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Classical and minimal models of the moduli space of curves of genus two


arXiv:math/0408338v1 [math.AG] 24 Aug 2004

Classical and minimal models of the moduli space of curves of genus two
Brendan Hassett August 2004
?

1

Introduction

This paper is an introduction to the minimal model program, as applied to the moduli space of curves. Our long-term goal is a geometric description of the canonical model of the moduli space when it is of general type. This entails proving that the canonical model exists and interpreting it as a parameter space in its own right. Work of Eisenbud, Harris, and Mumford shows that Mg is of general type when g ≥ 24 (see [HaMu] and subsequent papers). A standard conjecture of birational geometry–the ?nite generation of the canonical ring–would imply the canonical model is Proj ⊕n≥0 Γ(M g , nKM g ). Unfortunately, this has yet to be veri?ed in a single genus! There is some cause for optimism: Shepherd-Barron [SB] has recently shown that the canonical model of the moduli space of principally polarized abelian surfaces of dimension g ≥ 12 is the ?rst Voronoi compacti?cation. Another possible line of attack is to consider log canonical models of the moduli space. The moduli space is best regarded as a pair (M g , ?), where M g is Deligne-Mumford compacti?cation by stable curves and ? is its boundary. It is implicit in the work of Mumford [Mu1] that the moduli
? Partial support was provided by National Science Foundation grants 0196187 and 0134259 and the Sloan Foundation.

1

space of stable curves is its own log canonical model (see Theorem 4.7). Our basic strategy it to interpolate between the log canonical model and the (conjectural) canonical model by considering Proj ⊕n≥0 Γ(M g , n(KM g + α?)), where α ∈ Q ∩ [0, 1] is chosen so that K + α? is e?ective. This program is a subject of ongoing work, inspired by correspondence with S. Keel, in collaboration with D. Hyeon. Future papers will address the stable behavior of these spaces for successively smaller values of α. It is remarkable that their behavior is largely independent of the genus (see, for example, Remark 4.9.) However, for small values of g special complexities arise. When g = 2 or 3, the locus in M g of curves with automorphism has codimension ≤ 1. To include these spaces under our general framework, we must take into account the properties of the moduli stack Mg . In particular, it is necessary to use the canonical divisor of the moduli stack rather than its coarse moduli space. These di?er substantially, as the natural morphism Mg → M g is rami?ed at stable curves admitting automorphisms. Luckily, we have inherited a tremendously rich literature on curves of small genus. The invariant-theoretic properties of M2 were extensively studied by the 19th century German school [Cl], who realized it as an open subset of the weighted projective space P(1, 2, 3, 5). Theorem 4.10 reinterprets this classical construction using the modern language of stacks and minimal models. We work over an algebraically closed ?eld k of characteristic zero. We use the notation ≡ for Q-linear equivalence of divisors. Throughout, a curve is a connected, projective, reduced scheme of dimension one. The genus of a curve is its arithmetic genus. The moduli stack of smooth (resp. stable) curves of genus g is denoted Mg (resp. Mg ); the corresponding coarse moduli scheme is denoted Mg (resp. M g ). The boundary divisors in M g (resp. Mg ) are denoted ?0 , ?1 , . . . , ??g/2? (resp. δ0 , δ1 , . . . , δ?g/2? ). Let Q : Mg → M g denote the natural morphism from the moduli stack to the coarse moduli space, so that Q? ?i = δi , i = 1 2 Q? ?1 = 2δ1 .

We write δ = ing divisor on M g .

0≤i≤g/2 δi ;

abusing notation, we also use δ for the correspond-

?0 + 1/2?1 + ?2 + . . . + ??g/2?

Acknowledgments: The key ideas underlying this program were worked out in correspondence with S. Keel. They have been further developed in collaboration with D. Hyeon. The author also bene?ted from conversations with James Spencer about rigidi?cation, quotient stacks, and moduli of curves of genus two, and from comments on the manuscript by Michael van Opstall. Part of this paper was prepared during a visit to the Mathematisches Institut of the Georg-August-Universit¨ at, G¨ ottingen.

2
2.1

Classical geometry
Elementary facts about curves of genus two

We recall results from standard textbooks, e.g., [Ha] IV Ex 2.2 and §5. Let C denote a smooth curve of genus two with sheaf of di?erentials ωC . The global sections of ωC give the canonical morphism j : C → P(Γ(C, ωC )) ? P1 which is ?nite of degree two. The corresponding covering transformation ι : C → C is called the hyperelliptic involution. By the Hurwitz formula, j is branched over six distinct points {b1 , . . . , b6 } ? P(Γ(C, ωC )). On ?xing an identi?cation P(Γ(C, ωC )) ? P1 , we can write down a nontrivial binary sextic form vanishing at the branch points F ∈ Γ(P1 , OP1 (6)), determined by {b1 , . . . , b6 } ∈ P1 up to a scalar. Conversely, suppose we have a binary sextic form F with six distinct zeros b1 , . . . , b6 ∈ P1 . Then there is a unique degree-two cover of P1 branched over these points. This is a smooth curve of genus two and the map to P1 is the 3

canonical morphism. Moreover, the isomorphism class of C depends only on the orbit of F under the action of GL2 . To summarize: There is a one-to-one correspondence between isomorphism classes of curves of genus two and GL2 -orbits of binary sextic forms with distinct zeros. We will need a relative version of this dictionary, following [Vi]. Let π : C → S be a smooth morphism to a scheme of ?nite type over k , with ?bers curves of genus two. Since the relative dualizing sheaf ωπ is globally generated, there is a relative double cover C
π

?→ ? S ?ψ

j

P1 (π? ωπ ) := P

with associated involution ι. Using the trace we decompose j? OC ? OP ⊕ L, where L has relative degree ?3 on the ?bers of ψ . The OP -algebra structure on OC is thus determined by an isomorphism L2 → OP , i.e., by a nonvanishing section of L?2 with zeros along the branch locus of j . By relative duality j? ωπ = H om OP (j? OC , ωψ ) ? ωψ ⊕ (ωψ ? L?1 ) which yields π? ωπ ? ψ? (ωψ ? L?1 ). Using the identi?cations ψ? OP (+1) = π? ωπ , we ?nd
?1 L?1 ? OP (+1) ? ωψ = OP (3) ? (ψ ? det π? ωπ )?1 .

ωψ = (ψ ? det π? ωπ )(?2),

Pic(P) = Pic(S )⊕Zc1 (OP (+1)),

The class of the branch divisor is thus ?2c1 (L) = c1 (OP (6)) ? 2ψ ? c1 (det π? ωπ ). (1)

This has practical implications: An isomorphism of C induces a linear transformation on Γ(C, ωC ), which respects the binary sextic F up to a scalar. Formula (1) allows us to keep track of this scalar. For each M ∈ GL2 , we have the linear action (x, y ) → (x, y ) 4 m11 m12 m21 m22

which induces a natural left action on binary sextic forms F → F (xm11 + ym21 , xm12 + ym22 ). We normalize this action using formula (1) F → (M, F ) := (det M )?2 F (xm11 + ym21 , xm12 + ym22 ), (2)

so that (M, F ) = F if and only M is induced from an automorphism of C . A smooth curve is bielliptic if it admits a degree-two morphism i : C → E to an elliptic curve; the covering transformation is called a bielliptic involution. For curves of genus two any bielliptic involution commutes with the hyperelliptic involution, which yields a diagram C ↓ P1 → →
? i i

j

E ↓? j P1

where ? i and ? j are the double covers induced on quotients. The branch locus of j is preserved by the covering transformation for ? i, which is conjugate to [x, y ] → [y, x]. The resulting involution of the branch locus will also be called a bielliptic involution. Thus C is isomorphic to a double cover branched over {[α1 , 1], [1, α1], [α2 , 1, ][1, α2 ], [α3 , 1], [1, α3]} for some α1 , α2 , α3 ∈ k . Conversely, each such curve admits a diagram as above and thus is bielliptic.

2.2

Invariant theory of binary sextics

We observe the classical convention for normalizing the coe?cients of a binary sextic F = ax6 + 6bx5 y + 15cx4 y 2 + 20dx3 y 3 + 15ex2 y 4 + 6f xy 5 + gy 6. The action (2) induces an action of GL2 on k [a, b, c, d, e, f, g ]. Recall that a polynomial P ∈ k [a, b, c, d, e, f, g ] is SL2 -invariant if for each M ∈ SL2 , we have (M, P ) = P. 5

We write R := k [a, b, c, d, e, f, g ]SL2 for the ring of such invariants. If P is SL2 -invariant then each homogeneous component of P is as well, so R is a graded ring. Every homogeneous invariant satis?es the functional relation (M, P ) = (det M )deg(P ) P, M ∈ GL2 ; (3)

here it is essential that the action (2) include the factor (det M )2 . The transformation (x, y ) → (y, x) thus reverses the sign of invariants of odd degree. These are called skew invariants in the classical literature. Explicit generators for R were ?rst written down in the nineteenth century, e.g., [Cl], pp. 296, in symbolic notation, [Ca] and [Sa] as explicit polynomials–the second edition of Salmon’s Higher algebra has the most detailed information, and also [El] pp. 322. A nice early twentieth-century discussion is [Sc] pp. 90 and a modern account invoking the representation theory of SL2 is [Sp]. For our purposes, the symmetric function representation of the invariants in [Ig2] pp. 176 and 185 is the most useful. Let ξ1 , . . . , ξ6 denote the roots of the dehomogenized form F (x, 1), and write (ij ) as shorthand for ξi ? ξj . We write A = a2
?fteen

(12)2 (34)2 (56)2 (12)2(23)2 (31)2 (45)2 (56)2 (64)2
ten

B = a C = a

4

6

(12)2 (23)2 (31)2 (45)2(56)2 (64)2 (14)2 (25)2 (36)2
sixty

D = a

10 ij

(ij )2

where the summations are chosen to make the expressions S6 -symmetric. Consequently, A, B, C, D, and E can all be expressed as polynomials in 6

? ? 1 ξ1 + ξ2 ξ1 ξ2 det ?1 ξ3 + ξ4 ξ3 ξ4 ? = a15 E = a15 ((14)(36)(52) ? (16)(32)(54)) ?fteen ?fteen 1 ξ5 + ξ6 ξ5 ξ6

Q[a, b, c, d, e, f, g ], e.g., A = ?240(ag ? 6bf + 15ce ? 10d2 ) ? ? a b c d ?b c d e ? 2 2 ? B = ?162000 det ? ? c d e f ? + 1620(ag ? 6bf + 15ce ? 10d ) . d e f g

In classical terminology, (ag ? 6bf + 15ce ? 10d2 ) is the sixth transvectant of F over itself; transvection is one of the main operations in Gordan’s proof of ?niteness for invariants of binary forms. The determinantal expression is the catalecticant of F : It vanishes precisely when F can be expressed as a sum of three sixth powers [El] pp. 276. The following facts will be useful for subsequent analysis: Proposition 2.1 1. The expressions A, B, C, D, and E are invariant and generate R [Hi], pp. 100, [Cl], etc. 2. D is the discriminant and vanishes precisely when the binary form has a multiple root. 3. B, C, D, and E vanish whenever the binary form has a triple root; A vanishes when the form has a quadruple root. 4. E vanishes if and only if the form admits a bielliptic involution, as de?ned in §2.1 [El], pp. 327 and [Cl], pp. 457. 5. The unique irreducible relation among the invariants is E 2 = G(A, B, C, D), where G is weighted-homogeneous of degree 30 [Cl], pp. 299. The notation used for the generating invariants is not consistent among authors. Our notation is consistent with that of Igusa, but inconsistent with Clebsch’s and Salmon’s. Of course, the invariants of degree two and ?fteen are unique up to scalar.

7

2.3

The projective invariant-theory quotient
k [A, B, C, D, E ] . ? G(A, B, C, D)

We consider X := Proj R = Proj E2

If A = B = C = D = 0 then E = 0 as well, so X is covered by the distinguished a?ne open subsets { A = 0} { B = 0} { C = 0} { D = 0} . However, in each localization (k [A, B, C, D, E ][A?1])0 (k [A, B, C, D, E ][C ?1])0 (k [A, B, C, D, E ][B ?1 ])0 (k [A, B, C, D, E ][D ?1])0

only even powers of E appear, so all the functions over these distinguished open subsets can be expressed in terms of A, B, C, D. In light of Proposition 2.1, we ?nd Proposition 2.2 1. X ? Proj k [A, B, C, D] ? P(2, 4, 6, 10) ? P(1, 2, 3, 5) [Ig2], pp. 177. 2. A binary sextic with a zero of multiplicity three, admitting a nonvanishing invariant of positive degree, is mapped to p := [1, 0, 0, 0, 0] ∈ X . 3. All positive-degree invariants vanish at binary sextics with a zero of multiplicity four; they do not yield points of X . Geometric Invariant Theory gives an interpretation of the points of X : Proposition 2.3 1. a binary sextic is stable (resp. semistable) if and only if its zeros have multiplicity ≤ 2 (resp. ≤ 3) [GIT], ch. 4 §1; 2. X ? {[1, 0, 0, 0, 0]} is a geometric quotient for binary sextics with zeros of multiplicity ≤ 2 [GIT] 1.10. The ‘only if’ part of the ?rst assertion can be deduced from Proposition 2.2. As X ?{D = 0} is a geometric quotient for binary sextics with distinct zeros, our analysis of genus two curves in §2.1 yields 8

Proposition 2.4 The moduli scheme M2 can be identi?ed with X ?{D = 0}, where D is the discriminant. Remark 2.5 This construction de?nitely fails in characteristic two. If the double cover j : C → P1 is wildly rami?ed, the branch divisor may have multiplicities > 3. These curves correspond to unstable points under the SL2 -action, and thus are not represented in the invariant-theory quotient. [Ig1] has a detailed account of what must be done in this case.

2.4

Invariant-theory quotient as a contraction

We sketch the relationship between the invariant-theory quotient and the moduli space of stable curves. De?nition 2.6 A birational map of normal projective varieties β:Y X

is a contraction if β ?1 has no exceptional divisors, i.e., the proper transform of each codimension-one subset in X has codimension one in Y . X reProposition 2.7 There exists a birational contraction β : M 2 stricting to the identity along the open subset M2 . β is an isomorphism over M 2 ? ?1 and contracts ?1 to the point p. proof: To produce the birational contraction, we exhibit a morphism β ?1 : U ?→ M 2 where U ? X is open with complement of codimension ≥ 2 and β ?1 |M2 ∩U is the identity. We shall take U = X ? p, where p corresponds to the binary forms with a triple zero (cf. Proposition 2.2.) The universal binary sextic is a hypersurface W := {ax6 +6bx5 y +15cx4 y 2 +20dx3 y 3 +15ex2 y 4 +6f xy 5 +gy 6 = 0} ? A7 ×P1 . Its class in Pic(A7 × P1 ) is divisible by two, so there exists a double cover C ′ → A7 × P1 simply branched over W . Composing with the projection onto the ?rst factor, we obtain a morphism π ′ : C ′ → A7 . 9

Let S ? A7 denote the open subset corresponding to forms whose zeros all have multiplicity ≤ 2 and π:C→S the restriction of π ′ to S . Since π is a composition of ?at morphisms, it is also ?at. Consider the ?ber of π over a given binary sextic F : It is a double cover j : CF → P1 branched over the zeros of F . We claim CF is a stable curve of genus two, not contained in ?1 . Evidently CF is smooth and simply branched over the zeros with multiplicity one. Over the double zeros CF has local equation y 2 = x2 , which de?nes a node. We have j ? OP1 (+1) = ωCF , ν which is ample on CF , so CF is stable. The normalization ν : CF → CF is the double cover branched along F ′ = 0, where F ′ is the product of the factors of ν F with multiplicity one; CF is obtained from CF by gluing the pairs of points over the each double root of F . There are three possibilities:
ν 1. deg(F ′ ) = 4, in which case CF is connected of genus one; ν 2. deg(F ′ ) = 2, in which case CF is connected of genus zero; ν 3. deg(F ′ ) = 0, in which case CF has two connected components of genus zero.

Since CF cannot be expressed as the union of two subcurves of genus one meeting at a point, the resulting curve is not in ?1 . The classifying morphism S → M2 ? δ1 is equivariant with respect to the GL2 -action on binary sextics, and therefore descends to a morphism Φ : S/GL2 → M2 ? δ1 . We remark that this is a morphism of stacks. Since U is a geometric quotient for binary sextics with zeros of multiplicity ≤ 2 (see Proposition 2.3), U is also the coarse moduli space for S/GL2 . We de?ne β ?1 to be the induced morphism on coarse moduli spaces. It remains to show this is bijective onto its image. Suppose we are given a stable curve C of genus two not contained in ?1 . Quite generally, ωC is globally generated for any stable curve without disconnecting nodes; the only 10

curves in M 2 with disconnecting nodes lie in ?1 . Thus the sections of ωC give a double cover j : C → P1 branched along a sextic, with zeros of multiplicity ≤ 2 because C is nodal. The analysis above shows that every such sextic arises in this way.

2.5

Blowing up the invariant-theory quotient

We recall the principal result of [Ig2]. Let Ag denote the moduli space of principally polarized abelian varieties of dimension g , Ag its Satake compacti?cation. Recall that Ag = Proj S , where S is the ring of Sp(g, Z)-modular forms; we use λ to denote the resulting polarization on Ag . Let t : Mg ?→ Ag denote the Torelli morphism, associating to each curve its Jacobian. Now assume g = 2. Regarding M2 as an open subset of X (see Proposition 2.4), t extends to a rational map τ :X A2 .

The inclusion and the Torelli morphism induce M2 ?→ X := Graph(τ ) ? X × A2 . In particular, X compacti?es M2 . Theorem 2.8 [Ig2] The indeterminacy of τ is the point p = [1, 0, 0, 0, 0] ∈ X corresponding to binary sextic forms with a zero of multiplicity three. If we choose local coordinates at p x1 = 24 32 B/A2 x2 = 26 33 (3C ? AB )/A3 x3 = 2 · 35 D/A5 (4)

then τ is resolved by a weighted blow-up centered at p b:X →X with weights weight(x1 ) = 2 weight(x2 ) = 3 weight(x3 ) = 6.

The exceptional divisor of b is mapped isomorphically to the locus of principally polarized abelian surfaces that decompose as a product of two elliptic curves (with the induced product polarization). 11

Remark 2.9 Igusa’s result is considerably more precise: He explicitly computes the correspondence between the ring of invariants R and the ring of modular forms S . In particular, S is a polynomial ring with generators in degrees 4, 6, 10, 12 and the locus of products is given by the vanishing of a form of weight 10.

2.6

Comparing the blow-up with moduli space

Proposition 2.10 The open imbedding M2 ?→ X extends to a birational map γ : M2 X which is an isomorphism in codimension one. In particular, γ and γ ?1 are both birational contractions. In Proposition 4.2 we will prove that γ is an isomorphism. proof: The Torelli morphism admits an extension t : M g → Ag [Na1], Theorem 3. This is not an isomorphism for g > 1: The divisor ?0 ? M g is mapped to a boundary stratum of Ag , which has codimension ≥ 2. However, in genus two t is an isomorphism at the generic point of ?1 . Indeed, the Jacobian of a curve [E1 ∪q E2 ] ∈ ?1 , with E1 and E2 smooth of genus one, is the abelian surface E1 × E2 . The following diagram summarizes the various birational maps and morphisms: π A2 ← X t ↑ ↓b . β M2 X By Theorem 2.8, π is also an isomorphism over the generic point of t(?1 ), so γ is an isomorphism at the generic point of ?1 . β and b are both isomorphisms over the generic point of the divisor β (?0 ) (see Proposition 2.7), so γ is an isomorphism at the generic point of ?0 . Since γ is regular along M2 = M 2 ? ?0 ? ?1 , the result follows. The proper transforms of ?0 and ?1 in X are denoted ?0 and ?1 . Thus ?1 is the exceptional divisor of b : X → X . Remark 2.11 (Bibliographic note) There are a number of partial desingularizations of Ag through which t factors, e.g., the ‘Igusa monoidal transform’ [Ig3], [Na1] and the toroidal compacti?cation associated to the 2nd 12

Voronoi fan [Na2]. When g = 2, these approaches coincide [Na2] Remark 2.8 and yield a partial desingularization A2 → A2 . See [Ig3] Theorem 5 for a blow-up representation, expressed in terms of modular forms; the center of this blow-up is in the boundary A2 ? A2 . Namikawa [Na1] §9 has shown ? that the factorization M 2 → A2 is an isomorphism. Notwithstanding Igusa’s explicit formulas for τ : X A2 and A2 → A2 , it is not entirely obvious ? how to extract an isomorphism A2 → X.

3
3.1

Stack geometry
A stack-theoretic quotient

Proposition 2.2 might suggest that the invariant E is irrelevant to the geometry of the quotient. However, we have so far ignored possible stack structures on the quotient, which are intertwined with the geometry of E . There are a number of natural stacks to consider, including the GL2 -quotient stack. Our choice is dictated by pedagogical imperatives, i.e., to exhibit concretely the nontrivial inertia along the bielliptic locus where E vanishes. The ring of invariants R is graded by degree, so we have a natural Gm action on the a?ne variety Y = Spec R. Now Gm acts on the open subset Y ? (0, 0, 0, 0, 0) with ?nite stabilizers and closed orbits, so the quotient stack X := (Y ? (0, 0, 0, 0, 0)) /Gm is a separated Deligne-Mumford stack with coarse moduli space q : X → X (see [LM] 10.13.2,7.6,8.1 for more information). The points of Y ?(0, 0, 0, 0, 0) with nontrivial stabilizer map to the points of X with nontrivial inertia groups; this is the rami?cation locus of q . We collect some geometric properties of X : Proposition 3.1 1. The closed imbedding Spec R ?→ Spec k [A, B, C, D, E ] induces a closed imbedding i : X ?→ P (2, 4, 6, 10, 15), where P (2, 4, 6, 10, 15) := (Spec k [A, B, C, D, E ] ? (0, 0, 0, 0, 0)) /Gm , with Gm acting with weights (2, 4, 6, 10, 15) t · (A, B, C, D, E ) → (t2 A, t4 B, t6 C, t10 D, t15 E ); 13

2. the dualizing sheaf of X is given via adjunction ωX = i? ωP (2,4,6,10,15) (X ) ? i? OP (2,4,6,10,15) (?7), where OP (2,4,6,10,15) (+1) is the invertible sheaf associated with the principal Gm -bundle classi?ed by the identity character of Gm ; 3. the rami?cation divisor of q is E := {E = 0} ? X and we have q ? ωX ? ωX (?E ) ? i? OP (2,4,6,10,15) (?22). proof: The ?rst two assertions do not require proof. As for the third, it follows from the classi?cation of possible automorphisms of binary sextics [Bo], [Ig1] §8 The only automorphism type occuring in codimension one is the bielliptic involution (cf. Proposition 2.1). Remark 3.2 Notwithstanding Proposition 2.4, M2 is not contained in X as an open substack. Using the functional relation (3), the inertia group of [F ] ∈ X ? p is the quotient {M ∈ GL2 : (M, F ) = F }/{M ∈ SL2 : (M, F ) = F }, which is trivial for generic binary forms. The inertia group at [C ] ∈ M2 is Aut(C ), and the presence of the hyperelliptic involution ι means this is always nontrivial. Now Aut(C ) has a natural representation on Γ(C, ωC ) and an induced representation on ∧2 Γ(C, ωC ) that is not faithful: We do not see elements of Aut(C ) acting on Γ(C, ωC ) with determinant one, e.g., ι, which acts on Γ(C, ωC ) by ?I . The corresponding quotient of Aut(C ) is the inertia picked up by X .

3.2

Analysis of boundary divisors in the moduli stack

Lemma 3.3 Every stable curve of genus two admits a canonical hyperelliptic involution, which is central in its automorphism group.

14

proof: We claim that every stable curve of genus two is canonically a double cover of a nodal curve of genus zero j : C → R, R= P1 P1 ∪r P1 if [C ] ∈ ?1 if [C ] ∈ ?1

branched over six smooth points b1 , . . . , b6 ∈ R and the node r ∈ R. The covering transformation ι therefore commutes with each automorphism of C . The cover is induced by
2 C → P(Γ(C, ωC )) ? P2

which factors

C → R ? P2 , where R is a plane conic and j is ?nite of degree two. Indeed, for curves not in ?1 this is the double cover C → P1 discussed in the proof of Proposition 2.7. For curves C = E1 ∪q E2 ∈ ?1 with q = E1 ∩ E2 the disconnecting node joining the genus-one components E1 and E2 , we have a double cover j : E1 ∪q E2 ?→ P1 ∪r P1 , with j (q ) = r and j mapping the genus one components two-to-one onto rational components, with rami?cation at q along each component. Consider stable curves of genus two with automorphisms beyond ι. There are two possibilities: Either (R, b1 , . . . , b6 ) admits automorphisms permuting the bi or j : C → R admits covering transformations other than the canonical hyperelliptic involution. The classi?cation of automorphism groups ([Bo] or [Ig1]§8) yields the following possibilities in codimension one: 1. the curves C in ?1 ; here j : C → P1 ∪ P1 admits involutions ?xing each component of C ; 2. the closure of the locus of curves j : C → P1 branched over six points admitting a bielliptic involution. At each point [C ] ∈ M 2 , the moduli space is ? etale-locally isomorphic to the quotient T[C ] M2 /Aut(C ) at the origin, where T[C ] M2 = Ext1 (?1 C , OC )) 15

j

is the tangent space with the induced automorphism action. When C is smooth, Serre duality gives
2 ? T[C ] M2 = Γ(C, ωC ) ;

equation (2) shows that the hyperelliptic involution acts trivially and a bielliptic involution acts by re?ection. This local isomorphism can be chosen so the divisors ?0 and ?1 , correspond to the images of unions of distinguished hyperplanes in T[C ] M2 . The local-global spectral sequence gives an exact sequence
1 1 1 1 0 → H 1 (H om (?1 C , OC )) → Ext (?C , OC ) → Γ(E xt (?C , OC )) → 0.

(5)

A local computation implies Ext 1 (?1 C , OC ) ? ⊕nodes p∈C k. If C = E1 ∪q E2 is in ?1 , let δ1 ? T[C ] M2 denote the hyperplane corresponding to the node disconnecting the genus-one components of C , i.e., the kernel of the projection onto the direct summand corresponding to q . The extra covering involutions of j : C → P1 ∪ P1 mentioned above act on T[C ] M2 , trivially on δ1 and by multiplication by (?1) on Ext 1 (?1 C , OC )q . This corresponds to re?ection across δ1 . Let δ0 denote the union of the hyperplanes corresponding to each of the non-disconnecting nodes of C . The surjectivity of the last arrow in (5) means that δ = δ1 ∪ δ0 is normal crossings. De?nition 3.4 Let ξ ? M2 (resp. Ξ ? M 2 ) denote the closure of the smooth curves admitting a bielliptic involution. These are irreducible of codimension one, e.g., by Proposition 2.1 and the characterization of bielliptic curves. This can also be seen in?nitesimally: Under the local identi?cation with T[C ] M2 /Aut(C ), each branch of ξ is identi?ed with the hyperplane T[C ] M2 ?xed by the corresponding bielliptic involution (acting by re?ection on the tangent space). The union of these hyperplanes is denoted ξ . The bielliptic divisor has more complicated local geometry, as Ξ may have quite a few local branches. For example, F (x, y ) = xy (x ? ξy )(x + ξy )(x ? ξ ?1 y )(x + ξ ?1 y ) 16

has automorphism group isomorphic to the Klein four-group and admits two involutions [x, y ] → [y, x] [x, y ] → [?y, x]. The cyclotomic form F (x, y ) = x6 +y 6 = (x?ζy )(x?ζ 3 y )(x?ζ 5y )(x?ζ 7 y )(x?ζ 9y )(x?ζ 11 y ) ζ 12 = 1 has automorphism group isomorphic to the dihedral group with 12 elements and admits four distinct involutions [x, y ] → [y, x] [x, y ] → [?x, y ] [x, y ] → [ζ 4y, x] [x, y ] → [ζ 8y, x]. In particular, ξ is not normal crossings. Proposition 3.5 (Rami?cation formula for Q : M2 → M 2 ) 1 1+α ?1 + Ξ) 2 2 proof: Lemma 3.3 allows us to consider the rigidi?cation ([ACV] §5) of M2 with respect to the group ι generated by the canonical involution KM2 + αδ ≡ Q? (KM 2 + α?0 + r : M2 → M2
<ι>

.

Given a scheme T , T -valued points of correspond to families of genus two stable curves over T , where two families are identi?ed if they di?er by a <ι> ι -valued cocycle over T . The inertia group of M2 at [C ] is the quotient Aut(C )/ ι . We have a factorization M2 → M2
r <ι> q ?

<ι> M2

→ M2
<ι>

so that r is ? etale of degree two and M 2 is the coarse moduli space of M2 [ACV] Theorem 5.1.5. We therefore obtain the following formula for dualizing sheaves r ? ωM<ι> ? ωM2 .
2

Let and ξ denote the corresponding Cartier divisors in M2 As q ? has simple rami?cation along these divisors, we obtain
<ι> q ?? ?1 = 2δ1 q ?? Ξ = 2ξ <ι> <ι> ? ξ <ι> , q ?? KM 2 = KM<ι> ? δ1
2

<ι> δ1

<ι>

<ι>

.

which together imply the formula. 17

3.3

Analysis of b : X → X along the exceptional divisor

By Theorem 2.8, the exceptional divisor ?1 is mapped isomorphically to the locus in A2 parametrizing abelian surfaces decomposing into products of elliptic curves (as a principally polarized abelian variety); this is isomorphic to P(2, 3, 6). Proposition 3.6 Let Ξ denote the proper transform of Ξ in X . We have the following formulas: KX ≡ b? KX + 10?1 b? {G = 0} ≡ Ξ + 12?1 b? {D = 0} ≡ ?0 + 6?1 Ξ ≡ 3?0 + 12?1 . We pause to explore the geometry of ?1 . Naively, one might expect this to be the symmetric square of the moduli space of elliptic curves. However, in taking symmetric squares we should be mindful of the stack structures. The coarse moduli space of the symmetric square need not be isomorphic to the symmetric square of the coarse moduli space. The standard theory of modular forms implies M1,1 ? P (4, 6) = (Spec k [g2 , g3 ] ? (0, 0)) /Gm , with Gm acting with weights (4, 6) t · (g2 , g3 ) → (t4 g2 , t6 g3 ). The coarse moduli space is M 1,1 = A1 ? Proj k [g2 , g3 ] ? P1 . The symmetric square of the stack has the following quotient-stack presentation: (M1,1 × M1,1 )/S2 = (Spec k [g2 , g3 , h2 , h3 ] ? Z }/ H Z = {(g2 , g3 , h2 , h3 ) : g2 = g3 = 0 or h2 = h3 = 0}. Here H is the group generated by the torus (t, u) · (g2 , g3 , h2 , h3 ) → (t4 g2 , t6 g3 , u4 h2 , u6 h3 ) 18

and the involution (g2 , g3 , h2 , h3 ) → (h2 , h3 , g2 , g3 ),
2 i.e., H = S2 ? G2 m where S2 acts on Gm by interchanging the factors. The coarse moduli space of the stack is the invariant-theory quotient for the action of H . Consider the elements p ∈ k [g2 , g3 , h2 , h3 ] with the following properties:

1. p(g2 , g3 , h2 , h3 ) = p(h2 , h3 , g2 , g3 ); 2. p(t2 g2 , t3 g3 , u2 h2 , u3 h3 ) = (tu)N p(g2 , g3 , h2 , h3 ) for some N .
3 2 3 This ring is generated by g2 h2 , g3 h3 , and g2 h3 + g3 h2 and 3 2 2 3 Proj k [g2 h2 , g3 h3 , g2 h3 + g3 h2 ] ? P(4, 6, 12) ? P(2, 3, 6),

which explains why the weights of b are (2, 3, 6). See [Ig2], Theorem 3, for a discussion in terms of the modular forms for Sp(2, Z) (see Remark 2.9). proof of proposition: The ?rst equation follows because b : X → X has weights (2, 3, 6). As for the second, Ξ is the proper transform of the divisor {G = 0} ? X parametrizing forms admitting an bielliptic involution. When a bielliptic curve of genus two specializes to a stable curve in ?1 , the bielliptic involution specializes to a morphism exchanging the elliptic components. Therefore, Ξ ∩ ?1 ? ?1 is the diagonal in the symmetric square, which is cut out by a form of weighted-degree twelve. For the third equation, ?0 is the proper transform of {D = 0}. The 3 2 intersection ?0 ∩ ?1 ? ?1 is the locus where the discriminant ? = g2 ? 27g3 vanishes, and thus has weighted-degree six. The last equation follows because G has weighted degree thirty and D has weighted degree ten. One consequence of this analysis is worth mentioning. Proposition 3.7 For su?ciently small ? > 0, the divisor ?0 + (6 ? ?)?1 is ample on X . proof: The divisor can be expressed b? (ample divisor) ? ?(b-exceptional divisor), which yields a polarization of the blow-up b : X → X . 19

4
4.1

Birational geometry
Divisor classes and birational contractions of M 2

It is well known that the rational divisor class group of M 2 is freely generated by the boundary divisors ?0 and ?1 ; see [HaMo] for a nice account of divisors on M g for arbitrary g . When g = 2, Proposition 2.7 gives an elementary proof of this fact: Since X ? P(1, 2, 3, 5) its divisor class group has rank one and is generated by the discriminant divisor {D = 0}; the same holds true for X ? p. By Proposition 2.10 M 2 is isomorphic to X up to codimension ≤ 1, so these have isomorphic class groups. It follows that the rational divisor class group of M 2 is generated by ?1 and the proper transform of the discriminant, which is just ?0 . The nef cone of M 2 is also well-known. We will not give a self-contained proof here, but rather rely on the general result of Cornalba-Harris [CH]: Theorem 4.1 The line bundle aλ ? bδ is nef on M g , g ≥ 2, if and only if a ≥ 11b ≥ 0. Here λ is the pull-back of the polarization on Ag via the extended Torelli map t : M g → Ag (see §2.6). To apply this in our situation, we observe that λ≡ 1 (?0 + ?1 ) 10

over M 2 (see [HaMo] pp. 175). The factor 10 can be explained by the fact that t(?1 ) is de?ned by the vanishing of a modular form of weight ten (see Remark 2.9 and [Ig2]). Substitution gives the ?rst part of Proposition 4.2 The nef cone of M 2 is generated by the divisors ?0 + ?1 and ?0 +6?1 , respectively. These are both semiample, inducing the birational contractions t : M 2 → A2 β : M 2 → X. The rational map γ : M 2 X is an isomorphism.

remainder of proof: Of course, ?0 + ?1 is semiample and induces the birational contraction morphism t : M 2 → A2 . As X and M 2 are isomorphic in codimension one, Proposition 3.7 says that ?0 + (6 ? ?)?1 is ample on X 20

and the corresponding divisor ?0 + (6 ? ?)?1 is ample on M 2 . It follows that X and M 2 are each isomorphic to Proj of ⊕n≥0 Γ(OX (n(?0 + (6 ? ?)?1 ))) ? ⊕n≥0 Γ(OM 2 (n(?0 + (6 ? ?)?1 ))). In particular, the rational map γ is an isomorphism. Thus the contractions X coincide. b : X → X and β : M 2

4.2

Canonical class of M 2

The canonical class KM 2 can also be computed by elementary methods. We know that ωP(1,2,3,5) ? OP(1,2,3,5) (?11), which also follows from the third part of Proposition 3.1. Since the discriminant has degree ten, we ?nd 11 { D = 0} . 5 Applying the formulas of Proposition 3.6, we obtain KX = ? 16 11 ?0 ? ?1 5 5 1 1 7 3 KX + ?1 + Ξ ≡ ? ?0 + ?1 . 2 2 10 10 Since X and M 2 agree in codimension one, they have the same canonical class 11 16 KM 2 ≡ ? ?0 ? ?1 . 5 5 In particular, we obtain KX ≡ ? KM 2 + α?0 + 1 7 3 1+α ?1 + Ξ ≡ (? + α)?0 + ( + α/2)?1 . 2 2 10 10 (6)

Remark 4.3 The importance of this divisor stems from the rami?cation equation of Proposition 3.5. This divisor class pulls back to the class KM2 + αδ on the moduli stack. We shall interpret log canonical models of the moduli stack using this divisor. 21

Using the computations of Proposition 3.6, we obtain a discrepancy equations for β : M 2 → X 1 1 KM 2 + α?0 + Ξ = β ? (KX + α{D = 0} + {G = 0}) + (4 ? 6α)?1 2 2 (7)

KM 2 + α?0 +

1 1 9 ? 11α 1+α ?1 + Ξ = β ? (KX + α{D = 0} + {G = 0})+ ?1 . 2 2 2 2 (8)

4.3

Generalities on log canonical models

See [FA] §2 for de?nitions of the relevant terms and technical background. Let V be a normal projective variety, D = n i=1 ai Di a Q-divisor such that 0 ≤ ai ≤ 1 and KV + D is Q-Cartier. Abusing notation, we write V ? D for V ? ∪i Di . De?nition 4.4 (V, D) is a strict log canonical model if KV + D is ample, (V, D) has log canonical singularities, and V ? D has canonical singularities. The idea here is to realize (V, D) as a log canonical model without introducing boundary divisors over V ? D . This is natural if we want to respect the geometry of the open complement. The following recognition criterion for strict log canonical models is based on [FA] §2: Proposition 4.5 Consider birational projective contractions ρ : V → V ′ where the exceptional locus of ρ is divisorial, Di denotes the proper transform of Di , and Ej (resp. Fk ) denotes the exceptional divisors of ρ with ρ(Ej ) ? D (resp. ρ(Fk ) ? D ). The following are equivalent: 1. (V, D) is a strict log canonical model. 2. For some resolution of singularities ρ : V → V , with the union of the ′ exceptional locus and ∪i Di normal crossings, there exist bj ∈ Q ∩ [0, 1] so that ′ D= ai Di + bj Ej
i j

22

satis?es the formula KV + D ≡ ρ? (KV + D ) +
j

dj Ej +
k

ek Fk ,

dj , ek ≥ 0.

(9)

For each such choice of bj KV + D ?
j

dj Ej ?
k

ek Fk

is semiample and induces ρ. Furthermore, we may take the bj = 1. 3. For some contraction ρ : V → V , there exist bj ∈ Q ∩ [0, 1] so that D=
i ′ ai Di + j

bj Ej

satis?es the formula KV + D ≡ ρ? (KV + D ) +
j

dj Ej +
k

ek Fk ,

dj , ek ≥ 0,

(V , D ) is log canonical, and V ? D has canonical singularities. The divisor dj Ej ? ek Fk KV + D ?
j k

is semiample and induces ρ. Some general facts are worth mentioning before we indicate the proof. First, we can decide whether a pair is canonical or log canonical by computing discrepancies on any resolution. Second, discrepancies increase as the coef?cients of the log divisor are decreased [FA] 2.17.3. Third, in situations (2) and (3) the pair (V, D) is the log canonical model of (V , D); such models are unique [FA] 2.22.1. proof: It is trivial that the second statement implies the third. To see that the third implies the ?rst, take a resolution for (V , D) so that the union of ′ the exceptional locus and all the proper transforms of the Di , Ej , and Fk is normal crossings. Comparing discrepancies for (V , D ) and (V, D), using the fact the coe?cients of components in D are at least as large as the coe?cents of the corresponding components appearing in ρ? (KV + D ), we 23

?nd that (V, D) is log-canonical and has canonical singularities along V ? D . Since ρ? (KV + D ) induces ρ, KV + D must be ample on V . For the remaining implication, since (V, D) has log canonical singularities and canonical singularities away from D , the discrepancy equation (9) follows. Since KV + D is ample on V , its pull-back to V is semiample and induces ρ. We shall also need the following basic fact, a special case of [FA] 20.2, 20.3: Proposition 4.6 Let W be a smooth variety and h : W → V be a ?nite dominant morphism to a normal variety. Let D = i ai Di , 0 ≤ ai ≤ 1 be a Q-divisor on V containing all the divisorial components of the branch locus ? be a Q-divisor on W so that supp(h?1 (D )) = supp(D ? ) and of h. Let D ? ? . Then (V, D) has log canonical singularities along h ( KV + D ) = K W + D ? ?. D i? (W, D ) has log canonical singularities along D ? have multiplicity one at each component then h? (KV + D ) = If D and D ? follows from the other assumptions. KW + D

4.4

Example of M g

The standpoint of this section owes a great deal to Mumford [Mu1] [Mu2]: Theorem 4.7 For g ≥ 4, the pair (M g , ?) is a strict log canonical model. Remark 4.8 This is also the natural log canonical model from the point of view of the moduli stack. Indeed, for g ≥ 4 the locus in Mg of curves with automorphisms has codimension ≥ 2, so the branch divisor of Q : Mg → M g is just ?1 ; over ?1 , we have simple rami?cation. We therefore have ([HaMu] pp. 52) Q? KM g = KMg ? δ1 and thus Q? (KM g + ?) = KMg + δ. sketch proof: We ?rst check that KM g + ? is ample. The formula from [HaMu] §2 (or [HaMo]) KM g = 13λ ? 2?0 ? 3/2?1 ? 2
2≤i≤g/2

?i

gives KM g + δ = 13λ ? δ , which is ample by Theorem 4.1 (see also [Mu1]). 24

The singularity analysis follows [HaMu]. M g has canonical singularities by Theorem 1 of [HaMu]. To show that (M g , ?) has log canonical singularities, we use the fact that M g is ? etale-locally a quotient of a smooth variety by a ?nite group. At [C ] it has a local presentation h : T[C ] Mg ? T[C ] Mg /Aut(C ) in terms of its tangent space T[C ] Mg = Ext1 (?1 C , OC ). We analyze the quotient morphism using Proposition 4.6. The preimage of the boundary divisor corresponds to a union of hyperplanes δ ? T[C ] Mg , ? then has log canonical meeting in normal crossings. The pair (T[C ] Mg , ?) singularities. An application of Proposition 4.6, utilizing the rami?cation discussion in Remark 4.8, implies (KMg , ?) is log canonical. Remark 4.9 Using the full force of Theorem 4.1 we get a sharper statement. Consider the pair (M g , α?0 + 1+α ?1 + α(?2 + . . . + ??g/2? )), 2

with log canonical divisor pulling back to KMg + αδ on the moduli stack. The Q-divisor KMg + αδ is the pull-back of an ample line bundle if and only if 9/11 < α ≤ 1. Since M g is a locally a quotient of a smooth variety by a ?nite group, all divisors on M g are Q-Cartier. An easy computation with the discrepancy equation (9) then shows that the pair remains log canonical even as the coe?cients are reduced.

4.5

Application to M2

The source of the special di?culties in this case is the fact that KM 2 + ? is not e?ective. Indeed, in §4.1 we computed KM 2 ≡ ? 11?0 + 6?1 5

so KM 2 + ? = ?e?ective divisor. In order to recover a result analogous to Theorem 4.7, we must take the ‘log canonical model of the moduli stack’, as interpreted on M 2 via Proposition 3.5: 25

Theorem 4.10 Consider the log canonical model of M2 with respect to the KM2 + αδ , i.e., the log canonical model of M 2 with respect to KM 2 + α?0 + 1 1+α ?1 + Ξ. 2 2

1. For 9/11 < α ≤ 1, we recover M 2 . 2. For 7/10 < α ≤ 9/11 we recover the invariant theory quotient X ? P(1, 2, 3, 5). 3. For α = 7/10 we get a point; the log canonical divisor fails to be e?ective for α < 7/10. proof: The necessary ampleness results have already been stated. Proposition 4.2 and Equation (6) imply the log canonical divisor on M 2 is ample if and only if α > 9/11. Proposition 3.1 implies that KX + 1 / 2 { G = 0 } + α { D = 0 } is positive on X if and only if α > 7/10. When α = 7/10 it is zero and when α < 7/10 it is negative. It remains to verify the singularity conditions: First, we check that M 2 has canonical singularities away from ?0 , ?1 , and Ξ. Suppose that C is not in the boundary and does not admit admit a bielliptic involution. In Proposition 2.4 we saw M2 ? X ? {D = 0} ? P(1, 2, 3, 5), so we need to analyze the singularities of P(1, 2, 3, 5) ? {D = 0}. A point in weighted projective space is nonsingular when the weights corresponding to its non-vanishing coordinates are relatively prime, so the only possible singularity occurs when A = B = C = 0. The corresponding binary sextic form is x(x5 + y 5 ), the unique form with an automorphism group of order ?ve [Bo] pp. 51 [Ig1] pp. 645. At this point, P(1, 2, 3, 5) is locally isomorphic to the cyclic quotient singularity 1 (1, 2, 3), i.e., the quotient of A3 under the action 5 (a, b, c) → (ζa, ζ 2b, ζ 3 c) ζ = 1 ∈ ?5 . 26

This is canonical by the Reid-Tai criterion; see [HaMu] pp. 28 for a general result. Second, we address the singularities along the boundary. For α > 9/11 we need that 1+α KM 2 + α?0 + ?1 + 1/2Ξ 2 is log canonical. When α ≤ 9/11, Proposition 4.5 and the discrepancy computation (8) reduce us to showing that this is log canonical. Since M 2 is Q-factorial and discrepancies increase as coe?cients of log divisors decrease [FA] 2.17.3, it su?ces to verify that 1 KM 2 + ?0 + ?1 + Ξ 2 (10)

is log canonical. The proof relies on the description of the boundary divisors in terms of the local presentation T[C ] M2 /Aut(C ), as sketched in §3.2. The key observation is that Ξ does not play a r? ole in the analysis. Each bielliptic involution acts on T[C ] M2 by re?ection across the corresponding hyperplane in ξ , so the quotient h : T[C ] M2 → T[C ] M2 /H, H = Aut(C )/ ι ,

has simple rami?cation along ξ . Since Ξ has coe?cient 1/2 in (10), ξ does not appear in the pull-back of the log canonical divisor to T[C ] M2 . Thus (10) pulls back to KT[C] M2 + δ, and we have seen that δ is normal crossings. It follows that (T[C ] M2 , δ ) is log canonical and Proposition 4.6 gives the desired result.

References
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[Ca] A. Cayley, Tables for the binary sextic, Amer. J. Math. 4 (1881), 379– 384. Reprinted in: Collected Mathematical Papers, vol. XI, pp. 372-376, Cambridge University Press, 1896. [Cl] A. Clebsch, Theorie der bin¨ aren algebraischen Formen, Verlag von B.G. Teubner, Leipzig, 1872. [CH] M. Cornalba and J. Harris, Divisor classes associated to families of stable varieties, with applications to the moduli space of curves, Ann. Sci. ? Ecole Norm. Sup. (4) 21 (1988), no. 3, 455–475. [El] E.B. Elliott, An Introduction to the Algebra of Quantics, Oxford University/Clarendon Press, 1895. [FA] J. Koll? ar, Flips and abundance for algebraic threefolds, Ast? erisque 211, 1992. [HaMo] J. Harris and I. Morrison, Moduli of Curves, Springer-Verlag, New York, 1998. [HaMu] J. Harris and D. Mumford, On the Kodaira dimension of the moduli space of curves, Invent. math. 67 (1982), 23–86. [Ha] R. Hartshorne, Algebraic geometry, Springer-Verlag, New YorkHeidelberg, 1977. [Hi] D. Hilbert, Theory of algebraic invariants, Cambridge University Press, 1993. [Ig1] J. Igusa, Arithmetic variety of moduli for genus two, Ann. of Math. (2) 72 (1960) 612–649. [Ig2] J. Igusa, On Siegel modular forms of genus two, Amer. J. Math. 84 (1962) 175–200. [Ig3] J. Igusa, A desingularization problem in the theory of Siegel modular functions, Math. Ann. 168 (1967) 228–260 [LM] G. Laumon and L. Moret-Bailly, Champs alg? ebriques, Springer-Verlag, Berlin-Heidelberg, 2000.

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[Mu1] D. Mumford, Stability of projective varieties, Enseignement Math. (2) 23 (1977), no. 1-2, 39–110. [Mu2] D. Mumford, Hirzebruch’s proportionality theorem in the noncompact case, Invent. Math.42 (1977), 239–272. [GIT] D. Mumford, J. Fogarty, and F. Kirwan, Geometric Invariant Theory, third enlarged edition, Springer-Verlag, Berlin-Heidelberg, 1994. [Na1] Y. Namikawa, On the canonical holomorphic map from the moduli space of stable curves to the Igusa monoidal transform, Nagoya Math. J. 52 (1973), 197–259. [Na2] Y. Namikawa, A new compacti?cation of the Siegel space and degeneration of Abelian varieties I/II Math. Ann. 221 (1976), no. 2, 97–141/ no. 3, 201–241. [Sa] G. Salmon, Lessons introductory to the modern higher algebra, 2nd edition, Hodges Smith, Dublin, 1866. [SB] N. I. Shepherd-Barron, Canonical rings for moduli spaces of abelian varieties, preprint (2003). [Sc] I. Schur, Vorlesungen u ¨ber Invariententheorie, Bearbeitet und herausgegeben von Helmut Grunsky, Springer-Verlag, Berlin-New York 1968 [Sp] T.A. Springer, Invariant theory, Lecture Notes in Mathematics, Vol. 585, Springer-Verlag, Berlin-New York, 1977. [Vi] A. Vistoli, The Chow ring of M2 , Appendix to ”Equivariant intersection theory” by D. Edidin and W. Graham, Invent. Math. 131 (1998), no. 3, 635–644. Rice University, MS 136 Department of Mathematics PO Box 1892 Houston, TX 77251-1892 USA hassett@math.rice.edu

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