THE CLASSIFICATION OF UNIVERSAL JACOBIANS OVER THE MODULI SPACE OF CURVES

GAVRIL FARKAS AND ALESSANDRO VERRA

A well-established principle of Mumford asserts that all moduli spaces of curves of genus g > 2 (with or without marked points), are varieties of general type, except a finite number of cases occurring for relatively small genus, when these varieties tend to be unirational, or at least uniruled, see [HM], [EH1], [FL], [Log], [V] for illustrations of this fact. In all known cases, the transition from uniruledness to being of general type is quite sudden. The aim of this paper is to determine the Kodaira dimension of the universal Jacobian of degree g over \overline{M}_{g} for any genus g, in particular to highlight the surprising transition cases g = 10, 11. Let \overline{C}_{g,n} := \overline{M}_{g,n}/𝔖_{n} be the universal symmetric product of degree n. Since the fibre of the projection map φ : \overline{C}_{g,n} → \overline{M}_{g} over a smooth curve [C] ∈ M_{g} is birational to the n-th symmetric product C_{n}, it follows trivially that \overline{C}_{g,n} is uniruled when n > g. The global Abel-Jacobi map

𝔞_{g} : \overline{C}_{g,g} −−→ \overline{𝔓𝔦𝔠}_{g}^{g},

establishes a birational isomorphism between \overline{C}_{g,g} and (a compactification of) the degree g universal Picard variety α_{g} : \overline{𝔓𝔦𝔠}_{g}^{g} → \overline{M}_{g}. For a smooth curve [C] ∈ M_{g}, the map φ^{−1}([C]) → Pic^{g−1}(C) factors through C_{g}: the Abel-Jacobi map C_{g} → Pic^{g}(C) is the blow-up of Pic^{g}(C) along the Fitting ideal corresponding to the subscheme W_{g}^{1}(C), whereas φ^{−1}([C]) → C_{g} is an iterated blow-up along the diagonals. Thus, we may regard \overline{C}_{g,g} as a global blow-up of \overline{𝔓𝔦𝔠}_{g}^{g}. Applying the additivity of Kodaira dimensions for abelian fibrations [U] to the fibre space α_{g}, we obtain that κ(\overline{𝔓𝔦𝔠}_{g}^{g}) = 3g−3, whenever \overline{M}_{g} is of general type. It is natural to wonder, whether the equality κ(\overline{𝔓𝔦𝔠}_{g}^{g}) = κ(\overline{M}_{g}) holds for every g. We answer this question in the negative. Our most picturesque result, concerns the transition cases in the birational classification of universal Jacobians:

Theorem 0.1. The universal Jacobian \overline{𝔓𝔦𝔠}_{10}^{10} has Kodaira dimension zero. The Kodaira dimension of \overline{𝔓𝔦𝔠}_{11}^{11} equals 19.

It is well-known that both \overline{M}_{10} and \overline{M}_{11} are unirational varieties. We note that

when g ≤ 11, these are the only cases when \overline{C}_{g,n} has non-negative Kodaira dimension. Using the existence of certain Mukai models of the moduli space of curves of genus g < 10, it easily follows that \overline{𝔓𝔦𝔠}_{g}^{g} is unirational for g ≤ 9. In fact more can be said:

Theorem 0.2. The moduli space \overline{C}_{g,n} is unirational for all g < 10 and n ≤ g. Furthermore, \overline{C}_{10,n} is uniruled for all n ≠ 10; the space \overline{C}_{11,n} is uniruled for all n ≠ 11.

For higher g, we show that \overline{C}_{g,g} is of the maximal Kodaira dimension it could possibly have, in view of Iitaka’s easy addition inequality for fibre spaces

κ(\overline{C}_{g,g}) ≤ dim(\overline{M}_{g}) + κ(φ^{−1}([C])) = 3g − 3.

Theorem 0.3. For g > 11, the Kodaira dimension of \overline{𝔓𝔦𝔠}_{g}^{g} is equal to 3g − 3.

Theorems 0.1, 0.2, 0.3 highlight the fact that \overline{𝔓𝔦𝔠}_{g}^{g} does not capture the intricate transition of \overline{M}_{g} from uniruledness to general type that occurs in the range 17 ≤ g ≤ 21.

We describe the main steps in the proof of Theorems 0.1, 0.2 and 0.3. A key role is played by the effective divisor

D_{g} := {[C, x_{1}, …, x_{g}] ∈ M_{g,g} : h^{0}(C, O_{C}(x_{1} + ⋯ + x_{g})) ≥ 2}.

The class of the closure of D_{g} inside \overline{M}_{g,g} has been computed, see [Log] Theorem 5.4, or [F1], for an alternative proof:

\overline{D}_{g} ≡ −λ + ∑_{i=1}^{g} ψ_{i} − ∑_{i=0}^{[g/2]} ∑_{T ⊂\{1,...,g\}} \binom{|#(T) − i| + 1}{2}δ_{i:T} ∈ Pic(\overline{M}_{g,g}).

The divisor \overline{D}_{g} is 𝔖_{g}-invariant under the action permuting the marked points, thus if π : \overline{M}_{g,g} → \overline{C}_{g,g} is the quotient map, there exists an effective divisor D̃_{g} ∈ Eff(\overline{C}_{g,g}) such that \overline{D}_{g} = π^{∗}(D̃_{g}). Note that D̃_{g} is an exceptional divisor of the rational Abel-Jacobi map 𝔞_{g} : \overline{C}_{g,g} −−→ \overline{𝔓𝔦𝔠}_{g}^{g}, and as such, it is uniruled and an extremal point of Eff(\overline{C}_{g,g}).

In Theorem 1.1, we prove that for g ≥ 4 pluri-canonical forms on \overline{C}_{g,g,reg} extend to any desingularization of \overline{C}_{g,g}. Thus in order to bound κ(\overline{C}_{g,g}) from below, it suffices to exhibit sufficiently many global sections of K_{\overline{C}_{g,g}}. To that end, we choose an effective

divisor class D ≡ aλ − ∑_{i=0}^{[g/2]} b_{i}δ_{i} ∈ Eff(\overline{M}_{g}) of small slope s(D) := a/min_{i=0}^{[g/2]}b_{i}. For composite g + 1, one can take D = \overline{M}_{g,d}^{r} to be the closure of the Brill-Noether divisor of curves with a 𝔤_{d}^{r}, where ρ(g, r, d) = g − (r + 1)(g − d + r) = −1; there exists a constant c_{g,d,r} > 0 such that [EH2],

\overline{M}_{g,d}^{r} ≡ c_{g,d}^{r} · 𝔟𝔫_{g} :=≡ c_{g,d,r}((g + 3)λ − \frac{g + 1}{6}δ_{0} − ∑_{i=1}^{[g/2]} i(g − i)δ_{i}) ∈ Pic(\overline{M}_{g}).

In particular s(\overline{M}_{g,d}^{r}) = 6 + 12/(g + 1). By linear interpolation, we find an effective divisor E on \overline{C}_{g,g} supported along D̃_{g}, φ^{∗}(D) and the boundary of \overline{C}_{g,g}, such that

K_{\overline{C}_{g,g}} = (14 − 2s(D)) φ^{∗}(λ) + E.

Whenever s(D) < 7 (and such a divisor D ⊂ \overline{M}_{g} can be chosen exactly when g ≥ 12, see [FV] for the particularly difficult case g = 12), the following inequality holds:

κ(\overline{C}_{g}) ≥ κ(\overline{C}_{g}, (14 − 2s(D))φ^{∗}(λ)) = κ(\overline{M}_{g}, λ) = 3g − 3.

Since the opposite inequality is immediate, this proves Theorem 0.3. We summarize this discussion by linking \overline{C}_{g,g} to the slope s(\overline{M}_{g}) := inf\{s(D) : D ∈ Eff(\overline{M}_{g})\} of the moduli space of curves.

Proposition 0.4. Assume s(\overline{M}_{g}) < 7 for a given genus g. Then κ(\overline{C}_{g,g}) = 3g − 3.

This highlights that the birational geometry of \overline{C}_{g,g} is governed by a linear series on \overline{M}_{g} of slope 7, rather than the canonical linear series, for which s(K_{\overline{M}_{g}}) = 13/2.

We briefly discuss the cases g ≤ 11. In this range, apart from 𝔞_{g}∘π : \overline{M}_{g,g} −−→ \overline{𝔓𝔦𝔠}_{g}^{g}, there exists a second birational contraction of \overline{D}_{g}, under a birational map

f_{g} : \overline{M}_{g,g} −−→ 𝔐_{g,g},

where the target 𝔐_{g,g} is a GIT model of \overline{M}_{g,g} emerging from Mukai’s constructions [M1], [M3], [M4]. In genus 11, using this, we prove a stronger result, concerning the birational type of both \overline{C}_{11,11} and its covering \overline{M}_{11,11}:

Theorem 0.5. One has that κ(\overline{M}_{11,11}) = κ(\overline{C}_{11,11}) = 19.

Note that dim(\overline{M}_{11,11}) = 41, and \overline{M}_{11,11} is the first example of a moduli space \overline{M}_{g,n} with g ≥ 2, having intermediate Kodaira dimension. It is known [Log] that \overline{M}_{11,n} is uniruled for n ≤ 10 and of general type for n ≥ 12. To interpret such results, for a genus g ≥ 2, we define the invariant

ζ(g) := min\{n ∈ ℤ_{≥0} : κ(\overline{M}_{g,n}) ≥ 0\}.

We think of ζ(g) as measuring the complexity of the general curve of genus g. Since the relative dualizing sheaf of the forgetful map \overline{M}_{g,n} → \overline{M}_{g,n−1} is big, it follows that \overline{M}_{g,n} is of general type for n > ζ(g). Then results in [HM], [EH2], [F2] imply that ζ(g) = 0, for g ≥ 22. From [FP] Proposition 7.5 one obtains the value ζ(10) = 10, whereas Theorem 0.5 implies that ζ(11) = 11. This indicates, in precise terms, that counter-intuitively, algebraic curves of genus 10 are more complicated than curves of genus 11!

We make a few comments on Theorem 0.5. We note that K_{\overline{M}_{11,11}} is an effec-

tive combination of the pull-back to \overline{M}_{11,11} of the 6-gonal divisor \overline{M}_{11,6}^{1} on \overline{M}_{11}, the divisor \overline{D}_{11}, and certain boundary classes δ_{i:S}. Then we construct (cf. Proposition 1.4), rational curves R ⊂ \overline{M}_{11,11} passing through a general point of \overline{D}_{11}, such that (i) −R · \overline{D}_{11} > 0 equals precisely the multiplicity of \overline{D}_{11} in the above mentioned expression of K_{\overline{M}_{11,11}}, and (ii) R is disjoint from all boundary divisors Δ_{i:T}. Therefore, n\overline{D}_{11} is a fixed component of the pluri-canonical linear series |nK_{\overline{M}_{11,11}}| for all n ≥ 1. The equality κ(\overline{M}_{11,11}) = 19 is related to the Mukai fibration

q_{11} : \overline{M}_{11,11} −−→ \overline{F}_{11},

over the 19-dimensional moduli space \overline{F}_{11} of polarized K3 surfaces of degree 20. The map q_{11} associates to a general element [C, x_{1}, …, x_{11}] ∈ \overline{M}_{11,11} the unique K3 surface S containing C, see [M2]. According to Mukai, S is precisely the ”dual” K3 surface to the non-abelian Brill-Noether locus corresponding to vector bundles of rank 2

S^{∨} = SU_{C}(2, K_{C}, 6) := {E ∈ SU_{C}(2, K_{C}) : h^{0}(C, E) ≥ 7}.

An analysis of the fibration q_{11} shows that the difference K_{\overline{M}_{11,11}} − \overline{D}_{11} is essentially the pull-back of an ample class on \overline{F}_{11}. Eventually, this leads to the equality κ(\overline{M}_{11,11}) = κ(\overline{M}_{11}, 𝔟𝔫_{11}) = 19, where the last symbol stands for the Iitaka dimension of the linear system generated by the Brill-Noether divisors \overline{M}_{11,6}^{1} and \overline{M}_{11,9}^{2} on \overline{M}_{11}.

In the final section of this paper we study the uniruledness of \overline{M}_{g,n} when g ≤ 9.

Theorem 0.6. The space \overline{M}_{g,n} is uniruled for n ≤ f(g), where f(g) is given in the table below:

\begin{matrix}g & 5 & 7 & 8 & 9 & 10 \\ f(g) & 13 & 13 & 12 & 10 & 9 \\ h(g) & 15 & 15 & 14 & 13 & 11 \end{matrix}

New here is the statement regarding the uniruledness of \overline{M}_{g,n}. For the sake of comparison, we have copied from [Log] and [F2] Theorem 1.10, the bound h(g), for which \overline{M}_{g,n} is known to be of general type when n ≥ h(g). We note that moreover, κ(\overline{M}_{7,14}) ≥ 2, thus ζ(7) = 14. Nothing appears to be known about the Kodaira dimension of \overline{M}_{5,14} and \overline{M}_{8,13}, which are the missing cases from the classification, when g ≤ 8. The case g = 6, where a complete solution is known, cf. [Log], [CF], is omitted from the table.

In order to prove Theorem 0.6, it suffices to establish that K_{\overline{M}_{g,n}} is not pseudoeffective and then use [BDPP] to conclude that \overline{M}_{g,n} is uniruled. The non-effectiveness of K_{\overline{M}_{g,n}} is established by exhibiting one or two extremal uniruled divisors on \overline{M}_{g,n}, satisfying certain numerical properties, see Proposition 5.2.

1. An extremal effective divisor on \overline{M}_{g,g}

We begin by setting notation and terminology. If M is a Deligne-Mumford stack, we denote by M its associated coarse moduli space. Let X be a complex ℚ-factorial variety. A ℚ-Weil divisor D on X is said to be movable if codim(⋂_{m} Bs|mD|, X) ≥ 2, where the intersection is taken over all m which are sufficiently large and divisible. We say that D is rigid if |mD| = {mD}, for all m ≥ 1 such that mD is an integral Cartier divisor. The Kodaira-Iitaka dimension of a divisor D on X is denoted by κ(X, D). As usual, we set κ(X) := κ(X, K_{X}). We say that a curve Γ ⊂ X is a covering curve for a divisor D ⊂ X, when Γ deforms in a family of 1-cycles \{Γ_{t}\}_{t∈T}, such that \overline{∪_{t∈T} Γ_{t}} = D.

We recall the notation for boundary divisor classes on the moduli space \overline{M}_{g,n}, cf. [AC1]. For an integer 0 ≤ i ≤ [g/2] and a set of labels T ⊂ {1, …, n}, we denote by Δ_{i:T} the closure in \overline{M}_{g,n} of the locus of n-pointed curves [C_{1} ∪ C_{2}, x_{1}, …, x_{n}], where C_{1} and C_{2} are smooth curves of genera i and g − i respectively, and the marked points lying on C_{1} are precisely those labeled by T. As usual, we define δ_{i:T} := [Δ_{i:T}] ∈ Pic(\overline{M}_{g,n}). For 0 ≤ i ≤ [g/2] and 0 ≤ c ≤ g, we set

Δ_{i:c} := ∑_{#(T)=c} δ_{i:T} , δ_{i:c} := [Δ_{i:c}]_{ℚ} ∈ Pic(\overline{M}_{g,n}).

By convention, δ_{0:c} := ∅, for c < 2. If ϕ : \overline{M}_{g,n} → \overline{M}_{g} is the morphism forgetting the marked points, we set λ := ϕ^{∗}(λ) ∈ Pic(\overline{M}_{g,n}) and δ_{irr} := ϕ^{∗}(δ_{irr}) ∈ Pic(\overline{M}_{g,n}), where δ_{irr} := [Δ_{irr}] ∈ Pic(\overline{M}_{g}) denotes the class of the locus of irreducible nodal curves. Furthermore, ψ_{1}, …, ψ_{n} ∈ Pic(\overline{M}_{g,n}) are the cotangent classes corresponding to the marked points. The canonical class of \overline{M}_{g,n} can be computed by using Grothendieck-Riemann-Roch for the universal curve over \overline{M}_{g,n}:

(1)  K_{\overline{M}_{g,n}} ≡ 13λ − 2δ_{irr} + ∑_{i=1}^{n} ψ_{i} − 2 ∑_{{T ⊂\{1,...,n\} \atop i≥0}} δ_{i:T} − δ_{1:∅}.

On the universal symmetric product \overline{C}_{g,n}, we denote by λ̃, δ̃_{irr}, δ̃_{i:c} := [Δ̃_{i:c}] ∈ Pic(\overline{C}_{g,n}) the divisor classes corresponding to the same symbols on \overline{M}_{g,n}. The general point from the Δ̃_{0:2} corresponds to a marked curve with automorphism group isomorphic to ℤ/2ℤ, therefore δ̃_{0:2} = [Δ̃_{0:2}]_{ℚ} = [Δ̃_{0,2}]/2. Let π : \overline{M}_{g,n} → \overline{C}_{g,n} and φ : \overline{C}_{g,n} → \overline{M}_{g} be the quotient and forgetful maps respectively, thus ϕ = φ ∘ π. Clearly, π^{∗}(λ̃) = λ, π^{∗}(δ̃_{irr}) = δ_{irr}, π^{∗}(δ̃_{i:c}) = δ_{i:c}, where the last formula, in the case i = 0, c = 2, takes into account the branching of the map π along the divisor Δ̃_{0:2} ⊂ \overline{C}_{g,n}. We introduce the tautological line bundle 𝕃 on \overline{C}_{g,n}, having fibre

𝕃[C, x_{1} + ⋯ + x_{n}] = T_{x_{1}}^{∨}(C) ⊗ ⋯ ⊗ T_{x_{n}}^{∨}(C),

over a point [C, x_{1} + ⋯ + x_{n}] := π([C, x_{1}, …, x_{n}]) ∈ \overline{C}_{g,n}. We set ψ̃ := c_{1}(𝕃), and let π_{i} : \overline{M}_{g,n} → \overline{M}_{g,1} be the morphism forgetting all expect the i-th marked point. Then

(2) π^{∗}(ψ̃) = ∑_{i=1}^{n} π_{i}^{∗}(ψ) = ∑_{i=1}^{n}(ψ_{i} − ∑_{i∈T ⊂\{1,...,n\}} δ_{0:T} ) = ∑_{i=1}^{n} ψ_{i} − ∑_{c=2}^{n} c δ_{0:c} ∈ Pic(\overline{M}_{g,n}).

From the Riemann-Hurwitz formula K_{\overline{M}_{g,n}} = π^{∗}(K_{\overline{C}_{g},n}) + δ_{0:2} applied to π, we obtain

(3) K_{\overline{C}_{g,n}} ≡ 13λ̃ − 2δ̃_{irr} + ψ̃ − 2 ∑_{{i≥1,c≥0 \atop (i,c)≠(1,0)}} δ̃_{i:c} − 3δ̃_{1:0} − δ̃_{0:2} + ∑_{c=3}^{n}(c − 2) δ̃_{0:c} ∈ Pic(\overline{C}_{g,n}).

In order to obtain lower bounds on the Kodaira dimension of \overline{C}_{g,n}, we need to control its singularities. We fix a point [C, x_{1} + ⋯ + x_{n}] ∈ \overline{C}_{g,n}, and denote by Def(C, x_{1}, …, x_{n}) the versal deformation space of the n-pointed curve (C, x_{1}, …, x_{n}), viewed as an open neighborhood of the origin in the tangent space to the moduli stack T_{[C,x_{1},...,x_{n}]}(\overline{M}_{g,n}) = Ext^{1}(Ω_{C}^{1}, O_{C}(−x_{1} − ⋯ − x_{n})). We set

Aut(C, x̄) := {σ ∈ Aut(C) : σ({x_{1}, …, x_{n}}) = {x_{1}, …, x_{n}}}.

An analytic neighbourhood of [C, x_{1} + ⋯ + x_{n}] ∈ \overline{C}_{g,n} is isomorphic to the space

Def(C, x_{1}, …, x_{n})/Aut(C, x̄) ⊂ H^{0}(C, ω_{C} ⊗ Ω_{C}^{1}(x_{1} + ⋯ + x_{n}))^{∨}/Aut(C, x̄),

where the last identification uses Serre duality. To describe the action of Aut(C, x̄) on the tangent space of the moduli stack, we recall the concept of age.

Let (V, ρ) be a finite dimensional complex representation of a finite group G. If the eigenvalues of ρ(g) ∈ GL(V) are exp(2πir_{i}), where 0 ≤ r_{i} < 1 for i = 1, …, d, then following [R], we define the age of the element g ∈ G as

age(g) := r_{1} + ⋯ + r_{d}.

According to the Reid-Tai criterion, the singularities of V/G are canonical, if and only if for each element g ∈ G which does not act as a quasi-reflection, the inequality age(g) ≥ 1 holds, cf. [HM] pg. 27. Next we show that the singularities of \overline{C}_{g,n} are no worse that those of \overline{M}_{g}. In particular, one can bound the Kodaira dimension of \overline{C}_{g,n} by bounding the number of global sections of K_{\overline{C}_{g,n}}. This result concerning singularities, is also the main reason why we work with \overline{C}_{g,g} as a model for the universal Jacobian, rather than with Caporaso’s compactification P_{g,g} → \overline{M}_{g}, see [C].

Theorem 1.1. Fix integers g ≥ 4 and n, l ≥ 0, and let ϵ : C̃_{g,n} → \overline{C}_{g,n} be any resolution of singularities. Then l-canonical forms of \overline{C}_{g,n,reg} extend, that is, there are group isomorphisms

ϵ^{∗} : H^{0}(\overline{C}_{g,n,reg}, K\overline{C}{⊗l \atop g,n}) →^{≅} H^{0}(C̃_{g,n}, KC̃_{g,n}^{⊗l}).

Proof. We choose a point [C, x_{1} + ⋯ + x_{n}] ∈ \overline{C}_{g,n} which violates the Reid-Tai criterion, that is, there exists an automorphism σ ∈ Aut(C, x̄) which permutes the points x_{1}, …, x_{n}, such that with respect to the action of σ on H^{0}(C, ω_{C} ⊗ Ω_{C}^{1}(x_{1} + ⋯ + x_{n})), we have that age(σ) < 1. If {C_{α}}_{α} are the normalizations of the components of C and {p_{αβ}}_{β} are the points on C_{α} whose images in C are either nodes of C or marked points x_{1}, …, x_{n}, we recall that there exists an exact sequence:

(4) 0 → ⨁_{p∈Sing(C)} Tor_{p} → H^{0}(C, ω_{C} ⊗ Ω_{C}^{1}(x_{1} + … + x_{n})) → ⨁_{α} ω_{C_{α}}^{⊗2}(∑_{β} p_{αβ}) → 0,

where Tor_{p} ⊂ H^{0}(C, ω_{C} ⊗ Ω_{C}^{1}(x_{1} + ⋯ + x_{n})) is the 1-dimensional space of torsion differentials based at p ∈ Sing(C). It is proved in [HM] pg. 34, that Tor_{p} contributes at least 1/ord(σ) to age(σ), for each node p ∈ Sing(C). We distinguish two cases:

(i) σ acts non-trivially only on exceptional components R of C, which are smooth rational curves such that #(R ∩ \overline{(C − R)}) ≤ 2. In other words, σ induces the trivial automorphism on the stable model of C. Let R be an exceptional component, and for simplicity we assume that R meets the rest of C at only one point. We set {p} := R ∩ C and denote by P ⊂ R − {p} the marked points lying on R. Since σ ∈ Aut(R) has finite order, say l, one finds that σ has precisely two fixed points p = 0, ∞ ∈ R, and that σ(z) = ζ · z, where ζ ≠ 1 is an l-th root of unity. The points in P − {∞} can be grouped in orbits of l elements, and an immediate calculation shows that the contribution to age(σ) coming from H^{0}(R, ω_{R}^{⊗2}(p + P)) is at least (l − 1)/l + \binom{l−3}{2}. Since as mentioned above, there is a further contribution to age(σ) of at least 1/l, coming from Tor_{p}(ω_{C} ⊗ Ω_{C}^{1}(x_{1} + ⋯ + x_{n})), it follows that age(σ) ≥ 1, and this case corresponds to a canonical singularity. The case #(R ∩ \overline{(C − R)}) = 2 is analogous, also leading to a canonical singularity. (ii) There exists a non-exceptional component of C on which σ acts non-trivially. Since age(σ_{C}) (with respect to the action on H^{0}(C, ω_{C} ⊗ Ω_{C}^{1})) cannot exceed age(σ) < 1, the analysis from [HM] pg. 34-40, shows that in this case C = C_{1} ∪ E, C_{1} ∩ E = {p}, where C_{1} is a smooth curve of genus g − 1 and E is an elliptic curve. Moreover σ_{C_{1}} = Id_{C_{1}}, and one distinguishes between the cases when ord(σ_{E}) = 2, 4, 6. If at least one of the points x_{i} lies on E, an immediate calculation shows that age(σ) ≥ 1, thus this case too corresponds to a canonical singularity. When \{x_{1}, …, x_{n}\} ⊂ C_{1} − \{p\}, then if U ⊂ \overline{M}_{g} is the analytic neighbourhood of [C_{1} ∪_{p} E] constructed in [HM] pg. 41-43, any pluricanonical form defined on \overline{C}_{g,n,reg} extends over φ^{−1}(U). This completes the proof. □

We turn to the study of the divisor D̃_{g} ⊂ \overline{C}_{g,g} defined as the closure of the locus of points [C, x_{1} + ⋯ + x_{g}] ∈ C_{g,g} moving in a pencil. First we note that its class is given by

(5)  D̃_{g} ≡ −λ̃ + ψ̃ − ∑_{i≥1,c≥0} \binom{|c − i| + 1}{2}δ̃_{i:c} − ∑_{c=2}^{g} \binom{c}{2}δ̃_{0:c} ∈ Pic(\overline{C}_{g,g}).

We construct rational curves ℓ ⊂ \overline{C}_{g,g} sweeping-out the divisor D̃_{g}: We fix [C] ∈

M_{g}, a complete base point free pencil A ∈ W_{g}^{1}(C), and define the locus

ℓ := \{[C, x_{1} + ⋯ + x_{g}] ∈ \overline{C}_{g,g} : h^{0}(C, A(−x_{1} − ⋯ − x_{g})) ≥ 1\}^{−} ⊂ \overline{C}_{g,g}.

Proposition 1.2. One has that ℓ·ψ̃ = 2g−2, ℓ·δ̃_{0:2} = 2g−1, whereas ℓ has intersection number 0 with all remaining standard generators of Pic(\overline{C}_{g,g}). It follows that ℓ · K_{\overline{C}_{g,g}} = ℓ · D̃_{g} = −1.

Proof. Let ℓ̃⊂ C_{g}, be the isomorphic image of ℓ under the blow-up map φ^{−1}([C]) → C_{g} of the diagonals. Then, using e.g. [K] Proposition 2.6, we have that ℓ̃· K_{C_{g}} = −1. On the other side, ℓ̃· K_{C_{g}} = ℓ · K_{\overline{C}_{g,g}} = ℓ · ψ̃ − ℓ · δ̃_{0:2} (one may assume that A ∈ W_{g}^{1}(C)

has only simple ramification points, hence ℓ · δ̃_{0:c} = 0 for c ≥ 3). Furthermore, ℓ · δ̃_{0:2} equals the half of the number of ramification points of A, that is, 2g − 1, and the rest is immediate. □

As explained in the Introduction, for g ≥ 12 the estimate s(\overline{M}_{g}) < 7 holds, and from (5) one finds that

(6)  K_{\overline{C}_{g,g}} ∈ ℚ_{>0}⟨λ̃, [D̃_{g}], \{δ̃_{i:c}\}_{(i,c)≠(0,2)}, φ^{∗}Eff(\overline{M}_{g})⟩.

Coupled with Theorem 1.1, this implies that κ(\overline{C}_{g,g}) = 3g − 3. Furthermore, we note that D̃_{g} appears with multiplicity 1 in the stable base locus of K_{\overline{C}_{g}}.

Proposition 1.3. Set g ≥ 11. Then |nK_{\overline{C}_{g,g}}| = nD̃_{g} + |nK_{\overline{C}_{g,g}} − nD̃_{g}|, for all n ≥ 1.

Proof. The coefficient of D̃_{g} in the expression (6) is equal to 1. Since ℓ ⊂ D̃_{g} is a covering curve, such that ℓ · K_{\overline{C}_{g,g}} = ℓ · D̃_{g} = −1, whereas ℓ has intersection number zero with the remaining classes appearing in (6), the conclusion follows. □

We are thus left with the study of \overline{D}_{g} := π^{∗}(D̃_{g}) ⊂ \overline{M}_{g,g}, in the range g ≤ 11.

Proposition 1.4. For 3 ≤ g ≤ 11, the irreducible divisor \overline{D}_{g} is filled up by rational curves R ⊂ \overline{M}_{g,g} such that R · \overline{D}_{g} < 0. It follows that \overline{D}_{g} ∈ Eff(\overline{M}_{g,g}) is an extremal rigid divisor. Moreover, when g ≠ 10, one can assume that R · δ_{i:T} = 0 for all i ≥ 0 and T ⊂ {1, …, g}.

Proof. We first treat the case g ≠ 10, and start with a general point [C, x_{1}, …, x_{g}] ∈ D_{g}. We assume that the points x_{1}, …, x_{g} ∈ C are distinct and h^{0}(C, K_{C}(−x_{1} −⋯−x_{g})) = 1. Let us consider the (g − 2)-dimensional linear space

Λ := ⟨x_{1}, …, x_{g}⟩ ⊂ P(H^{0}(C, K_{C})^{∨}) = P^{g−1}.

Since ϕ(D_{g}) = M_{g}, we may assume that [C] ∈ M_{g} is a general curve. In particular, C

lies on a K3 surface S ↪^{|O_{S}(C)|} P^{g}, which admits the canonical curve C as a hyperplane section, cf. [M1]. We intersect S with the pencil of hyperplanes {H_{λ} ∈ (P^{g})^{∨}}_{λ∈P^{1}} such that Λ ⊂ H_{λ}. Since (i) the locus of hyperplanes H ∈ (P^{g})^{∨} such that the intersection S ∩ H is not nodal has codimension 2 in (P^{g})^{∨}, and (ii) the pencil {H_{λ}}_{λ∈P^{1}} can be viewed as a general pencil of hyperplanes containing P(H^{0}(C, K_{C})^{∨}) as a member, we may assume that all the curves H_{λ} ∩ S are nodal and that the nodes stay away from the fixed points x_{1}, …, x_{g}. In this way we obtain a family in \overline{M}_{g,g}

R := {[C_{λ} := H_{λ} ∩ S, x_{1}, …, x_{g}] : Λ ⊂ H_{λ}, λ ∈ P^{1}},

inducing a fibration f : S̃ := Bl_{2g−2}(S) → P^{1}, obtained by blowing-up the base points of the pencil, together with g sections given by the exceptional divisors E_{x_{i}} ⊂ S̃ corresponding to the base points x_{1}, …, x_{g}. The numerical parameters of R are computed using, for instance, [FP] Section 2. Precisely, one writes that

(7) R · λ = (ϕ_{∗}(R) · λ)_{\overline{M}_{g}} = g + 1, R · δ_{irr} = (ϕ_{∗}(R) · δ_{irr})_{\overline{M}_{g}} = 6g + 18, R · δ_{i:T} = 0,

for i ≥ 0 and T ⊂ {1, …, g}. Finally, from the adjunction formula, R · ψ_{i} = −(E_{x_{i}}^{2})S̃ = 1 for 1 ≤ i ≤ g. Thus, R · \overline{D}_{g} = −1. Since R is a covering curve for the divisor \overline{D}_{g}, it follows that \overline{D}_{g} is a rigid divisor on \overline{M}_{g,g}.

We turn to the case g = 10, when the previous argument breaks down because the general curve [C] ∈ M_{10} no longer lies on a K3 surface. More generally, we fix g < 11, g ≠ 9 and pick a general point [C, x_{1}, …, x_{g}] ∈ D_{g}. We denote by X := C_{ij} the nodal curve obtained from C by identifying x_{i} and x_{j}, where 1 ≤ i < j ≤ g. Since [X] ∈ Δ_{0} ⊂ \overline{M}_{g+1} is a general 1-nodal curve of genus g + 1, using e.g. [FKPS], there exists a smooth K3 surface S containing X. We denote by ν : C → X ⊂ S the normalization map and set ν(x_{i}) = ν(x_{j}) = p. The linear system |O_{S}(X)| embeds S in P^{g+1} and ν^{∗}(O_{S}(X)) = K_{C}(x_{i} + x_{j}). Let ϵ : S′ := Bl_{p}(S) → S be the blow-up of S at p and E ⊂ S′ the exceptional divisor. Note that C viewed as an embedded curve in S′ belongs to the linear system |ϵ^{∗}O_{S}(1) ⊗ O_{S′}(−2E)| and C · E = x_{i} + x_{j}. Let Z ⊂ S′ the reduced 0-dimensional scheme consisting of marked points of C with support {x_{i}, x_{j}}^{c}.

Since h^{0}(C, O_{C}(x_{1} + ⋯ + x_{g})) = 2, we find that Z together with the tangent plane 𝕋_{p}(X) = 𝕋_{p}(S) span a (g − 1)-dimensional linear space Λ ⊂ P^{g+1}. We obtain a 1-dimensional family in \overline{D}_{g} by taking the normalization of the intersection curves on S with hyperplanes H ∈ (P^{g+1})^{∨} passing through Λ. Equivalently, we note that

h^{0}(S′, I_{Z/S′}(C)) = h^{0}(S′, O_{S′}) + h^{0}(C, K_{C}(−x_{1} − ⋯ − x_{g})) = 2,

that is, |I_{Z/S′}(C)| is a pencil of curves on S′. We denote by ϵ̃: S̃ := Bl_{2g−4}(S′) → S′ the blow-up of S′ at the (ϵ^{∗}(H) − 2E)^{2} = 2g − 4 base points of |I_{Z/S′}(C)|, by f : S̃ → P^{1} the induced fibration with (g − 2) sections corresponding to the points of Z, as well as with a 2-section given by the divisor E := ϵ̃^{−1}(E). Since deg(f_{E}) = 2, there are precisely two fibres of f, say C_{1} and C_{2}, which are tangent to E. We make a base change or order 2 via the morphism f_{E} : E → P^{1}, and consider the fibration

q′ : Y′ := S̃ ×_{P}_{1} E → E.

Thus p : Y′ → S̃ is the double cover branched along C_{1} + C_{2}. Clearly q′ admits two sections E_{1}, E_{2} ⊂ Y′ such that p^{∗}(E) = E_{1} + E_{2} and E_{1} · E_{2} = 2. By direct calculation, it follows that E_{1}^{2} = E_{2}^{2} = −3. To separate the sections E_{1} and E_{2}, we blow-up the two points of intersection E_{1} ∩ E_{2} and we denote by q : Y := Bl_{2}(Y′) → E the resulting fibration, which possesses everywhere distinct sections σ_{i} : E → Y′ for 1 ≤ i ≤ g, given by the proper transforms of E_{1} and E_{2} as well as the proper transforms of the exceptional divisors corresponding to the points in Z. The numerical characters of the family Γ_{ij} := \{[q^{−1}(t), σ_{1}(t), …, σ_{g}(t)] : t ∈ E\} ⊂ \overline{M}_{g,g} are computed as follows:

Γ_{ij} · λ = 2(g + 1), Γ_{ij} · δ_{irr} = 2(6g + 17), Γ_{ij} · ψ_{l} = 2 for l ∈ {i, j}^{c},

Γ_{ij} · ψ_{i} = Γ_{ij} · ψ_{j} = −(E_{i}^{2})_{Y′} + 2 = 5, Γ_{ij} · δ_{0:{i,j}} = 2, Γ_{ij} · δ_{l:T} = 0 for l ≥ 0, T ⊂ {i, j}^{c}.

We take the 𝔖_{g}-orbit of the 1-cycle Γ_{ij} with respect to permuting the marked points,

Γ := \frac{1}{g(g − 1)} ∑_{i<j} Γ_{ij} ∈ NE_{1}(\overline{M}_{g,g}),

and note that Γ·\overline{D}_{g} = −1. Each component Γ_{ij} fills-up \overline{D}_{g}, which finishes the proof. □

We keep all the notation from the proof of Proposition 1.4 and set R̃ := π_{∗}(R) and Γ̃ := π_{∗}(Γ) ∈ NE_{1}(\overline{C}_{g,g}). Note that Γ̃ = π_{∗}(Γ_{ij})/2 for all 1 ≤ i < j ≤ g.

Corollary 1.5. The following intersection identities on \overline{C}_{g,g} hold true:

R̃ · λ̃ = g + 1, R̃ · δ̃_{irr} = 6g + 18, R̃ · ψ̃ = g and R̃ · δ̃_{i:c} = 0 for all pairs (i, c),

Γ̃ · λ̃ = g + 1, Γ̃ · δ̃_{irr} = 6g + 17, Γ̃ · ψ̃ = g + 1 and Γ̃ · δ̃_{0:2} = 1, Γ̃ · δ̃_{i:c} = 0 for (i, c) ≠ (0, 2).

It follows that R̃ · K_{\overline{C}_{g,g}} = 2g − 23 and Γ̃ · K_{\overline{C}_{g,g}} = 2g − 21.

2. The Mukai model of \overline{M}_{g,g}

Having showed that the divisor \overline{D}_{g} ∈ Eff(\overline{M}_{g,g}) is extremal when g ≤ 11, our next aim is to construct a ”modular” birational contraction of \overline{M}_{g,g}, such that \overline{D}_{g} appears among its exceptional divisors. We achieve this goal for g ≤ 9, using Mukai’s fundamental work on classification of Fano varieties. We recall that for g ≤ 9, there exists a n_{g}-dimensional Fano variety V_{g} ⊂ P^{N_{g}} of index n_{g} − 2 and ρ(V_{g}) = 1, where N_{g} := g+n_{g}−2, such that general 1-dimensional complete intersections of V_{g} are canonical curves [C] ∈ M_{g} with general moduli. For g ≤ 5, V_{g} is a complete intersection of hypersurfaces; for 6 ≤ g ≤ 9 one has the following table, see [M1], [M3]:

g n_{g} N_{g}  V_{g}
6 6 10 Quadric section of cone over G(2, 5)
7 10 15  Spinor variety OG(5, 10)
8 8 14  Grassmannian G(2, 6)
9 6 13 Symplectic Grassmannian SG(3, 6)

The automorphism group Aut(V_{g}) acts in a natural way on the product V_{g}^{g} and we choose the polarization L := O_{V_{g}}(1) ⊠ ⋯ ⊠ O_{V_{g}}(1) ∈ Pic(V_{g}^{g}). We call the GIT-quotient

𝔐_{g,g} := (V_{g}^{g})^{ss}(L)//Aut(V_{g})

the Mukai model of \overline{M}_{g,g}. There exists a birational rational map

f_{g} : \overline{M}_{g,g} −−→ 𝔐_{g,g}, f_{g}([C, x_{1}, …, x_{g}]) := (x_{1}, …, x_{g}) mod Aut(V_{g}).

The inverse map is given by f_{g}^{−1}(x_{1}, …, x_{g}) := [⟨x_{1}, …, x_{g}⟩ ∩ V_{g}, x_{1}, …, x_{g}]. It is easy to see that f_{g} contracts all boundary divisors Δ_{i:T}, where i ≥ 0. From [M1], [M4], it follows that f_{g} also blows-down the pull-back of the unique Brill-Noether divisor on \overline{M}_{g} when g ≠ 4, 6 (respectively the Petri divisor on \overline{M}_{4} and \overline{M}_{6}). By comparing Picard numbers, the exceptional divisor Exc(f_{g}) must contain one extra component:

Proposition 2.1. For g ≤ 9, the morphism f_{g} contracts the divisor \overline{D}_{g}.

Proof. It suffices to note that f_{g} blows-down the covering curves R ⊂ \overline{D}_{g} ⊂ \overline{M}_{g,g} constructed in the course of proving Proposition 1.4. □

We use the existence of the Mukai variety V_{g}, to establish Theorem 0.2. Proof of Theorem 0.2. In the range g ≤ 9, n ≤ g, the unirationality of \overline{C}_{g,n} follows from that of \overline{M}_{g,n}. Indeed, the parameter space

Σ := {((x_{1}, …, x_{n}), Λ) ∈ V_{g}^{n} × G(g, N_{g} + 1) : x_{i} ∈ Λ, for i = 1, …, n}

maps dominantly onto \overline{M}_{g,n} via the map ((x_{1}, …, x_{n}), Λ) ↦ [V_{g} ∩ Λ, x_{1}, …, x_{n}]. Since Σ is a Grassmann bundle over the rational variety V_{g}^{n}, the conclusion follows. It is proved in [FP] that \overline{M}_{11,n} (thus \overline{C}_{g,n} as well), is uniruled for n ≤ 10. Similarly, \overline{M}_{10,n} is uniruled for n ≤ 9, cf. loc. cit. □

These results can be improved when g ≤ 6 using plane models of minimal degree:

Proposition 2.2. \overline{C}_{g,n} is unirational for all g ≤ 6 and n ≥ 0.

Proof. One used the representation of the general curve [C] ∈ M_{g} as a plane model of degree d := [(2g + 8)/3] with δ := \binom{d−1}{2} − g nodes in general position, cf. [AC2]. We describe the details for the case g = d = 6. We choose general points p_{1}, …, p_{4} ∈ P^{2} and fix an integer l ≥ 3 such that 6l − 20 < n ≤ 6l − 14. On the normalization C of a curve Γ ∈ |O_{P}_{2}(6)(−2 ∑_{i=1}^{4} p_{i})|, the complete linear series K_{C}(l − 3) = 𝔤_{6l−8}^{6l−14} is cut out by degree l curves passing through p_{1}, …, p_{4}. We define the incidence correspondence

U := {(Γ, X_{l}, {a_{i}}_{i=1}^{6l−8−n}) ∈ |O_{P}_{2}(6)(−2 ∑_{i=1}^{4} p_{i})| × |O_{P}_{2}(l)(− ∑_{i=1}^{4} p_{i})| × (P^{2})^{6l−8−n} :

Γ · X_{l} ≥ 2(p_{1} + ⋯ + p_{4}) + a_{1} + ⋯ + a_{6l−8−n}}.

We note that U is rational. The residuation map r : U −−→ \overline{C}_{6,n} defined by

r(Γ, X_{l}, {a_{i}}_{i=1}^{6l−8−n}) := [C, D], where Γ · X_{l} = 2 ∑_{i=1}^{4} p_{i} + ∑_{i=1}^{6l−8−n} a_{i} + D,

and C → Γ is the normalization map, is dominant. Thus \overline{C}_{6,n} is unirational. □

3. The Kodaira dimension of \overline{M}_{11,11}

On \overline{M}_{11} there exist two divisors of Brill-Noether type consisting of curves with special linear series, namely the closure of the locus of 6-gonal curves

M_{11,6}^{1} := {[C] ∈ M_{11} : G_{6}^{1}(C) ≠ ∅}

and the closure of the locus M_{11,9}^{2} := \{[C] ∈ M_{11} : G_{9}^{2}(C) ≠ ∅\}. The divisors \overline{M}_{11,6}^{1} and

\overline{M}_{11,9}^{2} are irreducible, distinct, and their classes are proportional, cf. [EH2]. Precisely, there are explicit constants c_{11,6}^{1}, c_{11,9}^{2} ∈ ℤ_{>0}, such that

𝔟𝔫_{11} :≡ \frac{1}{c_{11,6}^{1}} \overline{M}_{11,6}^{1} ≡ \frac{1}{c_{11,9}^{2}} \overline{M}_{11,9}^{2} ≡ 7λ − δ_{0} − 5δ_{1} − 9δ_{2} − 12δ_{3} − 14δ_{4} − 15δ_{5} ∈ Pic(\overline{M}_{11}).

By interpolating, we find the following explicit canonical divisor:

(8)  K_{\overline{M}_{11,11}} ≡ \overline{D}_{11} + 2 · ϕ^{∗}(𝔟𝔫_{11}) + ∑_{i=0}^{5} ∑_{c=0}^{11} d_{i:c} δ_{i:c},

where

d_{0:c} = \frac{c^{2} + c − 4}{2} for c ≥ 2, d_{1:c} = 8 + \binom{|c − 1| + 1}{2} for c ≥ 1,

d_{1:0} = 7, d_{2:c} = 16 + \binom{|c − 2| + 1}{2}, d_{3:c} = 22 + \binom{|c − 3| + 1}{2},

d_{4:c} = 26 + \binom{|c − 4| + 1}{2}, and d_{5:c} = 28 + \binom{|c − 5| + 1}{2}.

Similarly, at the level of the universal symmetric product \overline{C}_{11,11} one has the relation

(9)  K_{\overline{C}_{11,11}} ≡ D̃_{11} + 2 · φ^{∗}(𝔟𝔫_{11}) + ∑_{(i,c)≠(0,2)} d_{i:c} δ̃_{i:c}.

One already knows that multiples of \overline{D}_{11} are non-moving divisors on \overline{M}_{11,11}. We show that \overline{D}_{11} does not move in any multiple of the canonical linear system on \overline{M}_{11,11}.

Proposition 3.1. For each integer n ≥ 1, one has an isomorphism

H^{0}(\overline{M}_{11,11}, O_{\overline{M}_{11,11}}(nK_{\overline{M}_{11,11}})) ≅ H^{0}(\overline{M}_{11,11}, O_{\overline{M}_{11,11}}(nK_{\overline{M}_{11,11}} − n\overline{D}_{11})).

In particular, κ(\overline{M}_{11,11}) = κ(\overline{M}_{11,11}, K_{\overline{M}_{11,11}} − \overline{D}_{11}). Furthermore, on \overline{C}_{11,11}, one has that κ(\overline{C}_{11,11}) = κ(\overline{C}_{11,11}, K_{\overline{C}_{11,11}} − D̃_{11}).

Proof. Using the notation and results from Proposition 1.4, we recall that we have constructed a curve R ⊂ \overline{M}_{11,11} moving in a family which fills-up the divisor \overline{D}_{11}, such that R · \overline{D}_{11} = −1 and R · δ_{i:S} = 0, for all i ≥ 0 and T ⊂ \{1, …, g\}. All points in R correspond to nodal curves lying on a fixed K3 surface S, which by the generality assumptions, can be chosen such that Pic(S) = ℤ. Applying [Laz], all underlying genus 11 curves corresponding to points in R satisfy the Brill-Noether theorem, in particular R · ϕ^{∗}(𝔟𝔫_{11}) = 0, that is, R · K_{\overline{M}_{11,11}} = R · \overline{D}_{11} = −1. It follows that for any effective divisor E on \overline{M}_{11,11} such that E ≡ nK_{\overline{M}_{11,11}}, one has that R · E = −n. Moreover, the class E − n\overline{D}_{11} is still effective and then |nK_{\overline{M}_{11,11}}| = n\overline{D}_{11} + |nK_{\overline{M}_{11,11}} − n\overline{D}_{11}|. The proof in the case of \overline{C}_{11,11} is similar. One uses that π^{∗}(D̃_{11}) = \overline{D}_{11}, hence R̃ · D̃_{11} = R · \overline{D}_{11} = −1, as well as R̃ · K_{\overline{C}_{11}} = −1. The rest of the argument is identical. □

We are in a position to complete the proof of Theorem 0.5:

Theorem 3.2. We have that

κ(\overline{M}_{11,11}, 2 · ϕ^{∗}(𝔟𝔫_{11}) + ∑_{i,c} d_{i:c} · δ_{i:c}) = κ(\overline{C}_{11,11}, 2 · φ^{∗}(𝔟𝔫_{11}) + ∑_{(i,c)≠(0,2)} d_{i:c} · δ̃_{i:c}) = 19.

It follows that the Kodaira dimension of both \overline{M}_{11,11} and \overline{C}_{11,11} equals 19.

Proof. To simplify the proof, we define a few divisors classes on \overline{M}_{11,11}:

A := 2 · ϕ^{∗}(𝔟𝔫_{11}) + ∑_{i≥0,c} d_{i:c} δ_{i:c} ≡ K_{\overline{M}_{11,11}} − \overline{D}_{11} and A′ := A − ∑_{c=2}^{11} d_{0:c} δ_{0:c},

as well as, B := 𝔟𝔫_{11} + 4δ_{3} + 7δ_{4} + 8δ_{5} ∈ Pic(\overline{M}_{11}).

We claim that for all integers n ≥ 1, one has isomorphisms,

H^{0}(\overline{M}_{11,11}, O_{\overline{M}_{11,11}}(nA)) ≅ H^{0}(\overline{M}_{11,11}, O_{\overline{M}_{11,11}}(nA′)).

Indeed, we fix a set of labels T ⊂ {1, …, 11} such that #(T) ≥ 2 and consider a pencil

\{[C_{t}, x_{i}(t), p(t) : i ∈ T^{c}]\}_{t∈P^{1}} ⊂ \overline{M}_{11,12−#(T)},

of (12 − #(T))-pointed curves of genus 11 on a general K3 surface S, with marked points being labeled by elements in T^{c} as well by another label p(t). The pencil is induced by a fibration obtained from a Lefschetz pencil of genus 11 curves on S, with regular sections given by (12 − #(T)) of the exceptional divisors obtained by blowingup S at the (2g −2) base points of the pencil. To each element in this pencil, we attach at the marked point labeled by p(t), a fixed copy of P^{1} together with fixed marked points x_{i} ∈ P^{1} − {∞}, for i ∈ T. The gluing identifies the point p(t) ∈ C_{t} with ∞ ∈ P^{1}. If R_{T} ⊂ \overline{M}_{11,11} denotes the resulting family, we compute:

R_{T} ·λ = g+1, R_{T} ·δ_{irr} = 6(g+3), R_{T} ·δ_{0:T} = −1, R_{T} ·ψ_{i} = 1 for i ∈ T^{c}, R_{T} ·ψ_{i} = 0 for i ∈ T.

Moreover, R_{T} is disjoint from all remaining boundary divisors of \overline{M}_{11,11}. One finds that R_{T} · ϕ^{∗}(𝔟𝔫_{11}) = 0. Thus for any effective divisor E ⊂ \overline{M}_{11,11} such that E ≡ nA, we find that R_{T} · E = −nd_{0,c}.

Since for all T, the pencil R_{T} fills-up the divisor Δ_{0:T}, we can deform the curves R_{T} ⊂ Δ_{0:T}, to find that E − ∑_{c=2}^{11} nd_{0:c} · δ_{0:c} is still an effective class, that is,

|nA| = ∑_{c=2}^{11} nd_{0:c} · Δ_{0:c} + |nA′|,

which proves the claim. Next, by direct calculation we observe that the class A′−2ϕ^{∗}(B) is effective. Zariski’s Main Theorem gives that ϕ_{∗}ϕ^{∗}O_{\overline{M}_{11}}(B) = O_{\overline{M}_{11}}(B), thus

κ(\overline{M}_{11,11}, A′) ≥ κ(\overline{M}_{11,11}, ϕ^{∗}(B)) = κ(\overline{M}_{11}, B) = 19.

The last equality comes from [FP] Proposition 6.2: The class B contains the pull-back of an ample class under the Mukai map [M2]

q_{11} : \overline{M}_{11,11} −−→ \overline{F}_{11}, [C, x_{1}, …, x_{11}] ↦ [S ⊃ C, O_{S}(C)],

to a compactification of the moduli space of polarized K3 surfaces of degree 20.

On the other hand, since ϕ^{∗}(δ_{i}) = ∑_{S} δ_{i:S} for 1 ≤ i ≤ 5, there is a divisor class on \overline{M}_{11} of type B′ := 2 · 𝔟𝔫_{11} + ∑_{i=1}^{5} a_{i}δ_{i} ∈ Pic(\overline{M}_{11}), with a_{i} ≥ 0, such that ϕ^{∗}(B′) − A′ is an effective divisor. It follows that

κ(\overline{M}_{11,11}, A′) ≤ κ(\overline{M}_{11,11}, ϕ^{∗}(B′)) = κ(\overline{M}_{11}, B′).

If R_{11} ⊂ \overline{M}_{11} is the family corresponding to a Lefschetz pencil of curves of genus 11 on a fixed K3 surface, then R_{11} · B′ = 0. The pencil R_{11} moves in a 11-dimensional family inside \overline{M}_{11} which is contracted to a point by any linear series |nB′| on \overline{M}_{11} with n ≥ 1 (in fact a general curve R_{11} is disjoint from the base locus of |nB′|). One finds that κ(\overline{M}_{11}, B′) ≤ 19, which completes the proof. The case of \overline{C}_{11,11} proceeds with obvious modifications. □

4. The Kodaira dimension of \overline{C}_{10,10}

The geometry of \overline{M}_{10} is governed to a large extent by the divisor \overline{K}_{10} of curves lying on K3 surfaces. It is shown in [FP] that \overline{K}_{10} is an irreducible divisor of class

\overline{K}_{10} ≡ 7λ − δ_{0} − 5δ_{1} − 9δ_{2} − 12δ_{3} − 14δ_{4} − b_{5}δ_{5} ∈ Pic(\overline{M}_{10}),

where b_{5} ≥ 6. Furthermore, s(\overline{K}_{10}) = 7 is the minimal slope of an effective divisor on \overline{M}_{10}. The irreducible divisor \overline{K}_{10} is rigid. Indeed, if R_{10} ⊂ \overline{K}_{10} is the covering family obtained by blowing-up the base points of a pencil of curves of genus 10 on a fixed K3 surface, then R_{10} · \overline{K}_{10} = −1 (see [FP] Lemma 2.1). Since R_{10} · δ_{i} = 0 for 1 ≤ i ≤ 5, this argument proves that any divisor D ∈ Eff(\overline{M}_{10}) with s(D) = s(\overline{K}_{10}), is rigid as well.

Comparing the expression of [\overline{K}_{10}] with that of K_{\overline{C}_{10,10}}, we obtain the formula

(10)  K_{\overline{C}_{10,10}} ≡ D̃_{10} + 2 · φ^{∗}(\overline{K}_{10}) + ∑_{(i,c)≠(0,2)} d_{i:c} δ̃_{i:c},

where remarkably, the coefficients d_{i:c} have exactly the same values as in formula (10) for 1 ≤ i ≤ 4, while d_{5:c} = 2b_{5} − 2 + \binom{|c−5|+1}{2}, for 0 ≤ c ≤ 10. We point out that the coefficient d_{0:2} of δ̃_{0:2} in formula (10), equals 0.

Theorem 4.1. For each n ≥ 1, there is an isomorphism of groups

H^{0}(\overline{C}_{10,10}, O_{\overline{C}_{10,10}}(nK_{\overline{C}_{10,10}})) ≅ H^{0}(\overline{C}_{10,10}, O_{\overline{C}_{10,10}}(nK_{\overline{C}_{10,10}} − nD̃_{10})).

Proof. We use Corollary 1.5. Through a general point of the divisor \overline{D}_{10} on \overline{M}_{10,10} there passes a curve Γ̃ ⊂ \overline{C}_{10,10} such that Γ̃ · D̃_{10} = Γ̃ · K_{\overline{C}_{10,10}} = −1 and Γ · δ̃_{i:c} = 0 for (i, c) ≠ (0, 2). One obtains that |nK_{\overline{C}_{10,10}}| = nD̃_{10} + |nK_{\overline{C}_{10,10}} − nD̃_{10}|. □

End of proof of Theorem 0.1 when g = 10. We define the following divisor classes on \overline{C}_{10,10}:

A := 2 · φ^{∗}(\overline{K}_{10}) + ∑_{(i,c)≠(0,2)} d_{i:c} δ̃_{i:c} ≡ K_{\overline{C}_{10,10}} − D̃_{10} and A′ := A − ∑_{c=3}^{10} d_{0:c} δ̃_{0:c}.

We claim that H^{0}(\overline{C}_{10,10}, O_{\overline{C}_{10,10}}(nA)) ≅ H^{0}(\overline{C}_{10,10}, O_{\overline{C}_{10,10}}(nA′)) for all n ≥ 1. Indeed, we fix a set of labels T ⊂ {1, …, 10} with c := #(T) ≥ 3, as well as two indexes i, j ∈ T^{c} and consider the 1-cycle Γ_{ij} ⊂ \overline{M}_{10,11−c} constructed in Proposition 1.4. We label by {p(t), x_{l}(t) : t ∈ P^{1}}_{l∈T^{c}} the sections of the family. We obtain a covering curve Γ′_{ij} for the divisor Δ_{0:T} ⊂ \overline{M}_{10,10}, by attaching along the section p(t) a fixed (c + 1)-pointed rational curve to each of the curves in Γ_{ij}, in a way that the marked points labeled by T are precisely those lying on the rational component. Then Γ̃_{0:c} := π_{∗}(Γ′_{ij}) ⊂ \overline{C}_{10,10} is a covering curve for Δ̃_{0:c}. From Corollary 1.5, Γ̃_{0:c} · δ̃_{0:c} < 0 and Γ̃_{0:c} has intersection number 0 with all the components of supp(A) − Δ̃_{0:c} (Note that Δ̃_{0:2} does not appear among these components). We repeat this argument for all divisors Δ̃_{0:c}, where 3 ≤ c ≤ g, and the claim becomes obvious. We finish the proof by using the same argument as at the end of the proof of Theorem 3.1: There exists an effective class B′ ∈ Eff(\overline{M}_{10}) such that B′ −\overline{K}_{10} is effective, s(B′) = s(\overline{K}_{10}) = 7, and such that φ^{∗}(B′)−A′ is effective. Then κ(\overline{C}_{10,10}) = κ(\overline{C}_{10,10}, A′) ≤ κ(\overline{C}_{10,10}, φ^{∗}(B′)) = κ(\overline{M}_{10}, B′) = 0. □

5. The uniruledness of \overline{M}_{g,n}

We formulate two general principles, which we use in proving the uniruledness of some moduli spaces \overline{M}_{g,n}. We begin with the following trivial remark:

Proposition 5.1. Let X be a projective ℚ-factorial variety and D ⊂ X an irreducible divisor filled-up by curves Γ ⊂ X, such that Γ · D ≥ 0 and Γ · K_{X} < 0. Then X is uniruled.

We can extend this principle to the case of several divisors as follows:

Proposition 5.2. Let X be a projective ℚ-factorial variety and suppose D_{1}, D_{2} ⊂ X are irreducible effective ℚ-divisors such that there exist covering curves Γ_{i} ⊂ D_{i}, with Γ_{i} · D_{i} < 0 for i = 1, 2 (in particular both D_{i} ∈ Eff(X) are non-movable divisors). Assume furthermore that

(11)  |\begin{matrix}Γ_{1} · D_{1} & Γ_{1} · D_{2} \\ Γ_{2} · D_{1} & Γ_{2} · D_{2} \end{matrix}| ≤ 0, |\begin{matrix}Γ_{1} · K_{X} & Γ_{1} · D_{1} \\ Γ_{2} · K_{X} & Γ_{2} · D_{1} \end{matrix}| < 0.

Then X is a uniruled variety.

Proof. According to [BDPP], it suffices to prove that K_{X} is not pseudo-effective. By contradiction, we choose α, β ∈ ℝ_{≥0} maximal such that K_{X} − αD_{1} − βD_{2} ∈ \overline{Eff}(X). Then we can write down the inequalities

Γ_{1} · K_{X} ≥ α(Γ_{1} · D_{1}) + β(Γ_{1} · D_{2}) and Γ_{2} · K_{X} ≥ α(Γ_{2} · D_{1}) + β(Γ_{2} · D_{2}).

Eliminating α, the resulting inequality contradicts the assumption β ≥ 0. □

We turn our attention to the proof of Theorem 0.6 which we split in three parts:

Theorem 5.3. \overline{M}_{5,n} is uniruled for n ≤ 13.

Proof. A general 2-pointed curve [C, x, y] ∈ M_{5,2} carries a finite number of linear series

L ∈ W_{6}^{2}(C), such that if ν_{L} : C ⟶^{|L|} Γ ⊂ P^{2} is the induced plane model, then ν_{L}(x) = ν_{L}(y) = p_{1}. Note that Γ has nodes, say p_{1}, …, p_{5}, and dim |O_{P}_{2}(Γ)(−2 ∑_{i=1}^{5} p_{i})| = 12.

We pick general points {x_{i}}_{i=1}^{11} and {p_{j}}_{j=1}^{5} ⊂ P^{2}, then consider the pencil of sextics passing with multiplicity 1 through x_{1}, …, x_{11} and having nodes (only) at p_{1}, …, p_{5}. The pencil induces a fibration f′ : S → P^{1}, where S := Bl_{21}(P^{2}) is obtained from P^{2} by blowing-up p_{1}, …, p_{5}, x_{1}, …, x_{11}, as well as the remaining unassigned base points of the pencil. The exceptional divisors E_{x_{i}} ⊂ S provide 11 sections of f′. The exceptional divisor E_{p_{1}} induces a 2-section. Making a base change via the map f′_{E_{p_{1}}} : E_{p_{1}} → P^{1}, the 2-section E_{p_{1}} splits into two sections E_{x} and E_{y} meeting at 2 points. Blowing these points up, we arrive at a fibration f : Y → E_{p_{1}}, carrying 13 everywhere disjoint sections, Ẽ_{x}, Ẽ_{y}, Ẽ_{x_{1}}, …, Ẽ_{x_{11}}, where Ẽ_{x_{i}} ⊂ Y denotes the inverse image of E_{x_{i}}, and Ẽ_{x}, Ẽ_{y} denote the proper transforms of E_{x} and E_{y} respectively. This induces a family of pointed stable curves

Γ := \{[C_{λ} := f^{−1}(λ), Ẽ_{x} · C_{λ}, Ẽ_{y} · C_{λ}, Ẽ_{x_{1}} · C_{λ}, …, Ẽ_{x_{11}} · C_{λ}] : λ ∈ E_{p_{1}}\} ⊂ \overline{M}_{5,13}.

We compute the numerical characters of Γ (see also the proof of Proposition 1.4):

Γ · λ = deg(f_{E_{p_{1}}} )(χ(S, O_{S}) + g − 1) = 10, Γ · δ_{irr} = deg(f_{E_{p_{1}}} )(c_{2}(S) + 4g − 4) = 80,

Γ · ψ_{x} = Γ · ψ_{y} = 5, Γ · ψ_{x_{1}} = ⋯ = Γ · ψ_{x_{11}} = 2, Γ · δ_{0:xy} = 2,

whereas Γ is disjoint from the remaining boundary divisors. One finds, Γ·K_{\overline{M}_{5,13}} = −2, which completes the proof. □

Remark 5.4. It is known that \overline{M}_{5,15} is of general type, [F2]. Using the fact that 12 general points in P^{4} determine a canonical curve of genus 5, it is proved in [CF] that \overline{M}_{5,n} is rational when n ≤ 12. Hence Theorem 5.3 settles the cases \overline{M}_{5,13}.

Theorem 5.5. \overline{M}_{8,n} is uniruled for n ≤ 12.

Proof. We apply Proposition 5.2 when D_{1} is a suitable multiple of the Brill-Noether divisor on \overline{M}_{8} consisting of curves with a 𝔤_{7}^{2}, that is,

D_{1} ≡ 2 · 𝔟𝔫_{8} := \frac{2}{c_{8,7}^{2}} \overline{M}_{8,7}^{2} ≡ 22λ − 3δ_{0} − 14δ_{1} − 24δ_{2} − 30δ_{3} − 32δ_{4} ∈ Pic(\overline{M}_{8}).

We also set D_{2} := Δ_{irr} ∈ Eff(\overline{M}_{8,n}). To construct a covering curve Γ_{1} ⊂ D_{1}, we lift to \overline{M}_{8,n} a Lefschetz pencil of 7-nodal plane septics. The fibration f : Bl_{28}(P^{2}) → P^{1} obtained by blowing-up the 21 + 7 base points of a general pencil of 7-nodal plane septics, induces a covering curve m : P^{1} → \overline{M}_{8} for the irreducible divisor \overline{M}_{8,7}^{2}. The numerical invariants of this pencil are

m^{∗}(λ) = χ(S, O_{S}) + g − 1 = 8 and m^{∗}(δ_{0}) = c_{2}(S) + 4(g − 1) = 59,

while m^{∗}(δ_{i}) = 0 for i = 1, …, 4. Moreover, for n as above, f carries n sections given by the exceptional divisors corresponding to n of the unassigned base points. If Γ_{1} ⊂ \overline{M}_{8,n} denotes the resulting, then

Γ_{1} · λ = ϕ_{∗}(Γ_{1}) · λ = 8, Γ_{1} · δ_{irr} = ϕ_{∗}(Γ_{1}) · δ_{irr} = 59, Γ_{1} · ψ_{i} = 1 for i = 1, …, n,

and Γ_{1} · δ_{i:T} = 0. It follows that Γ_{1} · D_{1} = −1, Γ_{1} · K_{\overline{M}_{8,n}} = n − 14 and Γ_{1} · D_{2} = 59.

We construct a covering curve Γ_{2} ⊂ D_{2} and start with a general pointed curve [C, x_{1}, …, x_{n+1}] ∈ \overline{M}_{7,n+1}. We identify x_{n+1} with a moving point y ∈ C, that is, take

Γ_{2} := \{[\frac{C}{y ∼ x_{n+1}} , x_{1}, …, x_{n}] : y ∈ C\} ⊂ \overline{M}_{8,n}.

It is easy to compute that Γ_{2} · λ = 0, Γ_{2} · δ_{irr} = −2g(C) = −14, Γ_{2} · δ_{1:∅} = 1, Γ_{2} · ψ_{i} = 1, for i = 1, …, n, and Γ_{2} · δ_{i:T} = 0 for (i, T) ≠ (1, ∅). Therefore Γ_{2} · D_{1} = 28 and Γ_{2} · K_{\overline{M}_{8,n}} = 25 + n. The assumptions of Proposition 5.2 are satisfied when n ≤ 12. □

Remark 5.6. The results of Theorem 5.5 are almost optimal. The space \overline{M}_{8,14} is of general type, [Log]. Note that it was already known [Log], [CF], that \overline{M}_{8,n} is unirational for n ≤ 11, thus the improvement here is the case \overline{M}_{8,12}.

Proposition 5.7. \overline{M}_{9,n} is uniruled for n ≤ 10 (in fact unirational for n ≤ 9).

Proof. We apply Proposition 5.1, when D := ϕ^{∗}(\overline{M}_{9,5}^{1}) is the pull-back of the 5-gonal locus inside \overline{M}_{9}. If [C] ∈ \overline{M}_{9,5}^{1} is a general 5-gonal curve and A ∈ W_{5}^{1}(C) is the (unique) 𝔤_{5}^{1}, then there exists an effective divisor D ∈ C_{3}, such that h^{0}(C, A ⊗ O_{C}(D)) ≥ 3, cf. [AC2]. In particular, A ⊗ O_{C}(D) ∈ W_{8}^{2}(C) induces a plane model of C having a 3-fold point, such that |A| can be retrieved by projecting from this point.

To obtain a covering curve for D, we start with general points p_{0}, p_{1}, …, p_{9} ∈ P^{2} and consider the surface ϵ : S := Bl_{p_{0},...,p_{9}}(P^{2}) → P^{2} together with the line bundle L := ϵ^{∗}O_{P}_{2}(8) ⊗ O_{S}(−3E_{p_{0}} − ∑_{i=1}^{9} E_{p_{i}}) ∈ Pic(S). Note that dim |L| = 11. We fix 10 general points x_{1}, …, x_{10} ∈ S, hence the pencil |I_{{x_{1},...,x_{10}}/S} ⊗ L| induces a curve Γ ⊂ \overline{M}_{9,10}. Standard calculations yield that Γ · λ = 9, Γ · δ_{irr} = 64 and Γ · ψ_{i} = 1 for i = 1, …, 10. Therefore Γ · K_{\overline{M}_{9,10}} = −1, while Γ · ϕ^{∗}(\overline{M}_{9,5}^{1}) > 0. Since Γ ⊂ ϕ^{∗}(\overline{M}_{9,5}^{1}) is a covering curve, this finishes the proof. □

Finally, we turn to the case of genus 7. In order to establish the uniruledness of \overline{M}_{7,n}, we consider the following effective divisors on \overline{M}_{7,n}:

D_{1} := {[C, x_{1}, …, x_{n}] ∈ M_{7,n} : ∃L ∈ W_{7}^{2}(C) with h^{0}(C, L(−x_{1} − x_{2})) ≥ 1},

and D_{2} := \frac{3}{2c_{7,4}^{1}} ϕ^{∗}(\overline{M}_{7,4}^{1}) ≡ 15λ − 2δ_{0} − 9δ_{1} − 15δ_{2} − 18δ_{3} ∈ Pic(\overline{M}_{7}) is (a rational

multiple of) the divisor of 4-gonal curves on \overline{M}_{7}. Before computing the class [\overline{D}_{1}], we need a calculation, which may be of independent interest:

Proposition 5.8. Let g ≡ 1 mod 3 be a fixed integer and set d := (2g + 7)/3, so that the Brill-Noether number ρ(g, 2, d) = 1. One considers the effective divisor of nodes of plane curves

𝔑𝔬𝔡𝔢_{g} := {[C, x, y] ∈ M_{g,2} : ∃L ∈ W_{d}^{2}(C) such thath^{0}(C, L(−x − y)) ≥ 2}.

The class of its closure in \overline{M}_{g,2} is given by the formula:

\overline{𝔑𝔬𝔡𝔢}_{g} ≡ c_{g}((g + 4)λ + \frac{g + 2}{6}(ψ_{1} + ψ_{2}) − \frac{g + 2}{6}δ_{irr} − gδ_{0:\{1,2\}} − ⋯) ∈ Pic(\overline{M}_{g,2}),

where c_{g} := \frac{24(g − 2)!}{(g − d + 5)! (g − d + 3)! (g − d + 1)!}.

Proof. We denote by ϕ_{1} : \overline{M}_{g,2} → \overline{M}_{g,1} the morphism forgetting the second marked point. The divisor \overline{ℭ𝔲}_{g} := (ϕ_{1})_{∗}(\overline{𝔑𝔬𝔡𝔢}_{g} · δ_{0:\{1,2\}}) coincides with the cusp locus in \overline{M}_{g,1}, that is, the locus of pointed curves [C, x] ∈ M_{g,1}, such that there exists L ∈ W_{d}^{2}(C) with h^{0}(C, L(−2x)) ≥ 2.

In order to compute its class, we fix a general elliptic curve [E, x] ∈ \overline{M}_{1,1} and consider the map j : \overline{M}_{g,1} → \overline{M}_{g+1}, given by j([C, x]) := [C ∪_{x} E]. Then

\overline{ℭ𝔲}_{g} = j^{∗}(\overline{M}_{g+1,d}^{2}),

where \overline{M}_{g+1,d}^{2} is the Brill-Noether divisor on \overline{M}_{g+1} consisting of curves with a 𝔤_{d}^{2} (Note that ρ(g + 1, 2, d) = −1). Since the class [\overline{M}_{g+1,d}^{2}] ∈ Pic(\overline{M}_{g+1}) is known, cf. [EH2], and j^{∗}(λ) = λ, j^{∗}(δ_{irr}) = δ_{irr}, j^{∗}(δ_{1}) = −ψ + δ_{g−1:1}, we find the following expression

\overline{ℭ𝔲}_{g} ≡ c_{g}((g + 4)λ + gψ − \frac{g + 2}{6}δ_{irr} − ∑_{i=1}^{g−1}(i + 1)(g − i)δ_{i:1}) ∈ Pic(\overline{M}_{g,1}).

Using the formulas (ϕ_{1})_{∗}(λ · δ_{0:{1,2}}) = λ, (ϕ_{1})_{∗}(δ_{0:{1,2}}^{2}) = −ψ, (ϕ_{1})_{∗}(δ_{irr} · δ_{0:{1,2}}) = δ_{irr} and (ϕ_{1})_{∗}(ψ_{i} · δ_{0:\{1,2\}}) = 0 for i = 1, 2, one finds that the δ_{0:\{1,2\}}-coefficient of \overline{𝔑𝔬𝔡𝔢}_{g} equals the ψ_{1}-coefficient of \overline{ℭ𝔲}_{g}, while the λ, δ_{irr}-coefficients coincide.

One still has to determine the ψ_{1}-coefficient in [\overline{𝔑𝔬𝔡𝔢}_{g}]. To this end, we fix a general point [C, q] ∈ M_{g,2} and consider the test curve C_{2} := \{[C, q, y] : y ∈ C\} ⊂ \overline{M}_{g,2}. Then, C_{2} · ψ_{1} = 1, C_{2} · ψ_{2} = 2g − 1 and obviously C_{2} · δ_{0:{1,2}} = 1. On the other hand, C_{2} · \overline{𝔑𝔬𝔡𝔢}_{g} equals the number of points y ∈ C, such that for some (necessarily complete and base point free) L ∈ W_{d}^{2}(C), the morphism

χ(L, y) : L_{|y+q}^{∨} → H^{0}(C, L)^{∨}

fails to be injective. The map χ(L, y) globalizes to a morphism of vector bundles over C × W_{d}^{2}(C), and the number in question is the Chern number of the top degeneracy locus of χ and is computed using [HT]. We omit the details. □

Proposition 5.9. The class of the closure of D_{1} in \overline{M}_{7,n} is given by the formula

\overline{D}_{1} ≡ 44λ + 6(ψ_{1} + ψ_{2}) − 6δ_{irr} − 28δ_{0:\{1,2\}} − 6 ∑_{j=3}^{n}(δ_{0:\{1,j\}} + δ_{0:\{2,j\}}) − ⋯ ∈ Pic(\overline{M}_{7,n}).

Proof. We denote by ϕ_{12} : \overline{M}_{7,n} → \overline{M}_{7,2} the morphism retaining the first two marked points. Then \overline{D}_{1} = ϕ_{12}^{∗}(\overline{𝔑𝔬𝔡𝔢}_{7}), and the conclusion follows from Proposition 5.8 using the pull-back formulas for generators of Pic(\overline{M}_{7,2}), see e.g. [Log] Theorem 2.3. □

Theorem 5.10. \overline{M}_{7,n} is uniruled for n ≤ 13.

Proof. We start by constructing a covering curve for \overline{D}_{1}. Choose general points p_{1}, …, p_{8}, x_{3}, …, x_{12} ∈ P^{2}, and a general line l ⊂ P^{2}. Then consider the pencil of plane septics of geometric genus 7 passing through x_{3}, …, x_{12} and having nodes at p_{1}, …, p_{8}. Blowingup the nodes as well as the base points of the pencil, we obtain a fibration f : S → P^{1}, where S := Bl_{25}(P^{2}). We observe that f has sections {E_{x_{i}}}_{i=3,...,12}, given by the respective exceptional divisors, a 2-section given by E_{p_{1}} and a 7-section induced by the proper transform of l. We make base changes of order 2 and 7 respectively, to arrive at the 1-cycle Γ_{1} := \{[C_{t}, x_{1}(t), …, x_{13}(t)] : t ∈ P^{1}\} ⊂ \overline{M}_{7,13}, where x_{1}(t) and x_{2}(t) map to the fixed node p_{1} ∈ P^{2}, whereas the image of x_{13}(t) lies on the line l. One finds that:

Γ_{1} · λ = 14 · g = 98, Γ_{1} · ψ_{1} = Γ_{1} · ψ_{2} = 35, Γ_{1} · ψ_{3} = ⋯ = Γ_{1} · ψ_{12} = 14, Γ_{1} · ψ_{13} = 22.

Furthermore, Γ_{1} · δ_{0:{1,2}} = 14, Γ_{1} · δ_{irr} = 14 · 52 = 728, and finally Γ_{1} · δ_{j:T} = 0 for all pairs (j, T) ≠ (0, \{1, 2\}). Clearly Γ_{1} is a covering curve for \overline{D}_{1}.

Next, we construct a covering curve for D_{2} and use that if [C] ∈ M_{7,4}^{1} and A ∈ W_{4}^{1}(C) is the corresponding pencil, then there exists a divisor D ∈ C_{3} such that A ⊗ O_{C}(D) ∈ W_{7}^{2}(C). One fixes general points p, {p_{i}}_{i=1}^{5}, {x_{j}}_{j=1}^{13} ∈ P^{2} and considers the pencil of genus 7 septics with a 3-fold point at p, nodes at p_{1}, …, p_{5} and passing through x_{1}, …, x_{13}. This induces a covering curve Γ_{2} ⊂ ϕ^{∗}(\overline{M}_{7,4}^{1}) with the following invariants:

Γ_{2} · λ = 7, Γ_{2} · δ_{irr} = 53, Γ_{2} · ψ_{i} = 1 for i = 1, …, 13, and Γ_{2} · δ_{j:T} = 0 for all (j, T).

Thus, Γ_{1} · \overline{D}_{1} = −28, Γ_{2} · D_{2} = −1, Γ_{1} · D_{2} = 14, Γ_{2} · \overline{D}_{1} = 2, as well as Γ_{1} · K_{\overline{M}_{7,13}} = 22 and Γ_{2} · K_{\overline{M}_{7,13}} = −2. The assumptions of Proposition 5.2 are thus fulfilled. □

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Humboldt-Universität zu Berlin, Institut Für Mathematik, Unter den Linden 6 10099 Berlin, Germany

      E-mail address: farkas@math.hu-berlin.de

Universitá Roma Tre, Dipartimenti di Matematica, Largo San Leonardo Murialdo 1-00146 Roma, Italy

      E-mail address: verra@mat.unirom3.it