Using Stacks to Impose Tangency Conditions on Curves

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1 Using Stacks to Impose Tangency Conditions on Curves arxiv:math/ v3 [math.ag] 5 Jul 2005 Charles Cadman Abstract We define a Deligne-Mumford stack X D,r which depends on a scheme X, an effective Cartier divisor D X, and a positive integer r. Then we show that the Abramovich-Vistoli moduli stack of stable maps into X D,r provides compactifications of the locally closed substacks of Mg,n (X,β) corresponding to relative stable maps. 1 Introduction The purpose of this paper is twofold. The first is to introduce a construction which takes as input a scheme X, an effective Cartier divisor D, and a positive integer r and produces a Deligne-Mumford stack X D,r. The only hypotheses required are that X is Noetherian and that r is invertible on X. The second is to compare twisted stable maps into X D,r with ordinary stable maps into X. Here we assume additionally that X is projective over a field containing all rth roots of unity. The theory of twisted stable maps has been developed by Chen and Ruan [CR] in the symplectic category and by Abramovich and Vistoli [AV] in the algebraic category. The latter showed that there is a Deligne-Mumford stack K g,n (X, β) parametrizing morphisms f : C X from twisted n-marked nodal curves C into a Deligne-Mumford stack X such that f [C] = β. In the case where X is a scheme, this is just the space of Kontsevich stable maps M g,n (X, β). The theory of relative stable maps has been developed by several symplectic geometers, and the algebraic definition was worked out partially by Andreas Gathmann [Ga] and in general by Jun Li [Li]. Let X be a scheme, D X an effective Cartier divisor, and β N 1 (X) such that D β > 0. Choose a partition D β = n, where each i is a nonnegative integer. Then a morphism f : C X from a smooth curve C having marked points x 1,...,x n such that f [C] = β and f D = ix i is a relative stable map. The space of such morphisms was compactified by [Ga] and [Li] and they defined relative Gromov-Witten invariants. In the genus 0 case, Gathmann defined the space of relative stable maps by taking the closure in M g,n (X, β) of the maps f : C X above. In this paper, we show that there is an open substack U g,n (X D,r, β, ) K g,n (X D,r, β) mapping isomorphically onto the locally closed substack of Mg,n (X, β) consisting of those maps. Here r is any integer larger than 1

2 every i. This provides evidence that relative Gromov-Witten invariants may be equal to Gromov-Witten invariants of stacks. One application of this idea is to count curves in the plane having certain tangency conditions with respect to a smooth cubic. If D P 2 C is the cubic, then the Gromov-Witten invariants of P 2 D,2 can be computed using the WDVV equations, as in [Ca1]. The invariants can then be used to count rational curves having a certain number of order 1 and 2 contacts with D at both specified and unspecified points [Ca2]. Section 2 is a very technical section in which the stack X D,r is introduced and shown to be a Deligne-Mumford stack. We also construct the families of smooth twisted curves C D, r S which are the source curves for twisted stable maps into X D,r. To classify these morphisms C D, r X D,r, we determine the coarse moduli scheme of C D, r and work out some results regarding invertible sheaves on C D, r. The reader is invited to skip to section 3 and refer to section 2 when needed. Here we give several examples of this construction which hopefully provide some intuition. We also give an important preliminary result relating morphisms C D, r X D,r to morphisms C X which have certain contact conditions with respect to D. Finally in section 4, we prove the main result which shows that there are isomorphisms between open substacks of K g,n (X, β) and locally closed substacks of Mg,n (X, β). These latter substacks are defined in terms of contact conditions with respect to D. Section 5 contains some important lemmas which are probably known. Notation and conventions By a family of curves, we mean a morphism of schemes C S which is flat and proper and whose geometric fibers are connected curves having at worst nodal singularities. Since S will always be Noetherian, we don t need the locally finite presentation hypothesis. To work in complete generality, we would have to allow C to be an algebraic space. However, since we are dealing with stable maps to projective schemes, it will always be the case that C is projective over S, and in particular a scheme. We use µ r to denote the scheme Spec Z[x]/(x r 1) with group structure (x, y) xy. In commutative diagrams, we use the label can. for arrows which are canonical isomorphisms. Acknowledgements This paper was derived from part of my Ph.D. thesis at Columbia University under the supervision of Michael Thaddeus. It has benefited from discussions with many people over the past few years, including Dan Abramovich, Linda Chen, Brian Conrad, William Fulton, Max Lieblich, Mircea Mustata, Rahul Pandharipande, Greg Smith, and Michael Thaddeus. This research was partially conducted during the period the author was employed by the Clay Mathematics Institute as a Liftoff Fellow. 2 The rth root construction 2.1 Application to a scheme Throughout this subsection, we fix a Noetherian scheme X, an invertible sheaf L on X, a global section s of L, and a positive integer r. We assume that the morphism X Spec Z 2

3 factors through Spec Z[r 1 ]. For example, X could be any scheme of finite type over C. We begin by introducing X (L,s,r), defining it as a category and then showing it is a Deligne-Mumford stack. In Definition 2.3, we introduce X D,r as a special case. The more general construction is useful because it is stable under base change (Proposition 2.4). Definition 2.1 The stack of rth roots of the pair (L, s) is denoted X (L,s,r), and has the following objects and morphisms. An object over a scheme S is a quadruple (f, M, t, ϕ), where f : S X is a morphism, M is an invertible sheaf on S, t is a global section of M, and ϕ : M r f L is an isomorphism such that ϕ(t r ) = f s. A morphism from (f, M, t, ϕ) (over S) to (g, N, u, ψ) (over T ) is a pair (h, ρ), where h : S T is a morphism such that g h = f, and ρ : M h N is an isomorphism such that ρ(t) = h u and the following diagram commutes. M r ρ r h N r ϕ h ψ f L = can. h g L Morphisms are composed in the obvious way. Remark Though we allow r to equal 1, it is easy to verify that X (L,s,1) = X. For notational simplicity, we will let r = 1 in our examples. Theorem 2.2 X (L,s,r) is a Deligne-Mumford stack. Proof: We prove this by going through the axioms. Existence and uniqueness of pullbacks. Given an object (f, M, t, ϕ) over a scheme S and a morphism h : T S, the object (f h, h M, h t, h ϕ) together with the morphism (h, id h M) defines a pullback. Given another object (g, N, u, ψ) over T and a morphism (h, ρ) to (f, M, t, ϕ), we must have g = f h, so (id T, ρ) defines an isomorphism between (g, N, u, ψ) and (g, h M, h t, h ψ) over id T. Moreover, this is the only isomorphism such that (h, id h M) (id T, ρ) = (h, ρ). Descent axiom. This follows from descent for morphisms to a scheme, descent for invertible sheaves, and descent for morphisms of invertible sheaves. Representability of Isom functors. For this axiom, one must show that given a scheme U and two objects x and y of X (L,s,r) over U, the functor Isom(x, y), which sends to an arbitrary U -scheme S the set of isomorphisms x S y S lying over id S, is representable by a U -scheme I x,y which is separated and quasi-compact over U. Given U, let x = (f, M, t, ϕ) and y = (g, N, u, ψ) be two objects. The objects of Isom(x, y) over a scheme S are pairs (h, ρ) where h : S U is a morphism such that f h = g h and ρ : h M h N is an isomorphism such that ρ(h t) = h (u) and h ϕ = h ψ ρ r. Let V = U X X X, where U X X is given by (f, g) and X X X is the diagonal. We will define a geometric line bundle E over V, and I x,y will be a closed subscheme of E, the complement of the zero section of E. 3

4 I x,y E π j V U X X X Let E be the geometric line bundle associated to (j M) 1 j N. Then E represents the functor of pairs (h, ρ) where h : S U is a morphism such that f h = g h and ρ : h M h N is an arbitrary isomorphism. The identity map E E corresponds to a universal such pair (π, σ). We define I x,y E to be the closed subscheme defined by the vanishing of the two sections and σ(π t) π u Γ(E, π N) σ r π ϕ π ψ 1 Γ(E, (π M 1 π N) r ). The morphisms to I x,y are precisely the pairs (h, ρ) satisfying the conditions we needed. The morphism V U is separated and quasi-compact because X X X is, and the morphism I x,y V has these properties because it is affine. This completes the proof of this axiom. Existence of an étale surjective morphism. Let g : Y X be an étale, surjective morphism, let N be an invertible sheaf on Y, and let ψ : N r g L be an isomorphism. For example, Y could be obtained by covering X by finitely many open affines on which L is trivial. Let U represent the functor of pairs (f, u), where f : S Y is a morphism and u Γ(S, f N) is a section such that f ψ(u r ) = f g s. For example, if π : E Y is the geometric line bundle associated to N, then E classifies pairs (f, u) with no condition on u, and U could be a closed subscheme of E defined by the vanishing of a section of π g L. Let U X (L,s,r) be the morphism given by (f, u) (g f, f N, u, f ψ), and let T X (L,s,r) be an arbitrary morphism given by an object (h, M, t, ϕ) over T. We must show that the projection P := U X(L,s,r) T T is étale and surjective. The objects of P over a scheme S are quadruples (f, u, j, ρ), where the following hold. 1. f : S Y is a morphism. 2. u Γ(S, f N) such that f ψ(u r ) = f g s. 3. j : S T is a morphism such that g f = h j. 4

5 4. ρ : j M f N is an isomorphism such that ρ(j t) = u and the following diagram commutes. (j M) r ρ r (f N) r j ϕ f ψ j h L = can. f g L Thus u is determined by ρ, and the pair (f, j) defines a morphism to Y X T. The second condition on ρ implies that P is the total space of the µ r torsor over Y X T associated to the invertible sheaf p 1 N (p 2 M) 1 and the isomorphism of its rth tensor power with the structure sheaf given by the two vertical arrows in the diagram above. µ r P Y X T Y T X It follows that P T is étale and surjective. Since this holds for any morphism T X (L,s,r), it follows that U X (L,s,r) is étale and surjective. Remark The third axiom is usually stated as the condition that the diagonal morphism X (L,s,r) X (L,s,r) X (L,s,r) is representable, separated, and quasi-compact. Note that for a suitable choice of Y, the étale surjective morphism U X (L,s,r) is of finite type and U is Noetherian. We are mainly interested in the case where (L, s) is associated to a divisor. Definition 2.3 Let D X be an effective Cartier divisor. We define the rth root of X along D to be the Deligne-Mumford stack X (OX (D),s D,r). Here s D is the tautological section of O X (D) which vanishes along D. We denote this stack by X D,r. 2.2 Application to a stack We begin with a useful observation. Let A be the CFG (category fibered in groupoids) whose objects over a scheme S are pairs (L, s), where L is an invertible sheaf on S and s is a global section of L. Given a positive integer r, there is a morphism M r : A A which sends a pair (L, s) to the pair (L r, s r ). If we are given such a pair (L, s) on a scheme X, then X (L,s,r) is isomorphic to X (L,s),A,Mr A. This implies a convenient base change property of the rth root construction. Proposition 2.4 If f : Y X is a morphism of schemes, and we are given (L, s) on X, then Y X X (L,s,r) = Y(f L,f s,r). 5

6 The CFG A is isomorphic to [A 1 /G m ], where G m acts on A 1 via the standard action. Indeed, an object of [A 1 /G m ] over a scheme S is a pair (P, f), where P is a G m torsor over S and f : P A 1 is a G m -equivariant map. This is equivalent to having the invertible sheaf L corresponding to P together with a global section of L. Definition 2.5 Let X be a Deligne-Mumford stack, let L be an invertible sheaf on X, let s be a global section of L, and let r be a positive integer. We define X (L,s,r) to be the (a priori) Artin stack X (L,s),A,Mr A. In the next subsection, we show how to get an étale presentation for X (L,s,r) given one for X. It follows from that construction that if X is a Deligne-Mumford stack which admits an étale, finite type, surjective morphism from a Noetherian scheme Y X, then the same holds for X (L,s,r). In particular, this holds for the stack X D, r defined as follows. Definition 2.6 Let X be a Noetherian scheme, let D = (D 1,...,D n ) be an n-tuple of effective Cartier divisors D i X, and let r = (r 1,...,r n ) be an n-tuple of positive integers which are invertible on X. We define X D, r to be the n-fold fiber product X D1,r 1 X X X Dn,r n. We could also have used Definition 2.5 and iterated the rth root construction. If we let X 0 = X and let X i = (X i 1 ) (Li,s i,r i ), where (L i, s i ) is the pullback of (O X (D i ), s Di ) to X i 1, then X D, r = Xn. Example 2.7 This is an important example for our main result. Let π : C S be a family of curves over a Noetherian scheme S and suppose we have an n-tuple D of disjoint Cartier divisors D i C which map isomorphically to S. Given an n-tuple of positive integers r = (r 1,...,r n ), we obtain a stack C := C D, r over S. This stack turns out to be a family of twisted n-pointed curves over S in the sense of [AV]. See Theorem 4.1. We need to classify morphisms C X D,r, where X D,r is another stack obtained by the rth root construction. Such a morphism is equivalent to a quadruple (f, M, t, ϕ), where f : C X is a morphism, M is an invertible sheaf on C, t is a global section of M, and ϕ : M r f L is an isomorphism sending t r to f s D. So we need to classify morphisms from C to schemes and invertible sheaves on C. Since C D, r is connected to C by a string of rth root constructions, it is enough to compare morphisms to schemes and invertible sheaves on a stack X with those on X (L,s,r). This is what we do for the remainder of the section. 2.3 Étale groupoid presentation Let X be a Deligne-Mumford stack and let L, s, and r be as in Definition 2.5. In this subsection we construct an étale groupoid presentation for X (L,s,r) given a nice enough presentation for X. 6

7 Let g : Y X be an étale surjective morphism, and assume that there is an invertible sheaf N on Y and an isomorphism σ : N r g L. For example, given any étale, finite type, surjective morphism V X from a Noetherian scheme V, we could construct Y by covering V by finitely many open affines such that the restriction of L V to each affine is trivial. We denote g L by L Y and g s by s Y. Let U represent the functor of pairs (f, u), where f : S Y is a morphism and u is a global section of f N such that f σ(u r ) = f s Y. There is a morphism U X (L,s,r) given by (f, u) (g f, f N, u, f σ). The proof that this is étale and surjective goes through as in the proof of Theorem 2.2. Let W = Y X Y, let p i : W Y, i = 1, 2, be the projections, and let N i = p i N. Then R := U X(L,s,r) U classifies quadruples (h, u 1, u 2, ρ), where h : S W is a morphism, u i is a global section of h N i, and ρ : h N 1 h N 2 is an isomorphism such that ρ(u 1 ) = u 2 and the following diagram commutes. h N r 1 ρ r h N r 2 (2.1) h p 1 σ h p 1 L Y h p 2 σ h p 2 L Y The bottom arrow is the isomorphism determined by the 2-morphism g p 1 g p 2. A scheme U classifying pairs (f, u) can be constructed explicitly as U = Spec (O Y N 1 N r+1 ), where the O Y -algebra structure comes from the composition N r L 1 Y O Y induced by σ and s Y. Moreover, R can be realized in two different ways as the spectrum of a sheaf of O W -algebras, which correspond to choosing either u 1 or u 2 to be the independent variable: R = Spec ( 0 i,j r 1 N i k (N 1 1 N 2 ) j ) for k = 1, 2. Here the O W -algebra structure is induced by the morphisms N r k O W and (N1 1 N 2 ) r = O W. The isomorphism between these two realizations of R is induced by the isomorphisms N1 i (N1 1 N 2 ) j = N i 2 (N1 1 N 2 ) i j. (2.2) We denote the sheaf of O Y -algebras defining U by A U, and either one of the sheaves of O W -algebras defining R by A R. Since U classifies pairs (f, u), the identity map U U corresponds to such a pair which we denote ( f, ū). Likewise, we have a quadruple ( h, ū 1, ū 2, ρ) on R. 2.4 Categorical quotient Recall that a scheme C is a categorical quotient for a groupoid in schemes A B if there is a morphism B C whose pullbacks to A agree such that for all schemes S and morphisms B S whose pullbacks to A agree, there is a unique morphism C S extending B S. The significance of this notion for us is that if C is a stack presented by the groupoid A B, then for all schemes S there is a natural bijection Mor(C, S) Mor(C, S). 7

8 Proposition 2.8 A categorical quotient for W Y is a categorical quotient for R U. Proof: We have a pair of commutative diagrams. s R h W U t p 1 p 2 Y f Let S be a scheme. If Y S is a morphism whose pullbacks to W agree, then by composition we obtain a morphism U S whose pullbacks to R agree. We need to find an inverse. Let j : U S be a morphism whose pullbacks to R agree. First we show that there is a continuous map ı : Y S induced by j. Since U Y is finite, it is closed, so it is enough to show that j is constant on the geometric fibers of U Y. Let y Y be a geometric point, and choose the point w = (y, y, id) of W lying over y. Let U y and R w be the fibers of U and R over y and w. If s Y vanishes at y, then U y consists of a single point and there is nothing to check. Otherwise, U y consists of r points corresponding to the rth roots of s Y y. Moreover, R w consists of r 2 points corresponding to pairs (u 1, u 2 ) of rth roots of s Y y. The morphisms s and t correspond to projections onto the two factors. Therefore, the condition that j s = j t implies that j is constant on U y. We have a morphism of sheaves of rings on S, j # : O S ı ( OY N 1 N r+1). To show that j induces a morphism Y S, it suffices to show that the image of j # is contained in the first summand, since the local homomorphism condition follows. Since j s = j t, it follows that ı p 1 = ı p 2 =: ĩ, and the morphism R S induces two morphisms ( O S ĩ 0 i,j r 1 N i k (N 1 1 N 2 ) j ), with k = 1, 2. These morphisms are related by the isomorphism of (2.2), and since they are induced by j #, their images are contained in summands having j = 0. It follows by (2.2) that i must also be 0, so the image of O S is contained in ĩ O W. It follows that the image of j # is contained in O Y, which is what we needed to check. It is clear from the construction that this gives a bijection between morphisms Y S whose pullbacks to W agree and morphisms U S whose pullbacks to R agree. It follows that if either categorical quotient exists, then both exist and are the same. Recall the stack C = C D, r of Example 2.7. The proposition implies that for any scheme X, any morphism C X comes from a unique morphism C X by composing with the projection C C. It is easy to check that for any algebraically closed field K, objects of C over Spec K modulo isomorphism are the same as morphisms Spec K C. This shows that C is the coarse moduli scheme of C. 8

9 It follows that C is the coarse moduli space for C. Indeed, in the proof of Theorem 2.2, I x,y is finite over V, which implies that C is separated over C. In this situation, it was shown by Keel and Mori that C has a coarse moduli space which is separated over C [LMB, 19.1], so it follows by [LMB, A.2] that the coarse moduli space is a scheme, and hence isomorphic to C. 2.5 Picard group Now we assume that R, U, W, and Y are Noetherian, and that s, t, p 1, and p 2 are morphisms of finite type. Recall that an invertible sheaf on a groupoid A B (with source and target maps s, t : A B and multiplication m : A t,b,s A A) is a pair (M, ϕ), where M is an invertible sheaf on B and ϕ : s M t M is an isomorphism satisfying the cocycle condition m ϕ = π 2ϕ π 1ϕ, where π i : A B A A are the projections. m ϕ m s M = can. π1s M π1 ϕ π1 t M = can. m t M = can. π2t M π π2 2 ϕ s M A morphism between invertible sheaves (M, ρ) and (N, σ) is a morphism γ : M N which makes the following diagram commute. s M ρ t M s γ t γ s N σ t N (2.3) Tensor products of invertible sheaves are defined in the obvious way, so we have a Picard group Pic(A B) associated to a groupoid A B. If A B is an étale presentation for a Deligne-Mumford stack C, then an invertible sheaf on A B is equivalent to one on C, and the same goes for morphisms and tensor product. Example 2.9 There is an invertible sheaf on R U given by the pair ( f N, ρ). The reader may check that this satisfies the cocycle condition. We denote this sheaf by T. As a sheaf on X (L,s,r), T corresponds to the functor (x, M, t, ϕ) M. Moreover, there is a morphism from the trivial sheaf (O U, id OR ) to T given by the section ū, which corresponds to the functor (s, M, t, ϕ) t. We call this section of T the tautological section. We clearly have a pullback homomorphism Pic(W Y ) Pic(R U), and we aim to use this to determine Pic(R U) given Pic(W Y ). It turns out that the remaining 9

10 generators of Pic(R U) correspond to connected components of the closed substack of X defined by s. For our purposes, it is sufficient to assume that there is only one such component. In terms of the groupoid W Y, this condition can be stated as follows. Let Z Y be the vanishing locus of s Y. Then p 1Z = p 2Z, so we have a subgroupoid p 1Z Z. We say that a groupoid A B is connected if for any two connected components C and D of B, there is a chain B 1,...,B n 1 of connected components of B such that p 1 B i 1 p 2 B i for i = 1,..., n, where we have set B 0 = C and B n = D. By abuse of notation, we use L to denote the invertible sheaf on W Y which corresponds to L. Lemma 2.10 Assume that the groupoid p 1Z Z is connected and nonempty. Then for any invertible sheaf (N, ϕ) on R U, there is an invertible sheaf (M, φ) on W Y and an integer n with 0 n r 1 such that (N, ϕ) = ( f M, h φ) T n. Moreover, n is unique and (M, φ) is unique up to isomorphism. Proof: Given (N, ϕ), let E = f N, which is a locally free sheaf of rank r since f is flat and finite of degree r. It follows from [Ei, Exercise 4.13] that as a sheaf of A U -modules on Y, E is locally isomorphic to A U. The A U -module structure gives us a multiplication map m : N 1 E E such that the r-fold composition factors into the natural maps N r E L 1 Y E O Y E. To simplify notation, let P = N 1 N2 1 and let E k = (p k E) ( r 1 i=0 P i ) for k = 1, 2. Since s, t : R U factor into R W Y U U, where the first arrow is an étale morphism associated to the pullback of P to W Y U, it follows that E 1 = h s N and E 2 = h t N since the direct image commutes with flat base change. The A R -module structures on E 1 and E 2 give us morphisms N 1 k E k E k, pulled back from m, and P E k E k, which amounts to rearranging the P i summands using the given isomorphism P r = O W. Therefore, ϕ induces an isomorphism ϕ : E 1 E 2 such that the following three diagrams commute. The first is a diagram of sheaves on W p2,y,p 1 W which expresses the cocyle condition. We let π k : W Y W W be the projections, m : W Y W W be the multiplication, δ k : W Y W = Y X Y X Y Y be the three projections for k = 1, 2, 3, and P k = πk P. (δ 1 E) ( P i 1 P i 2 ) m A R A R U R (δ 1 E) ( P i 1 ) π 1 A R A R U R π 1 ϕ (δ 2 E) ( P i 1 ) π 1 A R A R U R = m ϕ (δ 3 E) ( P i 1 P i 2 ) m A R A R U R = (δ 3 E) ( P i 2 ) π 2 A R A R U R π 2 ϕ = (δ 2 E) ( P i 2 ) π 2 A R A R U R (2.4) 10

11 The remaining two diagrams say that ϕ is a morphism of A R -modules. N 1 1 E 1 p 1 m ϕ E 1 ϕ P 1 N 1 2 E 2 p 2 m P 1 E 2 E 2 P E 1 E 1 (2.5) (2.6) P E 2 E 2 In order to apply this information, we pull everything back to Y via the diagonal morphism : Y W. Since P is canonically trivial, this gives us an isomorphism ψ : E r E r, which we view as r 2 endomorphisms of E. Diagram 2.6 tells us that many of these are redundant. It is enough to consider the r endomorphisms ψ i : E E obtained by composing ψ with the embedding E E r onto the 0th summand and the projection E r E onto the ith summand (0 i r 1). We count subscripts of ψ modulo r. Diagram 2.5 implies that the following diagram commutes and diagram 2.4 implies that N 1 E ψ i+1 N 1 E m m E E ψ i 0 if i j ψ j ψ i = if i = j This last equation implies that each ψ i projects E onto a locally free subsheaf F i E such that F i F j = 0 for i j. Since ψ is an isomorphism, it follows that E is spanned by the F i s. So we have a decomposition ψ i E = F 0 F 1 F r 1 (2.7) and by the above diagram, m sends N 1 F i+1 to F i. Here we count subscripts of F modulo r. Now we use the fact that E is locally isomorphic to A U as an A U -module. This implies that each F i is an invertible sheaf, since otherwise the image of m could have rank at most r 2. Choose an affine open set Spec S Y on which all the relevant sheaves are trivial. Then multiplication by a nonvanishing section of N 1 on E and A U respectively are given by matrices of the following forms β r β β β r s Y

12 Since the isomorphism between E and A U conjugates one matrix into the other, the determinants are equal and the ranks at each fiber are equal. It follows that away from Z, each map m i : N 1 F i+1 F i is an isomorphism, while at every point of Z, precisely one of the maps m i is not an isomorphism. This gives a locally constant function ı : Z Z associating to every point the integer 0 i r 1 for which m i is not an isomorphism. We show that this function is globally constant. Claim 1 ϕ maps p 1F i P j isomorphically onto p 2F i P i+j for all i and j. Since diagram 2.6 commutes, it is enough to show this for j = 0. We have already seen that this holds after pulling back to Y via. Let j : W W Y W be the morphism satisfying π 1 j = p 1 and π 2 j = id W. After pulling back diagram 2.4 to W via j, it follows that any local section of p 1 F i has to map to p 2 F i P i. Since ϕ is an isomorphism, the claim follows. Since p 1Z Z is connected, to show that ı is globally constant it suffices to show that for any z 1, z 2 Z such that there is a w W with p k (w) = z k, we have ı(z 1 ) = ı(z 2 ). But this follows from the claim along with diagram 2.5. We call this global constant n. It now follows that E = F n A U = F0 N n A U. Moreover, it follows from the claim that ψ makes F 0 into an invertible sheaf on W Y. Letting M = F 0, and φ be the isomorphism p 1 F 0 p 2 F 0 induced by ψ, we claim that (N, ϕ) = ( f M, h φ) T n. (2.8) First we work everything out for the sheaf T = ( f N, ρ). In this case, E T := f f N = N A U and the isomorphism p 1 N ( i,jn i 1 P j ) p 2 N ( i,jn i 2 P j ) is induced by the isomorphism N 1 N 2 P, and therefore the corresponding decomposition E T = G 0 G r 1 has G 1 = N and G 0 = O Y. The isomorphism (2.8) now follows from Claim 2. Note that A U is a Z/(r)-graded sheaf of O Y -algebras by giving N i the grading i, and that for any invertible sheaf (L, µ) on R U, f L is naturally a Z/(r)-graded sheaf of A U -modules using the decomposition of f L induced by µ as in (2.7). Claim 2 If (L, µ) and (M, ν) are two invertible sheaves on R U, then a morphism from (L, µ) to (N, ν) is equivalent to a morphism α : f L f M of Z/(r)-graded sheaves of A U -modules such that for each i, the following diagram commutes. p 1 ( f L) i p 1 α i p 1( f M) i p 2 ( f L) i P i p 2 α i p 2( f M) i P i The horizontal arrows are induced by µ and ν according to Claim 1. 12

13 Claim 2 follows from Claim 1 along with diagram 2.3. The uniqueness of (M, φ) follows from this claim since M is the degree 0 part of E. Finally, the uniqueness of n follows from the fact that it is an invariant of (N, ϕ) which for the sheaf T n equals n. Remark As in Claim 2, the tautological section of Example 2.9 corresponds to the graded morphism O Y N r+1 N r+2 N 1 O Y N N r+2 N 1 which is the identity on every factor except N r+1 N, which is multiplication by s Y. From this one may verify that any morphism (O U, id OR ) ( f M, h φ) T n gives rise to a section s of M such that 1 is sent to the section f s ū n. Theorem 2.11 Assume that the groupoid p 1Z Z is connected and nonempty. Then there is a surjection whose kernel is generated by (L, r). Pic(W Y ) Z Pic(R U) ((M, φ), n) ( f M, h φ) T n Proof: Surjectivity follows from the lemma. Note that (L, r) is indeed in the kernel, because we have an isomorphism f σ : f N r f L Y which makes diagram 2.1 commute. The rest follows from the uniqueness statement of the lemma. Applying this result to Example 2.7, we obtain the following classification of invertible sheaves on C = C D, r. For 1 i n, we let T i be the canonical invertible sheaf on C coming from the r i th root construction along D i by Example 2.9. Let γ : C C be the projection. Corollary 2.12 Let L be an invertible sheaf on C. Then there exist an invertible sheaf L on C and integers k i satisfying 0 k i r i 1 such that L = γ L n i=1 T k i i. Moreover, the integers k i are unique, L is unique up to isomorphism, and T r i i = γ O C (D i ). We also need to know something about global sections of invertible sheaves on C. This follows from the remark after the proof of the lemma. Corollary 2.13 Given the decomposition in the previous corollary, every global section of L is of the form γ s τ k 1 1 τn kn for a unique global section s of L, where τ i is the tautological section of T i. Remark It is also true that γ L = L, but we do not use this fact. We conclude the section with an important lemma, which is an easy application of these corollaries. 13

14 Lemma 2.14 Given a fiber square, C g B π S where C S and B T are two families of curves, and given two n-tuples of disjoint effective Cartier divisors D i C and E i B mapping isomorphically to the base such that g(d i ) = E i, any morphism G : C D, r B E, r over g satisfies G T i = Ti, where T i is the canonical invertible sheaf associated to E i and T i is the one associated to D i as above. Proof: By Corollary 2.12, we have a decomposition G T i, and by Corollary 2.13, the pullback of the tautological section τ i goes to a section of the form γ s τ k i T = γ L T k i i Raising both sides to the power r i and using the uniqueness statement of Corollary 2.12 and the fact that g(d i ) = E i, it follows that O C (D i ) = L r i O C ( j l jd j ) by an isomorphism sending s Di to s r i j sl j D j where l j = r i k j /r j. By comparing vanishing loci, it follows that k j = 0 for j i. Since the geometric fibers of π are smooth curves whose intersection with D i is a single point, it follows that s cannot vanish at any point of C and k i must equal 1. We assumed r i > 1, since the conclusion is obvious when r i = 1. Remark It follows that any two morphisms C D, r B E, r over g are 2-isomorphic. 3 Examples For this section, we fix a positive integer r and a field k whose characteristic does not divide r and which contains all rth roots of unity. We assume that every scheme is Noetherian and admits a morphism to Spec k. This additional restriction is only imposed because our proof of Lemma 5.3 requires it. The following example together with Proposition 2.4 shows that there exists an open covering of X such that the restriction of X (L,s,r) to each of the open sets is a quotient of a scheme by µ r. Example 3.1 Let X = Spec S and let L = O X. Then s corresponds to an element of S. We follow the construction of Section 2.3 with Y = X, N = O X, and ψ : N r O X the natural isomorphism. Then U = Spec A and R = Spec B, where A = S[x]/(x r s) and B = S[x, y]/(x r s, y r 1), and the source and target maps R U are given by (x, y) x and (x, y) xy. Thus R = U µ r and R U is the transformation groupoid determined by the action of µ r on U associated to the cyclic degree r covering U X. This implies that X (L,s,r) is isomorphic to the stack-theoretic quotient [U/µ r ]. In the next three examples we study two opposite extremes, one where the section s is nonvanishing and the other where s is identically 0. In general, this describes the restriction of X (L,s,r) to the open subscheme where s 0 and the closed subscheme where s = 0. i. 14

15 Example 3.2 If L = O X and s = 1, we claim that X (L,s,r) = X. Let X X(L,s,r) be the morphism which sends f : S X to the quadruple (f, O S, 1, c r ), where c r : O r S O S is the canonical isomorphism. We need to show that the composition X (L,s,r) X X (L,s,r) is 2-isomorphic to the identity morphism. If we are given a quadruple (f, M, t, ψ) over S, then t is a nonvanishing section of M, and therefore defines an isomorphism O S M which makes all the necessary diagrams commute. This is our required 2-isomorphism. Before doing the general case where s = 0, we ll start with something simple. Example 3.3 Let X = Spec K, where K is a field, and let s = 0. Example 3.1 implies that X (L,s,r) = [U/µr ], where U = Spec K[x]/(x r ). The embedding Spec K U determines an embedding ι : [Spec K/µ r ] X (L,s,r), where µ r acts trivially on Spec K. Here X (L,s,r) is nonreduced, but is an infinitesimal neighborhood of [Spec K/µ r ]. It is interesting to note that ι 1 O X(L,s,r) is isomorphic to the coherent sheaf which is determined by the regular representation of µ r (K). Example 3.4 Let X and L be arbitrary and let s = 0. The previous example shows that X (L,0,r) is not reduced; however, there is a closed substack X (L,r) X (L,0,r) which is reduced whenever X is. The objects of X (L,r) are those objects (f, M, t, ϕ) of X (L,0,r) for which t = 0 and the morphisms in X (L,r) are the same as morphisms in X (L,0,r). One could also define X (L,r) abstractly as the stack whose objects are triples (f, M, ϕ), imitating the definition of X (L,s,r) but leaving off all conditions on sections. This is called the moduli stack of rth roots of L in [AGV, Section 3.5.3]. If L = O X, we claim that X (L,r) = X Bµr. The objects of Bµ r are µ r -torsors and its morphisms are pullback diagrams which respect the group action. Since a µ r torsor on S is equivalent to a pair (M, ϕ), where M is an invertible sheaf on S and ϕ : M r O S is an isomorphism, the claim is easy to verify. Note that X Bµ r = [X/µr ], where µ r acts trivially on X. In general, we can cover X by finitely many affines so that the restriction of L to each affine is trivial. If U is the disjoint union of these affines, then we get an étale surjective morphism U X (L,r). Using this, one can show that X (L,r) X is an étale gerbe with band µ r. It is étale because U X is étale. To say it is a gerbe means that local sections exist and any two sections are locally isomorphic. Finally, to say that it has band µ r means that µ r acts in a compatible way on every section. Definition 3.5 Let D X be an effective Cartier divisor. We define the gerbe of X D,r to be the closed substack D (OD (D),r) X D,r. Example 3.6 Let X be a smooth variety over a field K and let D X be a divisor which has normal crossings. Given an integer r prime to the characteristic of K, Matsuki and Olsson [MO] constructed a smooth Deligne-Mumford stack X over X which has a divisor D X such that the pullback of O X ( D) to X is isomorphic to O X ( r D) as a subsheaf of O X. It follows that there is a morphism F : X X D,r given by a quadruple (f, M, t, ϕ), where M = O X ( D), t is its tautological section vanishing on D, and the rest is constructed from the above information. Since X is smooth, F cannot be an isomorphism unless D is 15

16 smooth, since otherwise X D,r will not be smooth. When D is smooth, it follows from the local descriptions of [MO, Lemma 4.3] and Example 3.1 that F is an isomorphism. An example of how these two stacks can differ is to let D be a reducible conic in X = P 2. If D 1 and D 2 are its irreducible components and one takes r 1 = r 2 = r, then X D, r (Definition 2.6) is isomorphic to the stack of [MO]. Over the node of D, there are two factors of µ r acting, whereas in X D,r there is only one. While X has the advantage of being smooth, X D,r has the advantage of behaving well under base change (Proposition 2.4). For example, this implies that a deformation of the pair (X, D) gives rise to a deformation of the stack X D,r. Example 3.7 Any complex orbicurve can be obtained from the r th root construction. Let X be a smooth curve, let D 1,...,D n be distinct points of X, and let r 1,..., r n be integers greater than 1. Then the complex orbicurve (X, D, r) defined in [CR, Definition 2.2.2] is isomorphic to X D, r. This follows from the local description of Example 3.1. Fix a scheme X with an effective Cartier divisor D X and let X = X D,r. Our goal for the remainder of the section is to classify representable morphisms C D, r X, where C is a smooth family of curves over a Noetherian scheme S, D is an n-tuple of disjoint effective Cartier divisors of C which map isomorphically to S, and r is an n-tuple of positive integers which are invertible on S. We first introduced the family C D, r S in Example 2.7. Let γ : C D, r C be the projection and fix a morphism F : C D, r X. In Proposition 2.8, we showed that every morphism C D, r X is of the form f γ for a unique morphism f : C X. Thus a morphism F is given by a quadruple (f, M, t, ϕ), where f : C X is a morphism, M is an invertible sheaf on C D, r with global section t, and ϕ : M r γ f O X (D) is an isomorphism such that ϕ(t r ) = γ f (s D ). F C D, r X γ C f X (3.1) By Corollary 2.12, the invertible sheaf M uniquely determines positive integers k i with 0 k i r i 1 and an invertible sheaf L on C so that M = γ L T k i i. Proposition 3.8 The morphism F is representable if and only if for every i, r i divides r and k i is relatively prime to r i. Proof: We apply the criterion of [AV, Lemma 4.4.3], which says that F is representable if and only if for every algebraically closed field K and every object ξ of C D, r over Spec K, the group homomorphism Aut(ξ) Aut(F(ξ)) is an injection. Such an object ξ is given by a morphism p : Spec K C together with n invertible sheaves M i, n sections t i, and n isomorphisms ϕ i. At most one of the sections t i can be zero because the divisors D i are disjoint. If all the sections are nonzero, then ξ has no nontrivial automorphisms, so it suffices to assume that p maps into some divisor D i, meaning that t i = 0. Then the automorphisms of ξ are given by r i th roots of unity in K. Since the automorphisms of 16

17 F(ξ) must be rth roots of unity, the condition that r i divides r for each i follows from injectivity of the homomorphism. By definition, the invertible sheaf M on C D, r determines an invertible sheaf M ξ on Spec K, and there are natural isomorphisms T iξ = Mi. Moreover, an automorphism ω K of M i goes under F to the automorphism ω k i of M ξ. This defines an injective homomorphism if and only if k i is relatively prime to r i. Given the n-tuples of integers r and k, define an n-tuple by i = k ir r i. (3.2) Under the hypotheses of the proposition, we have gcd(k i, r i ) = 1, and multiplying by r/r i we obtain gcd( i, r) = r/r i = i/k i. Thus in the representable case, k and r are determined from by the following formulas. r i = r gcd(r, i) k i = i gcd(r, i) (3.3) We call the n-tuple the contact type of the morphism F, a definition which is motivated by the theorem below. Let s be the global section of L determined by t according to Corollary 2.13, and let Z C be its vanishing locus. The above decomposition for M, together with the fact that T r i i = γ O C (D i ) imply that ϕ induces an isomorphism γ f O X (D) = γ (L r ( id i )). Moreover, this isomorphism sends γ f s D to γ (s r s i D i ). If we assume that f 1 D is an effective Cartier divisor, then it follows from Corollaries 2.12 and 2.13 that f D = rz + id i. By Lemma 5.2, this holds if f 1 D does not contain any fibers of C S. The second part of the following theorem has now been established. Theorem 3.9 Let π : C S be a smooth family of curves over a connected nonempty base S, and let D i C, i = 1,..., n, be an n-tuple of disjoint effective Cartier divisors which map isomorphically onto S. Fix an n-tuple of positive integers, and let r be determined from by (3.3). Given a morphism f : C X such that 1. f 1 D does not contain any fiber of π and 2. there exists an effective Cartier divisor Z C (necessarily unique by Lemma 5.3) such that f D = rz + id i, there is a unique (up to 2-isomorphism) representable morphism C D, r X D,r of contact type which makes diagram 3.1 commute. Conversely, any representable morphism F : C D, r X D,r of contact type such that F 1 G does not contain any fibers comes from such an f. Here G is the gerbe of X D,r (Definition 3.5). 17

18 Proof: It remains to prove only the first part. A representable morphism F : C D, r X D,r, if it existed, would be given by a quadruple (f, M, t, ϕ). We want to show that there is only one such quadruple up to isomorphism. The morphism f : C X is already given. Let L = O(Z) and let s be the canonical section of L vanishing on Z. Let M = γ L T k i i, and let t = γ s τ k i i. Then condition 2 implies that there is an isomorphism ϕ : M r γ f O(D) sending t r to γ f s D. This defines a morphism F of contact type. If we choose a different quadruple (f, N, u, ψ) giving rise to a morphism of contact type, then we have already shown that there is an invertible sheaf L on C and a section s and u = γ s τ k i i. If Z C is the vanishing locus of s, then so that N = γ L T k i i Lemma 5.3 implies that Z = Z, so there is an isomorphism L L sending s to s. This induces an isomorphism M N sending t to u, and it necessarily sends ϕ to ψ since t and u are nonzero on a dense open set. 4 Twisted stable maps As in the previous section, we fix a positive integer r and a field k whose characteristic does not divide r and contains all rth roots of unity, and we assume that every scheme admits a morphism to Spec k. We begin with an observation about nodal n-pointed curves over a Noetherian scheme S. In the standard definition, one has a flat morphism π : C S whose geometric fibers are nodal curves together with n sections σ i : S C whose images are disjoint and which don t pass through any singular points of the fibers. It is equivalent to replace the n sections σ i with n disjoint effective Cartier divisors D i C which map isomorphically to S. Indeed, given a section σ which doesn t pass through any nodes, it follows from Lemma 5.1 that its image is an effective Cartier divisor. Conversely, if it is effective Cartier and maps isomorphically onto S, then its restriction to each fiber is a point defined by a single equation, so it can t be a node of the fiber. Therefore, we consider a nodal n-pointed curve over S to be given by the data (π : C S, D 1,...,D n ). In the following theorem, we use the definition of a twisted nodal n-pointed curve over S from [AV, Definition 4.1.2]. We implicitly assume our curves are connected and proper. Theorem 4.1 To give a twisted smooth n-pointed genus g curve over a connected Noetherian scheme S is equivalent to giving a smooth n-pointed genus g curve C over S together with a choice, for each divisor D i C, of a positive integer r i which is invertible on S. Then the twisted curve is isomorphic to the stack C D, r of Example 2.7 in such a way that the marking Σ i C is sent to the gerbe over D i of Definition 3.5. Proof: What we have done in the paper so far shows that C D, r is a twisted curve over S. Given a twisted curve C S, with markings Σ i C, the coarse moduli scheme C S is a flat family of curves over S by [AV, 4.1.1]. Moreover, if D i is the coarse moduli space of Σ i, then it embeds into C as the image of Σ i. Let γ : C C be the projection. The local description for C given in [ACV, 2.1] says that at a marking Σ i, C looks étale locally like the 18

19 quotient of A 1 S by the cyclic action of µ r i which fixes the origin, for some integer r i which is invertible on S. Since Σ i is connected, the integer r i does not depend on the point where one takes the étale neighborhood. Since C is Noetherian, there exists an étale surjective morphism e : U C with U Noetherian. By Lemma 5.1, e 1 (Σ i ) is an effective Cartier divisor. Therefore, Σ i is an effective Cartier divisor, and it follows from the local description that r i Σ i = γ 1 D i. So γ, the invertible sheaves O C (Σ i ), their tautological sections vanishing on Σ i, and the natural isomorphisms O C (r i Σ i ) γ O C (D i ) define a morphism of stacks C C D, r. It follows from the local description that this is an isomorphism and that Σ i is sent to the gerbe over D i. Remark Martin Olsson has found another way to view twisted curves in terms of logarithmic structures [Ol] which works for arbitrary twisted nodal curves. Let X be a projective scheme over k and let D X be an effective Cartier divisor. Let X = X D,r, and let β N 1 (X). The moduli stack of twisted stable maps into X of class β, denoted K g,n (X, β), was constructed by [AV]. It is a proper Deligne-Mumford stack and admits a finite morphism to K g,n (X, β) (which is the same as M g,n (X, β)). Because K g,n (X, β) is Noetherian, it has a groupoid presentation involving Noetherian schemes and finite type morphisms. This means that anything we want to know about the stack can be learned by only considering objects of the stack over Noetherian schemes, so we will always assume our base schemes are Noetherian. Let be an n-tuple of integers such that 0 i r 1 and D β i is divisible by r. We call such an n-tuple of integers admissible. Now we introduce substacks of the spaces of stable maps forming a commutative diagram. U g,n (X, β, ) U g,n (X, β) K g,n (X, β) V g,n (X, β, ) V g,n (X, β) K g,n (X, β) Let U g,n (X, β) K g,n (X, β) be the open substack consisting of stable maps from smooth twisted curves which do not map into the gerbe G X of Definition 3.5. To see that this is open, it suffices to see that for any family (π : C S, Σ 1,...,Σ n, F : C X) of stable maps into X, the set of s S such that the fiber C s is smooth and does not map into G is open. This is an easy consequence of the fact that π is both open and closed. By Theorem 4.1, an object of U g,n (X, β) over a connected scheme S is equivalent to the data (π : C S, D, r, F : C D, r X), where (π : C S, D) is a smooth n-marked curve over S, r is an n-tuple of positive integers, and F is a stable morphism such that no fiber of π maps into D under the induced morphism f : C X. For F to be stable means that it is representable and f is stable. We will therefore denote objects of U g,n (X, β) by (π : C S, D, r, F). Equation 3.2 associates to any such morphism F an n-tuple of integers called the contact type. It follows from Lemma 2.14 that the contact type defines n locally constant functions on the moduli stack of stable maps. Let U g,n (X, β, ) U g,n (X, β) be the open and closed substack consisting of stable maps which have contact type. 19

20 Let V g,n (X, β) K g,n (X, β) be the open substack consisting of stable maps from smooth curves which do not map into D. Let V g,n (X, β, ) be the stack whose objects over a scheme S are quadruples (π : C S, D, f, Z), where (π : C S, D, f) is an object of V g,n (X, β) and Z C is an effective Cartier divisor such that f D = rz + id i, and whose morphisms are morphisms of stable maps which preserve Z. We have a morphism V g,n (X, β, ) V g,n (X, β) and we claim that this is a closed embedding. Let (π : C S, D, f) be an object of the second stack, and let T be the fiber product, as in the diagram below. V g,n (X, β, ) T V g,n (X, β) S Hilb d h C/S Hilb d C/S Let d = D β and let d satisfy d = rd + i, which is an integer since we assumed to be admissible. The stack T is isomorphic to the stack whose objects over a scheme U are pairs (g, Z), where g : U S is a morphism and Z g C is an effective Cartier divisor such that g f D = rz + i g D i. Here g : g C C is the projection. Such a Z is necessarily flat over S by Lemma 5.2, so there is a morphism from T to the Hilbert functor Hilb d C/S parametrizing length d subschemes of the fibers of C S. Since X is projective and f : C X is stable, C is projective over S. In this situation, the Hilbert functor is representable by a scheme which is projective over S [Gr1, 3.2]. We have a morphism h : Hilb d C/S Hilbd C/S sending Z to rz + id i which makes a fiber square as in the diagram. Moreover, it follows from Lemma 5.3 that h is a monomorphism and it is clearly proper, so it is a closed embedding by [Gr2, ]. The claim now follows. This brings us to our main theorem. Theorem 4.2 The morphism K g,n (X, β) K g,n (X, β) restricts to an isomorphism for any admissible n-tuple. U g,n (X, β, ) V g,n (X, β, ) Proof: As we remarked earlier, we only need to consider families of stable maps over Noetherian schemes. We may further restrict to connected schemes, since a Noetherian scheme is the disjoint union of its connected components. So let S be Noetherian, and let (π : C S, D, r i, F) be an object of U g,n (X, β, ) over S. In Theorem 3.9 we showed that there is a unique Z C so that under the induced morphism f : C X we have f D = rz + id i. This shows that U g,n (X, β, ) maps to V g,n (X, β, ). Theorem 3.9 also shows that any stable map in V g,n (X, β, ) comes from a stable map in U g,n (X, β, ). To show that we have an isomorphism of stacks, it now suffices to show that given any two stable maps ξ 1, ξ 2 U g,n (X, β, ) over S, any morphism over S between their images in V g,n (X, β, ) comes from a unique morphism ξ 1 ξ 2 over S. Let ξ i = (π i : C i 20

21 S, D i, r i, F i ), let f i : C i X be the coarse map of F i, and let g : C 1 C 2 be an S- isomorphism such that f 2 g = f 1 and g(dj 1) = D2 j. We saw in equation 3.3 that ri j is determined by j, so we drop the superscript i from r and let C i = (C i ) D i, r. Recall that a morphism ξ 1 ξ 2 is an equivalence class of pairs (G, α), where G : C 1 C 2 is an isomorphism of S-stacks which preserves the markings and α : F 2 G F 1 is a 2-morphism. Two such pairs (G 1, α 1 ) and (G 2, α 2 ) are equivalent if there is a 2-morphism β : G 1 G 2 such that α 2 (id F2 β) = α 1. Since g sends Dj 1 to D2 j, g induces an isomorphism G : C 1 C 2. If H : C 1 C 2 is any isomorphism inducing g : C 1 C 2, then Lemma 2.14 implies that there is a 2-morphism G H. It now suffices to show that there is a unique 2-morphism F 2 H F 1. In Theorem 3.9 we showed that there exists a 2-morphism. Since F 2 H and F 1 are representable and since the complement of F1 1 G is a dense open representable substack, it follows by [AV, 4.2.3] that the 2-morphism is unique. This finishes the proof. 5 Appendix Lemma 5.1 Let the following be a commutative diagram of Noetherian schemes, where D X is a closed subscheme and both π and π D are flat. D X π π D S If D s X s is an effective Cartier divisor for each s S, then D X is effective Cartier. Proof: We first show that the ideal sheaf of D is locally generated by a single element. For this, it suffices to assume that S = Spec A and X = Spec B, where A and B are local rings with maximal ideals m and M. Let I be the ideal of D. We have an exact sequence. 0 I B B/I 0 Since π D is flat, the local criterion for flatness [Ei, 6.8] implies that this remains exact after tensoring with A/m, so I/mI is the ideal of the restriction of D to Spec B/mB. This is generated by a single element by hypothesis, so I/MI is also, and Nakayama s Lemma implies that I is. We claim that any local generator of the ideal sheaf of D is a nonzerodivisor. If not, then there is a point x D which is an associated point of X, meaning that the maximal ideal of its local ring consists of zerodivisors. Let s = π(x). Since π is flat, it follows that x is an associated point of X s. For example, this follows from the fact that depth is additive for flat morphisms [Ma, 23.3]. But this contradicts the hypothesis that D s X s is an effective Cartier divisor. Lemma 5.2 Let π : X S be a flat morphism of Noetherian schemes all of whose fibers are integral. Let D X be a closed subscheme whose ideal sheaf is locally generated by a 21

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