THE INTERMEDIATE TYPE OF CERTAIN MODULI SPACES 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 or level structure), 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], [F3], [Log], [V] for illustrations of this fact. In all known cases, the transition from uniruledness to being of general type is quite sudden and until now no examples were known of naturally defined moduli spaces of curves of intermediate Kodaira dimension. The aim of this paper is to discuss the very surprising birational geometry of special moduli spaces of curves, which in particular have intermediate Kodaira dimension.

The moduli space S_{g} of smooth spin curves parameterizes pairs [C, η], where [C] ∈ M_{g} is a curve of genus g and η ∈ Pic^{g−1}(C) is a theta-characteristic. The map π : S_{g} → M_{g} is an étale covering of degree 2^{2g} and S_{g} is a disjoint union of two connected components S_{g}^{+} and S_{g}^{−} of relative degrees 2^{g−1}(2^{g} + 1) and 2^{g−1}(2^{g} − 1) corresponding to even and odd theta-characteristics respectively. We denote by \overline{S}_{g} the Cornalba compactification of S_{g}, that is, the coarse moduli space of the stack of stable spin curves of genus g, cf. [C]. The projection π : S_{g} → M_{g} extends to a finite covering π : \overline{S}_{g} → \overline{M}_{g} branched along the boundary divisor Δ_{0} of \overline{M}_{g}. It is known that \overline{S}_{g}^{+} is a variety of general type for g > 8 and uniruled for g < 8, cf. [F3]. We show that the only remaining case, that of \overline{S}_{8}^{+}, gives rise to a variety of Calabi-Yau type:

Theorem 0.1. The Kodaira dimension of \overline{S}_{8}^{+} is equal to zero.

We point out that the Kodaira dimension of the odd spin moduli space \overline{S}_{g}^{−} is known for all genera g, cf. [FV]. Thus \overline{S}_{g}^{−} is uniruled for g ≤ 11 (even unirational for g ≤ 9), and of general type for g ≥ 12. In particular, we observe the surprising phenomenon that \overline{S}_{8}^{−} is unirational, whereas \overline{S}_{8}^{+} is of Calabi-Yau type!

The proof of Theorem 0.1 relies on two main ideas: Following [F3], one finds an explicit effective representative for the canonical divisor K\overline{S}{+ \atop 8} as a ℚ-combination of the

divisor \overline{Θ}_{null} ⊂ \overline{S}_{8}^{+} of vanishing theta-nulls, the pull-back π^{∗}(\overline{M}_{8,7}^{2}) of the Brill-Noether divisor \overline{M}_{8,7}^{2} on \overline{M}_{8} of curves with a 𝔤_{7}^{2}, and boundary divisor classes corresponding to spin curves whose underlying stable model is of compact type. Each irreducible component of this particular representative of K\overline{S}{+ \atop 8} is rigid (see Section 1). Then we use

in an essential way the existence of a Mukai model of \overline{M}_{8} as a GIT quotient of a bundle over the Grassmannian G := G(2, 6) cf. [M2], in order to prove the following result:

Research of the first author partially supported by Sonderforschungsbereich ”Raum-Zeit-Materie”.

Proposition 0.2. The uniruled divisor \overline{Θ}_{null} ⊂ \overline{S}_{8}^{+} is swept by rational curves R ⊂ \overline{S}_{8}^{+} such that R·\overline{Θ}_{null} = −1 and R·π^{∗}(\overline{M}_{8,7}^{2}) = 0. Furthermore R is disjoint from all boundary divisors A_{i}, B_{i} ⊂ \overline{S}_{8}^{+} for i = 1, …, 4.

The pencil R corresponds to spin curves lying on special doubly elliptic K3 surfaces S, chosen in such a way that the rank 3 quadric containing the underlying canonical curve C ⊂ P^{7} corresponding to a general point [C, η] ∈ \overline{Θ}_{null}, lifts to a rank 4 quadric in P^{8} containing the K3 surface S ⊃ C. The existence of such K3 extensions of C follows from a precise description of quadrics containing the Plücker embedding of the Grassmannian G ⊂ P^{14} (see Sections 2 and 3). Proposition 0.2 implies that K\overline{S}{+ \atop 8}

expressed as a weighted sum of \overline{Θ}_{null}, the pull-back π^{∗}(\overline{M}_{8,7}^{2}) and boundary divisors A_{i}, B_{i}, i = 1, …, 4, is rigid as well. Equivalently, κ(\overline{S}_{8}^{+}) = 0.

Our next result concerns the moduli space \overline{M}_{g,n} of stable n-pointed curves of genus g. For a given genus g ≥ 2, we define the numerical 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). Clearly ζ(g) = 0 for g ≥ 22, cf. [HM], [EH2]. There exist explicit upper bounds for ζ(g) for 4 ≤ g ≤ 21, see [Log], [F2] Theorem 1.10. In particular, it is known that \overline{M}_{10,n} is uniruled for n ≤ 9, of general type for n ≥ 11, whereas κ(\overline{M}_{10,10}) ≥ 0, cf. [FP] Proposition 7.5. In other words, ζ(10) = 10. Similarly, it is known that \overline{M}_{11,n} is uniruled for n ≤ 10 and of general type for n ≥ 12. Until now, no example of a space \overline{M}_{g,n} (g ≥ 2) having intermediate type was known. Perhaps, the most picturesque finding of our study is the following:

Theorem 0.3. The moduli space \overline{M}_{11,11} has Kodaira dimension 19.

Note that dim(\overline{M}_{11,11}) = 41. In particular, Theorem 0.3 determines the value ζ(11) = 11, hence ζ(11) > ζ(10). This explains, in precise terms, that counter-intuitively, algebraic curves of genus 10 are more complicated than curves of genus 11!

The equality κ(\overline{M}_{11,11}) = 19 is related to the existence of 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 [M3]. 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, (i) the divisor n\overline{D}_{11} is a fixed component of the pluri-canonical linear series |nK_{\overline{M}_{11,11}}| for all n ≥ 1, and (ii) the difference K_{\overline{M}_{11,11}} − \overline{D}_{11} is essentially the pull-back of an ample class on \overline{F}_{11}.

The proof of Theorem 0.3 is similar in spirit to the proof of Theorem 0.1. An important 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} is the following, cf. [Log] Theorem 5.4:

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

Using [FP], as well as the expression of K_{\overline{M}_{g,n}} in terms of generators of Pic(\overline{M}_{g,n}), one finds an explicit representative of K_{\overline{M}_{11,11}} as an effective 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}. We then construct explicit curves R ⊂ \overline{M}_{11,11} passing through a general point of \overline{D}_{11}, such that −R·\overline{D}_{11} > 0 equals precisely the multiplicity of \overline{D}_{11} in the above mentioned expression of K_{\overline{M}_{11,11}}. More generally, we show the following:

Theorem 0.4. For g ≤ 11, the effective divisor \overline{D}_{g} ∈ Eff(\overline{M}_{g}) is extremal and rigid.

In genus 11, using the existence of the above mentioned Mukai fibration, this eventually leads to the equality κ(\overline{M}_{11,11}) = κ(\overline{M}_{11}, \overline{M}_{11,6}^{1}) = 19, where the last symbol stands for the Iitaka dimension of the linear system |\overline{M}_{11,6}^{1}| generated by the Brill-Noether divisors on \overline{M}_{11}.

In the final section of this paper we study the uniruledness of \overline{M}_{g,n} when g ≤ 8.

Theorem 0.5.

(i) \overline{M}_{5,n} is uniruled for n ≤ 14 and of general type for n ≥ 15. In particular, ζ(5) = 15. (ii) \overline{M}_{7,n} is uniruled for n ≤ 13, of general type for n ≥ 15, while κ(\overline{M}_{7,14}) ≥ 0. In

particular, ζ(7) = 14. (iii) \overline{M}_{8,n} is uniruled for n ≤ 12 and of general type for n ≥ 14. Thus ζ(8) ∈ \{13, 14\}.

New here is the statement about the uniruledness of \overline{M}_{g,n}. For the sake of completeness, we have copied from [Log] and [F2] Theorem 1.10, the range in which \overline{M}_{g,n} was known to be of general type when g = 5, 7, 8. The problem of determining ζ(g) is thus completely solved for g ≤ 7. In order to prove Theorem 0.5, it suffices to establish that K_{\overline{M}_{g,n}} is not pseudo-effective. This is carried out by exhibiting two extremal uniruled divisors on \overline{M}_{g,n}, satisfying certain numerical properties, cf. Proposition 5.1. This simple uniruledness principle seems to be of particular use when studying the birational geometry of moduli spaces, where one typically has an ample supply of explicit extremal effective divisors.

1. Spin curves and the divisor \overline{Θ}_{null}

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}).

If D = m_{1}D_{1} + ⋯ + m_{s}D_{s} is an effective ℚ-divisor on X, with irreducible components D_{i} ⊂ X and m_{i} > 0 for i = 1, …, s, a (trivial) way of showing that κ(X, D) = 0 is by exhibiting for each 1 ≤ i ≤ s, a curve Γ_{i} ⊂ X passing through a general point of D_{i}, such that Γ_{i} · D_{i} < 0 and Γ_{i} · D_{j} = 0 for i ≠ j.

We recall basic facts about the moduli space \overline{S}_{g}^{+} of even spin curves of genus g, see [C], [F3] for details. An even spin curve of genus g consists of a triple (X, η, β), where X is a genus g quasi-stable curve, η ∈ Pic^{g−1}(X) is a line bundle of degree g − 1 such that η_{E} = O_{E}(1) for every rational component E ⊂ X such that #(E ∩ (\overline{X − E})) = 2 (such a component is called exceptional), and h^{0}(X, η) ≡ 0 mod 2, and finally, β : η^{⊗2} → ω_{X} is a sheaf homomorphism which is generically non-zero along each non-exceptional component of X. Even spin curves of genus g form a smooth Deligne-Mumford stack π : \overline{S}_{g}^{+} → \overline{M}_{g}. At the level of coarse moduli schemes, the morphism π : \overline{S}_{g}^{+} → \overline{M}_{g} is the stabilization map π([X, η, β]) := [st(X)], which associates to a quasi-stable curve its stable model.

We explain the boundary structure of \overline{S}_{g}^{+}: If [X, η, β] ∈ π^{−1}([C ∪_{y} D]), where [C, y] ∈ M_{i,1}, [D, y] ∈ M_{g−i,1} and 1 ≤ i ≤ [g/2], then necessarily X = C ∪_{y_{1}} E ∪_{y_{2}} D, where E is an exceptional component such that C ∩ E = {y_{1}} and D ∩ E = {y_{2}}. Moreover η = (η_{C}, η_{D}, η_{E} = O_{E}(1)) ∈ Pic^{g−1}(X), where η_{C}^{⊗2} = K_{C}, η_{D}^{⊗2} = K_{D}. The condition h^{0}(X, η) ≡ 0 mod 2, implies that the theta-characteristics η_{C} and η_{D} have the same parity. We denote by A_{i} ⊂ \overline{S}_{g}^{+} the closure of the locus corresponding to pairs

([C, y, η_{C}], [D, y, η_{D}]) ∈ S_{i,1}^{+} × S_{g−i,1}^{+}

and by B_{i} ⊂ \overline{S}_{g}^{+} the closure of the locus corresponding to pairs

([C, y, η_{C}], [D, y, η_{D}]) ∈ S_{i,1}^{−} × S_{g−i,1}^{−}.
We set α_{i} := [A_{i}] ∈ Pic(\overline{S}_{g}^{+}), β_{i} := [B_{i}] ∈ Pic(\overline{S}_{g}^{+}), and then one has that
(1)  π^{∗}(δ_{i}) = α_{i} + β_{i}.

We recall the description of the ramification divisor of the covering π : \overline{S}_{g}^{+} → \overline{M}_{g}. For a point [X, η, β] ∈ \overline{S}_{g}^{+} corresponding to a stable model st(X) = C_{yq} := C/y ∼ q, with [C, y, q] ∈ M_{g−1,2}, there are two possibilities depending on whether X possesses an exceptional component or not. If X = C_{yq} (i.e. X has no exceptional component) and η_{C} := ν^{∗}(η) where ν : C → X denotes the normalization map, then η_{C}^{⊗2} = K_{C}(y + q). For each choice of η_{C} ∈ Pic^{g−1}(C) as above, there is precisely one choice of gluing the fibres η_{C}(y) and η_{C}(q) such that h^{0}(X, η) ≡ 0 mod 2. We denote by A_{0} the closure in \overline{S}_{g}^{+} of the locus of spin curves [C_{yq}, η_{C} ∈ \sqrt{K_{C}(y + q)}] as above.

If X = C ∪_{{y,q}} E, where E is an exceptional component, then η_{C} := η ⊗ O_{C} is a theta-characteristic on C. Since H^{0}(X, ω) ≅ H^{0}(C, ω_{C}), it follows that [C, η_{C}] ∈ S_{g−1}^{+}. We denote by B_{0} ⊂ \overline{S}_{g}^{+} the closure of the locus of spin curves

[C ∪_{\{y,q\}} E, E ≅ P^{1}, η_{C} ∈ \sqrt{K_{C}}, η_{E} = O_{E}(1)] ∈ S_{g}^{+}.

If α_{0} := [A_{0}], β_{0} := [B_{0}] ∈ Pic(\overline{S}_{g}^{+}), we have the relation, see [C]:

(2)  π^{∗}(δ_{0}) = α_{0} + 2β_{0}.

In particular, B_{0} is the ramification divisor of π. An important effective divisor on \overline{S}_{g}^{+} is the locus of vanishing theta-nulls

Θ_{null} := {[C, η] ∈ S_{g}^{+} : H^{0}(C, η) ≠ 0}.

The class of its compactification inside \overline{S}_{g}^{+} is given by the formula, cf. [F3]:

(3)  \overline{Θ}_{null} ≡ \frac{1}{4}λ − \frac{1}{16}α_{0} − \frac{1}{2} ∑_{i=1}^{[g/2]} β_{i} ∈ Pic(\overline{S}_{g}^{+}).

It is also useful to recall the formula for the canonical class of \overline{S}_{g}^{+}:

K\overline{S}{+ \atop g} ≡ π^{∗}(K_{\overline{M}_{g}}) + β_{0} ≡ 13λ − 2α_{0} − 3β_{0} − 2 ∑_{i=1}^{[g/2]}(α_{i} + β_{i}) − (α_{1} + β_{1}).

An argument involving spin curves on certain singular canonical surfaces in P^{6}, implies that for g ≤ 9, the divisor \overline{Θ}_{null} is uniruled and a rigid point in the cone of effective divisors Eff(\overline{S}_{g}^{+}):

Theorem 1.1. For g ≤ 9 the divisor \overline{Θ}_{null} ⊂ \overline{S}_{g}^{+} is uniruled and rigid. Precisely, through a general point of \overline{Θ}_{null} there passes a rational curve Γ ⊂ \overline{S}_{g}^{+} such that Γ·\overline{Θ}_{null} < 0. In particular, if D is an effective divisor on \overline{S}_{g}^{+} with D ≡ n\overline{Θ}_{null} for some n ≥ 1, then D = n\overline{Θ}_{null}. Proof. We assume 7 ≤ g ≤ 9, the other cases being similar and simpler. A general point [C, η_{C}] ∈ Θ_{null} corresponds to a canonical curve C ↪^{|K_{C}|} P^{g−1} lying on a rank 3 quadric Q ⊂ P^{g−1} such that C ∩ Sing(Q) = ∅. The pencil η_{C} is recovered from the ruling of Q.

Let V ∈ G(7, H^{0}(C, K_{C})) be a general subspace such that if π_{V} : P^{g−1} −−→ P(V^{∨}) is the projection, then Q̃ := π_{V} (Q) is a quadric of rank 3. Let C′ := π_{V} (C) ⊂ P(V^{∨}) be the projection of the canonical curve C. By counting dimensions we find that

dim{I_{C′/P(V^{∨})}(2) := Ker{Sym^{2}(V) → H^{0}(C, K_{C}^{⊗2})}} ≥ 31 − 3g ≥ 4,

that is, the embedded curve C′ ⊂ P^{6} lies on at least 4 independent quadrics, namely the rank 3 quadric Q̃ and Q_{1}, Q_{2}, Q_{3} ∈ |I_{C′/P(V^{∨})}(2)|. By choosing V sufficiently general we make sure that S := Q̃ ∩ Q_{1} ∩ Q_{2} ∩ Q_{3} is a canonical surface in P(V^{∨}) with 8 nodes corresponding to the intersection ⋂_{i=1}^{3} Q_{i} ∩ Sing(Q̃) (This transversality statement can also be checked with Macaulay by representing C as a section of the corresponding Mukai variety). From the exact sequence on S,

0 ⟶ O_{S} ⟶ O_{S}(C) ⟶ O_{C}(C) ⟶ 0,

coupled with the adjunction formula O_{C}(C) = K_{C} ⊗ K_{S|C}^{∨} = O_{C}, as well as the fact H^{1}(S, O_{S}) = 0, it follows that dim |C| = 1, that is, C ⊂ S moves in its linear system. In particular, \overline{Θ}_{null} is a uniruled divisor for g ≤ 9.

We determine the numerical parameters of the family Γ ⊂ \overline{S}_{g}^{+} induced by varying C ⊂ S. Since C^{2} = 0, the pencil |C| is base point free and gives rise to a fibration f : S̃ → P^{1}, where S̃ := Bl_{8}(S) is the blow-up of the nodes of S. This in turn induces a moduli map m : P^{1} → \overline{S}_{g}^{+} and Γ =: m(P^{1}). We have the formulas

Γ · λ = m^{∗}(λ) = χ(S, O_{S}) + g − 1 = 8 + g − 1 = g + 7, and

Γ · α_{0} + 2Γ · β_{0} = m^{∗}(π^{∗}(δ_{0})) = m^{∗}(α_{0}) + 2m^{∗}(β_{0}) = c_{2}(S̃) + 4(g − 1).

Noether’s formula gives that c_{2}(S̃) = 12χ(S̃, OS̃) − K2 ; S̃ = 12χ(S, O_{S}) − K_{S}^{2} = 80, hence m^{∗}(α_{0}) + 2m^{∗}(β_{0}) = 4g + 76. The singular fibres corresponding to spin curves lying in B_{0} are those in the fibres over the blown-up nodes and all contribute with multiplicity 1, that is, Γ · β_{0} = 8 and then Γ · α_{0} = 4g + 60. It follows that Γ · \overline{Θ}_{null} = −2 < 0 (independent of g!), which finishes the proof.

To illustrate one of the cases g < 7, we discuss the situation on \overline{S}_{4}^{+}. We denote by S = 𝔽_{2} the blow-up of the vertex of a cone Q ⊂ P^{3} over a conic in P^{3} and write Pic(S) = ℤ · F + ℤ · C_{0}, where F^{2} = 0, C_{0}^{2} = −2 and C_{0} · F = 1. We choose a Lefschetz pencil of genus 4 curves in the linear system |3(C_{0} + 2F)|. By blowing-up the 18 = 9(C_{0} + 2F)^{2} base points, we obtain a fibration f : S̃ := Bl_{18}(S) → P^{1} which induces a family of spin curves m : P^{1} → \overline{S}_{4}^{+} given by m(t) := [f^{−1}(t), O_{f^{−1}(t)}(F)]. We have the formulas

m^{∗}(λ) = χ(S̃, OS̃) + g − 1 = 4, and
m^{∗}(π^{∗}(δ_{0})) = m^{∗}(α_{0}) + 2m^{∗}(β_{0}) = c_{2}(S̃) + 4(g − 1) = 34.

The singular fibres lying in B_{0} correspond to curves in the Lefschetz pencil on Q passing through the vertex of the cone, that is, when f^{−1}(t_{0}) splits as C_{0} + D, where D ⊂ S̃ is the residual curve. Since C_{0} · D = 2 and O_{C_{0}}(F) = O_{C_{0}}(1), it follows that m(t_{0}) ∈ B_{0}. One finds that m^{∗}(β_{0}) = 1, hence m^{∗}(α_{0}) = 32 and m^{∗}(\overline{Θ}_{null}) = −1. Since Γ := m(P^{1}) fills-up the divisor \overline{Θ}_{null}, we obtain that [\overline{Θ}_{null}] ∈ Eff(\overline{S}_{4}^{+}) is rigid. □

2. Spin curves of genus 8

The moduli space M_{8} carries one Brill-Noether divisor, the locus of plane septics

M_{8,7}^{2} := {[C] ∈ M_{8} : G_{7}^{2}(C) ≠ ∅}.

The locus \overline{M}_{8,7}^{2} is irreducible and for a known constant c_{8,7}^{2} ∈ ℤ_{>0}, one has, cf. [EH2],

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

In particular, s(\overline{M}_{8,7}^{2}) = 6 + 12/(g + 1) and this is the minimal slope of an effective divisor on \overline{M}_{8}. The following fact is probably well-known:

Proposition 2.1. Through a general point of \overline{M}_{8,7}^{2} there passes a rational curve R ⊂ \overline{M}_{8} such that R · \overline{M}_{8,7}^{2} < 0. In particular, the class [\overline{M}_{8,7}^{2}] ∈ Eff(\overline{M}_{8}) is rigid.

Proof. One takes a Lefschetz pencil of nodal plane septic curves with 7 assigned nodes in general position (and 21 unassigned base points). After blowing up the 21 unassigned base points as well as the 7 nodes, we obtain a fibration f : S := Bl_{28}(P^{2}) → P^{1}, and the corresponding moduli map m : P^{1} → \overline{M}_{8} is a covering curve 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. We find m^{∗}([\overline{M}_{8,7}^{2}]) = c_{8,7}^{2}(8·22−3·59) = −c_{8,7}^{2} < 0. □

Using (3) we find the following explicit representative for the canonical class K\overline{S}{+ \atop 8}:

(4)  K\overline{S}{+ \atop 8} ≡ \frac{1}{2}π^{∗}(𝔟𝔫_{8}) + 8\overline{Θ}_{null} + ∑_{i=1}^{4}(a_{i} α_{i} + b_{i} β_{i}),

where a_{i}, b_{i} > 0 for i = 1, …, 4. The multiples of each irreducible component appearing in (4) are rigid divisors on \overline{S}_{8}^{+}, but in principle, their sum could still be a movable class. Assuming for a moment Proposition 0.2, we explain how this implies Theorem 0.1:

Proof of Theorem 0.1. The covering curve R ⊂ \overline{Θ}_{null} constructed in Proposition 0.2, satisfies R · \overline{Θ}_{null} < 0 as well as R · π^{∗}(\overline{M}_{8,7}^{2}) = 0 and R · α_{i} = R · β_{i} = 0 for i = 1, …, 4. It follows from (4) that for each n ≥ 1, one has an equality of linear series on \overline{S}_{8}^{+}

|nK\overline{S}{+ \atop 8}| = 8n\overline{Θ}_{null} + |n(K\overline{S}{+ \atop 8} − 8\overline{Θ}_{null})|.

Furthermore, from (4) one finds constants a′_{i} > 0 for i = 1, …, 4, such that if

D ≡ 22λ − 3δ_{0} − ∑_{i=1}^{4} a′_{i} δ_{i} ∈ Pic(\overline{M}_{8}),

then the difference \frac{1}{2}π^{∗}(D) − (K\overline{S}{+ \atop 8} − 8\overline{Θ}_{null}) is still effective on \overline{S}_{8}^{+}. We can thus write

0 ≤ κ(\overline{S}_{8}^{+}) = κ(\overline{S}_{8}^{+}, K\overline{S}{+ \atop 8} − 8\overline{Θ}_{null}) ≤ κ(\overline{S}_{8}^{+}, \frac{1}{2}π^{∗}(D)) = κ(\overline{S}_{8}^{+}, π^{∗}(D)).

We claim that κ(\overline{S}_{8}^{+}, π^{∗}(D)) = 0. Indeed, in the course of the proof of Proposition 2.1 we have constructed a covering family B ⊂ \overline{M}_{8} for the divisor \overline{M}_{8,7}^{2} such that B · \overline{M}_{8,7}^{2} < 0 and B · δ_{i} = 0 for i = 1, …, 4. We lift B to a family R ⊂ \overline{S}_{8}^{+} of spin curves by taking

B̃ := B ×_{\overline{M}_{8}} \overline{S}_{8}^{+} = \{[C_{t}, η_{C_{t}}] ∈ \overline{S}_{8}^{+} : [C_{t}] ∈ B, η_{C_{t}} ∈ \overline{Pic}^{7}(C_{t}), t ∈ P^{1}\} ⊂ \overline{S}_{8}^{+}.

One notes that B̃ is disjoint from the boundary divisors A_{i}, B_{i} ⊂ \overline{S}_{8}^{+} for i = 1, …, 4, hence B̃ · π^{∗}(D) = 2^{g−1}(2^{g} + 1)(B · \overline{M}_{8,7}^{2})_{\overline{M}_{8}} < 0. Thus we write that

κ(\overline{S}_{8}^{+}, π^{∗}(D)) = κ(\overline{S}_{8}^{+}, π^{∗}(D − (22λ − 3δ_{0})) = κ(\overline{S}_{8}^{+}, ∑_{i=1}^{4} a′_{i}(α_{i} + β_{i})) = 0.

□

3. A family of spin curves R ⊂ \overline{S}_{8}^{+} with R · π^{∗}(\overline{M}_{8,7}^{2}) = 0 and R · \overline{Θ}_{null} = −1

The aim of this section is to prove Proposition 0.2, which is the key ingredient in the proof of Theorem 0.1. We begin by reviewing facts about the geometry of \overline{M}_{8}, in particular the construction of general curves of genus 8 as complete intersections in a rational homogeneous variety, cf. [M2].

We fix V ≅ ℂ^{6} and denote by G := G(2, V) ⊂ P(∧^{2}V) the Grassmannian of lines. Noting that smooth codimension 7 linear sections of G are canonical curves of genus 8, one is led to consider the Mukai model of the moduli space of curves of genus 8

𝔐_{8} := G(8, ∧^{2}V)//SL(V).

There is a birational map f : \overline{M}_{8} −−→ 𝔐_{8}, whose inverse is given by f^{−1}(H) := G ∩ H, for a general H ∈ G(8, ∧^{2}V). The map f is constructed as follows: Starting with a curve [C] ∈ M_{8} − M_{8,7}^{2}, one notes that C has a finite number of pencils 𝔤_{5}^{1}. We choose A ∈ W_{5}^{1}(C) and set L := K_{C} ⊗ A^{∨} ∈ W_{9}^{3}(C). There exists a unique rank 2 vector bundle E ∈ SU_{C}(2, K_{C}) (independent of A!), sitting in an extension

0 ⟶ A ⟶ E ⟶ L ⟶ 0,

such that h^{0}(E) = h^{0}(A) + h^{0}(L) = 6. Since E is globally generated, we define the map

ϕ_{E} : C → G(2, H^{0}(C, E)^{∨}), ϕ_{E}(p) := E(p)^{∨} (↪ H^{0}(C, E)^{∨}),

and let ℘ : G(2, H^{0}(C, E)^{∨}) → P(∧^{2}H^{0}(C, E)^{∨}) be the Plücker embedding. The determinant map u : ∧^{2}H^{0}(E) → H^{0}(K_{C}) is surjective, that is, H^{0}(K_{C})^{∨} ∈ G(8, ∧^{2}H^{0}(E)^{∨}), see [M2] Theorem C. We set

f([C]) := [C ⟶^{℘∘ϕ_{E}} P(∧^{2}H^{0}(E)^{∨}), P(H^{0}(K_{C})^{∨})] mod SL(V) ∈ 𝔐_{8}.

It follows from [M2] that the exceptional divisors of f are the Brill-Noether locus \overline{M}_{8,7}^{2} and the boundary divisors Δ_{1}, …, Δ_{4}. The map f^{−1} does not contract any divisors.

Inside the moduli space F_{8} of polarized K3 surfaces [S, h] of degree h^{2} = 14, we consider the following Noether-Lefschetz divisor

𝔑𝔏 := {[S, O_{S}(C_{1} + C_{2})] ∈ F_{8} : Pic(S) ⊃ ℤ · C_{1} ⊕ ℤ · C_{2}, C_{1}^{2} = C_{2}^{2} = 0, C_{1} · C_{2} = 7},

of doubly-elliptic K3 surfaces. For a general element [S, O_{S}(C)] ∈ 𝔑𝔏, the embedded

surface S ↪^{|O_{S}(C)|} P^{8} lies on a rank 4 quadric whose rulings induce the elliptic pencils |C_{1}| and |C_{2}| on S. We denote by 𝔑𝔏′ ⊂ 𝔑𝔏 the open subset corresponding to polarized surfaces [S, O_{S}(C_{1} + C_{2})] such that Pic(S) = ℤ · C_{1} ⊕ ℤ · C_{2}. Then we consider the P^{3}-bundle U → 𝔑𝔏′ classifying pairs ([S, O_{S}(C_{1} + C_{2})], C ⊂ S), where

C ∈ |H^{0}(S, O_{S}(C_{1})) ⊗ H^{0}(S, O_{S}(C_{2}))| ⊂ |H^{0}(S, O_{S}(C_{1} + C_{2}))|.

An element of U corresponds to a hyperplane section C ⊂ S ⊂ P^{8} of a doubly-elliptic K3 surface, such that the intersection of C with the rank 4 quadric induced by the elliptic pencils, has rank 3. There exists a rational map

q : U −−→ \overline{Θ}_{null}, q([S, O_{S}(C_{1} + C_{2})], C) := [C, O_{C}(C_{1}) = O_{C}(C_{2})].

Clearly U is irreducible and dim(U) = 21(= 3 + dim(𝔑𝔏)). We shall show that the morphism q is dominant, by explicitly describing its generic fibre. This produces a parametrization of the divisor \overline{Θ}_{null}, in particular it provides an explicit covering curve.

We fix a general point [C, η] ∈ \overline{Θ}_{null} ⊂ \overline{S}_{8}^{+}, with η a vanishing theta-null. Then

C ⊂ Q ⊂ P^{7} := P(H^{0}(C, K_{C})^{∨}),

where Q ∈ H^{0}(P^{7}, I_{C/P^{7}}(2)) is the rank 3 quadric such that the ruling of Q cuts out on C precisely η. As explained, there exists a linear embedding P^{7} ⊂ P^{14} := P(∧^{2}H^{0}(E)^{∨}) such that P^{7} ∩ G = C. The restriction map yields an isomorphism between spaces of quadrics, cf. [M2],

res_{C} : H^{0}(G, I_{G/P^{14}}(2)) ⟶^{≅} H^{0}(P^{7}, I_{C/P^{7}}(2)).

In particular there is a unique quadric G ⊂ Q̃ ⊂ P^{14} such that Q̃ ∩ P^{7} = Q.

There are three possibilities for the rank of any quadric Q̃ ∈ H^{0}(P^{14}, I_{G/P^{14}}(2)):

(a) rk(Q̃) = 15, (b) rk(Q̃) = 6 and then Q̃ is a Plücker quadric, or (c) rk(Q̃) = 10, in which case Q̃ is a sum of two Plücker quadrics, see [M2].

Proposition 3.1. For a general [C, η] ∈ \overline{Θ}_{null}, the quadric Q̃ is smooth, that is, rk(Q̃) = 15.

Proof. We may assume that dim G_{5}^{1}(C) = 0 (in particular C has no 𝔤_{4}^{1}’s), and G_{7}^{2}(C) = ∅. The space P(Ker(u)) ⊂ P(∧^{2}H^{0}(E)) is identified with the space of hyperplanes H ∈ (P^{14})^{∨} containing the canonical space P^{7}. Claim: If rk(Q̃) < 15, there exists a pencil of 8-dimensional planes P^{7} ⊂ Ξ ⊂ P^{14}, such that S := G ∩ Ξ is a K3 surface containing C as a hyperplane section, and

rk{Q_{Ξ} := Q̃ ∩ Ξ ∈ H^{0}(Ξ, I_{S/Ξ(2)})} = 3.

The conclusion of the claim contradicts the assumption that [C, η] ∈ \overline{Θ}_{null} is general. Indeed, we pick such an 8-plane Ξ and corresponding K3 surface S. Since Sing(Q) ∩ C = ∅, where Q_{Ξ} ∩ P^{7} = Q, it follows that S ∩ Sing(Q_{Ξ}) is finite. The ruling of Q_{Ξ} cuts out an elliptic pencil |E| on S. Furthermore, S has nodes at the points S ∩ Sing(Q_{Ξ}). For numerical reasons, #Sing(S) = 7, and then on the surface S̃ obtained from S by resolving the 7 nodes, one has the linear equivalence

C ≡ 2E + Γ_{1} + ⋯ + Γ_{7},

where Γ_{i}^{2} = −2, Γ_{i} · E = 1 for i = 1, …, 7 and Γ_{i} · Γ_{j} = 0 for i ≠ j. In particular rk(Pic(S̃)) ≥ 8. A standard parameter count, see e.g. [Do], shows that

dim{(S, C) : C ∈ |O_{S}(2E + Γ_{1} + ⋯ + Γ_{7})|} ≤ 19 − 7 + dim|OS̃(C)| = 20.

Since dim(\overline{Θ}_{null}) = 20 and a general curve [C] ∈ \overline{Θ}_{null} lies on infinitely many such K3 surfaces S, one obtains a contradiction.

We are left with proving the claim made in the course of the proof. The key point is to describe the intersection P(Ker(u)) ∩ Q̃^{∨}, where we recall that the linear span ⟨Q̃^{∨}⟩ classifies hyperplanes H ∈ (P^{14})^{∨} such that rk(Q̃ ∩ H) ≤ rk(Q̃) − 1. Note also that dim ⟨Q̃⟩ = rk(Q̃) − 2.

If rk(Q̃) = 6, then Q̃^{∨} is contained in the dual Grassmannian G^{∨} := G(2, H^{0}(E)), cf. [M2] Proposition 1.8. Points in the intersection P(Ker(u)) ∩ G^{∨} correspond to decomposable tensors s_{1} ∧ s_{2}, with s_{1}, s_{2} ∈ H^{0}(C, E), such that u(s_{1} ∧ s_{2}) = 0. The image of the morphism O_{C}^{⊕2} ⟶^{(s_{1},s_{2})} E is thus a subbundle 𝔤_{5}^{1} of E and there is a bijection

P(Ker(u)) ∩ G(2, H^{0}(E)) ≅ W_{5}^{1}(C).

It follows, there are at most finitely many tangent hyperplanes to Q̃ containing the space P^{7} = ⟨C⟩, and consequently, dim(P(Ker(u)) ∩ ⟨Q̃^{∨}⟩) ≤ 1. Then there exists a codimension 2 linear space W^{12} ⊂ P^{14} such that rk(Q̃ ∩ W) = 3, which proves the claim (and much more), in the case rk(Q̃) = 6.

When rk(Q̃) = 10, using the explicit description of the dual quadric Q̃^{∨} provided in [M2] Proposition 1.8, one finds that dim(P(Ker(u)) ∩ ⟨Q̃^{∨}⟩) ≤ 4. Thus there exists a codimension 5 linear section W^{9} ⊂ P^{14} such that rk(Q̃ ∩ W) = 3, which implies the claim when rk(Q̃) = 10 as well.

□

We consider an 8-dimensional linear extension P^{7} ⊂ Λ^{8} ⊂ P^{14} of the canonical space P^{7} = ⟨C⟩, such that S_{Λ} := Λ ∩ G is a smooth K3 surface. The restriction map

res_{C/S_{Λ}} : H^{0}(Λ, I_{S_{Λ}/Λ}(2)) → H^{0}(P^{7}, I_{C/P^{7}}(2))

is an isomorphism, cf. [SD]. Thus there exists a unique quadric S_{Λ} ⊂ Q_{Λ} ⊂ Λ with Q_{Λ} ∩ P^{7} = Q. Since rk(Q) = 3, it follows that 3 ≤ rk(Q_{Λ}) ≤ 5 and it is easy to see that for a general Λ, the corresponding quadric Q_{Λ} ⊂ Λ is of rank 5. We show however, that one can find K3-extensions of the canonical curve C, which lie on quadrics of rank 4:

Proposition 3.2. For a general [C, η] ∈ \overline{Θ}_{null}, there exists a pencil of 8-dimensional extensions

P(H^{0}(C, K_{C})^{∨}) ⊂ Λ ⊂ P^{14}

such that rk(Q_{Λ}) = 4. It follows that there exists a smooth K3 surface S_{Λ} ⊂ Λ containing C as a transversal hyperplane section, such that rk(Q_{Λ}) = 4.

Proof. We pass from projective to vector spaces and view the rank 15 quadric

Q̃ : ∧^{2}H^{0}(C, E)^{∨} ⟶^{∼} ∧^{2}H^{0}(C, E)

as an isomorphism, which by restriction to H^{0}(C, K_{C})^{∨} ⊂ ∧^{2}H^{0}(C, E)^{∨}, induces the rank 3 quadric Q : H^{0}(C, K_{C})^{∨} → H^{0}(C, K_{C}). The map u ∘ Q̃ : ∧^{2}H^{0}(E)^{∨} → H^{0}(K_{C}) being surjective, its kernel Ker(u ∘ Q̃) is a 7-dimensional vector space containing the 5-dimensional subspace Ker(Q). We choose an arbitrary element

[v̄ := v + Ker(Q)] ∈ P(\frac{Ker(u ∘ Q̃)}{Ker(Q)}),

inducing a subspace H^{0}(C, K_{C})^{∨} ⊂ Λ := H^{0}(C, K_{C})^{∨} + ℂv ⊂ ∧^{2}H^{0}(C, E)^{∨}, with the property that Ker(Q_{Λ}) = Ker(Q), where Q_{Λ} : Λ → Λ^{∨} is induced from Q̃ by restriction and projection. It follows that rk(Q_{Λ}) = 4. Moreover, we have shown that dim q^{−1}([C, η]) ≤ 1, in particular q is dominant. □

Now we can begin the proof of Proposition 0.2. Let C ⊂ Q ⊂ P^{7} be a general canonical curve endowed with a vanishing theta-null, where Q ∈ H^{0}(P^{7}, I_{C/P^{7}}(2)) is the corresponding rank 3 quadric. We choose a general 8-plane P^{7} ⊂ Λ ⊂ P^{14} such that S := Λ ∩ G is a smooth K3 surface, and the lift of Q to Λ

Q_{Λ} ∈ H^{0}(Λ, I_{S/Λ}(2))

has rank 4. Moreover, we can assume that S ∩ Sing(Q_{Λ}) = ∅. The linear projection f_{Λ} : Λ −−→ P^{3} with center Sing(Q_{Λ}), induces a regular map f : S → P^{3} with image the smooth quadric Q_{0} ⊂ P^{3}. Then S is endowed with two elliptic pencils |C_{1}| and |C_{2}| corresponding to the projections of Q_{0} ≅ P^{1} × P^{1} onto the two factors. Since C ∈ |O_{S}(1)|, one has a linear equivalence C ≡ C_{1} + C_{2}, on S. As already pointed out, deg(f) = C_{1} · C_{2} = C^{2}/2 = 7. The condition rk(Q_{Λ} ∩ P^{7}) = rk(Q_{Λ}) − 1, implies that the hyperplane P^{7} ∈ (Λ)^{∨} is the pull-back of a hyperplane from P^{3}, that is, P^{7} = f_{Λ}^{−1}(Π_{0}), where Π_{0} ∈ (P^{3})^{∨}.

We choose a general line l_{0} ⊂ Π_{0} and denote by {q_{1}, q_{2}} := l_{0} ∩ Q_{0}. We consider the pencil {Π_{t}}_{t∈P^{1}} ⊂ (P^{3})^{∨} of planes through l_{0} as well as the induced pencil of curves of genus 8

{C_{t} := f^{−1}(Π_{t}) ⊂ S}_{t∈P^{1}},

each endowed with a vanishing theta-null induced by the pencil f_{t} : C_{t} → Q_{0} ∩ Π_{t}.

This pencil contains precisely two reducible curves, corresponding to the planes Π_{1}, Π_{2} in P^{3} spanned by the rulings of Q_{0} passing through q_{1} and q_{2} respectively. Precisely, if l_{i}, m_{i} ⊂ Q_{0} are the rulings passing through q_{i} such that l_{1} · l_{2} = m_{1} · m_{2} = 0, then it follows that for Π_{1} = ⟨l_{1}, m_{2}⟩, Π_{2} = ⟨l_{2}, m_{1}⟩, the fibres f^{−1}(Π_{1}) and f^{−1}(Π_{2}) split into two elliptic curves f^{−1}(l_{i}) and f^{−1}(m_{j}) meeting transversally in 7 points. The half-canonical 𝔤_{7}^{1} specializes to a degree 7 admissible covering

f^{−1}(l_{i}) ∪ f^{−1}(m_{j}) →^{f} l_{i} ∪ m_{j}, i ≠ j,

such that the 7 points in f^{−1}(l_{i}) ∩ f^{−1}(m_{j}) map to l_{i} ∩ m_{j}. To determine the point in \overline{S}_{8}^{+} corresponding to the admissible covering (f^{−1}(l_{i}) ∪ f^{−1}(m_{j}), f_{|f^{−1}(l_{i})∪f^{−1}(m_{j})}), one must insert 7 exceptional components at all the points of intersection of the two components. We denote by R ⊂ \overline{Θ}_{null} ⊂ \overline{S}_{8}^{+} the pencil of spin curves obtained via this construction.

Lemma 3.3. Each member C_{t} ⊂ S in the above constructed pencil is nodal. Moreover, each curve C_{t} different from f^{−1}(l_{1}) ∪ f^{−1}(m_{2}) and f^{−1}(l_{2}) ∪ f^{−1}(m_{1}) is irreducible. It follows that R · α_{i} = R · β_{i} = 0 for i = 1, …, 4.

Proof. This follows since f : S → Q_{0} is a regular morphism and the base line l_{0} ⊂ H_{0} of the pencil {Π_{t}}_{t∈P^{1}} is chosen to be general. □

Lemma 3.4. R · π^{∗}(\overline{M}_{7,8}^{2}) = 0.

Proof. We show instead that π_{∗}(R) · \overline{M}_{8,7}^{2} = 0. From Lemma 3.3, the curve R is disjoint from the divisors A_{i}, B_{i} for i = 1, …, 4, hence π_{∗}(R) has the numerical characteristics of a Lefschetz pencil of curves of genus 8 on a fixed K3 surface. In particular, π_{∗}(R) · δ/π_{∗}(R) · λ = 6 + 12/(g + 1) = s(\overline{M}_{8,7}^{2}) and π_{∗}(R) · δ_{i} = 0 for i = 1, …, 4. This implies the statement. □

Lemma 3.5. T · \overline{Θ}_{null} = −1.

Proof. We have already determined that R · λ = π_{∗}(R) · λ = χ(S̃, OS̃) + g − 1 = 9, where S̃ := Bl_{2g−2}(S) is the blow-up of S at the points f^{−1}(q_{1}) ∪ f^{−1}(q_{2}). Moreover,

(5)  R · α_{0} + 2R · β_{0} = π_{∗}(R) · δ_{0} = c_{2}(X̃) + 4(g − 1) = 38 + 28 = 66.

To determine R · β_{0} we study the local structure of \overline{S}_{8}^{+} in a neighbourhood of one of the two points, say t^{∗} ∈ R corresponding to a reducible curve, say f^{−1}(l_{1}) ∪ f^{−1}(m_{2}), the situation for f^{−1}(l_{2})∪f^{−1}(m_{1}) being of course identical. We set {p} := l_{1} ∩m_{2} ∈ Q_{0} and {x_{1}, …, x_{7}} := f^{−1}(p) ∈ S. We insert exceptional components E_{1}, …, E_{7} at the nodes x_{1}, …, x_{7} of f^{−1}(l_{1}) ∪ f^{−1}(m_{2}) and denote by X the resulting quasi-stable curve. If

μ : f^{−1}(l_{1}) ∪ f^{−1}(m_{2}) ∪ E_{1} ∪ … ∪ E_{7} → f^{−1}(l_{1}) ∪ f^{−1}(m_{2})

is the stabilization morphism, we set {y_{i}, z_{i}} := μ^{−1}(x_{i}), where y_{i} ∈ E_{i} ∩ f^{−1}(l_{1}) and z_{i} ∈ E_{i} ∩ f^{−1}(m_{2}) for i = 1, …, 7. If t^{∗} = [X, η, β], then η_{f^{−1}(l_{1})} = O_{f^{−1}(l_{1})}, η_{f^{−1}(m_{2})} = O_{f^{−1}(m_{2})}, and of course η_{E_{i}} = O_{E_{i}}(1). Moreover, one computes that Aut(X, η, β) = ℤ_{2} and Aut(f^{−1}(l_{1}) ∪ f^{−1}(m_{2})) = {Id}, cf. [C] Lemma 2.2.

If ℂ_{τ}^{3g−3} denote the versal deformation space of [X, η, β] ∈ \overline{S}_{g}^{+}, then there are local parameters (τ_{1}, …, τ_{3g−3}), such that for i = 1, …, 7, the locus (τ_{i} = 0) ⊂ ℂ_{τ}^{3g−3} parameterizes spin curves for which the exceptional component E_{i} persists. It particular, the pull-back ℂ_{τ}^{3g−3} ×\overline{S}{+ \atop g} B_{0} of the boundary divisor B_{0} ⊂ \overline{S}_{g}^{+} is given by the equation

(τ_{1} ⋯ τ_{7} = 0) ⊂ ℂ_{τ}^{3g−3}. The group Aut(X, η, β) acts on ℂ_{τ}^{3g−3} by

(τ_{1}, …, τ_{7}, τ_{8}, …, τ_{3g−3}) ↦ (−τ_{1}, …, −τ_{7}, τ_{8}, …, τ_{3g−3}),

and since an étale neighbourhood of t^{∗} ∈ \overline{S}_{g}^{+} is isomorphic to ℂ_{τ}^{3g−3}/Aut(X, η, β), we find that B_{0} is not Cartier around t^{∗} (though 2B_{0} is Cartier). It follows that the intersection multiplicity of R ×\overline{S}{+ \atop g} ℂ_{τ}^{3g−3} with the locus (τ_{1} ⋯ τ_{7}) = 0 equals 7, that is, the intersection multiplicity of R ∩ β_{0} at the point t^{∗} equals 7/2, hence

R · β_{0} = (R · β_{0})_{f^{−1}(l_{1})∪f^{−1}(m_{2})} + (R · β_{0})_{f^{−1}(l_{2})∩f^{−1}(m_{1})} = \frac{7}{2} + \frac{7}{2} = 7.

Then using (5) we find that R · β_{0} = 66 − 14 = 52, and finally

R · \overline{Θ}_{null} = \frac{1}{4}R · λ − \frac{1}{16}R · α_{0} = \frac{9}{4} − \frac{52}{16} = −1.

□

Remark 3.6. The final argument in the previous proof, namely that the reducible curve f^{−1}(l_{1})∪f^{−1}(m_{2}) contributes with multiplicity 7/2 to R·β_{0}, can also be derived by interpreting \overline{Θ}_{null} as a space of admissible coverings of degree 7 over the versal deformation space ℂ_{τ}^{3g−3} and then making a local analysis similar to the one in [D] pg. 47-50.

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

We begin by recalling the notation for boundary divisor classes on the moduli space \overline{M}_{g,n}. 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} .

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} has been computed, cf. [Log] Theorem 2.6:

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

We show that, at least for small g, the divisor \overline{D}_{g} of curves with g marked points moving in a pencil, is an extremal point in the effective cone of \overline{M}_{g,g}:

Proposition 4.1. 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 a 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:ij} = 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 now specialize to the case of genus 11: On \overline{M}_{11} there exist two divisors of Brill-Noether type, 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} = 7 + \binom{|c − 1| + 1}{2}, 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}, d_{5:c} = 28 + \binom{|c − 5| + 1}{2}.

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 4.2. 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}).

Proof. Using the notation and results from Proposition 4.1, 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, thus 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}|. □

We are in a position to complete the proof of Theorem 0.3:

Theorem 4.3. We have that κ(\overline{M}_{11,11}, 2 · ϕ^{∗}(𝔟𝔫_{11}) + ∑_{i,c} d_{i:c} · δ_{i:c}) = 19. It follows that the Kodaira dimension of \overline{M}_{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 [M3]

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. □

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

We formulate a general principle, somewhat similar to the one used in the proof of Theorem 0.1, which we use in proving the uniruledness of some moduli spaces \overline{M}_{g,n}. The next result, although simple, can be applied to determine all uniruled moduli spaces \overline{M}_{g,n} for g ≤ 8:

Proposition 5.1. 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

(9)  |\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 0.5, which we split in three parts:

Theorem 5.2. \overline{M}_{5,n} is uniruled for n ≤ 14.

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}, {p_{j}}_{j=1}^{5} ⊂ P^{2}, and a general line l ⊂ P^{2}. One considers the pencil of sextics passing simply 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 4.1):

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. Furthermore, f admits a 6-section given by the points of intersection l · C_{λ}. Making a base change of order 6 via f_{l} : l → E_{p_{1}}, we obtain a family U ⊂ \overline{M}_{5,14}, with marked points labeled by (x(t), y(t), x_{1}(t), …, x_{11}(t), l(t))_{t∈l}. Clearly (U · α)_{\overline{M}_{5,14}} = deg(f_{l})(Γ · α)_{\overline{M}_{5,13}}, for every class α ∈ \{λ, ψ_{x}, ψ_{y}, ψ_{x_{1}}, …, ψ_{x_{11}}, δ_{irr}, δ_{0:xy}\} ⊂ Pic(\overline{M}_{5,14}). Finally, one has that

U · ψ_{l} = −(l^{2})_{Y} + (2deg(f_{l}) − 2) = 10.

By direct calculation one finds, U · K_{\overline{M}_{5,14}} = −2, which completes the proof. □

Remark 5.3. It is known, cf. [F2] Section 5, that \overline{M}_{5,15} is of general type. Thus Theorem 5.2 settles completely the classification problem for \overline{M}_{5,n}. In particular, ζ(5) = 15.

Theorem 5.4. \overline{M}_{8,n} is uniruled for n ≤ 12.

Proof. We apply Proposition 5.1 for D_{1} := 𝔟𝔫_{8} (see Section 2), and 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} constructed in the course of proving Proposition 2.1, carries n sections given by the exceptional divisors corresponding to n 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} = −1/3, 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} = 28/3 and Γ_{2} · K_{\overline{M}_{8,n}} = 25 + n. The conditions of Proposition 5.1 are satisfied for n ≤ 12. □

Remark 5.5. The results of Theorem 5.4 are almost optimal. The space \overline{M}_{8,14} is of general type, cf. [Log]. The Kodaira dimension of \overline{M}_{8,13} is still unknown. Note that it was already known [Log], [CF], that \overline{M}_{8,n} is unirational for n ≤ 11.

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

linear system spanned by the unique Brill-Noether divisor on \overline{M}_{7}. Before computing the class [\overline{D}_{1}], we need a calculation, which may be of independent interest:

Proposition 5.6. 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_{0:12} − ⋯) ∈ 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_{0:12}) coincides with the cusp locus in \overline{M}_{g,1}, that is, the closure of 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 obtain 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 obvious formulas (ϕ_{1})_{∗}(λ·δ_{0:12}) = λ, (ϕ_{1})_{∗}(δ_{0:12}^{2}) = −ψ, (ϕ_{1})_{∗}(δ_{irr} ·δ_{0:12}) = δ_{irr} and (ϕ_{1})_{∗}(ψ_{i} · δ_{0:12}) = 0 for i = 1, 2, one finds that the δ_{0:12}-coefficient of \overline{𝔑𝔬𝔡𝔢}_{g} equals the ψ_{1}-coefficient of \overline{ℭ𝔲}_{g}, while the λ, δ_{irr}-coefficients coincide.

We determine the ψ_{1}-coefficient in [\overline{𝔑𝔬𝔡𝔢}_{g}]. To this end, we fix a general point [C, q] ∈ M_{g,1} 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_{0:12} = 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) linear series L ∈ W_{d}^{2}(C), the morphism

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

fails to be injective. The map χ(y, L) globalizes to a morphism χ between vector bundles over C × W_{d}^{2}(C), and the number in question is expressed as the Chern number of the top degeneracy locus of χ. Precisely, if P denotes a Poincaré bundle on C × Pic^{d}(C) and ν : C × W_{d}^{2}(C) → W_{d}^{2}(C) is the second projection, then one has that

C_{2} · \overline{𝔑𝔬𝔡𝔢}_{g} = −2θ · [W_{d}^{2}(C)] + (d − 1)c_{1}(ν_{∗}P^{∨}),

where the Chern number c_{1}(ν_{∗}P^{∨}) can be computed using [HT]. After a determinantal calculation, one finds that c_{1}(ν_{∗}P^{∨}) = 3gc_{g}/4 and θ · [W_{d}^{2}(C)] = c_{g}(g − d + 5)/4. □

Proposition 5.7. 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_{0:12} − 6 ∑_{j=3}^{n}(δ_{0:1j} + δ_{0:2j}) − ⋯ ∈ 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 5.6 using the pull-back formulas for generators of Pic(\overline{M}_{7,2}), see e.g. [Log] Theorem 2.3. □

Theorem 5.8. \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 genus 7 passing through x_{3}, …, x_{12} and having nodes at p_{1}, …, p_{8}. Blowing-up 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 effective 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. Then one computes 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} = 24.

Furthermore, Γ_{1} · δ_{0:12} = 14, Γ_{1} · δ_{0: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 \overline{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 point p ∈ C such that A ⊗ O_{C}(3p) ∈ W_{7}^{2}(C). To reverse this construction, 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} ⊂ \overline{D}_{2} whose numerical invariants are as follows:

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

One computes that Γ_{1} · \overline{D}_{1} = −28, Γ_{2} · \overline{D}_{2} = −14, Γ_{1} · \overline{D}_{2} = 14, Γ_{2} · \overline{D}_{1} = 28, as well as Γ_{1} · K_{\overline{M}_{7,13}} = 24, Γ_{2} · K_{\overline{M}_{7,13}} = −28. The assumptions of Proposition 5.1 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