxn in a neighbourhood U as above. This sheaf is called the sheaf of differential 1-forms on X with logarithmic poles along Y . (See [GH94] and especially the papers of Deligne such as [Del71] for more on this structure.) Now, let X denote the algebraic variety AnC, and let ΩX/C be its sheaf of Kähler dif- ferentials over C. ΩX/C is locally free; the (geometric) vector bundle associated to ΩX/C, TX := Spec (SymΩX/C), is called the total tangent space of X. It is straightforward to show that J1(X) ∼= TX . We refer to a section v of the structure morphism TX → X as a vector field on X; note that such a v is an X-valued 1-jet in X. Let D be the (effective) Cartier divisor on X defined by the global section X1 · · ·Xr. There is a sheaf ΩX/C(logD) onX associated toD analogous to the sheaf Ω 1 X(logY ) described above in the analytic context. This is the sheaf B̃ of OX-modules associated to the free C[X] = C[X1, . . . , Xn]-module B = C[X] · dX1 X1 ⊕ . . .⊕ C[X] · dXr Xr ⊕ C[X] · dXr+1 ⊕ . . .⊕ C[X] · dXn; that is, ΩX/k(logD) = B̃. Further, there is a vector bundle TX(logD) := Spec (Sym ΩX/C(logD)) 2 over X called the logarithmic total tangent space of X with respect to the divisor D. There is a canonical injection ΩX/C → ΩX/C(logD) from which we obtain a morphism TX(logD)→ TX factoring the structure morphism TX(logD)→ X. One refers to a section of TX(logD)→ X as a vector field on X with logarithmic poles along D (though we may simply use vector field logarithmic along D, or logarithmic vector field). A vector field v on X is written v = f1 · ∂∂X1 +· · ·+fn · ∂∂Xn , with fi ∈ C[X] for each i. Here ∂ ∂Xi is the homomorphism ΩX/C → C[X] determined by the Kronecker delta: ∂∂Xi (dXj) = δij for 1 ≤ i, j ≤ n. Supposing that D is a divisor with normal crossings as above, a vector field logarithmic along D has the form v = g1 ·X1 · ∂ ∂X1 + · · ·+ gr ·Xr · ∂ ∂Xr + fr+1 · ∂ ∂Xr+1 + · · ·+ fn · ∂ ∂Xn , with gi ∈ C[X]; that is for such a vector field Xi|fi for every 1 ≤ i ≤ r. The isomorphism between J1(X) and TX lets us reformulate this in terms of jets. Namely, the vector field v is anX-valued point of J1(X) corresponding to a homomorphism C[X]→ C[X][t]/(t2). Since v is a section of the projection pi1, this homomorphism is the one determined by Xi 7→ Xi+fit. When v is logarithmic along D this becomes Xi 7→ Xi+(gi ·Xi)t = Xi(1+git) for 1 ≤ i ≤ r; that is, Xi maps to Xi times a unit of the ring C[X][t]/(t2). Equivalently, the equation X1 · · ·Xr defining D maps to X1 · · ·Xr times a unit. We can restate this last condition as follows: an equation of the (normally crossing, effective Cartier) divisor D pulls back to an equation for the divisor D1 := {(X × Spec C[t]/(t2), X1 · · ·Xr)}. That is, if vγ : X × Spec C[t]/(t2) → X is the X-valued jet corresponding to v, then the morphism of sheaves (vγ) ∗ : OX → (vγ)∗OX×Spec C[t]/(t2) takes a local equation for D to a local equation for D1. We observe here that the condition “v is a logarthmic vector field along D” can be stated functorially. That is, given any schemes X and Z of finite type over any (algebraically closed) field, and any m ≥ 0 we may ask which Z-valued m-jets on X pull back some fixed divisor D on X. These m-jets will comprise the Z-valued points of a scheme that we call the mth logarithmic jet scheme of X with respect to the divisor D. In fact, we can work in more generality, parametrizing the m-jets in X that pull back a fixed family of effective Cartier divisors (D1, . . . , Dr). In the case of a normally crossing divisor on affine space X, it will then follow that a section of the logarithmic jet scheme of X with respect to D corresponds exactly to a vector field on X logarithmic along the divisor. In the proceeding section we will recall some notions that are basic in algebraic geometry, but essential to our study. Namely we will recall the functor of points of a scheme, and Cartier divisors on schemes. Following this we will fix a framework, the categories of pairs, for working with divisors on schemes and morphisms pulling their local equations back to the domain. Following these two sections we will move on to use this framework to prove the existence of logarithmic jet schemes associated to any pair (X, (D1, . . . , Dr)) as above. Here, results about the jet schemes generalize to such pairs, allowing a constructive proof; we obtain equations for the logarithmic jet schemes similarly to the method above in the 3 case of ordinary jet schemes. Finally, we will conclude with a short summary of our work, and then make some closing remarks on likely improvements to the choice of categories in which we can prove our results. 4 2 Preliminaries In this section we collect some definitions and results that we will use in the following sections. We will begin by recalling the definition of the functor of points of an object in a general category, and stating Yoneda’s Lemma. This notion is basic, but fundamental; our main result uses Yoneda’s lemma to prove that a certain functor is representable. We also include the definitions of Cartier and effective Cartier divisors on schemes, as we will use these throughout. 2.1 Functors of Points and Yoneda’s Lemma From the formulation of scheme-theoretic algebraic geometry in terms of prime ideal spectra and their Zariski topologies on arbitrary commutative rings, situations arise in which not all the information encapsulated by a scheme is captured by the underlying point-sets. As a simple example, the points of a (fibered) product scheme are not necessarily in direct correspondence with the points of the Cartesian product of the underlying sets. Further, the Zariski-topology on a scheme is defined in such a way that generic points of a scheme do not relate exactly the geometry of what one usually considers to be a point. Thanks to Grothendieck, the present language of algebraic geometry includes an alterate notion of points on a scheme. Namely to any scheme we have an associated “functor of points”. Though seemily set at a high level of abstraction (defined as a process rather than as an object), this functorial notion of points retains, and effectively describes in terms of sets, (universal) properties we expect in geometry. With regards to the example above, the functor of points of a (fibered) product of schemes is canonically isomorphic to the fibered product of the functors of points of the factors, which is in essence a Cartesian product of sets. Let us recall the definition of functors of points. It is possible, and it will be more efficient for us, to define functors of points of objects in a general category. In particular, we will talk of the functor of points of a “pair” in the next chapter. Let X and Y denote objects in a category Γ. Recall that the functor of points of X, denoted hX , is the (contravariant) functor defined as follows: let hX : (Γ) ◦ → Sets take any Y to the set hX(Y ) = HomΓ(Y,X), where (Γ) ◦ denotes the opposite category to Γ. In this context, a morphism φ : Y → X is referred to as a Y -valued point of X. Further, recall that the mapping h : Γ → Fun((Γ)◦,Sets) taking X to hX is a (covariant) functor. We include the following fundamental fact from category theory: Lemma (Yoneda) 2.1. Let X and Y be objects in a category Γ as above. Then, (i) if F : (Γ)◦ → Sets is a functor, the natural transformations from hX = HomΓ(−, X) to F are in natural correspondence with the elements of F(X). (ii) if hX = HomΓ(−, X) ∼= HomΓ(−, Y ) = hY , X ∼= Y . That is, h : X → hX is fully faithful. 5 When Γ is the category of k-schemes, this result can be refined to the following: Proposition 2.2. The functor h : k − Schemes→ Fun(k −Algebras,Sets) is a fully faithful functor from the category of schemes over k to the category of functors from k-Algebras to Sets . That is, a scheme over k is determined by the restriction of its functor of points to the category of affine schemes over k. One may actually replace k with any commutative ring R here; however, we do not need this generality. We shall not provide proofs here as they are easily found elsewhere (see e.g. [EH00] or [FmI+05]). These results are crucial in obtaining a suitable parameter space via a concise functorial definition (e.g. the jet scheme Jm(X) parametrizing jets on a scheme X). The first part of Yoneda’s lemma tells us in particular that natural transformations from hX to hY naturally correspond to morphisms from X to Y . The second part tells us that X is uniquely deter- mined by hX . Hence, rather than simply studying the objects X and Y and the morphisms between them in Γ, we may alternatively study their functors of points, transferring our inquiry into the broader context of natural transformations between functors. Finally, the notion of a representable functor will be important for us. Recall that a functor F : (Γ)◦ → Sets is a representable functor if there is some object X in Γ such that hX ∼= F. Such an object X is unique by the second part of Yoneda’s lemma. In this case we also say that X represents the functor F. We shall return to functors of points later on; in particular, proposition 2.2 has an analogue in the categories of pairs to be defined. 2.2 Divisors In the first part of 2.2 we will collect some necessary definitions, particularly those of Cartier divisors and predivisors on a scheme. The second part of this section will link these two notions, and state a useful result relating divisors and predivisors on X to those on a scheme Y , given a morphism Y → X. 2.2.1 Definitions of Divisors We recall here the notion of a Cartier divisor. We will almost exclusively work with effective Cartier divisors in the following sections, and we define these as well. For a thorough introduction to divisors, see [Har06]. Let us first agree on the notation a ∈ nzd(R) for “a is a non-zerodivisor in the ring R”. Given a scheme X and an open neighbourhood U ⊆ X, let SX(U) ⊆ Γ(U,OX) denote the set of sections over U that are non-zerodivisors in every local ring OX,p with p ∈ U . The mapping U 7→ SX(U)−1Γ(U,OX) is a pre-sheaf on X; its associated sheaf is named the sheaf of total quotient rings of X, denoted by MX . Further, let the sheaf of multiplicative groups of invertible elements of a sheaf of rings G be denoted by G∗. Cartier divisors are defined as follows: 6 Definition 2.2.1. A Cartier divisor D on X is a global section of the sheaf M∗X/O∗X . Thus, we may specify a Cartier divisor D on X with an open covering {Uα : α ∈ A} of X and an element fα ∈ Γ(Uα,M∗X) for each α, such that for every α, β ∈ A the quotient fα/fβ ∈ Γ(Uα ∩ Uβ,O∗X). We write D = {(Uα, fα) : α ∈ A} for such an object. Remark 2.1. Let us unravel this definition for the case X is of finite type, and in particular is locally Noetherian. Remember that if R is a Noetherian ring, then r ∈ nzd(R) if and only if r/1 ∈ nzd(Rp) for all prime ideals p ≤ R. When specifying a Cartier divisor D = {(Uα, fα) : α ∈ A} on a locally Noetherian X, we may assume that every Uα = Spec Rα is an affine open subscheme such that Rα is a Noetherian ring. Then, fα ∈ Γ(Uα,M∗X) is equivalent to fα = gα/hα for gα, hα ∈ nzd(Rα). From the definition, we see that D = {(Uα, fα) : α ∈ A} and D′ = {(Wγ, hγ) : γ ∈ G} determine the same Cartier divisor if for any α and γ such that Uα∩Wγ 6= ∅ the restrictions of fα and hγ differ by a unit in the ring Γ(Uα ∩Wγ,OX); that is, fα|Uα∩Wγ = uα,γ · hγ|Uα∩Wγ for some uα,γ ∈ Γ(Uα ∩Wγ,O∗X). We may restrict a Cartier divisor D to any open subscheme U of X. The restriction of D to U is the Cartier divisor {(Uα ∩ U, fα|Uα∩U) : α ∈ A} on U , denoted D|U . Effective Cartier divisors, defined as follows, are those Cartier divisors that correspond to closed subschemes whose sheaf of ideals can locally be generated by a single section that is a non-zerodivisor (i.e. locally principal proper closed subschemes): Definition 2.2.2. A Cartier divisor D = {(Uα, fα) : α ∈ A} on X is called effective if fα ∈ Γ(Uα,OX) for every α ∈ A. In other words, D is an effective Cartier divisor on X if and only if D defines a closed subscheme XD → X whose sheaf of ideals JD is invertible (i.e. locally isomorphic to OX). The set of Cartier divisors on a scheme X forms a group Div(X), and in “nice” situations, such as for subschemes of projective space over a field, Div(X) is generated by the effective Cartier divisors. We will not dwell on properties of divisors here; for a detailed treatment see [EH00], [Har06], [Gro67] or any of the litany of references that exists touching on the subject. Though we will work mainly with effective Cartier divisors in the following sections, the condition that fα,p ∈ nzd(OX,p) for every p ∈ Uα, on an effective Cartier divisor D as above, is quite restrictive. Since it may be desirable from a geometric perspective to loosen this condition, we include the following definition: Definition 2.2.3. A predivisor D′ on X is a collection {(Uα, fα) : α ∈ A} satisfying the following: • {Uα : α ∈ A} is an open cover of X, • fα ∈ Γ(Uα,OX) for all α ∈ A, and • for every α, β ∈ A such that Uα ∩ Uβ 6= ∅, there exists some uα,β ∈ Γ(Uα ∩ Uβ,O∗X) such that fα|Uα∩Uβ = uα,β · fβ|Uα∩Uβ . 7 Remark 2.2. It is important to note that every effective Cartier divisorD onX defines many predivisors on X; to every equivalent presentation of D, there is an associated predivisor. Conversely, given a predivisor D′ = {(Uα, fα) : α ∈ A} such that fα,p ∈ nzd(OX,p) for all p ∈ Uα and all α ∈ A, there is an effective Cartier divisor associated toD′, and this Cartier divisor is associated to any predivisor whose presentation satisfies the natural equivalence condition with D′. In the following subsection, we will elaborate on the link between predivisors on X and effective Cartier divisors on X. 2.2.2 Pullbacks of Divisors In this subsection we will describe the effects of morphisms on predivisors and Cartier divisors (in the case of Cartier divisors, we refer the reader to [Gro67] for further details). Let us agree that, when unmodified, divisor shall mean effective Cartier divisor in all that follows. First, let φ : Y → X be a k-morphism between k-schemes Y and X; by definition this is a continuous map φ : Y → X of topological spaces, coupled with a morphism φ∗ : OX → φ∗OY of sheaves on X (which behaves “nicely” with regard to localisation). Further, let D′ = {(Uα, fα) : α ∈ A} be a predivisor on X. Let φ∗α def = φ∗Uα : Γ(Uα,OX)→ Γ(φ−1(Uα),OY ), so that φ∗α(fα) ∈ Γ(φ−1(Uα),OY ). We find that the collection {(φ−1(Uα), φ∗α(fα)) : α ∈ A} is a predivisor on Y . In particular note that the third condition for predivisors holds, as φ∗α(fα)|φ−1(Uα)∩φ−1(Uβ) = φ∗Uα∩Uβ(uα,β) · φ∗β(fβ)|φ−1(Uα)∩φ−1(Uβ), for every α, β ∈ A such that φ−1(Uα) ∩ φ−1(Uβ) 6= ∅. Note that φ∗Uα∩Uβ(uα,β) ∈ Γ(φ−1(Uα) ∩ φ−1(Uβ),O∗Y ) is invertible, as uα,β ∈ Γ(Uα ∩ Uβ,O∗X) is. We make the following definition: Definition 2.2.1. Given a predivisor D′ on X, and a morphism Y → X as above, the predivisor {(φ−1(Uα), φ∗α(fα)) : α ∈ A} on Y is called the pullback of D′ by φ, denoted φ∗(D′). We say that φ pulls back D′ to φ∗(D′). Recall that for every p ∈ X and any q ∈ φ−1(p) there is an induced morphism φ∗q : OX,p → OY,q of local rings. Suppose that p ∈ Uα ⊆ X; this morphism takes fα,p to the element φ∗q(fα,p) = (φ ∗ α(fα))q. Hence, if φ ∗ q(fα,p) ∈ nzd(OY,q) for every such p, q, and α, then the pullback φ∗(D′) = {(φ−1(Uα), φ∗α(fα)) : α ∈ A} will define an effective Cartier divisor on Y , by remark 2.2. We obtain the following lemma: Lemma 2.1. Suppose that D′ = {(Uα, fα) : α ∈ A} and D′′ = {(Wγ, hγ) : γ ∈ G} are predivisors obtained from equivalent presentations of the effective Cartier divisor D. More- over, suppose that D′ satisfies the condition above, so that φ∗(D′) yields an effective Cartier divisor on Y . Then, the pullback φ∗(D′′) also yields an effective Cartier divisor on Y , equal to the one obtained from φ∗(D′). 8 Proof. To verify this, we must show that the pullback φ∗(D′′) is locally defined by sections that differ by a unit from those defining φ∗(D′) on the intersections of their respective domains. Now, as φ∗α(fα) is locally a non-zerodivisor, so is φ ∗ α,γ(fα|Uα∩Wγ ) for all α and γ such that Uα ∩Wγ 6= ∅, where φ∗α,γ = φ∗Uα∩Wγ . Then, φ∗α,γ(fα|Uα∩Wγ ) = φ∗α,γ(uα,γ · hγ|Uα∩Wγ ) = φ∗α,γ(uα,γ) · φ∗α,γ(hγ|Uα∩Wγ ), for some uα,γ ∈ Γ(Uα ∩ Wγ,O∗X). Since uα,γ ∈ Γ(Uα ∩ Wγ,O∗X) is invertible, we have φ∗α,γ(uα,γ) ∈ Γ(φ−1(Uα ∩Wγ),O∗Y ) = Γ(φ−1(Uα)∩ φ−1(Wγ),O∗Y ); that is, φ∗α,γ(uα,γ) is invert- ible. Hence φ∗α,γ(hγ|Uα∩Wγ ) is locally a non-zerodivisor, showing that φ∗γ(hγ) is itself locally a non-zerodivisor. Also, we see that on φ−1(Uα ∩Wγ) = φ−1(Uα) ∩ φ−1(Wγ) the local equa- tions φ∗α(fα) and φ ∗ γ(hγ) differ by a unit. Hence, the predivisors φ ∗(D′) and φ∗(D′′) yield equivalent effective Cartier divisors. We formalize this case with the following definition: Definition 2.2.2. Let D be an effective Cartier divisor on the scheme X and let φ : Y → X be a morphism, as in lemma 2.1. We define the pullback of D by φ to be the effective Cartier divisor on Y obtained from the pullback φ∗(D′) of the predivisor D′ = {(Uα, fα) : α ∈ A}, where {(Uα, fα) : α ∈ A} is any presentation of D. We denote the pullback of D as φ∗(D), and say in this case that φ pulls back the effective Cartier divisor D to φ∗(D). The following definition is convenient in the context of the categories Pairsr, described in the following section: Definition 2.2.3. Let r ≥ 1 be an integer, and let D′ = (D′1, D′2, . . . , D′r) be an r-tuple of predivisors on the scheme X. Given a morphism φ : Y → X as above, we define the pullback of D′ by φ to be the r-tuple (φ∗(D′1), φ∗(D′2), . . . , φ∗(D′r)), denoted φ∗(D′). In the case that each D′i is a presentation of a divisor Di, then as above each pullback φ ∗(D′i) is a presentation of a divisor determined by Di. Letting D denote the r-tuple of divisors (D1, D2, . . . , Dr), we define the pullback of D by φ to be the r-tuple (φ∗(D1), φ∗(D2), . . . , φ∗(Dr)) of divisors on Y . In this case, we say that φ pulls back the family D of divisors on X to the family φ∗(D) on Y . We will end the section with a description of how, by “removing components” from a scheme X of finite type over k, one can force a predivisor D′ on X to describe an effective Cartier divisor D on some maximal closed immersion X ′ → X. Thus, any morphism Y → X of schemes of finite type such that φ∗(D′) defines an effective Cartier divisor on Y will factor through X ′. Lemma 2.2. Given a predivisor D′ = {(Uα, fα) : α ∈ A} on a scheme X of finite type over k, there exists a unique closed immersion i : X ′ → X such that (i) i∗(D′) yields an effective Cartier divisor on X ′, and 9 (ii) for any k-morphism φ : Y → X such that φ∗(D′) yields an effective Cartier divisor on Y , there exists a morphism φ′ : Y → X ′ such that φ = i ◦ φ′. Consequently, the pullback by φ′ of the effective Cartier divisor defined by i∗(D′) is the effective Cartier divisor defined by φ∗(D′). Proof. Let giα = fα|Viα ∈ Riα , where {Viα = Spec Riα : iα ∈ Iα} is an open cover of Uα by affines for each α ∈ A. Suppose that giα is a zerodivisor in Riα . We begin by letting Siα = {gγiα : γ ≥ 0} be the multiplicative subset in Riα generated by giα , and we consider the canonical ring homomorphism hiα : Riα → Riα [S−1iα ]. Let Kiα = ker(hiα). Notice that a ∈ Kiα if and only if ∃ γ such that gγiα · a = 0 in Riα . Thus, the ring Riα/Kiα is the largest quotient ring of Riα in which every zerodivisor of giα equals zero; that is, Kiα is the minimal ideal containing all such elements. Now let φ be a morphism as in the statement of the lemma. The homomorphism (φ|φ−1(Viα ))∗ : Riα → Biα = Γ(φ−1(Viα),OY ) must factor through Riα/Kiα ; i.e. the mor- phism φ|φ−1(Viα ) : φ−1(Viα)→ Viα = SpecRiα factors through SpecRiα/Kiα . Thus the ideals Kiα indexed over all α ∈ A and iα ∈ Iα define a closed immersion i : X ′ → X through which φ : Y → X must factor. By construction, we see that the pullback i∗(D) yields an effective Cartier divisor on X ′. Moreover, it is immediate that the pullback of this effective Cartier divisor is defined by φ∗(D). 10 3 Categories of Pairs In the first part of this section, we define the categories of pairs. This allows us to use some categorical arguments in studying scheme morphisms that pull back divisors. Following this, we collect some definitions and examples in the categories of pairs that will be used in the next section. 3.1 Defining the Categories of Pairs The categories of pairs over k provide us primarily with a convenient framework for studying jets on a scheme that pull back divisors. In any category of pairs we will define the “jet pairs” analogously to jet schemes, in terms of the representability of a certain functor. For any r ≥ 0 there is a category of pairs k-Pairsr, though we will usually work in a general category Pairs, specifying r and k only as needed. An object in Pairs consists of a scheme X of finite type over k coupled with an ordered r-tuple D = (D1, D2, . . . , Dr) of its effective Cartier divisors Di, 1 ≤ i ≤ r. This forms the pair X = (X,D) (again, one may wish specify r-pair, pair over k, r-pair over k, etc.). We must describe the morphisms in Pairs. Given two pairs X = (X,D) and Y = (Y, E) ∈ Pairs, we define HomPairs(Y,X) = {φ ∈ Homk−Schemes(Y,X) : φ∗(D) = E}. Thus, for φ : Y → X to be considered as a morphism of pairs, the pullback of D by φ must exist and equal E . It is clear from the definitions that the identity idX : (X,D) → (X,D) is a morphism of pairs; simply note that id∗X = idOX . Further, pullbacks behave well with regards to composition; that is, given two morphisms of pairs ψ : (Z,F) → (Y, E) and φ : (Y, E)→ (X,D), their composition φ◦ψ is a morphism of pairs. To verify this, note that if Di = {(Uαi , fαi) : αi ∈ Ai} is a presentation of Di, then Ei = {(φ−1(Uαi), φ∗αi(fαi)) : αi ∈ Ai} is a presentation of Ei, and so Fi = {(ψ−1(φ−1(Uαi)), ψ∗φ−1(Uαi )(φ ∗ αi (fαi))) : αi ∈ Ai} = {((φ ◦ ψ)−1(Uαi), (φ ◦ ψ)∗αi(fαi)) : αi ∈ Ai}, is a presentation of Fi, proving that the pullback of Di under φ ◦ ψ exists and equals Fi. Hence k-Pairsr is a category. We choose to define k-Pairs0 to be the category of schemes of finite type over k, k-Schemes. At times we will want to focus our attention exclusively on pairs with affine underlying schemes, and (families of) divisors defined by global sections. Just as the affine schemes form a category, we let k-Aff Pairsr denote the category whose objects are pairs X = (X,D) in k-Pairsr such that X is an affine scheme, and D = ({(X, f1)}, . . . , {(X, fr)}) for some choice of presentations Di = {(X, fi)}. The morphisms between two objects are all those between the objects in Pairs. Hence, k-Aff Pairsr forms a full subcategory of k-Pairsr; we call this the category of affine pairs. Again, we will almost exclusively write Aff Pairs and work in the general category. 11 3.2 Definitions and Examples in the Categories of Pairs Here we collect some examples of morphisms and constructions in the categories of pairs that will be useful to us later on. 3.2.1 Open Subpairs and a Gluing Construction for Pairs The simplest examples of morphisms of pairs come from open subschemes. Let X = (X,D) be a pair, and U ⊆ X an open subscheme. The divisors Di for 1 ≤ i ≤ r naturally restrict to U (as mentioned in section 2.2) as the local equations for Di are non-zerodivisors in the stalks OU,p = OX,p for all p ∈ U . So the open immersion U → X naturally defines an open immersion of pairs U→ X, where U = (U,D|U) and D|U = (D1|U , . . . , Dr|U). We will refer to U as an open subpair of the pair X. We will define the intersection of two open subpairs U and V of X, whose underlying schemes U, V ⊆ X have non-empty intersection U ∩V 6= ∅, to be the open subpair U∩V = (U ∩ V,D|U∩V ) of X. Suppose that {(Xα,Dα) : α ∈ A} is a collection of open subpairs of (X,D) indexed by the set A. We will call {(Xα,Dα) : α ∈ A} an open cover of (X,D) if {Xα : α ∈ A} is an open cover of X. Note that by assumption the families of divisors Dα|Xα∩Xβ and Dβ|Xα∩Xβ are equivalent on the open subscheme Xα ∩Xβ for all α, β ∈ A, as they are both equivalent to D|Xα∩Xβ . Given another pair Y = (Y, E), and a morphism φ : Y → X, we would also like to speak of the preimage of an open subpair U ⊆ X. We let the preimage of U under φ refer to the open subpair φ−1(U) = (φ−1(U), E|φ−1(U)) of Y. Notice that we have restricted our attention especially to open subschemes here. Suppose that Y → X is a closed immersion, and that (X,D) is a pair. In general it is certainly not true that the local sections defining a Cartier divisor on X will pull back to non-zerodivisors in the stalks of the structure sheaf of Y . Hence working with closed immersions is considerably more subtle; in general we must “remove components” in order to ensure we are always working in the category Pairs (see lemma 2.2). Example (Gluing Construction) 3.1. We now remind the reader of the gluing construc- tion for schemes, giving the analogous construction in Pairs. The reason that gluing works in Pairs is simple; gluing together a scheme from a collection of schemes with divisors, whose local equations coincide (up to multiplication by an invertible section) via the local isomorphisms defining the gluing, yields divisors on the new scheme that are locally defined by the original equations. That is, suppose first that we are given a collection of pairs {Xα = (Xα,Dα) : α ∈ A} indexed by a set A and for every α, β ∈ A such that β 6= α an open subpair Uαβ of Xα. Suppose further that we have isomorphisms ψαβ : Uαβ → Uβα for all such α, β, that satisfy ψβα = ψ −1 αβ for all α, β; that ψαβ(Uαβ ∩Uαγ) = Uβα ∩Uβγ ∀α, β, γ; and that ψαβ ◦ ψβγ|Uαβ∩Uαγ = ψαγ|Uαβ∩Uαγ ∀α, β, γ. 12 Then we may glue together a pair along the isomorphisms ψαβ analogously to the gluing of schemes. In fact, the underlying scheme is obtained by gluing along the morphisms ψαβ considered as morphisms of schemes, while the r-tuple of divisors on the glued scheme will have local equations exactly those on any Xα. The following example shows that flat morphisms of schemes are morphisms of pairs: Example 3.2. Let X = (X,D) be a pair, and let φ : Y → X be a morphism of schemes, such that Y is flat over X. This means that for every y ∈ Y and x = φ(y) the morphism φ∗y : OX,x → OY,y makes OY,y into a flat OX,x-module. Let U ⊆ X be an open subscheme, and suppose that for each 1 ≤ i ≤ r the local equation of Di on U is fi. We pull the local equations for D back to φ∗(fi) ∈ φ−1(U) ⊆ Y . Since φ makes Y flat over X, for every point y ∈ φ−1(U) we deduce that (φ∗(fi))y = φ∗y(fi,x) is a non-zerodivisor in OY,y. (This follows from a basic property of flat modules; namely, since fi,x ∈ nzd(OX,x) and the morphism φ∗y makes OY,y into a flat OX,x-module, fi,x remains a non-zerodivisor on OY,y (see e.g. [Eis04]).) Hence the pullback φ∗(D) exists on Y , and so any flat morphism from a scheme to a scheme with attached family of effective Cartier divisors automatically pulls back the family, yielding a morphism of pairs. Of course, this implies that smooth and étale morphisms also always pull back families of effective Cartier divisors. We shall refer to a morphism Y → X of pairs as flat (resp. étale, smooth) if the underlying scheme-morphism is flat (resp. étale, smooth). 3.2.2 m-jets in Pairs The reason for formulating Pairs as we have done stems from the following example. Let Y and X be schemes of finite type over k. Morphisms from the fibred product Y × Spec k[t]/(tm+1) to X are referred to as Y -valued m-jets, and are thought of as order m germs of arcs on X. We would like to study m-jets that pull back divisors on X. That is, given r-tuples D on X and E ′ on Y × Spec k[t]/(tm+1) we will study the morphisms of pairs (Y × Spec k[t]/(tm+1), E ′)→ (X,D). As a particular example, let us consider the projection Y × Spec k[t]/(tm+1) → Y . Let E = (E1, . . . , Er) be an r-tuple of effective Cartier divisors on Y . For any 1 ≤ i ≤ r suppose that Vi = Spec Bi ⊆ Y is an open affine subscheme on which Ei is defined by gi ∈ Bi. The preimage of Vi under the projection is isomorphic to Vi × Spec k[t]/(tm+1) = Spec Bi[t]/(t m+1), and the morphism of structure sheaves takes the section gi on Vi to gi as an element in the ring Bi[t]/(t m+1). Since gi is a non-zerodivisor in Bi, it is a non- zerodivisor in Bi[t]/(t m+1). Further, as the rings Bi and Bi[t]/(t m+1) are finitely generated k- algebras, and hence Noetherian, this implies that gi/1 is an element inM∗Y×Spec k[t]/(tm+1)(Vi× Spec k[t]/(tm+1)). Thus gi is locally the equation of an effective Cartier divisor on Vi × Spec k[t]/(tm+1) = Spec Bi[t]/(t m+1) ⊆ Y × Spec k[t]/(tm+1). We denote this divisor Emi , and let Em def= (Em1 , Em2 , . . . , Emr ). Examining further, we find that gi ∈ nzd(Bi) if and only if gi + gi1t + · · · + gimtm ∈ nzd(Bi[t]/(t m+1)) for any gi1, . . . , gim ∈ Bi. Therefore a divisor E ′ on Vi × Spec k[t]/(tm+1) 13 defined by gi + gi1t+ · · ·+ gimtm is of the form above if and only if gi|gil for all 1 ≤ l ≤ m. Referring back to the introduction, this condition matches exactly the one we derived for 1-jets that guarantees they are logarithmic along a divisor with normal crossings. Thus, we make the following definition: Definition 3.2.1. Given pairs (Y, E) and (X,D) as above, a morphism of pairs from (Y × Spec k[t]/(tm+1), Em) to (X,D) is called a (Y, E)-valued m-jet in (X,D). Alternatively, we may use the terms m-jets logarithmic along D, or simply m-jets when the context is clear. Obviously, in Pairs0 = k-Schemes, a logarithmic m-jet is the same thing as a usual m-jet. Often, the way we work in the category Pairs is as follows: we begin with some pair (X,D) and a scheme-morphism Y → X, and then examine whether the pullback of D exists on Y . The way we defined logarithmic m-jets provides an example of this. This process seems to be in contrast to the way the category is defined; namely, the morphisms of pairs are those that a priori have well-defined pullbacks that coincide with an r-tuple of divisors on the domain. This is not a problem; the real advantage of the categorical formalism lies in the arguments and results it enables us to use, as in the next section. Now, consider the truncation homomorphism k[t]/(tm ′+1) → k[t]/(tm+1), where m′ and m are integers such that m′ ≥ m ≥ 0. This morphism induces a morphism of affine schemes Spec k[t]/(tm+1) → Spec k[t]/(tm′+1). Given a scheme Y consider the following diagram, which commutes: Y × Spec k[t]/(tm+1) - Spec k[t]/(tm+1) Y × Spec k[t]/(tm′+1) - ηm ′,mY - Spec k[t]/(tm ′+1) truncation ? Y ? - - Spec k ? So ηm ′,m Y is the unique morphism of schemes guaranteed by the universal mapping property of the fibred product Y ×Speck[t]/(tm′+1) making the diagram commute. As in the previous example, suppose that (Y, E) is a pair. Then we may pull E back to both fibred products Y × Spec k[t]/(tm+1) and Y × Spec k[t]/(tm′+1), making ηm′,mY into a morphism of pairs. (Note that the local equations of the r-tuples remain essentially unchanged; only the ring they live in changes.) Thus, denoting (Y, E) by Y, let us write ηm′,mY for the morphism of pairs (Y ×Speck[t]/(tm+1), Em)→ (Y ×Speck[t]/(tm′+1), Em′). We refer to such a morphism as a truncation morphism; note that we can pull back (Y, E)-valued m′-jets to (Y, E)-valued m-jets via ηm ′,m Y . We will write η m′ Y rather than η m′,0 Y when m = 0. Later on, we will use these morphisms to define “projection morphisms” between jet pairs. Moving on, suppose that φ : Z → Y is a morphism, where Z = (Z,F) and Y = (Y, E). Then there is a morphism φm : Z × Spec k[t]/(tm+1) → Y × Spec k[t]/(tm+1) of schemes 14 guaranteed by the universal mapping property for the fibred product Y × Spec k[t]/(tm+1). φm is in fact a morphism of pairs φm : (Z×Speck[t]/(tm+1),Fm)→ (Y×Speck[t]/(tm+1), Em). This follows since pulling back the local equations of Em by φm is equivalent to pulling back the local equations of E by the composition Z × Spec k[t]/(tm+1)→ Z φ−→ Y . Thus for every φ as above we obtain an induced morphism φm such that the following commutes: (Z × Spec k[t]/(tm+1),Fm) φm- (Y × Spec k[t]/(tm+1), Em) (Z,F) projection ? φ - (Y, E) projection ? We will use morphisms of this form in the results in the following section. To finish this section, let us explicitly give the definition of the functor of points of a pair, as we will work with these immediately in what follows. Example 3.3. Recall that in section 2.1 we defined the functor of points of an object in a general category. Let X ∈ Pairs; the functor of points of the pair X, denoted hX, is defined to be the functor hX : (Pairs) ◦ → Sets such that hX(Y) = HomPairs(Y,X) for Y ∈ Pairs. 15 4 Jet Pairs and Logarithmic Jet Schemes We prove some results about pairs that enable us to demonstrate the existence of a parameter space for certain (Y, E)-valued m-jets on a pair (X,D). We call this parameter space, which lives in a category of pairs, the “jet pair” associated to the pair (X,D). The underlying scheme of the jet pair is referred to as the “logarithmic jet scheme of X with respect to D”. The preliminary results generalize analogous results in the category of schemes, which are employed to give a constructive proof of the existence of the jet schemes associated to a chosen scheme. Indeed, the scheme case is subsumed within ours by the case Pairs0. 4.1 The Main Construction Let X = (X,D) ∈ Pairs, and let jm denote Spec k[t]/(tm+1). The mapping LXm : (Y, E) 7→ HomPairs((Y × jm, Em), (X,D)) defines a covariant functor (Pairs)◦ → Sets (in other words a contravariant functor from Pairs to Sets). We will prove that this functor is representable for all m > 0, i.e. that there exists a pair Jm(X) = (J D m(X), Jm(D)) such that hJm(X) ∼= LXm. (Note in addition that LX0 is always representable by X). To do this we will use two helpful facts; first, that one can determine whether such a functor is representable from its action on affine pairs, and second, that the functors hJm(X) can be obtained by “gluing together affine pieces” in the sense of the gluing construction of example 3.1. Once these facts are established we will only need to work in the category of affine pairs to prove representability; it will then be true for all pairs. We begin with the first claim, noting that this is only an adjustment of the analogous fact, lemma 2.2 on page 6, in which the roles of the categories of pairs and affine pairs (with arrows reversed) are taken by the categories of k-schemes and k-algebras respectively. Proposition 4.1. Let X = (X,D) be a pair over k. The restriction of the functor of points hX of X to the category of affine pairs over k determines X. That is, the func- tor h : Pairs - Func((Aff Pairs)◦, (Sets)) X - hX|(Aff Pairs)◦ is fully faithful. Proof. Let Y = (Y, E) also be an element of Pairs, and let hX and hY denote the restrictions hX|(Aff Pairs)◦ and hY|(Aff Pairs)◦ respectively for the remainder of the proof. Any morphism t : Y → X defines a natural transformation ht : hY → hX by composition of t with morphisms φ : Z→ Y for any Z = (Z,F) ∈ Aff Pairs. That is, ht(Z) : hY(Z) - hX(Z) HomPairs(Z,Y) - HomPairs(Z,X) φ - t ◦ φ 16 Hence, it is sufficient to prove that any natural transformation τ : hY → hX is equal to ht for some unique morphism t : Y → X. Let τ be such a natural transformation. We shall obtain the desired morphism t from τ . First, let {Yα = (Yα, Eα) : α ∈ A} be an open cover of Y by affines. Let ıα : Yα → Y denote the inclusion for each α. Then, there is a unique morphism tα def = τYα(ıα) : Yα → X corresponding to each inclusion. We claim that the tα’s glue together to define the desired morphism t : Y → X. To show this, first let Yαβ = Yα ∩Yβ for every α, β ∈ A. Further, let ıαβ : Yαβ → Yα denote the inclusion of the intersection into Yα. Then, by naturality of τ we see that τYαβ(ıα ◦ ıαβ) = tα ◦ ıαβ, and that τYβα(ıα ◦ ıαβ) = tβ ◦ ıβα, using the fact that ıβ ◦ ıβα = ıα ◦ ıαβ. But Yαβ = Yβα, therefore tα ◦ ıαβ = tβ ◦ ıβα; that is, the restrictions of the tα’s to the intersections Yα ∩Yβ are equal, and we may glue the morphisms to define t : Y → X. Now we will prove ht = τ by showing that ht(Z)(φ) = τZ(φ) for any Z and any element φ : Z → Y of hY(Z). Letting Zα = φ−1(Yα), and remebering that ht(Z)(φ) = t ◦ φ, we see that it suffices to prove (t ◦ φ)|Zα = τZ(φ)|Zα for all α ∈ A. Let φα = φ|Zα : Zα → Yα and let α : Zα → Z be the inclusion for each α ∈ A. Then by naturality of τ we find that τZα(ıα ◦ φα) = tα ◦ φα, and using ıα ◦ φα = φ ◦ α that τZα(ıα ◦ φα) = τZ(φ) ◦ α. Thus, (t ◦ φ)|Zα = φα ◦ tα = τZ(φ) ◦ α = τZ(φ)|Zα for any α. Hence indeed t ◦ φ = τZ(φ), and we are done. So supposing that there is a pair Jm(X) and an isomorphism of functors hJm(X)|(Aff Pairs)◦ ∼= LXm|(Aff Pairs)◦ , by this proposition we may conclude that hJm(X) ∼= LXm as functors on (Pairs)◦. Before we move on to the second claim, let us suppose that given any pair X and any m > 0, there is a pair Jm(X) = (J D m(X), Jm(D)) that represents LXm. Recall from an example in 3.2.2 that for a pair Z = (Z,F) and any m′ > m, the truncation homomorphism k[t]/(tm′)→ k[t]/(tm) induces a morphism ηm ′,m Z : (Z× jm,Fm)→ (Z× jm′ ,Fm ′ ) of pairs. Using this, we can define a mapping pim ′,m Z : hJm′ (X)(Z)→ hJm(X)(Z) by pulling back Z-valued points of Jm′(X) to Z-valued points of Jm(X) via η m′,m Z . That is, the Z-valued point γ̃ of Jm′(X) corresponds to a unique m ′-jet γ that we pull back with ηm ′,m Z to an m-jet. This m-jet corresponds uniquely to a Z-valued point of Jm(X) that will be the image pim ′,m Z (γ̃) of γ̃. We would like to show that these mappings on Z-valued points define a morphism from Jm′(X) → Jm(X) in the category Pairs. By Yoneda’s lemma 2.1 this is equivalent to the following fact, which we prove: Lemma 4.2. The mappings pim ′,m Z : hJm′ (X)(Z) → hJm(X)(Z) over all Z define a natural transformation pim ′,m between the functors hJm′ (X) and hJm(X). 17 Proof. Suppose that φ : Z → Y is a morphism of pairs. We must show that the following diagram commutes: hJm′ (X)(Z) ff hJm′ (X)(φ) hJm′ (X)(Y) hJm(X)(Z) pim ′,m Z ? ffhJm(X)(φ) hJm(X)(Y) pim ′,m Y ? This diagram will commute if and only if the next diagram commutes, since we have supposed that the pairs Jm(X) and Jm′(X) represent L X m and L X m′ respectively: HomPairs((Z × jm′ ,Fm′),X) ff(−)◦φ m′ HomPairs((Y × jm′ , Em′),X) HomPairs((Z × jm,Fm),X) (−)◦ηm′,mZ ? ff(−)◦φ m HomPairs((Y × jm, Em),X) (−)◦ηm′,mY ? Recall that φm : (Z × jm,Fm)→ (Y × jm, Em) is the morphism induced by φ, as we defined in section 3.2.2. Hence for any γ : (Y × jm′ , Em′)→ X, we must show that γ ◦ φm′ ◦ ηm′,mZ = γ ◦ ηm ′,m Y ◦ φm, which holds if φm ′ ◦ ηm′,mZ = ηm ′,m Y ◦φm. Now, both φm ′ ◦ ηm′,mZ and ηm ′,m Y ◦φm are morphisms from Z × jm to Y × jm′ , which are compatible with the projections Y × jm′ → Y and Y × jm′ → jm′ . Hence by the universal mapping property of the fibred product Y × jm′ , φm ′ ◦ ηm′,mZ = ηm ′,m Y ◦ φm. So pim ′,m uniquely determines a morphism, which we call the projection morphism from Jm′(X) to Jm(X), and denote pim′,m. We may denote pi m′,m by piX,m ′,m, and pim′,m by pi X m′,m to avoid confusion when working with more than one projection. We will also denote pim ′,0 by pim ′ , and pim′,0 by pim′ . Note, of course, that this implies that we may alternatively express the pair (JDm(X), Jm(D)) as (JDm(X), pi∗m(D)). Our next result will help to prove the second claim made at the beginning of this section. Proposition 4.3. Let V = (V,DV ) be an open subpair of the pair U = (U,DU). If there exists a pair Jm(U) representing L U m, then there exists a pair Jm(V) representing L V m and (piUm) −1(V) = Jm(V). Proof. We will show that for any Z = (Z,F)-valued m-jet γ : (Z × jm,Fm)→ U, γ factors through V if and only if the morphism γ̃ : Z→ Jm(U) corresponding to γ under the repre- sentation of LUm factors through (pi U m) −1(V). First of all, we may suppose that Z = SpecA is affine, given our proposition 4.1. Assume that γ factors through V. The truncation homo- morphism A[t]/(tm+1)→ A induces the morphism of pairs ηmZ : Z→ (SpecA[t]/(tm+1),Fm) with which we pull back γ to pim(γ). Note that, by definition, pulling back γ to pim(γ) yields 18 the same result as composing γ̃ with pim. Hence, the composition pim ◦ γ̃ factors through V . From this we see that γ̃ must factor through (piUm) −1(V). Conversely, let γ̃ be a Z-valued point of Jm(U) that factors through (pi U m) −1(V). Noting that this implies that pim ◦ γ̃ factors through V, we obtain the following commutative square: Spec A[t]/tm+1 γ - U Spec A ηmZ 6 pim(γ) - V immersion ∪ 6 - where γ is the m-jet corresponding to γ̃. We wish to show that γ factors through V. But this is true because the open immersion V → U is formally étale, this property ensuring us a scheme morphism SpecA[t]/tm+1 → V commuting with the square. Further, the pullback of DV by this morphism exists and equals Fm (we are just pulling back DU restricted to V). This result yields the local isomorphisms needed to glue together the jet pair Jm(X) of an arbitrary X from the jet pairs Jm(Xα), where {Xα : α ∈ A} is an open cover of X. Before we apply this result we will prove two more results. The second of these generalizes the statement of proposition 4.3 to the case of (formally) étale morphisms, hence provides another proof of proposition 4.3. Proposition 4.4. Let φ : Y → X be a morphism of pairs, and suppose that LYm and LXm are represented by Jm(Y) and Jm(X) respectively. Then φ induces a morphism φm : Jm(Y) → Jm(X) that commutes with the projections as in the following diagram: Jm(Y) φm- Jm(X) Y piYm ? φ - X piXm ? Proof. Similarly to the jet scheme case, we begin by choosing the Jm(Y)-valued point of Jm(Y) given by idJm(Y), which corresponds to ιY : (J E m(Y )× jm, Jm(E)m)→ Y. Composing ιY with φ corresponds to the Jm(Y)-valued point of Jm(X) that we denote φm. Then, because φ ◦ ιY and φm correspond to each other under the representation of LXm, we know that piXm ◦ φm = φ ◦ ιY ◦ ηmJm(Y) (the correspondence is trivial when m = 0). By the same reasoning piYm ◦ idJm(Y) = ιY ◦ ηmJm(Y), as ιY corresponds to idJm(Y). Thus, piXm ◦ φm = φ ◦ piYm ◦ idJm(Y) = φ ◦ piYm . So the diagram above commutes. Proposition 4.5. Let φ : Y → X be an étale morphism of pairs, and suppose that LYm and LXm are represented by Jm(Y) and Jm(X) respectively. Then, Jm(Y) ∼= Jm(X)×X Y. Proof. We will show that for any pair Z, and every commutative square Z eγ- Jm(X) Y ψ ? φ - X piXm ? 19 there exists a unique morphism γ̄ : Z→ Jm(Y) making the following diagram commutative: Z Jm(Y) φm- γ̄ - Jm(X) eγ - Y piYm ? φ - ψ - X piXm ? Now, piXm ◦ γ̃ = γ ◦ ηmZ , where γ is the jet corresponding to the point γ̃. Without loss of generality, assume that Z = (Spec A,F) is affine; the following diagram commutes: (Spec A,F) η m Z- (Spec A[t]/(tm+1),Fm) Y ψ ? φ - bγ ff X γ ? Since φ : Y → X is an étale morphism of schemes, γ factors through Y ; i.e. there is a unique “scheme-jet” γ̂ : Spec A[t]/(tm+1) → Y commuting with the square. This jet does indeed define a “pair-jet”, since the local equations of the pullback commute around the bottom triangle in the opposite direction, and since φ and γ pull back the divisors. This jet γ̂ corresponds to the Z-valued point we desire, γ̄. To verify that φm ◦ γ̄ = γ̃, note that the jet corresponding to φm ◦ γ̄ is φ ◦ γ̂. But this is exactly γ, hence γ̃ = φm ◦ γ̄. Similarly, piYm ◦ γ̄ = γ̂ ◦ ηmZ because γ̂ is the jet corresponding to γ̄. The latter equals ψ, so piYm ◦ γ̄ = ψ. Thus, the second diagram is commutative; we conclude that Jm(Y) ∼= Jm(X)×X Y. Now, let us suppose momentarily that given any affine pair Xα = (Xα,Dα) and any m > 0 the functor LXαm is represented by Jm(Xα). For X = (X,D) ∈ Pairs let {Xα : α ∈ A} be an open cover by affine pairs. Then, according to proposition 4.3, for every α and β such that Xα∩Xβ 6= ∅, both (piXαm )−1(Xα∩Xβ) and (piXβm )−1(Xα∩Xβ) yield the pair Jm(Xα∩Xβ) representing L Xα∩Xβ m ; that is, the preimages are canonically isomorphic to each other. These isomorphisms satisfy the conditions necessary to glue together a pair Jm(X) from the various Jm(Xα). We claim that the pair we obtain this way represents LXm. Indeed, letting Z = (Z,F), given any m-jet γ : (Z × jm,Fm) → (X,D), we can break up the jet into its restric- tions γ−1(Xα) → Xα. Then, we can break up the 0th truncation of γ into morphisms (ηmZ ) −1(γ−1(Xα)) = (γ ◦ ηmZ )−1(Xα) → Xα. Since (γ ◦ ηmZ )−1(Xα) is an open subpair of Z, we know that its preimage p−1Z ((γ ◦ ηmZ )−1(Xα)) under the projection pZ : (Z × jm,Fm)→ Z is isomorphic to ((γ ◦ ηmZ )−1(Xα) × jm,Fm|(γ◦ηmZ )−1(Xα)×jm). Thus we get a correspond- ing (γ ◦ ηmZ )−1(Xα)-valued point of Jm(Xα). Since these points must agree on overlaps Xα ∩Xβ 6= ∅ and the preimages (γ ◦ ηmZ )−1(Xα) cover Z, we obtain a unique Z-valued point of Jm(X) corresponding to γ by gluing the domain of these morphisms. Note that all the 20 divisors pulled back locally throughout, and that funtoriality follows from the fact that these correspondences are functorial locally on the pairs. We can now prove that LXm is a representable functor for every m > 0 when X is any pair. Theorem 4.6. Let X = (X,D) ∈ Pairs. For every m ≥ 0 the contravariant functor LXm : (Y, E) 7→ HomPairs((Y × jm, Em), (X,D)) from Pairs to Sets is representable, represented by a pair Jm(X) = (J D m(X), Jm(D)). Proof. The case m = 0 is trivial, so let m > 0. By our previous results we may restrict to the case X = (SpecA,D) is affine and the domain of LXm is Aff Pairs. So let (Y, E) = (SpecB, E) and let γ : (Y × jm, Em) → (X,D) be an m-jet in X. Thus γ : Spec B[t]/(tm+1) → Spec A has corresponding homomorphism γ∗ : A→ B[t]/(tm+1). We wish to describe a scheme with a B-valued point corresponding uniquely to γ. We will break up the remainder of the proof into two cases. Case 1. Assume that A = k[X1, . . . , Xn] is affine n-space for some n > 0, that r ≤ n, and that Di = {(Spec A,Xi)} for 1 ≤ i ≤ r. Then the homomorphism γ∗ is determined exactly by the values γ∗(X1), . . . , γ∗(Xn). Let γ∗(Xi) = bi0 + bi1t+ · · ·+ bimtm. Since Xi is the local equation for Di when 1 ≤ i ≤ r, γ∗(Xi) is the local equation for Emi . But this local equation is bi0, hence bi0 · ui = bi0 + bi1t + · · · + bimtm for some invertible regular section ui. Writing ui = ui0+ui1t+ · · ·+uimtm we see that ui0 = 1 and bi0 ·uil = bil for each 1 ≤ l ≤ m. Hence, the value γ∗(Xi) is determined by the values bi0, ui1, ui2, . . . , uim. For r + 1 ≤ i ≤ n, the value of γ∗(Xi) is simply determined by bi0, bi1, bi2, . . . , bim. Now let us write the affine coordinate ring in n(m+ 1) variables as C = k[X (0) 1 , . . . , X (0) n , X (1) 1 X1 , . . . , X (1) r Xr , X (1) r+1, . . . , X (1) n , . . . , X (m) 1 X1 , . . . , X (m) r Xr , X (m) r+1, . . . , X (m) n ]. Then γ∗ determines a unique homomorphism γ̃∗ : C → B sending X (0) i 7→ bi0, ∀1 ≤ i ≤ n, X (l) i Xi 7→ uil, ∀1 ≤ i ≤ r, 1 ≤ l ≤ m, and X (l) i 7→ bil, ∀r + 1 ≤ i ≤ n, 1 ≤ l ≤ m. Thus we let JDm(X) = Spec C, and we let Jm(D) = ({(Spec C,X1)}, . . . , {(Spec C,Xr)}). It is immediate that γ̃ : (SpecB, E)→ (JDm(X), Jm(D)) is indeed a morphism of pairs and that this correspondence is functorial; note that for any morphism φ : (Spec S,F)→ (SpecB, E) the homomorpism (φm)∗ maps B[t]/(tm+1)→ S[t]/(tm+1) such that b0 + b1t+ · · ·+ bmtm 7→ φ∗(b0) + φ∗(b1)t + · · · + φ∗(bm)tm. This guarantees functoriality, and so we see that LXm ∼= h(JDm(X),Jm(D)) when X = (A n k , ({(Ank , X1)}, . . . , {(Ank , Xr)})). 21 Case 2. Let A = k[X1, . . . , Xn]/(f1, . . . , fs) and suppose that for every i, 1 ≤ i ≤ r, Di is defined by gi on X, where gi ∈ k[X1, . . . , Xn]. A homomorphism from the polynomial ring in n+ r variables R = k[X1, . . . , Xn,W1, . . . ,Wr] to A sending each Xi to Xi is onto. Consider the homomorphism k[X1, . . . , Xn,W1, . . . ,Wr]→ k[X1, . . . , Xn]/(f1, . . . , fs) such that X1 7→ X1, . . . , Xn 7→ Xn and W1 7→ g1, . . . ,Wr 7→ gr. This makes X = Spec A into a closed immersion in SpecR = An+r cut out by the ideal I = (f1, . . . , fs,W1 − g1, . . . ,Wr − gr) (here is it important to notice that R/(f1, . . . , fs) ∼= A[W1, . . . ,Wr]). Under this k-algebra isomorphism R/I ↔ A, the local equations g1, . . . , gr map to W1, . . . ,Wr respectively, hence this isomorphism of schemes defines an isomorphism of pairs (SpecR/I, ({(SpecR/I,W1)}, . . . , {(SpecR/I,Wr)}))→ (X,D). Thus, in this case we will define the desired parameter space for X as a closed immersion in the parameter space for (An+r, ({(An+r,W1)}, . . . , {(An+r,Wr)})). By the arguments made in the first case this latter space is the pair consisting of the scheme A(n+r)(m+1) with the r-tuple of divisors defined on A(n+r)(m+1) by the W (0)i ’s. To find the equations for the ideal of JDm(X) we must consider m-jets in An+r that factor through X. A (SpecB, E)-valued m-jet in (X,D) is determined by a homomorphism γ∗ : k[X1, . . . , Xn,W1, . . . ,Wr]→ B[t]/(tm+1) such that γ∗(Wi) is the non-zerodivisor locally defining the “ith” effective Cartier divisor Emi of Spec B[t]/(tm+1), and such that γ∗(fj) = 0 and γ∗(Wi − gi) = 0 for every 1 ≤ j ≤ s and 1 ≤ i ≤ r. Such a homomorphism is completely determined by the coefficients of γ∗(X1), . . . , γ∗(Xn) and γ∗(W1) . . . , γ∗(Wr), hence γ∗ defines a homomorphism as we expect from k[X (0) 1 , . . . , X (0) n ,W (0) 1 , . . . ,W (0) r , X (1) 1 , . . . , X (1) n ,W (1) 1 , . . . ,W (1) r , . . . , X (m) 1 , . . . ,W (m) r ]→ B. Given the condition on pullbacks, just as in the first case the degree 0 coefficient of γ∗(Wi) divides the coefficients of the higher degree terms, hence the m-jet is equivalently determined by a homomorphism k[X (0) 1 , . . . , X (0) n ,W (0) 1 , . . . ,W (0) r , X (1) 1 , . . . , X (1) n , W (1) 1 W1 , . . . , W (1) r Wr , . . . , X (m) 1 , . . . , W (m) r Wr ]→ B. 22 Let us denote the domain of this homomorphism as S. We write γ∗(Xi) = bi0 + bi1t+ · · ·+ bimt m for 1 ≤ i ≤ n. Then γ∗(fj) = fj0 + fj1t + · · · + fjmtm for 1 ≤ j ≤ s, where for each 0 ≤ l′ ≤ m the coefficient fjl′ is a polynomial in (bil)1≤i≤n,1≤l≤m. Thus, we consider each fjl′ as a polynomial in (X (l) i )i,l; the condition γ ∗(fj) translates in terms of the homomorphism from S to B into fjl′ 7→ 0 for all 1 ≤ j ≤ s and 0 ≤ l′ ≤ m. Similarly, writing γ∗(Wi) = ci0+ci0·ui1t+· · ·+ci0·uimtm and γ∗(gi) = gi0+gi1t+· · ·+gimtm for 1 ≤ i ≤ r, the condition on γ∗ indicates that ci0 = gi0 and ci0 · uil′ = gil′ for every 1 ≤ l′ ≤ m. This time considering gil′ as a polynomial in (X(l)i )1≤i≤n,1≤l≤m, we must have gil′ −W (0)i · W (l′) i W (0) i 7→ 0 for all 1 ≤ i ≤ r and 0 ≤ l′ ≤ m. Hence, the (SpecB, E)-valued m-jets on (X,D) are parametrized by points in the closed immersion of schemes Spec S/(fjl, gi0 −W (0)i , gil′ −W (0)i · W (l′) i W (0) i : 1 ≤ j ≤ s, 0 ≤ l ≤ m, 1 ≤ i ≤ r, 1 ≤ l′ ≤ m) in A(n+r)(m+1). Thus, for every Y-valued point of Jm(An+r) corresponding to an m-jet that factors through X, its underlying scheme morphism factors through the closed immersion we have just described. However, the equations Wi on A(n+r)(m+1) may not pull back to non-zerodivisors in this closed immersion. In order to obtain the pair (JDm(X), Jm(D)) repre- senting LXm we may need to remove components, as in lemma 2.2; it is immediate that doing this yields a pair with the appropriate functor of points. We make the following definition: Definition 4.1. We call the pair Jm(X) associated to X from theorem 4.6 the jet pair associated to the pair X. We refer to the scheme JDm(X) underlying Jm(X) as the logarithmic jet scheme of X with respect to D. Remark 4.1. Let (X,D) be a pair over the field k, and suppose that char(k) = 0. We can describe the equations of the ideal of JDm(X) more explicitly. First, let S be the ring S = k[X (0) 1 , . . . ,W (0) r , X (1) 1 , . . . , W (1) r Wr , . . . , X (m) 1 , . . . , W (m) r Wr ]. As in the jet scheme case outlined in the introduction, there is a k-derivation d on S deter- mined as follows: dX (l) i = X (l+1) i where X (l) i = 0 for all l > m, and dW (l) i = W (l+1) i where W (l) i def = W (0) i · W (l) i Wi for all 1 ≤ l ≤ m and W (l)i = 0 for all l > m. By the same arguments as in the jet scheme case we find that the equations fjl ′ all map to 0 if and only if the equations dl ′ fj all map to 0, where we consider fj as a polynomial in X (0) 1 , . . . , X (0) n . Similarly, since dl ′ (gi −W (0)i ) = dl′gi − dl′W (0)i we find that gi0 −W (0)i and gil′ −W (0)i all map to 0 if and only if the equations dl ′ (gi −W (0)i ) all map to 0. Thus we have an equality of ideals (fjl′ , gi0−W (0)i , gil′−W (0)i · W (l′) i W (0) i ) = (fj, dfj, . . . , d mfj, gi−W (0)i , d(gi−W (0)i ), . . . , dm(gi−W (0)i )). 23 5 Conclusion 5.1 Summary Following the constructive method of providing a proof for the existence of jet schemes Jm(X) associated to a scheme X of finite type over an algebraically closed field k as in [EM08], [Mus01], [Ish07], we have provided a constructive proof of the existence of logarithmic jet schemes JDm(X) associated to X and its family of effective Cartier divisors D = (D1, . . . , Dr). This was carried out in four major steps as follows: after providing basic definitions for the objects we would work with throughout our paper, we first formulated the categories of pairs Pairsr, whose objects (X,D) consist of a scheme X of finite type over a fixed algebraically closed field k and its r-tuple of effective Cartier divisors, and whose morphisms (X,D) → (Y, E) are those scheme morphisms φ : Y → X that “pull back” D to E ; second, we defined the functors L (X,D) m taking a “pair” (Y, E) to the set of “(Y, E)-valued m-jets in (X,D)”; third, we proved that the representability of such functors can be determined by the case of affine (X,D) and (Y, E); finally, we gave explicit equations for a pair representing the functor L (X,D) m . The question of representability of such functors, or equivalently of parametrizability of such families of m-jets, was motivated by the construction of the sheaf of differential 1-forms with logarithmic poles along a normally crossing divisor on a complex-analytic variety, and the possibility of framing such a construction functorially, as the sheaf of differetial 1-forms finds expression in jet schemes. 5.2 Discussion and Further Research Referring back to the definition 2.2.3, notice that a predivisor on the scheme X is, by definition, a presentation of a global section of the quotient sheaf of commutative monoids OX/O∗X . It is straightforward to define the pullback φ∗(D) of a global section D of this sheaf by a morphism Y → X of schemes (of finite type over k) as we have done for effective Cartier divisors. Let us refer to such an object D temporarily as an effective divisor. With this notion at hand, we might choose to work in a category whose objects are pairs (X,D), where now D = (D1, . . . , Dr) is an r-tuple of effective divisors on X, and whose morphisms (Y, E) φ→ (X,D) pull back D to φ∗(D) def= (φ∗(D1), . . . , φ∗(Dr)) = E . One may verify that the proofs supplied in section 4 carry over to this category word-for-word, with the exception of omitting some justifications that certain pullbacks of sections do not locally divide zero. The geometric significance in this choice of a category lies in that rather than only parametrizing jets that “avoid” the family D of effective Cartier divisors, we parametrize also the jets that are “tangential along” the family D of effective divisors. For example, letting D consist of the single effective Cartier divisor defined on X = A2 globally by X1 ·X2, one shows in the first case that the fibre of the projection pi1 : J D 1 (A2)→ A2 above the origin (0, 0) (or above any point on the X1 or X2 axis) is empty, whereas in the second case we have pi−11 ((0, 0)) ∼= A2 (while pi−11 ((a, b)) ∼= A1 for any (a, b) with a = 0, b 6= 0 or a 6= 0, b = 0). 24 Though this adjustment to the categories Pairsr immediately yields some interesting geometric objects, it is likely that there is an even better category in which to formulate our results and construct such objects. Namely, we expect that the natural context for the logarithmic jet schemes lies in the category of “schemes with logarithmic structures”, on which foundational material was developed by Fontaine-Illusie and Kato (see for example [Kat94] and especially [Kat89]). In particular it seems that this formalism supplies a language for working with (sheaves of) monoids attached to schemes, and will hopefully carry over the idea we have just mentioned. Once the transfer to this language is complete, we hope in particular to apply the geometry of the logarithmic jet schemes to the study of singularities; we allude in particular to such results as contained in the work of Mustaţǎ in [Mus01]. Of course, the jet schemes are fundamental to the theory of motivic integration, and we also hope to study the logarithmic jets in this context. 25 References [Del71] Pierre Deligne, Théorie de Hodge : II, Publications mathématiques de l’I.H.É.S. 40 (1971), 5 – 57. [EH00] David Eisenbud and Joe Harris, The Geometry of Schemes, GTM, vol. 197, Springer-Verlag New York, Inc., 2000. [Eis04] David Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, GTM, vol. 150, Springer Science+Business Media, Inc., 2004. [EM08] Lawrence Ein and Mircea Mustaţǎ, Jet Schemes and Singularities, arXiv:math/0612862v2 [math.AG], 2008. [FmI+05] Barbara Fantechi, Lothar Göttsche, Luc Illusie, Steven L. Kleiman, Nitin Nit- sure, and Angelo Vistoli, Fundamental Algebraic Geometry: Grothendieck’s FGA Explained, Mathematical Surveys and Monographs, vol. 123, The American Math- ematical Society, 2005. [GH94] Philip Griffiths and Joe Harris, Principles of Algebraic Geometry, Wiley Classics Library, John Wiley & Sons, Inc., 1994. [Gro67] Alexander Grothendieck, Éléments de géométrie algébrique (rédigés avec la collab- oration de Jean Dieudonné) : IV. Étude locale des schémas et des morphismes de schémas, Quatriéme partie, Publications mathématiques de l’I.H.É.S. 32 (1967), 5 – 361. [Har06] Robin Hartshorne, Algebraic Geometry, GTM, vol. 52, Springer Science+Business Media, LLC, 2006. [Ish07] Shihoko Ishii, Jet Schemes, Arc Spaces and the Nash Problem, arXiv:0704.3327v1 [math.AG], 2007. [Kat89] Kazuya Kato, Logarithmic Structures of Fontaine-Illusie, Algebraic Analysis, Ge- ometry, and Number Theory (J. Igusa, ed.), Johns Hopkins University Press, 1989, pp. 191 – 224. [Kat94] , Toric singularities, American Journal of Mathematics 116 (1994), no. 5, 1073 – 1099. [Mus01] Mircea Mustaţǎ, Jet Schemes of Locally Complete Intersection Canonical Singu- larities, Inventiones Mathematicae 145 (2001), no. 3, 397 – 424. 26