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 M_{g} is of general type when g ≥ 24 (see [HaMu] and subsequent papers). A standard conjecture of birational geometry–the finite generation of the canonical ring–would imply the canonical model is

Proj ⊕_{n≥0} Γ(\overline{M}_{g}, nK_{\overline{M}_{g}}).

Unfortunately, this has yet to be verified in a single genus!

Another possible line of attack is to consider log canonical models of the moduli space. The moduli space is best regarded as a pair (\overline{M}_{g}, Δ), where \overline{M}_{g} is Deligne-Mumford compactification by stable curves and Δ is its boundary. It is implicit in the work of Mumford [Mu1] that the moduli 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

∗Partial support was provided by National Science Foundation grants 0196187 and 0134259 and the Sloan Foundation.

Proj ⊕_{n≥0} Γ(\overline{M}_{g}, n(K_{\overline{M}_{g}} + αΔ)),

where α ∈ ℚ ∩ [0, 1] is chosen so that K + αΔ is effective.

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 \overline{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 \overline{M}_{g}. In particular, it is necessary to use the canonical divisor of the moduli stack rather than its coarse moduli space. These differ substantially, as the natural morphism \overline{M}_{g} → \overline{M}_{g} is ramified at stable curves admitting automorphisms.

Luckily, we have inherited a tremendously rich literature on curves of small genus. The invariant-theoretic properties of M_{2} 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 field 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 M_{g} (resp. \overline{M}_{g}); the corresponding coarse moduli scheme is denoted M_{g} (resp. \overline{M}_{g}). The boundary divisors in \overline{M}_{g} (resp. \overline{M}_{g}) are denoted

Δ_{0}, Δ_{1}, …, Δ_{⌊g/2⌋}(resp. δ_{0}, δ_{1}, …, δ_{⌊g/2⌋}).

Let Q : \overline{M}_{g} → \overline{M}_{g} denote the natural morphism from the moduli stack to the coarse moduli space, so that

Q^{∗}Δ_{i} = δ_{i}, i ≠ 1 Q^{∗}Δ_{1} = 2δ_{1}.

We write δ = ∑_{0≤i≤g/2} δ_{i}; abusing notation, we also use δ for the corresponding divisor

Δ_{0} + 1/2Δ_{1} + Δ_{2} + … + Δ_{⌈g/2⌉}

on \overline{M}_{g}.

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 benefited from conversations with James Spencer about rigidification, 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-Universität, Göttingen.

2 Classical geometry

2.1 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 differentials ω_{C}. The global sections of ω_{C} give the canonical morphism

j : C → ℙ(Γ(C, ω_{C})) ≃ ℙ^{1}

which is finite 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

{b_{1}, …, b_{6}} ⊂ ℙ(Γ(C, ω_{C})).

On fixing an identification ℙ(Γ(C, ω_{C})) ≃ ℙ^{1}, we can write down a nontrivial binary sextic form vanishing at the branch points

F ∈ Γ(ℙ^{1}, O_{ℙ^{1}}(6)),

determined by {b_{1}, …, b_{6}} ∈ ℙ^{1} up to a scalar.

Conversely, suppose we have a binary sextic form F with six distinct zeros b_{1}, …, b_{6} ∈ ℙ^{1}. Then there is a unique degree-two cover of ℙ^{1} branched over these points. This is a smooth curve of genus two and the map to ℙ^{1} is the canonical morphism. Moreover, the isomorphism class of C depends only on the orbit of F under the action of GL_{2}.

To summarize: There is a one-to-one correspondence between isomorphism classes of curves of genus two and GL_{2}-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 finite type over k, with fibers curves of genus two. Since the relative dualizing sheaf ω_{π} is globally generated, there is a relative double cover

C  ⟶^{j}  ℙ^{1}(π_{∗}ω_{π}) := ℙ
π ↘  ↙ ψ
S

with associated involution ι. Using the trace we decompose j_{∗}O_{C} ≃ O_{ℙ} ⊕ L, where L has relative degree −3 on the fibers of ψ. The O_{ℙ}-algebra structure on O_{C} is thus determined by an isomorphism L^{2} → O_{ℙ}, i.e., by a nonvanishing section of L^{−2} with zeros along the branch locus of j. By relative duality

j_{∗}ω_{π} = ℋ om_{O_{ℙ}}(j_{∗}O_{C}, ω_{ψ}) ≃ ω_{ψ} ⊕ (ω_{ψ} ⊗ L^{−1})

which yields

π_{∗}ω_{π} ≃ ψ_{∗}(ω_{ψ} ⊗ L^{−1}).

Using the identifications

ψ_{∗}O_{ℙ}(+1) = π_{∗}ω_{π}, ω_{ψ} = (ψ^{∗} det π_{∗}ω_{π})(−2), Pic(ℙ) = Pic(S)⊕ℤc_{1}(O_{ℙ}(+1)),

we find

L^{−1} ≃ O_{ℙ}(+1) ⊗ ω_{ψ}^{−1} = O_{ℙ}(3) ⊗ (ψ^{∗} det π_{∗}ω_{π})^{−1}.

The class of the branch divisor is thus

−2c_{1}(L) = c_{1}(O_{ℙ}(6)) − 2ψ^{∗}c_{1}(det π_{∗}ω_{π}).  (1)

This has practical implications: An isomorphism of C induces a linear transformation on Γ(C, ω_{C}), which respects the binary sextic F up to a scalar. Formula (1) allows us to keep track of this scalar. For each M ∈ GL_{2}, we have the linear action

(x, y) ↦ (x, y) (\begin{matrix}m_{11} & m_{12} \\ m_{21} & m_{22} \end{matrix})

which induces a natural left action on binary sextic forms

F ↦ F (xm_{11} + ym_{21}, xm_{12} + ym_{22}).

We normalize this action using formula (1)

F ↦ (M, F) := (det M)^{−2}F (xm_{11} + ym_{21}, xm_{12} + ym_{22}),  (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 →^{i} E
j ↓  ↓ j̄
ℙ^{1}  →^{ī} ℙ^{1}

where ī and j̄ are the double covers induced on quotients. The branch locus of j is preserved by the covering transformation for ī, 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

{[α_{1}, 1], [1, α_{1}], [α_{2}, 1,][1, α_{2}], [α_{3}, 1], [1, α_{3}]}

for some α_{1}, α_{2}, α_{3} ∈ 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 coefficients of a binary sextic

F = ax^{6} + 6bx^{5}y + 15cx^{4}y^{2} + 20dx^{3}y^{3} + 15ex^{2}y^{4} + 6fxy^{5} + gy^{6}.

The action (2) induces an action of GL_{2} on k[a, b, c, d, e, f, g]. Recall that a polynomial P ∈ k[a, b, c, d, e, f, g] is SL_{2}-invariant if for each M ∈ SL_{2}, we have

(M, P) = P.

We write

R := k[a, b, c, d, e, f, g]^{SL^{2}}

for the ring of such invariants. If P is SL_{2}-invariant then each homogeneous component of P is as well, so R is a graded ring. Every homogeneous invariant satisfies the functional relation

(M, P) = (det M)^{deg(P)}P, M ∈ GL_{2};  (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 first 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 SL_{2} is [Sp].

For our purposes, the symmetric function representation of the invariants in [Ig2] pp. 176 and 185 is the most useful. Let ξ_{1}, …, ξ_{6} denote the roots of the dehomogenized form F (x, 1), and write (ij) as shorthand for ξ_{i} − ξ_{j}. We write

A = a^{2} ∑_{fifteen}(12)^{2}(34)^{2}(56)^{2}
B = a^{4} ∑_{ten}(12)^{2}(23)^{2}(31)^{2}(45)^{2}(56)^{2}(64)^{2}
C = a^{6} ∑_{sixty}(12)^{2}(23)^{2}(31)^{2}(45)^{2}(56)^{2}(64)^{2}(14)^{2}(25)^{2}(36)^{2}
D = a^{10} ∏_{ij}(ij)^{2}

E = a^{15} ∏_{fifteen} det (\begin{matrix}1 & ξ_{1} + ξ_{2} & ξ_{1}ξ_{2} \\ 1 & ξ_{3} + ξ_{4} & ξ_{3}ξ_{4} \\ 1 & ξ_{5} + ξ_{6} & ξ_{5}ξ_{6} \end{matrix}) = a^{15} ∏_{fifteen} ((14)(36)(52) − (16)(32)(54))

where the summations are chosen to make the expressions 𝔖_{6}-symmetric. Consequently, A, B, C, D, and E can all be expressed as polynomials in ℚ[a, b, c, d, e, f, g], e.g.,

A = −240(ag − 6bf + 15ce − 10d^{2})

B = −162000 det (\begin{matrix}a & b & c & d \\ b & c & d & e \\ c & d & e & f \\ d & e & f & g \end{matrix}) + 1620(ag − 6bf + 15ce − 10d^{2})^{2}.

In classical terminology, (ag − 6bf + 15ce − 10d^{2}) is the sixth transvectant of F over itself; transvection is one of the main operations in Gordan’s proof of finiteness 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 defined 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 fifteen are unique up to scalar.

2.3 The projective invariant-theory quotient

We consider

X := Proj R = Proj \frac{k[A, B, C, D, E]}{⟨E^{2} − G(A, B, C, D)⟩}.

If A = B = C = D = 0 then E = 0 as well, so X is covered by the distinguished affine 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][B^{−1}])_{0}
(k[A, B, C, D, E][C^{−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 find

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 first 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 Proposition 2.4 The moduli scheme M_{2} can be identified with X−{D = 0}, where D is the discriminant.

Remark 2.5 This construction definitely fails in characteristic two. If the double cover j : C → ℙ^{1} is wildly ramified, the branch divisor may have multiplicities > 3. These curves correspond to unstable points under the SL_{2}-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.

Definition 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.

Proposition 2.7 There exists a birational contraction β : \overline{M}_{2} −−→ X restricting to the identity along the open subset M_{2}. β is an isomorphism over \overline{M}_{2} − Δ_{1} and contracts Δ_{1} to the point p.

proof: To produce the birational contraction, we exhibit a morphism

β^{−1} : U ↪ \overline{M}_{2}

where U ⊂ X is open with complement of codimension ≥ 2 and β^{−1}|_{M_{2}∩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 := {ax^{6}+6bx^{5}y+15cx^{4}y^{2}+20dx^{3}y^{3}+15ex^{2}y^{4}+6fxy^{5}+gy^{6} = 0} ⊂ 𝔸^{7}×ℙ^{1}.

Its class in Pic(𝔸^{7} × ℙ^{1}) is divisible by two, so there exists a double cover C′ → 𝔸^{7} × ℙ^{1} simply branched over W. Composing with the projection onto the first factor, we obtain a morphism

π′ : C′ → 𝔸^{7}.

Let S ⊂ 𝔸^{7} 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 flat morphisms, it is also flat.

Consider the fiber of π over a given binary sextic F: It is a double cover j : C_{F} → ℙ^{1} branched over the zeros of F. We claim C_{F} is a stable curve of genus two, not contained in Δ_{1}. Evidently C_{F} is smooth and simply branched over the zeros with multiplicity one. Over the double zeros C_{F} has local equation y^{2} = x^{2}, which defines a node. We have j^{∗}O_{ℙ^{1}}(+1) = ω_{C_{F}}, which is ample on C_{F}, so C_{F} is stable. The normalization ν : C_{F}^{ν} → C_{F} is the double cover branched along F′ = 0, where F′ is the product of the factors of F with multiplicity one; C_{F} is obtained from C_{F}^{ν} by gluing the pairs of points over the each double root of F. There are three possibilities:

1. deg(F′) = 4, in which case C_{F}^{ν} is connected of genus one;

2. deg(F′) = 2, in which case C_{F}^{ν} is connected of genus zero;

3. deg(F′) = 0, in which case C_{F}^{ν} has two connected components of genus zero.

Since C_{F} cannot be expressed as the union of two subcurves of genus one meeting at a point, the resulting curve is not in Δ_{1}.

The classifying morphism

S → \overline{M}_{2} − δ_{1}

is equivariant with respect to the GL_{2}-action on binary sextics, and therefore descends to a morphism

Φ : S/GL_{2} → \overline{M}_{2} − δ_{1}.

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/GL_{2}. We define 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 Δ_{1}. Quite generally, ω_{C} is globally generated for any stable curve without disconnecting nodes; the only curves in \overline{M}_{2} with disconnecting nodes lie in Δ_{1}. Thus the sections of ω_{C} give a double cover

j : C → ℙ^{1}

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 A_{g} denote the moduli space of principally polarized abelian varieties of dimension g, \overline{A}_{g} its Satake compactification. Recall that \overline{A}_{g} = Proj S, where S is the ring of Sp(g, ℤ)-modular forms; we use λ to denote the resulting polarization on \overline{A}_{g}. Let t : M_{g} ↪ A_{g} denote the Torelli morphism, associating to each curve its Jacobian.

Now assume g = 2. Regarding M_{2} as an open subset of X (see Proposition 2.4), t extends to a rational map

τ : X −−→ \overline{A}_{2}.

The inclusion and the Torelli morphism induce

M_{2} ↪ X̃ := \overline{Graph(τ)} ⊂ X × \overline{A}_{2}.  (4)

In particular, X̃ compactifies M_{2}.

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

x_{1} = 2^{4}3^{2}B/A^{2} x_{2} = 2^{6}3^{3}(3C − AB)/A^{3} x_{3} = 2 · 3^{5}D/A^{5}

then τ is resolved by a weighted blow-up centered at p

b : X̃ → X

with weights

weight(x_{1}) = 2 weight(x_{2}) = 3 weight(x_{3}) = 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).

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 M_{2} ↪ X̃ extends to a birational map

γ : \overline{M}_{2} −−→ X̃

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 \overline{t} : \overline{M}_{g} → \overline{A}_{g} [Na1], Theorem 3. This is not an isomorphism for g > 1: The divisor Δ_{0} ⊂ \overline{M}_{g} is mapped to a boundary stratum of \overline{A}_{g}, which has codimension ≥ 2. However, in genus two \overline{t} is an isomorphism at the generic point of Δ_{1}. Indeed, the Jacobian of a curve [E_{1} ∪_{q} E_{2}] ∈ Δ_{1}, with E_{1} and E_{2} smooth of genus one, is the abelian surface E_{1} × E_{2}.

The following diagram summarizes the various birational maps and morphisms:

\begin{matrix}\overline{A}_{2} & ←^{π} & X̃ \\ \overline{t} ↑ & & ↓ b \\ \overline{M}_{2} & −−→^{β} & X \end{matrix} .

By Theorem 2.8, π is also an isomorphism over the generic point of \overline{t}(Δ_{1}), so γ is an isomorphism at the generic point of Δ_{1}. β and b are both isomorphisms over the generic point of the divisor β(Δ_{0}) (see Proposition 2.7), so γ is an isomorphism at the generic point of Δ_{0}. Since γ is regular along M_{2} = \overline{M}_{2} − Δ_{0} − Δ_{1}, the result follows. □

The proper transforms of Δ_{0} and Δ_{1} in X̃ are denoted Δ̃_{0} and Δ̃_{1}. Thus Δ̃_{1} is the exceptional divisor of b : X̃ → X.

Remark 2.11 (Bibliographic note) There are a number of partial desingularizations of \overline{A}_{g} through which \overline{t} factors, e.g., the ‘Igusa monoidal transform’ [Ig3], [Na1] and the toroidal compactification associated to the 2nd Voronoi fan [Na2]. When g = 2, these approaches coincide [Na2] Remark 2.8 and yield a partial desingularization Ã_{2} → \overline{A}_{2}. 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 \overline{A}_{2} − A_{2}. Namikawa [Na1] §9 has shown that the factorization \overline{M}_{2} →^{∼} Ã_{2} is an isomorphism. Notwithstanding Igusa’s explicit formulas for τ : X −−→ \overline{A}_{2} and Ã_{2} → \overline{A}_{2}, it is not entirely obvious how to extract an isomorphism Ã_{2} →^{∼} X̃.

3 Stack geometry

3.1 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 GL_{2}-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 𝔾_{m}- action on the affine variety Y = Spec R. Now 𝔾_{m} acts on the open subset Y −(0, 0, 0, 0, 0) with finite stabilizers and closed orbits, so the quotient stack

X := (Y − (0, 0, 0, 0, 0)) /𝔾_{m}

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 ramification 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)) /𝔾_{m},

with 𝔾_{m} acting with weights (2, 4, 6, 10, 15)

t · (A, B, C, D, E) ↦ (t^{2}A, t^{4}B, t^{6}C, t^{10}D, t^{15}E);

2. the dualizing sheaf of X is given via adjunction

ω_{X} = i^{∗}ω_{P(2,4,6,10,15)}(X) ≃ i^{∗}O_{P(2,4,6,10,15)}(−7),

where O_{P(2,4,6,10,15)}(+1) is the invertible sheaf associated with the principal 𝔾_{m}-bundle classified by the identity character of 𝔾_{m};

3. the ramification divisor of q is

E := {E = 0} ⊂ X

and we have

q^{∗}ω_{X} ≃ ω_{X} (−E) ≃ i^{∗}O_{P(2,4,6,10,15)}(−22).

proof: The first two assertions do not require proof. As for the third, it follows from the classification 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, M_{2} 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 ∈ GL_{2} : (M, F) = F }/{M ∈ SL_{2} : (M, F) = F },

which is trivial for generic binary forms.

The inertia group at [C] ∈ M_{2} 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, ω_{C}) and an induced representation on ∧^{2}Γ(C, ω_{C}) that is not faithful: We do not see elements of Aut(C) acting on Γ(C, ω_{C}) with determinant one, e.g., ι, which acts on Γ(C, ω_{C}) 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. 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 = \begin{cases}ℙ^{1} & if [C] ∉ Δ_{1} \\ ℙ^{1} ∪_{r} ℙ^{1} & if [C] ∈ Δ_{1} \end{cases}

branched over six smooth points b_{1}, …, b_{6} ∈ R and the node r ∈ R. The covering transformation ι therefore commutes with each automorphism of C.

The cover is induced by

C → ℙ(Γ(C, ω_{C}^{2})) ≃ ℙ^{2}

which factors

C →^{j} R ⊂ ℙ^{2},

where R is a plane conic and j is finite of degree two. Indeed, for curves not in Δ_{1} this is the double cover C → ℙ^{1} discussed in the proof of Proposition 2.7. For curves C = E_{1} ∪_{q} E_{2} ∈ Δ_{1} with q = E_{1} ∩ E_{2} the disconnecting node joining the genus-one components E_{1} and E_{2}, we have a double cover

j : E_{1} ∪_{q} E_{2} ⟶ ℙ^{1} ∪_{r} ℙ^{1},

with j(q) = r and j mapping the genus one components two-to-one onto rational components, with ramification at q along each component. □

Consider stable curves of genus two with automorphisms beyond ι. There are two possibilities: Either (R, b_{1}, …, b_{6}) admits automorphisms permuting the b_{i} or j : C → R admits covering transformations other than the canonical hyperelliptic involution. The classification of automorphism groups ([Bo] or [Ig1]§8) yields the following possibilities in codimension one:

1. the curves C in Δ_{1}; here j : C → ℙ^{1} ∪ℙ^{1} admits involutions fixing each component of C;

2. the closure of the locus of curves j : C → ℙ^{1} branched over six points admitting a bielliptic involution.

At each point [C] ∈ \overline{M}_{2}, the moduli space is étale-locally isomorphic to the quotient T_{[C]}\overline{M}_{2}/Aut(C) at the origin, where

T_{[C]}\overline{M}_{2} = Ext^{1}(Ω_{C}^{1}, O_{C}))

is the tangent space with the induced automorphism action. When C is smooth, Serre duality gives

T_{[C]}\overline{M}_{2} = Γ(C, ω_{C}^{2})^{∗};

equation (2) shows that the hyperelliptic involution acts trivially and a bielliptic involution acts by reflection.

This local isomorphism can be chosen so the divisors Δ_{0} and Δ_{1}, correspond to the images of unions of distinguished hyperplanes in T_{[C]}\overline{M}_{2}. The local-global spectral sequence gives an exact sequence

0 → H^{1}(ℋ om(Ω_{C}^{1}, O_{C})) → Ext^{1}(Ω_{C}^{1}, O_{C}) → Γ(ℰxt^{1}(Ω_{C}^{1}, O_{C})) → 0. (5)

A local computation implies

ℰxt^{1}(Ω_{C}^{1}, O_{C}) ≃ ⊕_{nodes p∈C}k.

If C = E_{1} ∪_{q} E_{2} is in Δ_{1}, let δ̂_{1} ⊂ T_{[C]}\overline{M}_{2} 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 → ℙ^{1} ∪ ℙ^{1} mentioned above act on T_{[C]}\overline{M}_{2}, trivially on δ̂_{1} and by multiplication by (−1) on ℰxt^{1}(Ω_{C}^{1}, O_{C})_{q}. This corresponds to reflection across δ̂_{1}. Let δ̂_{0} 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 δ̂ = δ̂_{1} ∪ δ̂_{0} is normal crossings.

Definition 3.4 Let ξ ⊂ \overline{M}_{2} (resp. Ξ ⊂ \overline{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 infinitesimally: Under the local identification with T_{[C]}\overline{M}_{2}/Aut(C), each branch of ξ is identified with the hyperplane T_{[C]}\overline{M}_{2} fixed by the corresponding bielliptic involution (acting by reflection on the tangent space). The union of these hyperplanes is denoted ξ̂.

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)

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) = x^{6}+y^{6} = (x−ζy)(x−ζ^{3}y)(x−ζ^{5}y)(x−ζ^{7}y)(x−ζ^{9}y)(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^{4}y, x] [x, y] ↦ [ζ^{8}y, x].

In particular, ξ̂ is not normal crossings.

Proposition 3.5 (Ramification formula for Q : \overline{M}_{2} → \overline{M}_{2})

K_{\overline{M}_{2}} + αδ ≡ Q^{∗}(K_{\overline{M}_{2}} + αΔ_{0} + \frac{1 + α}{2}Δ_{1} + \frac{1}{2}Ξ)

proof: Lemma 3.3 allows us to consider the rigidification ([ACV] §5) of \overline{M}_{2} with respect to the group ⟨ι⟩ generated by the canonical involution

r : \overline{M}_{2} → \overline{M}_{2}^{<ι>}.

Given a scheme T, T -valued points of \overline{M}_{2}^{<ι>} correspond to families of genus two stable curves over T, where two families are identified if they differ by a ⟨ι⟩-valued cocycle over T. The inertia group of \overline{M}_{2}^{<ι>} at [C] is the quotient

Aut(C)/ ⟨ι⟩ .

We have a factorization

\overline{M}_{2} →^{r} \overline{M}_{2}^{<ι>} →^{q̄} \overline{M}_{2}

so that r is étale of degree two and \overline{M}_{2} is the coarse moduli space of \overline{M}_{2}^{<ι>} [ACV] Theorem 5.1.5. We therefore obtain the following formula for dualizing sheaves

r^{∗}ω_{\overline{M}}_{2}^{<ι>} ≃ ω_{\overline{M}_{2}}.

Let δ_{1}^{<ι>} and ξ^{<ι>} denote the corresponding Cartier divisors in \overline{M}_{2}^{<ι>}. As q̄ has simple ramification along these divisors, we obtain

q̄^{∗}Δ_{1} = 2δ_{1}^{<ι>}  q̄^{∗}Ξ = 2ξ^{<ι>}
q̄^{∗}K_{\overline{M}_{2}} = K_{\overline{M}}_{2}^{<ι>} − δ_{1}^{<ι>} − ξ^{<ι>},

which together imply the formula.□

3.3 Analysis of b : X̃ → X along the exceptional divisor

By Theorem 2.8, the exceptional divisor Δ̃_{1} is mapped isomorphically to the locus in \overline{A}_{2} 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 Ξ̃ denote the proper transform of Ξ in X̃. We have the following formulas:

KX̃ ≡ b^{∗}K_{X} + 10Δ̃_{1}
b^{∗}{G = 0} ≡ Ξ̃ + 12Δ̃_{1}
b^{∗}{D = 0} ≡ Δ̃_{0} + 6Δ̃_{1}
Ξ̃ ≡ 3Δ̃_{0} + 12Δ̃_{1}.

We pause to explore the geometry of Δ̃_{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

\overline{M}_{1,1} ≃ P(4, 6) = (Spec k[g_{2}, g_{3}] − (0, 0)) /𝔾_{m},

with 𝔾_{m} acting with weights (4, 6)

t · (g_{2}, g_{3}) ↦ (t^{4}g_{2}, t^{6}g_{3}).

The coarse moduli space is

\overline{M}_{1,1} = \overline{A}_{1} ≃ Proj k[g_{2}, g_{3}] ≃ ℙ^{1}.

The symmetric square of the stack has the following quotient-stack presentation:

(\overline{M}_{1,1} × \overline{M}_{1,1})/𝔖_{2} = (Spec k[g_{2}, g_{3}, h_{2}, h_{3}] − Z\}/ H
Z = {(g_{2}, g_{3}, h_{2}, h_{3}) : g_{2} = g_{3} = 0 or h_{2} = h_{3} = 0}.

Here H is the group generated by the torus

(t, u) · (g_{2}, g_{3}, h_{2}, h_{3}) ↦ (t^{4}g_{2}, t^{6}g_{3}, u^{4}h_{2}, u^{6}h_{3})

and the involution

(g_{2}, g_{3}, h_{2}, h_{3}) ↦ (h_{2}, h_{3}, g_{2}, g_{3}),

i.e., H = 𝔖_{2} ⋉ 𝔾_{m}^{2} where 𝔖_{2} acts on 𝔾_{m}^{2} 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[g_{2}, g_{3}, h_{2}, h_{3}] with the following properties:

1. p(g_{2}, g_{3}, h_{2}, h_{3}) = p(h_{2}, h_{3}, g_{2}, g_{3});

2. p(t^{2}g_{2}, t^{3}g_{3}, u^{2}h_{2}, u^{3}h_{3}) = (tu)^{N}p(g_{2}, g_{3}, h_{2}, h_{3}) for some N.

This ring is generated by g_{2}h_{2}, g_{3}h_{3}, and g_{2}^{3}h_{3} + g_{3}^{2}h_{2}^{3} and

Proj k[g_{2}h_{2}, g_{3}h_{3}, g_{2}^{3}h_{3}^{2} + g_{3}^{2}h_{2}^{3}] ≃ 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 first equation follows because b : X̃ → X has weights (2, 3, 6). As for the second, Ξ̃ 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 Δ_{1}, the bielliptic involution specializes to a morphism exchanging the elliptic components. Therefore, Ξ̃ ∩ Δ̃_{1} ⊂ Δ̃_{1} is the diagonal in the symmetric square, which is cut out by a form of weighted-degree twelve.

For the third equation, Δ̃_{0} is the proper transform of {D = 0}. The intersection Δ̃_{0} ∩ Δ̃_{1} ⊂ Δ̃_{1} is the locus where the discriminant Δ = g_{2}^{3} − 27g_{3}^{2} 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 sufficiently small ϵ > 0, the divisor

Δ̃_{0} + (6 − ϵ)Δ̃_{1}

is ample on X̃.

proof: The divisor can be expressed

b^{∗}(ample divisor) − ϵ(b-exceptional divisor),

which yields a polarization of the blow-up b : X̃ → X. □

4 Birational geometry

4.1 Divisor classes and birational contractions of \overline{M}_{2}

It is well known that the rational divisor class group of \overline{M}_{2} is freely generated by the boundary divisors Δ_{0} and Δ_{1}; see [HaMo] for a nice account of divisors on \overline{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 \overline{M}_{2} is isomorphic to X̃ up to codimension ≤ 1, so these have isomorphic class groups. It follows that the rational divisor class group of \overline{M}_{2} is generated by Δ_{1} and the proper transform of the discriminant, which is just Δ_{0}.

The nef cone of \overline{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 \overline{M}_{g}, g ≥ 2, if and only if a ≥ 11b ≥ 0.

Here λ is the pull-back of the polarization on \overline{A}_{g} via the extended Torelli map \overline{t} : \overline{M}_{g} → \overline{A}_{g} (see §2.6).

To apply this in our situation, we observe that

λ ≡ \frac{1}{10}(Δ_{0} + Δ_{1})

over \overline{M}_{2} (see [HaMo] pp. 175). The factor 10 can be explained by the fact that \overline{t}(Δ_{1}) is defined by the vanishing of a modular form of weight ten (see Remark 2.9 and [Ig2]). Substitution gives the first part of

Proposition 4.2 The nef cone of \overline{M}_{2} is generated by the divisors Δ_{0} + Δ_{1} and Δ_{0}+6Δ_{1}, respectively. These are both semiample, inducing the birational contractions

\overline{t} : \overline{M}_{2} → \overline{A}_{2} β : \overline{M}_{2} → X.

The rational map γ : \overline{M}_{2} −−→ X̃ is an isomorphism.

remainder of proof: Of course, Δ_{0} + Δ_{1} is semiample and induces the birational contraction morphism \overline{t} : \overline{M}_{2} → \overline{A}_{2}. As X̃ and \overline{M}_{2} are isomorphic in codimension one, Proposition 3.7 says that Δ̃_{0} + (6 − ϵ)Δ̃_{1} is ample on X̃ and the corresponding divisor Δ_{0} +(6−ϵ)Δ_{1} is ample on \overline{M}_{2}. It follows that X̃ and \overline{M}_{2} are each isomorphic to Proj of

⊕_{n≥0}Γ(OX̃(n(Δ̃_{0} + (6 − ϵ)Δ̃_{1}))) ≃ ⊕_{n≥0}Γ(O_{\overline{M}_{2}}(n(Δ_{0} + (6 − ϵ)Δ_{1}))).

In particular, the rational map γ is an isomorphism. Thus the contractions b : X̃ → X and β : \overline{M}_{2} −−→ X coincide. □

4.2 Canonical class of \overline{M}_{2}

The canonical class K_{\overline{M}_{2}} can also be computed by elementary methods. We know that

ω_{ℙ(1,2,3,5)} ≃ O_{ℙ(1,2,3,5)}(−11),

which also follows from the third part of Proposition 3.1. Since the discriminant has degree ten, we find

K_{X} = −\frac{11}{5}\{D = 0\}.

Applying the formulas of Proposition 3.6, we obtain

KX̃ ≡ −\frac{11}{5}Δ̃_{0} − \frac{16}{5}Δ̃_{1}

KX̃ + \frac{1}{2}Δ̃_{1} + \frac{1}{2}Ξ̃ ≡ −\frac{7}{10}Δ̃_{0} + \frac{3}{10}Δ̃_{1}.

Since X̃ and \overline{M}_{2} agree in codimension one, they have the same canonical class

K_{\overline{M}_{2}} ≡ −\frac{11}{5}Δ_{0} − \frac{16}{5}Δ_{1}.

In particular, we obtain

K_{\overline{M}_{2}} + αΔ_{0} + \frac{1 + α}{2}Δ_{1} + \frac{1}{2}Ξ ≡ (−\frac{7}{10} + α)Δ_{0} + (\frac{3}{10} + α/2)Δ_{1}. (6)

Remark 4.3 The importance of this divisor stems from the ramification equation of Proposition 3.5. This divisor class pulls back to the class

K_{\overline{M}_{2}} + αδ

on the moduli stack. We shall interpret log canonical models of the moduli stack using this divisor.

Using the computations of Proposition 3.6, we obtain a discrepancy equations for β : \overline{M}_{2} → X

K_{\overline{M}_{2}} + αΔ_{0} + \frac{1}{2}Ξ = β^{∗}(K_{X} + α\{D = 0\} + \frac{1}{2}\{G = 0\}) + (4 − 6α)Δ_{1} (7)

K_{\overline{M}_{2}} +αΔ_{0}+ \frac{1 + α}{2}Δ_{1}+ \frac{1}{2}Ξ = β^{∗}(K_{X} +α\{D = 0\}+ \frac{1}{2}\{G = 0\})+ \frac{9 − 11α}{2}Δ_{1}.

(8)

4.3 Generalities on log canonical models

See [FA] §2 for definitions of the relevant terms and technical background. Let V be a normal projective variety, D = ∑_{i=1}^{n} a_{i}D_{i} a ℚ-divisor such that 0 ≤ a_{i} ≤ 1 and K_{V} + D is ℚ-Cartier. Abusing notation, we write V − D for V − ∪_{i}D_{i}.

Definition 4.4 (V, D) is a strict log canonical model if K_{V} + 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 : Ṽ → V where the exceptional locus of ρ is divisorial, D′_{i} denotes the proper transform of D_{i}, and E_{j} (resp. F_{k}) denotes the exceptional divisors of ρ with ρ(E_{j}) ⊂ D (resp. ρ(F_{k}) ⊄ D).

The following are equivalent:

1. (V, D) is a strict log canonical model.

2. For some resolution of singularities ρ : Ṽ → V, with the union of the exceptional locus and ∪_{i}D′_{i} normal crossings, there exist b_{j} ∈ ℚ ∩ [0, 1] so that

D̃ = ∑_{i} a_{i}D′_{i} + ∑_{j} b_{j}E_{j}

satisfies the formula

KṼ + D̃ ≡ ρ^{∗}(K_{V} + D) + ∑_{j} d_{j}E_{j} + ∑_{k} e_{k}F_{k}, d_{j}, e_{k} ≥ 0. (9)

For each such choice of b_{j}

KṼ + D̃ − ∑_{j} d_{j}E_{j} − ∑_{k} e_{k}F_{k}

is semiample and induces ρ. Furthermore, we may take the b_{j} = 1.

3. For some contraction ρ : Ṽ → V, there exist b_{j} ∈ ℚ ∩ [0, 1] so that

D̃ = ∑_{i} a_{i}D′_{i} + ∑_{j} b_{j}E_{j}

satisfies the formula

KṼ + D̃ ≡ ρ^{∗}(K_{V} + D) + ∑_{j} d_{j}E_{j} + ∑_{k} e_{k}F_{k}, d_{j}, e_{k} ≥ 0,

(Ṽ, D̃) is log canonical, and Ṽ − D̃ has canonical singularities. The divisor

KṼ + D̃ − ∑_{j} d_{j}E_{j} − ∑_{k} e_{k}F_{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 coefficients 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 (Ṽ, D̃); 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 first, take a resolution for (Ṽ, D̃) so that the union of the exceptional locus and all the proper transforms of the D′_{i}, E_{j}, and F_{k} is normal crossings. Comparing discrepancies for (Ṽ, D̃) and (V, D), using the fact the coefficients of components in D̃ are at least as large as the coefficents of the corresponding components appearing in ρ^{∗}(K_{V} + D), we find that (V, D) is log-canonical and has canonical singularities along V − D. Since ρ^{∗}(K_{V} + D) induces ρ, K_{V} + 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 K_{V} + D is ample on V, its pull-back to Ṽ 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 finite dominant morphism to a normal variety. Let D = ∑_{i} a_{i}D_{i}, 0 ≤ a_{i} ≤ 1 be a ℚ-divisor on V containing all the divisorial components of the branch locus of h. Let D̄ be a ℚ-divisor on W so that supp(h^{−1}(D)) = supp(D̄) and h^{∗}(K_{V} + D) = K_{W} + D̄. Then (V, D) has log canonical singularities along D iff (W, D̄) has log canonical singularities along D̄.

If D and D̄ have multiplicity one at each component then h^{∗}(K_{V} + D) = K_{W} + D̄ follows from the other assumptions.

4.4 Example of \overline{M}_{g}

The standpoint of this section owes a great deal to Mumford [Mu1] [Mu2]:

Theorem 4.7 For g ≥ 4, the pair (\overline{M}_{g}, Δ) 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 M_{g} of curves with automorphisms has codimension ≥ 2, so the branch divisor of Q : \overline{M}_{g} → \overline{M}_{g} is just Δ_{1}; over Δ_{1}, we have simple ramification. We therefore have ([HaMu] pp. 52) Q^{∗}K_{\overline{M}_{g}} = K_{\overline{M}_{g}} − δ_{1} and thus

Q^{∗}(K_{\overline{M}_{g}} + Δ) = K_{\overline{M}_{g}} + δ.

sketch proof: We first check that K_{\overline{M}_{g}} + Δ is ample. The formula from [HaMu] §2 (or [HaMo])

K_{\overline{M}_{g}} = 13λ − 2Δ_{0} − 3/2Δ_{1} − 2 ∑_{2≤i≤g/2} Δ_{i}

gives K_{\overline{M}_{g}} + δ = 13λ − δ, which is ample by Theorem 4.1 (see also [Mu1]).

The singularity analysis follows [HaMu]. \overline{M}_{g} has canonical singularities by Theorem 1 of [HaMu]. To show that (\overline{M}_{g}, Δ) has log canonical singularities, we use the fact that \overline{M}_{g} is étale-locally a quotient of a smooth variety by a finite group. At [C] it has a local presentation

h : T_{[C]}\overline{M}_{g} ↠ T_{[C]}\overline{M}_{g}/Aut(C)

in terms of its tangent space

T_{[C]}\overline{M}_{g} = Ext^{1}(Ω_{C}^{1}, O_{C}).

We analyze the quotient morphism using Proposition 4.6. The preimage of the boundary divisor corresponds to a union of hyperplanes δ̂ ⊂ T_{[C]}\overline{M}_{g}, meeting in normal crossings. The pair (T_{[C]}\overline{M}_{g}, Δ̄) then has log canonical singularities. An application of Proposition 4.6, utilizing the ramification discussion in Remark 4.8, implies (K_{\overline{M}_{g}}, Δ) is log canonical. □

Remark 4.9 Using the full force of Theorem 4.1 we get a sharper statement. Consider the pair

(\overline{M}_{g}, αΔ_{0} + \frac{1 + α}{2}Δ_{1} + α(Δ_{2} + … + Δ_{⌊g/2⌋})),

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

4.5 Application to \overline{M}_{2}

The source of the special difficulties in this case is the fact that K_{\overline{M}_{2}} + Δ is not effective. Indeed, in §4.1 we computed

K_{\overline{M}_{2}} ≡ −\frac{11Δ_{0} + 6Δ_{1}}{5}

so K_{\overline{M}_{2}} + Δ = −effective 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 \overline{M}_{2} via Proposition 3.5:

Theorem 4.10 Consider the log canonical model of \overline{M}_{2} with respect to the K_{\overline{M}_{2}} + αδ, i.e., the log canonical model of \overline{M}_{2} with respect to

K_{\overline{M}_{2}} + αΔ_{0} + \frac{1 + α}{2}Δ_{1} + \frac{1}{2}Ξ.

1. For 9/11 < α ≤ 1, we recover \overline{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 effective 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 \overline{M}_{2} is ample if and only if α > 9/11. Proposition 3.1 implies that

K_{X} + 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 \overline{M}_{2} has canonical singularities away from Δ_{0}, Δ_{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

M_{2} ≃ 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(x^{5} + y^{5}),

the unique form with an automorphism group of order five [Bo] pp. 51 [Ig1] pp. 645. At this point, ℙ(1, 2, 3, 5) is locally isomorphic to the cyclic quotient singularity \frac{1}{5}(1, 2, 3), i.e., the quotient of 𝔸^{3} under the action

(a, b, c) ↦ (ζa, ζ^{2}b, ζ^{3}c) ζ ≠ 1 ∈ μ_{5}.

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

K_{\overline{M}_{2}} + αΔ_{0} + \frac{1 + α}{2}Δ_{1} + 1/2Ξ

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 \overline{M}_{2} is ℚ-factorial and discrepancies increase as coefficients of log divisors decrease [FA] 2.17.3, it suffices to verify that

K_{\overline{M}_{2}} + Δ_{0} + Δ_{1} + \frac{1}{2}Ξ  (10)

is log canonical.

The proof relies on the description of the boundary divisors in terms of the local presentation

T_{[C]}\overline{M}_{2}/Aut(C),

as sketched in §3.2. The key observation is that Ξ does not play a rôle in the analysis. Each bielliptic involution acts on T_{[C]}\overline{M}_{2} by reflection across the corresponding hyperplane in ξ̂, so the quotient

h : T_{[C]}\overline{M}_{2} → T_{[C]}\overline{M}_{2}/H, H = Aut(C)/ ⟨ι⟩ ,

has simple ramification along ξ̂. Since Ξ has coefficient 1/2 in (10), ξ̂ does not appear in the pull-back of the log canonical divisor to T_{[C]}\overline{M}_{2}.

Thus (10) pulls back to

K_{T_{[C]}\overline{M}_{2}} + δ̂,

and we have seen that δ̂ is normal crossings. It follows that (T_{[C]}\overline{M}_{2}, δ̂) is log canonical and Proposition 4.6 gives the desired result.□

References

[ACV] D. Abramovich, A. Corti, and A. Vistoli, Twisted bundles and admissible covers, Comm. Algebra 31 (2003), no. 8, 3547–3618.

[Bo] O. Bolza, On binary sextics with linear transformations into themselves, Amer. J. Math. 10 (1887) 47-70.

[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 binären 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. École 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. Kollár, Flips and abundance for algebraic threefolds, Astérisque 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 York-Heidelberg, 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 algébriques, Springer-Verlag, Berlin-Heidelberg, 2000.

[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 compactification 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.

[Sc] I. Schur, Vorlesungen über 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 M_{2}, 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