MODULI OF THETA-CHARACTERISTICS VIA NIKULIN SURFACES

GAVRIL FARKAS AND ALESSANDRO VERRA

The importance of the locus K_{g} := {[C] ∈ M_{g} : C lies on a K3 surface} has been recognized for some time. Fundamental results in the theory of algebraic curves like the Brill-Noether Theorem [Laz], or Green’s Conjecture for generic curves [Vo] have been proved by specialization to a general point [C] ∈ K_{g}. The variety K_{g} viewed as a subvariety of M_{g} serves as an obstruction for effective divisors on \overline{M}_{g} to having small slope [FP] and thus plays a significant role in determining the cone of effective divisors on \overline{M}_{g}.

The first aim of this paper is to show that at the level of the the Prym moduli space R_{g} classifying étale double covers of curves of genus g, the locus of curves lying on a Nikulin K3 surfaces plays a similar role. The analogy is far-reaching: Nikulin surfaces furnish an explicit unirational parametrization of R_{g} in small genus, see Theorem 0.2, just like ordinary K3 surfaces do the same for M_{g}; numerous results involving curves on K3 surfaces have a Prym-Nikulin analogue, see Theorem 0.4, and even exceptions to uniform statements concerning curves on K3 surfaces carry over in this analogy!

Our other aim is to complete the birational classification of the moduli space \overline{S}_{g}^{+} of even spin curves of genus g. It is known [F] that \overline{S}_{g}^{+} is of general type when g ≥ 9. Using Nikulin surfaces we show that \overline{S}_{g}^{+} is uniruled for g ≤ 7, see Theorem 0.7, which leaves \overline{S}_{8}^{+} as the only case missing from the classification. We prove the following:

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

Theorems 0.1 and 0.7 highlight the fact that the birational type of \overline{S}_{g}^{+} is entirely governed by the world of K3 surfaces, in the sense that \overline{S}_{g}^{+} is uniruled precisely when a general even spin curve of genus g moves on a special K3 surface. This is in contrast to \overline{M}_{g} which is known to be uniruled at least for g ≤ 16, whereas the general curve of genus g ≥ 12 does not lie on a K3 surface.

A Nikulin surface [Ni] is a K3 surface S endowed with a non-trivial double cover

f : S̃ → S

with a branch divisor N := N_{1} + ⋯ + N_{8} consisting of 8 disjoint smooth rational curves N_{i} ⊂ S. Blowing down the (−1)-curves E_{i} := f^{−1}(N_{i}) ⊂ S̃, one obtains a minimal K3 surface σ : S̃ → Y, together with an involution ι ∈ Aut(Y) having 8 fixed points corresponding to the images σ(E_{i}) of the exceptional divisors. The class O_{S}(N) is divisible by 2 in Pic(S) and we set e := \frac{1}{2}O_{S}(N_{1} + ⋯ + N_{8}) ∈ Pic(S). Assume that C ⊂ S is a smooth curve of genus g such that C · N_{i} = 0 for i = 1, …, 8. We say that the triple (S, e, O_{S}(C)) is a polarized Nikulin surface of genus g and denote by F_{g}^{𝔑} the 11-dimensional moduli space of such objects. Over F_{g}^{𝔑} we consider the P^{g}-bundle

P_{g}^{𝔑} := {(S, e, C) : C ⊂ S is a smooth curve such that [S, e, O_{S}(C)] ∈ F_{g}^{𝔑}},

which comes equipped with two maps

\xymatrix{ & P_{g}^{𝔑} \ar@{-->}[dl]_{p_{g}} \ar[dr]^{χ_{g}} \\ F_{g}^{𝔑} &  & R_{g}}

where p_{g}([S, e, C]) := [S, e, O_{S}(C)] and χ_{g}([S, e, C]) := [C, e_{C} := e ⊗ O_{C}]. Since C · N = 0, it follows that e_{C}^{⊗2} = O_{C}. The étale double cover induced by e_{C} is precisely the restriction f_{C} := f_{|C̃} : C̃ → C, where C̃ := f^{−1}(C). Note that dim(P_{g}^{𝔑}) = 11 + g and it is natural to ask when is χ_{g} dominant and induces a uniruled parametrization of R_{g}.

Theorem 0.2. The general Prym curve [C, e_{C}] ∈ R_{g} lies on a Nikulin surface if and only if g ≤ 7 and g ≠ 6, that is, the morphism χ_{g} : P_{g}^{𝔑} → R_{g} is dominant precisely in this range.

In contrast, the general Prym curve [C, e_{C}] ∈ R_{6} lies on an Enriques surface [V1] but not on a Nikulin surface. Since P_{g}^{𝔑} is a uniruled variety being a P^{g}-bundle over F_{g}^{𝔑}, we derive from Theorem 0.2 the following immediate consequence:

Corollary 0.3. The Prym moduli space R_{g} is uniruled for g ≤ 7.

The discussion in Sections 2 and 3 implies the stronger result that F_{g}^{𝔑} (and thus N_{g} := Im(χ_{g})) is unirational for g ≤ 6. It was known that R_{g} is rational for g ≤ 4, see [Do2], [Ca], and unirational for g = 5, 6, see [Do], [ILS], [V1], [V2]. Apart from the result in genus 7 which is new, the significance of Corollary 0.3 is that Nikulin surfaces provide an explicit uniform parametrization of R_{g} that works for all genera g ≤ 7.

Before going into a more detailed explanation of our results on F_{g}^{𝔑}, it is instructive to recall Mukai’s work on the moduli space F_{g} of polarized K3 surfaces of genus g: \underline{Mukai’s results [M1], [M2], [M3]}:

(1) A general curve [C] ∈ M_{g} lies on a K3 surface if and only if g ≤ 11 and g ≠ 10, that is, the equality K_{g} = M_{g} holds precisely in this range.

(2) M_{11} is birationally isomorphic to the tautological P^{11}-bundle P_{11} over the moduli space F_{11} of polarized K3 surfaces of genus 11. There is a commutative diagram

[figure: M_{11}; ≅; q_{11}; P_{11}; p_{11}; F_{11}]

with q_{11}^{−1}([C]) = [S, C], where S is the unique K3 surface containing a general [C] ∈ M_{11}.

(3) The locus K_{10} is a divisor on M_{10} which has the following set-theoretic incarnation:

K_{10} = {[C] ∈ M_{10} : ∃L ∈ W_{12}^{4}(C) such that μ_{0}(L) : Sym^{2}H^{0}(C, L) ⟶^{≇} H^{0}(C, L^{⊗2})}.

(4) There exists a rational variety X ⊂ P^{13} with K_{X} = O_{X}(−3) and dim(X) = 5, such that the general K3 surface of genus 10 appears as a 2-dimensional linear section of X. Such a realization is unique up to the action of Aut(X) and one has birational isomorphisms:

F_{10} −−→^{≅} G(P^{10}, P^{13})^{ss}//Aut(X) and K_{10} −−→^{≅} G(P^{9}, P^{13})^{ss}//Aut(X).

To this list of well-known results, one could add the following statement from [FP]:

(5) The closure \overline{K}_{10} of K_{10} inside \overline{M}_{10} is an extremal point in the effective cone Eff(\overline{M}_{10}); its class \overline{K}_{10} ≡ 7λ−δ_{0}−5δ_{1}−9δ_{2}−12δ_{3}−14δ_{4}−⋯ ∈ Pic(\overline{M}_{10}) has minimal slope among all effective divisors on \overline{M}_{10} and provides a counterexample to the Slope Conjecture [HMo].

Quite remarkably, each of the statements (1)-(5) has a precise Prym-Nikulin analogue. Theorem 0.2 is the analogue of (1). For the highest genus when the Prym-Nikulin condition is generic, the moduli space acquires a surprising Mori fibre space structure:

Theorem 0.4. The moduli space R_{7} is birationally isomorphic to the tautological P^{7}-bundle P_{7}^{𝔑} and there is a commutative diagram:

[figure: R_{7}; ≅; χ_{7}; P_{7}^{𝔑}; p_{7}; F_{7}^{𝔑}]

Furthermore, χ_{7}^{−1}([C, η]) = [S, C], where the unique Nikulin surface S containing C is given by the base locus of the net of quadrics containing the Prym-canonical embedding ϕ_{K_{C}⊗η} : C → P^{5}.

Just like in Mukai’s work, the genus next to maximal from the point of view of Prym-Nikulin theory, behaves exotically.

Theorem 0.5. The Prym-Nikulin locus N_{6} := Im(χ_{6}) is a divisor on R_{6} which can be identified with the ramification locus of the Prym map Pr_{6} : R_{6} → A_{5}:

N_{6} = {[C, η] ∈ R_{6} : μ_{0}(K_{C} ⊗ η) : Sym^{2}H^{0}(C, K_{C} ⊗ η) ⟶^{≇} H^{0}(C, K_{C}^{⊗2})}.

Observe that both divisors K_{10} and N_{6} share the same Koszul-theoretic description. Furthermore, they are both extremal points in their respective effective cones, cf. Proposition 3.6. Is there a Prym analogue of the genus 10 Mukai G_{2}-variety X := G_{2}/P ⊂ P^{13}? The answer to this question is in the affirmative and we outline the construction of a Grassmannian model for F_{6}^{𝔑} while referring to Section 3 for details.

Set V := ℂ^{5} and U := ℂ^{4} and view P^{3} = P(U) as the space of planes inside P(U^{∨}). Let us choose a smooth quadric Q ⊂ P(V). The quadratic line complex W_{Q} ⊂ G(2, V) ⊂ P(∧^{2}V) consisting of tangent lines to Q is singular along the codimension 2 subvariety V_{Q} of lines contained in Q. One can identify V_{Q} with the Veronese 3-fold

ν_{2}(P^{3}) ⊂ P(Sym^{2}(U)) = P(∧^{2}V) = P^{9}.

The projective tangent bundle P_{Q} of Q, viewed as the blow-up of W_{Q} along V_{Q}, is endowed with a double cover branched along V_{Q} and induced by the map

P^{3} × P^{3} ⟶^{2:1} P(Sym^{2}(U)), (H_{1}, H_{2}) ↦ H_{1} + H_{2}.

We show in Theorem 3.4 that codimension 3 linear sections of W_{Q} are Nikulin surfaces of genus 6 with general moduli. Moreover there is a birational isomorphism

F_{6}^{𝔑} −−→^{≅} G(7, ∧^{2}V)^{ss}//Aut(Q).

Taking codimension 4 linear sections of W_{Q} one obtains a similar realization of N_{6}, which should be viewed as the Prym counterpart of Mukai’s construction of K_{10}.

The subvariety K_{g} ⊂ M_{g} is intrinsic in moduli, that is, its generic point [C] admits characterizations that involve C alone and the K3 surface containing C is a result of some peculiarity of the canonical curve. For instance [BM], if [C] ∈ K_{g} then the Wahl map

ψ_{K_{C}} : ∧^{2}H^{0}(C, K_{C}) → H^{0}(C, K_{C}^{⊗3}),

is not surjective. It is natural to ask for similar intrinsic characterizations of the Prym-Nikulin locus N_{g} ⊂ R_{g} in terms of Prym curves alone, without making reference to Nikulin surfaces. In this direction, we prove in Section 1 the following result:

Theorem 0.6. Set g := 2i + 6. Then K_{i,2}(C, K_{C} ⊗ η) ≠ 0 for any [C, η] ∈ N_{g}, that is, the

Prym-canonical curve C ⟶^{|K_{C}⊗η|} P^{g−2} of a Prym-Nikulin section fails to satisfy property (N_{i}).

It is the content of the Prym-Green Conjecture [FL] that K_{i,2}(C, K_{C} ⊗ η) = 0 for a general Prym curve [C, η] ∈ R_{2i+6}. This suggests that curves on Nikulin surfaces can be recognized by extra syzygies of their Prym-canonical embedding.

Our initial motivation for considering Nikulin surfaces was to use them for the birational classification of moduli spaces of even theta-characteristics and we propose to turn our attention to the moduli space S_{g}^{+} of even spin curves classifying pairs [C, η], where [C] ∈ M_{g} is a smooth curve of genus g and η ∈ Pic^{g−1}(C) is an even thetacharacteristic. Let \overline{S}_{g}^{+} be the coarse moduli space associated to the Deligne-Mumford stack of even stable spin curves of genus g, cf. [Cor]. 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 shown in [F] that \overline{S}_{g}^{+} is a variety of general type as soon as g ≥ 9.

The existence of the dominant morphism χ_{g} : P_{g}^{𝔑} → R_{g} when g ≤ 7 and g ≠ 6, leads to a straightforward uniruled parametrization of \overline{S}_{g}^{+}, which we briefly describe. Let us start with a general even spin curve [C, η] ∈ S_{g}^{+} and a non-trivial point of order two e_{C} ∈ Pic^{0}(C) in the Jacobian, such that h^{0}(C, e_{C} ⊗ η) ≥ 1. Since the curve [C] ∈ M_{g} is general, it follows that h^{0}(C, e_{C} ⊗ η) = 1 and Z := supp(e_{C} ⊗ η) consists of g − 1 distinct points. Applying Theorem 0.2, if g ≠ 6 there exists a Nikulin K3 surface (S, e) containing C such that e_{C} = e ⊗ O_{C}. When g = 6, there exists an Enriques surface (S, e) satisfying the same property, see [V1], and the construction described below goes through in that case as well. In the embedding ϕ_{|O_{S}(C)|} : S → P^{g}, the span ⟨Z⟩ ⊂ P^{g} is a codimension 2 linear subspace and h^{0}(S, I_{Z/S}(1)) = 2. Let

P := PH^{0}(S, I_{Z/S}(1)) ⊂ |O_{S}(C)|

be the corresponding pencil of curves on S. Each curve D ∈ P is endowed with the odd theta-characteristic O_{D}(Z). Twisting this line bundle with e ⊗ O_{D} ∈ Pic^{0}(D), we obtain an even theta-characteristic on D. This procedure induces a rational curve in moduli

m : P → \overline{S}_{g}^{+}, P ∋ D ↦ [D, e ⊗ O_{D}(Z)],

which passes through the general point [C, η] ∈ \overline{S}_{g}^{+}. This proves the following result:

Theorem 0.7. The moduli space \overline{S}_{g}^{+} is uniruled for g ≤ 7.

It is known [F] that \overline{S}_{g}^{+} is of general type when g ≥ 9. We complete the birational classification of \overline{S}_{g}^{+} and wish to highlight the following result, see Theorem 0.1:

\overline{S}_{8}^{+} is a variety of Calabi-Yau type.
We observe the curious fact that \overline{S}_{8}^{−} is unirational [FV] whereas \overline{S}_{8}^{+} is not even

uniruled. In contrast to the case of \overline{S}_{g}^{∓}, the birational classification of other important classes of moduli spaces is not complete. The Kodaira dimension of \overline{M}_{g} is unknown for 17 ≤ g ≤ 21, see [HM], [EH1], the birational type of \overline{R}_{g} is not understood in the range 8 ≤ g ≤ 13, see [FL], whereas finding the Kodaira dimension of A_{6} is a notorious open problem. Settling these outstanding cases is expected to require genuinely new ideas.

The proof of Theorem 0.1 relies on two main ideas: Following [F], 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. This already implies the inequality κ(\overline{S}_{8}^{+}) ≥ 0. Each irreducible component of this particular representative of K\overline{S}{+ \atop 8} is rigid (see Section 3), and the goal is to show that K\overline{S}{+ \atop 8} is rigid as well. To that end,

we use the existence of a birational model 𝔐_{8} of \overline{M}_{8} inspired by Mukai’s work [M2]. The space 𝔐_{8} is realized as the following GIT quotient

𝔐_{8} := G(8, ∧^{2}V)^{ss}//SL(V),

where V = ℂ^{6}. We note that ρ(𝔐_{8}) = 1 and there exists a birational morphism

f : \overline{M}_{8} −−→ 𝔐_{8},

which contracts all the boundary divisors Δ_{1}, …, Δ_{4} as well as \overline{M}_{8,7}^{2}. Using the geometric description of f, we establish a geometric characterization of points inside \overline{Θ}_{null}:

Proposition 0.8. Let C be a smooth curve of genus 8 without a 𝔤_{7}^{2}. The following are equivalent:

• There exists a vanishing theta-null L on C, that is, [C, L] ∈ \overline{Θ}_{null}.

• There exists a smooth K3 surface S together with elliptic pencils |F_{1}| and |F_{2}| on S, such that C ∈ |F_{1} + F_{2}| and L = O_{C}(F_{1}) = O_{C}(F_{2}).

The existence of such a doubly elliptic K3 surface S is equivalent to stating that there exists a smooth K3 extension S ⊂ P^{8} of the canonical curve C ⊂ P^{7}, such that the rank three quadric C ⊂ Q ⊂ P^{7} which induces the theta-null L, lifts to a rank 4 quadric S ⊂ Q_{S} ⊂ P^{8}. Having produced S, the pencils |F_{1}| and |F_{2}| define a product map

ϕ : S → P^{1} × P^{1},

such that each smooth member D ∈ I := |ϕ^{∗}O_{P^{1}×P^{1}}(1, 1)| is a canonical curve contained in a rank 3 quadric. A general pencil in I passing through C induces a rational curve R ⊂ \overline{S}_{8}^{+}, and after intersection theoretic calculations on the stack \overline{S}_{8}^{+}, we prove the following:

Proposition 0.9. The theta-null divisor \overline{Θ}_{null} ⊂ \overline{S}_{8}^{+} is uniruled and swept by rational curves R ⊂ \overline{S}_{8}^{+} such that R · \overline{Θ}_{null} < 0 and R · π^{∗}(\overline{M}_{8,7}^{2}) = 0. Furthermore R is disjoint from all boundary divisors π^{∗}(Δ_{i}) for i = 1, …, 4.

Proposition 0.9 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 π^{∗}(Δ_{i}) for i = 1, …, 4, is rigid as well. Equivalently, κ(\overline{S}_{8}^{+}) = 0. Note that since K\overline{S}{+ \atop 8} consists of 10 uniruled base components which

can be blown-down, the variety \overline{S}_{8}^{+} is not minimal and there exists a birational model S of \overline{S}_{8}^{+} which is a genuine Calabi-Yau variety in the sense that K_{S} = 0. Finding an explicit modular interpretation of this Calabi-Yau 21-fold (or perhaps even its equations!) is a very interesting question.

1. Prym-canonical curves on Nikulin surfaces

Let us start with a smooth K3 surface Y. A Nikulin involution on Y is an automorphism ι ∈ Aut(Y) of order 2 which is symplectic, that is, ι^{∗}(ω) = ω, for all ω ∈ H^{2,0}(Y). A Nikulin involution has 8 fixed points, see [Ni] Lemma 3, and the quotient Ȳ := Y/⟨ι⟩ has 8 ordinary double point singularities. Let σ : S̃ → Y be the blow-up of the 8 fixed points and denote by E_{1}, …, E_{8} ⊂ S̃ the exceptional divisors and by ι̃ ∈ Aut(S̃) the automorphism induced by ι. Then S := S̃/⟨ι̃⟩ is a smooth K3 surface and if f : S̃ → S is the projection, then N_{i} := f(E_{i}) are (−2)-curves on S. The branch divisor of f is equal to N := ∑_{i=1}^{8} N_{i}. We summarize the situation in the following diagram:

(1)  \begin{matrix}S̃ & ⟶^{σ} & Y \\ ^{f}↓ & & ↓ \\ S & ⟶ & Ȳ \end{matrix}

Sometimes we shall refer to the pair (Y, ι) as a Nikulin surface, while keeping the previous diagram in mind. We refer to [Mo], [vGS] for a lattice-theoretic study on the action of the Nikulin involution on the cohomology H^{2}(Y, ℤ) = U^{3} ⊕ E_{8}(−1) ⊕ E_{8}(−1), where U is the standard rank 2 hyperbolic lattice and E_{8} is the unique even, negative-definite unimodular lattice of rank 8. It follows from [Mo] Theorem 5.7 that the orthogonal complement E_{8}(−2) ≅ (H^{2}(Y, ℤ)^{ι})^{⊥} is contained in Pic(Y), hence Y has Picard number at least 9. The class O_{S}(N_{1} + ⋯ + N_{8}) is divisible by 2, and we denote by e ∈ Pic(S) the class such that e^{⊗2} = O_{S}(N_{1} + ⋯ + N_{8}).

Definition 1.1. The Nikulin lattice is an even lattice 𝔑 of rank 8 generated by elements \{𝔫_{i}\}_{i=1}^{8} and 𝔢 := \frac{1}{2} ∑_{i=1}^{8} 𝔫_{i}, with the bilinear form induced by 𝔫_{i}^{2} = −2 for i = 1, …, 8 and 𝔫_{i} · 𝔫_{j} = 0 for i ≠ j.

Note that 𝔑 is the minimal primitive sublattice of H^{2}(S, ℤ) containing the classes N_{1}, …, N_{8} and e. For any Nikulin surface one has an embedding 𝔑 ⊂ Pic(S). Assuming that (Y, ι) defines a general point in an irreducible component of the moduli space of Nikulin involutions, both Y and S have Picard number 9 and there is a decomposition Pic(S) = ℤ · [C] ⊕ 𝔑, where C is an integral curve of genus g ≥ 2. According to [vGS] Proposition 2.2, only two cases are possible: either C · e = 0 so that the previous decomposition is an orthogonal sum, or else, C · e ≠ 0, this second case being possible only when g is odd. In this paper we consider only Nikulin surfaces of the first kind.

We fix an integer g ≥ 2 and consider the lattice Λ_{g} := ℤ · 𝔠 ⊕ 𝔑, where 𝔠 · 𝔠 = 2g − 2.

Definition 1.2. A Nikulin surface of genus g is a K3 surface S together with a primitive embedding of lattices j : Λ_{g} ↪ Pic(S) such that C := j(𝔠) is a nef class.

The coarse moduli space F_{g}^{𝔑} of Nikulin surfaces of genus g is the quotient of the 11-dimensional domain

D_{Λ_{g}} := {ω ∈ P(Λ_{g} ⊗_{ℤ} ℂ) : ω^{2} = 0, ω · ω̄ > 0}

by an arithmetic subgroup of O(Λ_{g}). Its existence follows e.g. from [Do1] Section 3.

We now consider a Nikulin surface f : S̃ → S, together with a smooth curve C ⊂ S of genus g such that C · N = 0. If C̃ := f^{−1}(C), then f_{C} := f_{|C̃} : C̃ → C is an étale double covering. By the Hodge index theorem, C̃ cannot split in two disjoint connected components, hence f_{C} is non-trivial and e_{C} := O_{C}(e) ∈ Pic^{0}(C) is the non trivial 2torsion element defining the covering f_{C}. We set H ≡ C − e ∈ NS(S), hence H^{2} = 2g − 6 and H · C = 2g − 2. For further reference we collect a few easy facts:

Lemma 1.3. Let [S, e, O_{S}(C)] ∈ F_{g}^{𝔑} be a Nikulin surface such that Pic(S) = Λ_{g}. The following statements hold:

(i) H^{i}(S, e) = 0 for all i ≥ 0.

(ii) Cliff(C) = [\frac{g−1}{2}].

(iii) The line bundle O_{S}(H) is ample for g ≥ 4 and very ample for g ≥ 6. In this range, it defines an embedding ϕ_{H} : S → P^{g−2} such that the images ϕ_{H}(N_{i}) are lines for all i = 1, …, 8.

(iv) If g ≥ 7, the ideal of the surface Φ_{H}(S) ⊂ P^{g−2} is cut out by quadrics.

Proof. Recalling that e^{⊗2} = O_{S}(N_{1} + ⋯ + N_{8}) and that the curves {N_{i}}_{i=1}^{8} are pairwise disjoint, it follows that H^{0}(S, e) = 0 and clearly H^{2}(S, e) = 0. Since e^{2} = −4, by Riemann-Roch one finds that H^{1}(S, e) = 0 as well.

In order to prove (ii) we assume that Cliff(C) < [\frac{g−1}{2}]. From [GL2] it follows that there exists a divisor D ∈ Pic(S) such that h^{i}(S, O_{C}(D)) ≥ 2 for i = 0, 1 and C ·D ≤ g −1, such that O_{C}(D) computes the Clifford index of C, that is, Cliff(C) = Cliff(O_{C}(D)). But C · ℓ ≡ 0 mod 2g − 2 for every class ℓ ∈ Pic(S), hence no such divisor D can exist.

Moving to (iii), the ampleness (respectively very ampleness) of O_{S}(H) is proved in

[GS] Proposition 3.2 (respectively Lemma 3.1). From the exact sequence

0 ⟶ O_{S}(−H) ⟶ O_{S}(e) ⟶ O_{C}(e) ⟶ 0,

one finds that h^{1}(S, O_{S}(H)) = 0 and then dim|H| = g − 2. Furthermore H · N_{i} = 1 for i = 1, …, 8 and the claim follows.

To prove (iv), following [SD] Theorem 7.2, it suffices to show that there exists no irreducible curve Γ ⊂ S with Γ^{2} = 0 and H · Γ = 3. Assume by contradiction that Γ ≡ aC−b_{1}N_{1}−⋯−b_{8}N_{8} is such a curve, where necessarily a, b_{i} ∈ ℤ_{≤0}. Then ∑_{i=1}^{8} b_{i} = 2ag− 2a − 3 and ∑_{i=1}^{8} b_{i}^{2} = a^{2}(g − 1). Applying the Cauchy-Schwarz inequality (∑_{i=1}^{8} b_{i})^{2} ≤ 8(∑_{i=1}^{8} b_{i}^{2}), we obtain an immediate contradiction. □

We consider the P^{g}-bundle p_{g} : P_{g}^{𝔑} → F_{g}^{𝔑}, as well as the map
χ_{g} : P_{g}^{𝔑} → R_{g}, χ_{g}([S, e, C]) := [C, e_{C} := e ⊗ O_{C}]

defined in the introduction. We fix a Nikulin surface [S, e, O_{S}(C)] ∈ P_{g}^{𝔑}. A Lefschetz pencil of curves {C_{λ}}_{λ∈P^{1}} inside |O_{S}(C)| induces a rational curve

Ξ_{g} := \{[C_{λ}, e_{C_{λ}} := e ⊗ O_{C_{λ}}] : λ ∈ P^{1}\} ⊂ \overline{R}_{g}.

In the range where χ_{g} is a dominant map, Ξ_{g} is a rational curve passing through a general point of \overline{R}_{g}, and it is of some interest to compute its numerical characters. If π : \overline{R}_{g} → \overline{M}_{g} denotes the projection map, we recall the formula [FL] Example 1.4

(2)  π^{∗}(δ_{0}) = δ0^{′} + δ0^{′′} + 2δ_{0}^{ram},

where δ0^{′} := [Δ0^{′}], δ0^{′′} := [Δ0^{′′}] and δ_{0}^{ram} := [Δ_{0}^{ram}] are boundary divisor classes on \overline{R}_{g} whose meaning we recall. Let us fix a general point [C_{xy}] ∈ Δ_{0} induced by a 2-pointed curve [C, x, y] ∈ M_{g−1,2} and the normalization map ν : C → C_{xy}, where ν(x) = ν(y). A general point of Δ0^{′} (respectively of Δ0^{′′}) corresponds to a stable Prym curve [C_{xy}, η], where η ∈ Pic^{0}(C_{xy})[2] and ν^{∗}(η) ∈ Pic^{0}(C) is non-trivial (respectively, ν^{∗}(η) = O_{C}). A general point of Δ_{0}^{ram} is of the form [X, η], where X := C ∪_{{x,y}} P^{1} is a quasi-stable curve, whereas η ∈ Pic^{0}(X) is characterized by η_{P}_{1} = O_{P}_{1}(1) and η_{C}^{⊗2} = O_{C}(−x − y).

Proposition 1.4. If Ξ_{g} ⊂ \overline{R}_{g} is the curve induced by a pencil on a Nikulin surface, then

Ξ_{g} · λ = g + 1, Ξ_{g} · δ0^{′} = 6g + 2, Ξ_{g} · δ0^{′′} = 0 and Ξ_{g} · δ_{0}^{ram} = 8.

It follows that Ξ_{g} · K_{\overline{R}_{g}} = g − 15.

Proof. We use [FP] Lemma 2.4 to find that Ξ_{g} · λ = π_{∗}(Ξ_{g}) · λ = g + 1 and Ξ_{g} · π^{∗}(δ_{0}) = π_{∗}(Ξ_{g}) · δ_{0} = 6g + 18, as well as Ξ_{g} · π^{∗}(δ_{i}) = 0 for 1 ≤ i ≤ [g/2]. For each 1 ≤ i ≤ 8, the sublinear system P H^{0}(O_{S}(C − N_{i})) ⊂ P H^{0}(O_{S}(C)) intersects Ξ_{g} transversally in one point which corresponds to a curve N_{i} + C_{i} ∈ |O_{S}(C)|, where N_{i} · C_{i} = −N_{i}^{2} = 2 and C_{i} ≡ C − N_{i}. Furthermore e ⊗ O_{N_{i}} = O_{N_{i}}(1) and e_{C_{i}}^{⊗2} = O_{C_{i}}(−N_{i} · C_{i}). Each of these points lie in the intersection Ξ_{g} ∩ Δ_{0}^{ram}. All remaining curves in Ξ_{g} are irreducible, hence Ξ_{g} · δ_{0}^{ram} = 8. Since Ξ_{g} · δ0^{′′} = 0, from (2) we find that Ξ_{g} · δ0^{′} = 6g + 2. Finally, according to [FL] Theorem 1.5 the formula K_{\overline{R}_{g}} ≡ 13λ − 2(δ0^{′} + δ0^{′′}) − 3δ_{0}^{ram} − ⋯ ∈ Pic(\overline{R}_{g}) holds, therefore putting everything together, Ξ_{g} · K_{\overline{R}_{g}} = g − 15. □

The calculations in Proposition 1.4 are applied now to show that syzygies of Prymcanonical curves on Nikulin surfaces are exceptional when compared to those of general Prym-canonical curves. To make this statement precise, let us recall the Prym-Green Conjecture, see [FL] Conjecture 0.7: If g := 2i + 6 with i ≥ 0, then the locus

U_{g,i} := {[C, η] ∈ R_{2i+6} : K_{i,2}(C, K_{C} ⊗ η) ≠ 0}

is a virtual divisor, that is, the degeneracy locus of two vector bundles of the same rank defined over R_{2i+6}. The statement of the Prym-Green Conjecture is that this vector bundle morphism is generically non-degenerate: Prym-Green Conjecture: K_{i,2}(C, K_{C} ⊗ η) = 0 for a general Prym curve [C, η] ∈ R_{2i+6}.

The conjecture is known to hold in bounded genus and has been used in [FL] to show that \overline{R}_{g} is of general type when g ≥ 14 is even.

Theorem 1.5. For each [S, e, C] ∈ P_{2i+6}^{𝔑} one has K_{i,2}(C, K_{C} ⊗ e_{C}) ≠ 0. In particular, the Prym-Green Conjecture fails along the locus N_{2i+6}.

Proof. If the non-vanishing K_{i,2}(C, K_{C} ⊗η) ≠ 0 holds for a general point [C, η] ∈ R_{g}, then there is nothing to prove, hence we may assume that U_{g,i} is a genuine divisor on R_{g}. The class of its closure inside \overline{R}_{g} has been calculated [FL] Theorem 0.6:

\overline{U}_{g,i} ≡ \binom{2i + 2}{i} (\frac{3(2i + 7)}{i + 3}λ − \frac{3}{2}δ_{0}^{ram} − δ0^{′} − α δ0^{′′} − ⋯) ∈ Pic(\overline{R}_{2i+6}).

From Proposition 1.4, by direct calculation one finds that Ξ_{g} · \overline{U}_{g,i} = −\binom{2i+3}{i} < 0, thus Ξ_{g} ⊂ \overline{U}_{g,i}. By varying Ξ_{g} inside \overline{R}_{g}, we obtain that N_{g} ⊂ \overline{U}_{g,i}, which ends the proof. □

Remark 1.6. A geometric proof of Theorem 1.5 using the Lefschetz hyperplane principle for Koszul cohomology is given in [AF] Theorem 3.5. The indirect proof presented here is however shorter and illustrates how cohomology calculations on \overline{R}_{g} can be used to derive geometric consequences for individual Prym curves.

Remark 1.7. One might ask whether similar applications to R_{g} can be obtained using Enriques surfaces. There is a major difference between Prym curves [C, η] ∈ R_{g} lying on a Nikulin surface and those lying on an Enriques surface. For instance, if C ⊂ S is a curve of genus g lying on an Enriques surface S, then from [CD] Corollary 2.7.1

gon(C) ≤ 2 inf\{F · C : F ∈ Pic(S), F^{2} = 0, F ≢0\} ≤ 2\sqrt{2g − 2}.

In particular, for g sufficiently high, C is far from being Brill-Noether general. On the other hand, we have seen that for [S, e, C] ∈ P_{g}^{𝔑} such that Pic(S) = Λ_{g}, one has that gon(C) = [\frac{g+3}{2}]. For this reason, the Prym-Nikulin locus N_{g} := Im(χ_{g}) ⊂ R_{g} appears as a more promising and less constrained locus than the Prym-Enriques locus in R_{g}, being transversal to stratifications of R_{g} coming from Brill-Noether theory.

2. The Prym-Nikulin locus in R_{g} for g ≤ 7

In this section we give constructive proofs of Theorems 0.2 and 0.4. Comparing the dimensions dim(P_{g}^{𝔑}) = 11 + g and dim(R_{g}) = 3g − 3, one may inquire whether the morphism χ_{g} : P_{g}^{𝔑} → R_{g} is dominant when g ≤ 7. The similar question for ordinary K3 surfaces has been answered by Mukai [M1]. Let F_{g} denote the 19-dimensional moduli space of polarized K3 surfaces of genus g and consider the associated P^{g}-bundle

P_{g} := {[S, C] : C ⊂ S is a smooth curve such that [S, O_{S}(C)] ∈ F_{g}}.

The map q_{g} : P_{g} → M_{g} forgetting the K3 surface is dominant if and only if g ≤ 11 and g ≠ 10. The result for g = 10 is contrary to untutored expectation since the general fibre of q_{10} is 3-dimensional, hence dim(Im(q_{10})) = dim(P_{10}) − 3 = 26. A strikingly similar picture emerges for Nikulin surfaces and Prym curves. The morphism χ_{g} : P_{g}^{𝔑} → R_{g} is dominant when g ≤ 7 and g ≠ 6. For each genus we describe a geometric construction that furnishes a Nikulin surface in the fibre χ_{g}^{−1}([C, η]) over a general point [C, η] ∈ R_{g}.

2.1. Nikulin surfaces of genus 7. We start with a general element [C, η] ∈ R_{7} and construct a Nikulin surface containing C. One may assume that gon(C) = 5 and that the line bundle η does not lie in the difference variety C_{2} − C_{2} ⊂ Pic^{0}(C), or equivalently, the linear series L := K_{C} ⊗ η ∈ W_{12}^{5}(C) is very ample. It is a consequence of [GL1]

Theorem 2.1 that the Prym-canonical image C ⟶^{|L|} P^{5} is quadratically normal, that is, h^{0}(P^{5}, I_{C/P^{5}}(2)) = 3.

Lemma 2.1. For a general [C, η] ∈ R_{7}, the base locus of |I_{C/P^{5}}(2)| is a smooth K3 surface.

Proof. The property that the base locus of |I_{C/P^{5}}(2)| is smooth, is open in R_{7} and it suffices to exhibit a single Prym-canonical curve [C, η] ∈ R_{7} satisfying it. Let us fix an element (S, e, C) ∈ P_{7}^{𝔑} such that Pic(S) = Λ_{7} and set H ≡ C − e. Then according to Lemma 1.3, ϕ_{H} : S → P^{5} is an embedding whose image ϕ_{H}(S) is ideal-theoretically cut out by quadrics. Moreover gon(C) = 5, hence K_{C} ⊗ e_{C} ∈ W_{12}^{5}(C) is quadratically normal. This implies that H^{0}(S, O_{S}(2H − C)) = H^{1}(S, O_{S}(2H − C)) = 0, and then H^{0}(P^{5}, I_{S/P^{5}}(2)) ≅ H^{0}(P^{5}, I_{C/P^{5}}(2)), therefore the quadrics in |I_{C/P^{5}}(2)| cut out precisely the surface S. □

Remark 2.2. This proof shows that if [S, e, C] ∈ P_{7}^{𝔑} is general then χ_{7}^{−1}([C, e_{C}]) = [S, e, C] and in particular the fibre χ_{7}^{−1}([C, e_{C}]) is reduced. Indeed, let [S′, e′, C] ∈ P_{7}^{𝔑} be an arbitrary Nikulin surface containing C. Set H′ ≡ C − e′ ∈ NS(S′). We may assume that Pic(S′) = Λ_{7}, therefore the map ϕ_{H′} : S′ → P^{5} is an embedding whose image is cut out by quadrics. Since Cliff(C) = 3, from Lemma 1.3 we find that K_{C} ⊗ e_{C} is quadratically normal and then S′ is cut out by the quadrics contained in Prym-canonical embedding of C ⊂ P^{5}.

Since both P_{7}^{𝔑} and R_{7} are irreducible varieties of dimension 18, Remark 2.2 shows that χ_{7} : P_{7}^{𝔑} → R_{7} is a birational morphism and we now describe χ_{7}^{−1}.

Proposition 2.3. For a general [C, η] ∈ R_{7}, the surface S := bs |I_{C/P^{5}}(2)| is a polarized Nikulin surface of genus 7.

Proof. We show that Pic(S) ⊃ ℤ · C ⊕ 𝔑. Denote by H ⊂ S the hyperplane class and let N :≡ 2(C − H), thus N^{2} = −16, N · H = 8 and N · C = 0. We aim to prove that N is linearly equivalent to a sum of 8 pairwise disjoint integral (−2) curves on S. We consider the following exact sequence

0 ⟶ O_{S}(N − C) ⟶ O_{S}(N) ⟶ O_{C}(N) ⟶ 0.

Note that O_{C}(N) is trivial because e_{C} = O_{C}(C − H) and that h^{1}(S, O_{S}(N − C)) = h^{1}(S, O_{S}(C − 2H)) = 0, because C ⊂ P^{5} is quadratically normal. Passing to the long exact sequence, it follows that h^{0}(S, O_{S}(N)) = 1. Using Remark 2.2 it follows that N ≡ N_{1} + ⋯ + N_{8}, where N_{i} · N_{j} = −2δ_{ij}. Finally, to conclude that [S, ℤ · C ⊕ 𝔑] ∈ F_{7}^{𝔑} we must show that there is a primitive embedding ℤ · C ⊕ 𝔑 ↪ Pic(S). We apply [vGS] Proposition 2.7. Since H^{0}(S̃, O_{S}(C̃)) = H^{0}(S, O_{S}(C)) ⊕ H^{0}(S, O_{S}(C) ⊗ e^{∨}) and sections in the second summand vanish on the exceptional divisor of the morphism σ : S̃ → Y, it follows that this is precisely the decomposition of H^{0}(Y, O_{Y} (C̃)) into ι_{Y}^{∗} -eigenspaces. Invoking loc. cit., we finish the proof. □

2.2. The symmetric determinantal cubic hypersurface and Prym curves. We provide a general set-up that allows us to reconstruct a Nikulin surface from a Prym curve of genus g ≤ 5. Let us start with a curve [C, η] ∈ R_{g} inducing an étale double cover f : C̃ → C together with an involution ι : C̃ → C̃ such that f ∘ ι = f. For each integer r ≥ −1, the Prym-Brill-Noether locus is defined as the locus

V^{r}(C, η) := {L ∈ Pic^{2g−2}(C̃) : Nm_{f}(L) = K_{C}, h^{0}(L) ≥ r + 1 and h^{0}(L) ≡ r + 1 mod 2}.

Note that V^{−1}(C, η) = Pr(C, η). For each line bundle L ∈ V^{r}(C, η), the Petri map

μ_{0}(L) : H^{0}(C̃, L) ⊗ H^{0}(C̃, KC̃ ⊗ L^{∨}) → H^{0}(C̃, KC̃)

splits into an ι-anti-invariant part

μ_{0}^{−}(L) : Λ^{2}H^{0}(C̃, L) → H^{0}(C, K_{C} ⊗ η), s ∧ t ↦ s · ι^{∗}(t) − t · ι^{∗}(s),

and an ι-invariant part respectively

μ_{0}^{+}(L) : Sym^{2}H^{0}(C̃, L) → H^{0}(C, K_{C}), s ⊗ t + t ⊗ s ↦ s · ι^{∗}(t) + t · ι^{∗}(s).

For a general [C, η] ∈ R_{g}, the Prym-Petri map μ_{0}^{−}(L) is injective for every L ∈ V^{r}(C, η) and V^{r}(C, η) is equidimensional of dimension g − 1 − \binom{r+1}{2}, see [We]. We introduce the universal Prym-Brill-Noether variety

R_{g}^{r} := {([C, η], L) : [C, η] ∈ R_{g}, L ∈ V^{r}(C, η)}.

When g − 1 − \binom{r+1}{2} ≥ 0, the variety R_{g}^{r} is irreducible of dimension 4g − 4 − \binom{r+1}{2}. We propose to focus on the case r = 2 and g ≥ 4 and choose a general triple (f : C̃ → C, L) ∈ R_{g}^{2}, such that L is base point free and h^{0}(C̃, L) = 3.

Setting P^{2} := P(H^{0}(L)^{∨}), we consider the quasi-étale double cover q : P^{2}×P^{2} → P^{5} obtained by projecting via the Segre embedding to the space of symmetric tensors. Note that q is ramified along the diagonal Δ ⊂ P^{2} × P^{2} and V_{4} := q(Δ) ⊂ P^{5} is the Veronese surface. Moreover Σ := Im(q) is the determinantal symmetric cubic hypersurface isomorphic to the secant variety of V_{4}. We have the following commutative diagram:

\xymatrix{C̃ \ar[r]^{(L,ι^{∗}L)} \ar[dd]_{f} & P^{2} × P^{2} \ar[ddr]_{q} \ar@{-->}[drr] \\  &  &  & P^{8} = P(H^{0}(L)^{∨} ⊗ H^{0}(L)^{∨}) \ar@{-->}[dl] \\ C \ar[rr]^{μ_{0}^{+}(L)} &  & P^{5} = P(Sym^{2}H^{0}(L)^{∨})}

Observe that the involution ι : P^{8} → P^{8} given by ι[v⊗w] := [w⊗v] where v, w ∈ H^{0}(L), is compatible with ι : C̃ → C̃. To summarize, giving a point (C̃ → C, L) ∈ R_{g}^{2} is equivalent to specifying a symmetric determinantal cubic hypersurface Σ ∈ H^{0}(P^{g−1}, I_{C/P^{g−1}}(3)) containing the canonical curve.

2.3. A birational model of F_{4}^{𝔑}. As a warm-up, we indicate how the set-up described above is a generalization of the construction that Catanese [Ca] used to prove that R_{4} is rational. For a general point [C, η] ∈ R_{4} we find that V^{2}(C, η) = {L, ι^{∗}L}, that is, the pair (L, ι^{∗}L) is uniquely determined. The map μ_{0}(L) has corank 2 and P6 ; C̃ := P(Im μ_{0}(L)) ⊂ P^{8} has codimension 2. The intersection T̃ := (P^{2}×P^{2})∩P6 ; C̃ is a del Pezzo surface of degree 6, whereas T := Σ ∩ P_{+}^{3} is a 4-nodal Cayley cubic. Here we set P_{+}^{3} := P(H^{0}(K_{C})^{∨}). The double cover q : T̃ → T is ramified at the singular points of T.

To obtain a Nikulin surface containing [C, η], we reverse this construction and start with a quartic rational normal curve R ⊂ P^{4} and denote by Ȳ := Sec(R) ⊂ P^{4} its secant variety, which we view as a hyperplane section of Σ ⊂ P^{5}. Retaining the notation of diagram (1), for a general quadric Q ∈ |O_{P}_{4}(2)|, the intersection Ȳ := Ȳ ∩ Q is a K3 surface with 8 rational double points at R ∩ Q. There exists a cover q : Y →^{2:1} Ȳ ramified at the singular points of Y, induced by restriction from the map q : P^{2} × P^{2} → Σ. Clearly q : Y → Ȳ is a Nikulin covering, and a hyperplane section in |OȲ (1)| induces a Prym curve [C, η] ∈ R_{4} having general moduli. Moreover we have a birational isomorphism

F_{4}^{𝔑} −−→^{≅} P(H^{0}(O_{P}_{4}(2)))^{ss}//SL_{2},

where P GL_{2} = Aut(R) ⊂ P GL_{5}. An immediate consequence is that F_{4}^{𝔑} is unirational.

2.4. Nikulin surfaces of genus 3. We prove that χ_{3} : P_{3}^{𝔑} → R_{3} is dominant and fix a complete intersection of 3 quadrics Y ⊂ P^{5} invariant with respect to an involution fixing a line L ⊂ P^{5} and a 3-dimensional linear subspace Λ ⊂ P^{5}. The projection π_{L} : P^{5} −−→ Λ induces a quartic Ȳ := π_{L}(Y) with 8 nodes, which is a Nikulin surface. We check that a general Prym curve [C, η] ∈ R_{3} corresponding to an étale cover f : C̃ → C embeds in such a surface.

Indeed, the canonical model C̃ ⊂ P^{4} is a complete intersection of 3 quadrics. Fixing projective coordinates on P^{4}, we can assume that the involution ι : C̃ → C̃ is induced by the projective involution [x : y : u : v : t] ↔ [−x : −y : u : v : t]. Note that the ι^{∗}anti-invariant quadratic forms are vectors q = ax + by, where a, b are linear forms in u, v, t. Since C̃ is complete intersection of 3 quadrics, no non-zero quadric q = ax + by vanishes on C̃, for not, C̃ would intersect the plane {x = y = 0} and then ι would have fixed points. Thus ι acts as the identity on the space H^{0}(P^{4}, I_{C̃/P^{4}}(2)). Hence it follows C̃ = {a_{1} + b_{1} = a_{2} + b_{2} = a_{3} + b_{3} = 0}, where a_{i}, b_{i} are quadratic forms in x, y and u, v, t. Passing to P^{5} by adding one coordinate h, we can choose quadratic forms a_{i} + b_{i} + hl_{i}, where l_{i} is a general linear form in h, u, v, t. Consider the surface Y ⊂ P^{5} defined by the latter 3 equations. Then [x : y : h : u : v : t] ↔ [−x : −y : h : u : v : t] induces a Nikulin involution on Y. Let π_{L} : Y → P^{3} be the projection of center L = {h = u = v = t = 0}. Then \overline{Y} := π_{L}(Y) is a quartic Nikulin surface and C = π_{L}(C̃) is a plane section of it.

2.5. Nikulin surfaces of genus 5. To describe the morphism χ_{5} : P_{5}^{𝔑} → R_{5} more geometrically, we use the set-up introduced in Subsection 2.2. If [C, η] ∈ R_{5} is general, then dim V^{2}(C, η) = 1, the ι-invariant Petri map μ_{0}^{−}(L) is injective, μ_{0}^{+}(L) surjective, thus dim(Coker μ_{0}(L)) = 1. We consider the hyperplane

P7 ; C̃ := P(Im(μ_{0}(L)) ⊂ P(H^{0}(L)^{∨} ⊗ H^{0}(L)^{∨})

and also set P_{+}^{4} := P(H^{0}(K_{C})^{∨}) ⊂ P^{5}. Then we further denote

T̃ := (P^{2} × P^{2}) ∩ P7 ; C̃ and T := Σ ∩ P_{+}^{4}.

Note that T̃ is a degree 6 threefold in P7 ; C̃. Since the hyperplane P7 ; C̃ is ι-invariant, it follows T̃ is also endowed with the involution ιT̃ ∈ Aut(T̃) such that Fix(ιT̃) = Δ∩T̃ is a rational quartic curve in P_{+}^{4}. Furthermore T ⊂ P_{+}^{4} is the secant variety of R.

Proposition 2.4. For a general point [C, η, L] ∈ R_{5}^{2} the following statements hold:

(i) The threefold T̃ ⊂ P^{2} × P^{2} is smooth, while T ⊂ P_{+}^{4} is singular precisely along R. (ii) h^{0}(T̃, I_{C̃/T̃}(2)) = 3. Moreover H^{i}(T̃, I_{C̃/T̃}(2)) = 0 for i = 1, 2. (iii) Every quadratic section in the linear system |I_{C̃/T̃}(2)| is ι-invariant, that is,

H^{0}(T̃, I_{C̃/T̃}(2)) = q^{∗}H^{0}(T, I_{C/T} (2)).

(iv) A general quadratic section Y ∈ |I_{C̃/T̃}(2)| is a smooth K3 surface endowed with an involution ι_{Y} with fixed points precisely at the 8 points in the intersection R ∩ Y.

Proof. We take cohomology in the following exact sequence

0 ⟶ I_{C̃/P^{2}×P^{2}}(2) ⟶ O_{P^{2}×P^{2}}(2) ⟶ K⊗2 ; C̃ ⟶ 0,

to note that h^{0}(I_{C̃/T̃}(2)) = 3(⇔ H^{1}(I_{C̃/P^{2}×P^{2}}(2)) = 0), if and only if the composed map

Sym^{2}H^{0}(C̃, L) ⊗ Sym^{2}H^{0}(C̃, ι^{∗}L) → H^{0}(C̃, L^{⊗2}) ⊗ H^{0}(C̃, ι^{∗}(L^{⊗2})) → H^{0}(C̃, K⊗2 ; C̃)

is surjective. This is an open condition and a triple (C̃ →^{f} C, L) ∈ R_{g}^{2} satisfying it, and for which moreover T̃ ⊂ P^{2} × P^{2} is smooth, has been constructed in [V2] Section 4. Finally, from the exact sequence

0 ⟶ I_{T/P_{+}^{4}}(2) ⟶ I_{C/P_{+}^{4}}(2) ⟶ I_{C/T} (2) → 0,

we compute that h^{0}(T, I_{C,T} (2)) = 3, therefore q^{∗} : H^{0}(T, I_{C/T} (2)) → H^{0}(T̃, I_{C̃/T̃}(2)) is an isomorphism, based on dimension count. Part (iv) is a consequence of (i)-(iii). Assume that Ȳ = T ∩ Q, where Q ∈ H^{0}(I_{C/P_{+}^{4}}(2)). Then Y = T̃ ∩ q^{∗}(Q) and Ȳ is the quotient of Y by the involution ι_{Y} obtained by restriction from ι ∈ Aut(P^{2} × P^{2}). It follows that the covering q : Y → Ȳ is a Nikulin surface such that C ⊂ Ȳ ⊂ P_{+}^{4}. To conclude, we must check that for a general choice of Y ∈ |I_{C̃/T̃}(2)|, the point [Y, ι_{Y}] gives rise to an element of F_{5}^{𝔑}, that is, using the notation of diagram (1), that Pic(S) = Λ_{5}. Proposition 2.7 from [vGS] picks out two possibilities for Pic(Y) (or equivalently for Pic(S)), and we must check that ℤ · O_{Y} (C̃) ⊕ E_{8}(−2) has index 2 in Pic(Y), see also [GS] Corollary 2.2. ^{1}

This is achieved by finding the decomposition of H^{0}(O_{Y} (C̃)) into ι_{Y}^{∗} -eigenspaces. In the course of the proof of [V2] Proposition 5.2 an example of a smooth quadratic section Y ∈ |I_{C̃/T̃}(2)| is constructed such that

H^{0}(Y, O_{Y} (C̃))^{+} = q^{∗}H^{0}(Ȳ , OȲ (C)).

In particular the (+1)-eigenspace of H^{0}(Y, O_{Y} (C̃)) is 6-dimensional and invoking once more [vGS] Proposition 2.7, we conclude that [Y, ι_{Y}] ∈ F_{5}^{𝔑}. □

We close this subsection with an amusing result on a geometric divisor on R_{5}. For a Prym curve [C, η] ∈ R_{5} and L := K_{C} ⊗ η ∈ W_{8}^{3}(C), we observe that the vector spaces entering the multiplication map ν_{3}(L) : Sym^{3}H^{0}(C, L) → H^{0}(C, L^{⊗3}) have the same dimension. The condition that ν_{3}(L) be not an isomorphism is divisorial in R_{5}. We have not been able to find a direct proof of the following equality of cycles on R_{5}, even though one inclusion is straightforward:

Theorem 2.5. Let [C, η] ∈ R_{5} be a Prym curve such that the Prym-canonical line bundle K_{C} ⊗η is very ample. Then ϕ_{K_{C}⊗η} : C → P^{3} lies on a cubic surface if and only if C is trigonal.

Proof. Let 𝔇_{1} be the locus of Prym curves whose Prym-canonical model lies on a cubic

𝔇_{1} := \{[C, η] ∈ \overline{R}_{5} : ν_{3}(ω_{C} ⊗ η) : Sym^{3}H^{0}(C, ω_{C} ⊗ η) ⟶^{≇} H^{0}(C, ω_{C}^{⊗3} ⊗ η^{⊗3})\},

1We are grateful to the referee for raising this point that he have initially overlooked.

and 𝔇_{2} the closure inside \overline{R}_{5} of the divisor \{[C, η] ∈ R_{5} : η ∈ C_{2} − C_{2}\} of smooth Prym curves for which L := K_{C} ⊗ η ∈ W_{8}^{3}(C) is not very ample. Obviously, 𝔇_{1} − 𝔇_{2} ≥ 0, for if L is not very ample, then the multiplication map ν_{3}(L) : Sym^{3}H^{0}(C, L) → H^{0}(C, L^{⊗3}) cannot be an isomorphism. The class of 𝔇_{2} can be read off [FL] Theorem 5.2:

𝔇_{2} ≡ 14λ − 2(δ0^{′} + δ0^{′′}) − \frac{5}{2}δ_{0}^{ram} − ⋯ ∈ Pic(\overline{R}_{5}).

For i ≥ 1, let 𝔼_{i} be the vector bundle over \overline{R}_{5} with fibre 𝔼_{i}[C, η] = H^{0}(C, ω_{C}^{⊗i} ⊗ η^{⊗i}) for every [C, η] ∈ \overline{R}_{g}. One has the following formulas from [FL] Proposition 1.7:

c_{1}(𝔼_{i}) = \binom{i}{2}(12λ − δ0^{′} − δ0^{′′} − 2δ_{0}^{ram}) + λ − \frac{i^{2}}{4} δ_{0}^{ram} ∈ Pic(\overline{R}_{g}).

As a consequence, 𝔇_{1} ≡ c_{1}(𝔼_{3}) − c_{1}(Sym^{3}𝔼_{1}) ≡ 37λ − 3(δ_{0} + δ0^{′′}) − \frac{33}{4}δ_{0}^{ram} − ⋯ ∈ Pic(\overline{R}_{5}),

therefore 𝔇_{1} − 𝔇_{2} ≡ 8λ − (δ0^{′} + δ0^{′′}) − 2δ_{0}^{ram} − ⋯ = π^{∗}(8λ − δ_{0} − ⋯) ≥ 0, where the

terms left out are combinations of boundary divisors π^{∗}(δ_{i}) with i ≥ 1, corresponding to reducible curves. The only effective divisors D ≡ aλ − b_{0}δ_{0} − b_{1}δ_{1} − b_{2}δ_{2} on \overline{M}_{5} such that \frac{a}{b_{0}} ≤ 8 and satisfying Δ_{i} ⊈ supp(D) for i = 1, 2, are multiples of the trigonal locus \overline{M}_{5,3}^{1} (the proof is identical to that of Proposition 5.1). This proves that if [C, η] ∈ 𝔇_{1} − 𝔇_{2}, with C being a smooth curve, then necessarily [C] ∈ M_{5,3}^{1}, which finishes the proof. □

3. A singular quadratic complex and a birational model for F_{6}^{𝔑}

Let us set V := ℂ^{n+1} and denote by G := G(2, V) ⊂ P(∧^{2}V) the Grassmannian of lines in P(V). We fix once and for all a smooth quadric Q ⊂ P(V). The projective tangent bundle P_{Q} := P(T_{Q}) can be realized as the incidence correspondence

P_{Q} = {(x, ℓ) ∈ Q × G : x ∈ ℓ ⊂ P(T_{x}Q)}.

For each point x ∈ Q, the fibre P_{Q}(x) is the space of lines tangent to Q at x. We introduce the projections p : P_{Q} → G and q : P_{Q} → Q, then set

W_{Q} := p(P_{Q}) = {ℓ ∈ G : ℓ is tangent to the quadric Q}.

Note that W_{Q} contains the Hilbert scheme of lines in Q, which we denote by V_{Q} ⊂ W_{Q}. It is well-known that V_{Q} is smooth, irreducible and dim(V_{Q}) = 2n − 5. The restriction p_{|p^{−1}(W_{Q}−V_{Q})} is an isomorphism and E_{Q} := p^{−1}(V_{Q}) ⊂ P_{Q} is the exceptional divisor of p.

Proposition 3.1. The variety W_{Q} is a quadratic complex of lines in G. Its singular locus is equal to V_{Q} and each point of V_{Q} is an ordinary double point of W_{Q}.

Proof. Let Q : V → ℂ be the quadratic form whose zero locus is the quadric hypersurface also denoted by Q ⊂ P(V), and Q̃ : V × V → ℂ the associated bilinear map. We define the bilinear map ν_{2}(Q̃) : ∧^{2}V × ∧^{2}V → ℂ by the formula

ν_{2}(Q̃)(u ∧ v, s ∧ t) := Q̃(u, s)Q̃(v, t) − Q̃(v, s)Q̃(u, t)

for u, v, s, t ∈ V, and denote by ν_{2}(Q) : ∧^{2}V → ℂ the induced quadratic form.

For fixed points x = [u] ∈ Q and y = [v] ∈ P(V), we observe that the line ℓ = ⟨x, y⟩ is tangent to Q if and only if Q̃(u, v) = 0 ⇔ ν_{2}(Q)(u ∧ v) = 0. Therefore W_{Q} = G ∩ ν_{2}(Q) is a quadratic line complex in G, being the vanishing locus of ν_{2}(Q).

Keeping the same notation, a point ℓ = [u ∧ v] ∈ W_{Q} is a singular point, if and only if the linear form ν_{2}(Q̃)(u ∧ v, −) vanishes along P(T_{ℓ} G). Since P(T_{ℓ} G) is spanned by the Schubert cycle {m ∈ G : m ∩ ℓ ≠ ∅}, any tangent vector in T_{ℓ}(G) has a representative of the form u ∧ a − v ∧ b, where a, b ∈ V. We obtain that [u ∧ v] ∈ Sing(W_{Q}) if and only if Q(v, v) = 0, that is, ℓ = [u ∧ v] ∈ V_{Q}. Since W_{Q} is a quadratic complex, each point ℓ ∈ V_{Q} has multiplicity 2. □

The map p : P_{Q} → W_{Q} appears as a desingularization of the quadratic complex W_{Q}. We shall compute the class of the exceptional divisor E_{Q} of P_{Q}. Let H := p^{∗}(O_{G}(1)) be the class of the family of tangent lines to Q intersecting a fixed (n − 2)-plane in P(V) and B := q^{∗}(O_{Q}(1)) ∈ Pic(P_{Q}). Furthermore, we consider the class h ∈ NS_{1}(P_{Q}) of the pencil of tangent lines to Q with center a given point x ∈ Q. It is clear that

h · H = 1 and h · B = 0.

If ℓ ∈ V_{Q} is a fixed line, let s ∈ NS_{1}(P_{Q}) be the class of the family {(x, ℓ) : x ∈ ℓ}. Then

s · H = 0 and s · B = 1.

Lemma 3.2. The linear equivalence E_{Q} ≡ 2H − 2B in Pic(P_{Q}) holds. In particular, the class E_{Q} is divisible by 2 and it is the branch divisor of a double cover

f : P̃_{Q} → P_{Q}.

Proof. To compute the class of E_{Q} it suffices to compute h · E_{Q} and s · E_{Q}. First we note that h · E_{Q} = 2. Indeed a pencil of tangent lines to Q through a fixed point x ∈ Q has two elements which are in Q. Finally, recalling that V_{Q} = Sing(W_{Q}) consists of ordinary double points, we obtain that s · E_{Q} = −2, since p^{−1}(ℓ) is a conic inside P(N_{V_{Q}/G}(ℓ)). □

3.1. A birational model for F_{6}^{𝔑}. Let us now specialize to the case n = 4, that is,

Q ⊂ P^{4}, G = G(2, 5) ⊂ P^{9} and dim(W_{Q}) = 5.

The class of V_{Q} equals 4σ_{2,1} ∈ H^{6}(G, ℤ), therefore deg(V_{Q}) = 4σ_{2,1} · σ_{1}^{3} = 8. This can also be seen by recalling that V_{Q} is isomorphic to the Veronese 3-fold ν_{2}(P^{3}) ⊂ P^{9}.

The double covering f : P̃_{Q} → P_{Q} constructed above has a transparent projective interpretation. For (x, ℓ) ∈ P_{Q}, we denote by Π_{ℓ} ∈ G(3, V) the polar space of ℓ defined as the base locus of the pencil of polar hyperplanes {z ∈ P(V) : Q̃(y, z) = 0}_{y∈ℓ}. Clearly x ∈ Π_{ℓ} ⊂ P(T_{x}Q) and Q ∩ Π_{ℓ} is a conic of rank at most 2 in Π_{ℓ}. When ℓ ∈ W_{Q} − V_{Q}, the quadric has rank exactly 2 which corresponds to a pair of lines ℓ_{1}+ℓ_{2} with ℓ_{1}, ℓ_{2} ∈ V_{Q}. The double cover is induced by the map from the parameter space of the lines themselves.

In the next statement we shall keep in mind the notation of diagram (1):

Proposition 3.3. A general codimension 3 linear section Ȳ := Λ ∩ W_{Q} of the quadratic complex W_{Q} where Λ ∈ G(7, ∧^{2}V), is a 8-nodal K3 surface with desingularization

p : S := p^{−1}(Ȳ) → Ȳ .

The triple [S, O_{S}(H − B), O_{S}(H)] ∈ F_{6}^{𝔑} is a Nikulin surface of genus 6 and the induced double cover is the restriction f : S̃ := f^{−1}(S) → S.

Proof. We fix a general 6-plane Λ ∈ G(7, ∧^{2}V). Since K_{W_{Q}} = O_{W_{Q}}(−3H), by adjunction we obtain that Ȳ := Λ ∩ W_{Q} is a K3 surface. From Bertini’s theorem, Ȳ has ordinary double points at the 8 points of intersection Λ ∩ V_{Q}. General hyperplane sections of C ∈ |OȲ (H)|, viewed as codimension 4 linear sections of W_{Q}, are canonical curves of genus 6, endowed with a line bundle of order 2 given by O_{C}(H − B). The remaining statements are immediate. □

It turns out that the general Nikulin surface of genus 6 arises in this way:

Theorem 3.4. Let V := ℂ^{5} and Q ⊂ P(V) be a smooth quadric. One has a dominant map

φ : G(7, ∧^{2}V)^{ss}//Aut(Q) −−→ F_{6}^{𝔑},

given by φ(Λ) := [S := p^{−1}(Λ ∩ W_{Q}), O_{S}(H − B), O_{S}(H)].

Proof. Via the embedding Aut(Q) ⊂ P GL(V) ↪ P GL(∧^{2}V), we observe that every automorphism of Q induces an automorphism of P(∧^{2}V) that fixes both W_{Q} and V_{Q}. Since (i) the moduli space F_{6}^{𝔑} is irreducible and (ii) polarized Nikulin surfaces have finite automorphism groups, it suffices to observe that dim G(7, ∧^{2}V)//Aut(Q) = 21 − 10 = 11 and dim(F_{6}^{𝔑}) = 11 as well. □

Corollary 3.5. The Prym-Nikulin locus N_{6} ⊂ R_{6} is an irreducible unirational divisor, which is set-theoretically equal to the ramification locus of the Prym map Pr : R_{6} → A_{5}

U_{6,0} = {[C, η] ∈ R_{6} : K_{0,2}(C, K_{C} ⊗ η) ≠ 0}.

Furthermore, the exists a dominant rational map G(6, ∧^{2}V)^{ss}//Aut(Q) −−→ N_{6}.

Proof. Just observe that ⟨C⟩ = P^{5} and that this has codimension 4 in P(∧^{2}V), hence there is a P^{3} of Nikulin sections of W_{Q} containing C. □

The divisor \overline{K}_{10} ⊂ \overline{M}_{10} of sections of K3 surfaces is known to be an extremal point of the effective cone Eff(\overline{M}_{10}). An analogous result holds for the closure of N_{6}:

Proposition 3.6. The Prym-Nikulin divisor \overline{N}_{6} is extremal in the effective cone Eff(\overline{R}_{6}):

Proof. It follows from [FL] Theorem 0.6 that \overline{N}_{6} ≡ 7λ − \frac{3}{2}δ_{0}^{ram} − (δ0^{′} + δ0^{′′}) − ⋯ ∈ Pic(\overline{R}_{6}). The divisor \overline{KN}_{6} is filled-up by the rational curves Ξ_{6} ⊂ \overline{R}_{6} constructed in the course of proving Theorem 1.4. We compute that Ξ_{6} · \overline{N}_{6} = −1, which completes the proof. □

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

We turn our attention to the moduli space of spin curves and begin by setting notation and terminology. If M is a Deligne-Mumford stack, we denote by M its associated coarse moduli space. A ℚ-Weil divisor D on a normal ℚ-factorial projective variety 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).

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, an irreducible 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}^{+} and refer to [Cor], [F] for details.

Definition 4.1. 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 with |E ∩ (\overline{X − E})| = 2 (such a component is called exceptional), h^{0}(X, η) ≡ 0 mod 2, and β : η^{⊗2} → ω_{X} is a morphism of sheaves 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 the relation
(3)  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 [Cor]:

(4)  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. [F]:

(5)  \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 4.2. 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. 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. We construct the pencil Γ ⊂ \overline{S}_{g}^{+} by representing C as a section of a nodal canonical surface S ⊂ Q and noting that dim |O_{S}(C)| = 1. The construction of S depends on the genus and we describe the various cases separately.

(i) 7 ≤ g ≤ 9. We choose V ∈ G(7, H^{0}(C, K_{C})) such that if π_{V} : P^{g−1} −−→ P(V^{∨}) denotes 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. (ii) g = 5. In the case C ⊂ Q ⊂ P^{4} and we choose a general quartic X ∈ H^{0}(P^{4}, I_{C/P^{4}}(4)) and set S := Q ∩ X. Then S is a canonical surface with nodes at the 4 points X ∩ Sing(Q). As in the previous case dim |C| = 1, and the numerical characters of the induced family Γ ⊂ \overline{S}_{5}^{+} can be readily computed:

Γ · λ = g + 5 = 10, Γ · β_{0} = |Sing(S)| = 4, and Γ · α_{0} = 4g + 52,

where the last equality is a consequence of Noether’s formula Γ·(α_{0}+2β_{0}) = 12χ(S, O_{S})− K_{S}^{2} +4(g−1) = 4g+60. By direct calculation, we obtain once more that Γ·\overline{Θ}_{null} = −2. The case g = 6 is similar, except that the canonical surface S is a (2, 2, 3) complete intersection in P^{5}, where one of the quadrics is the rank 3 quadric Q. (iii) g = 4. In this last case we proceed slightly differently and denote by S = 𝔽_{2} the blowup 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. □

5. 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. [EH1],

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

(6)  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 (6) are rigid divisors on \overline{S}_{8}^{+}, but in principle, their sum could still be a movable class. Assuming for a moment Proposition 0.9, we explain how this implies Theorem 0.1: Proof of Theorem 0.1. The covering curve R ⊂ \overline{Θ}_{null} constructed in Proposition 0.9, 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 (6) 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 (6) 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 5.1 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.

□

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

The aim of this section is to prove Proposition 0.9, 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, see [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)^{st}//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}(E)^{∨}), ϕ_{E}(p) := E(p)^{∨} (↪ H^{0}(E)^{∨}),

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

f([C]) := H^{0}(K_{C})^{∨} mod SL(H^{0}(E)^{∨}) ∈ 𝔐_{8},

that is, we assign to C its linear span ⟨C⟩ under the Plücker map ℘∘ϕ_{E} : C → P(∧^{2}H^{0}(E)^{∨}).

Even though this is not strictly needed for our proof, 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 ϕ_{O_{S}(C)} : S ↪ P^{8} lies on a rank 4 quadric whose rulings induce the elliptic pencils |C_{1}| and |C_{2}| on S.

Let U → 𝔑𝔏 be the space 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 at most 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})].

Since U is birational to a P^{3}-bundle over an open subvariety of 𝔑𝔏, we obtain that U is irreducible and dim(U) = 21(= 3 + dim(𝔑𝔏)). We shall show that the morphism q is dominant (see Corollary 6.3) and begin with some preparations.

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 1.4.

Proposition 6.1. 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. [Do1], 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, see [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 6.2. 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 5dimensional subspace Ker(Q). We choose an arbitrary element

[v̄ := v + Ker(Q)] ∈ P(\frac{Ker(u ∘ Q̃)}{Ker(Q)}) = P^{1},

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 and there is a pencil of 8-planes Λ ⊃ P^{7} with this property. □

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 (cf. Proposition 6.2). 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})^{∨}. This proves the following:

Corollary 6.3. The rational morphism q : U −−→ \overline{Θ}_{null} is dominant.

Proof. Keeping the notation from above, if [C] ∈ \overline{Θ}_{null} is a general point corresponding to the rank 3 quadric Q ∈ H^{0}(P^{7}, I_{C/P^{7}}(2)), then [S, O_{S}(C_{1} + C_{2}), C] ∈ q^{−1}([C]). □

We begin the proof of Proposition 0.9 while retaining the set-up described above. Let us 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 6.4. 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 6.5. R · π^{∗}(\overline{M}_{7,8}^{2}) = 0.

Proof. We show instead that π_{∗}(R) · \overline{M}_{8,7}^{2} = 0. From Lemma 6.4, 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 6.6. R · \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,

(7)  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}, see [Cor] Lemma 2.2, while clearly Aut(f^{−1}(l_{1}) ∪ f^{−1}(m_{2})) = {Id}.

If ℂ_{τ}^{3g−3} denotes 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 the Weil divisor 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 (7) we find that R · α_{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.

□

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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, Dipartimento di Matematica, Largo San Leonardo Murialdo 1-00146 Roma, Italy

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