SYZYGIES OF TORSION BUNDLES AND THE GEOMETRY OF THE LEVEL ℓ MODULAR VARIETY OVER \overline{M}_{g}

ALESSANDRO CHIODO, DAVID EISENBUD, GAVRIL FARKAS, AND FRANK-OLAF SCHREYER

Abstract. We formulate, and in some cases prove, three statements concerning the purity or, more generally, the naturality of the resolution of various modules one can attach to a generic curve of genus g and a torsion point of ℓ in its Jacobian. These statements can be viewed as analogues of Green’s Conjecture and we verify them computationally for bounded genus. We then compute the cohomology class of the corresponding nonvanishing locus in the moduli space R_{g,ℓ} of twisted level ℓ curves of genus g and use this to derive results about the birational geometry of R_{g,ℓ}. For instance, we prove that R_{g,3} is a variety of general type when g > 11 and the Kodaira dimension of R_{11,3} is greater than or equal to 19. In the last section we explain probabilistically the unexpected failure of the Prym-Green conjecture in genus 8 and level 2.

One way of proving that the moduli space of curves of odd genus g ≫ 0 has general type is via the divisor of curves whose canonical ring has “extra” syzygies. In this paper we apply the same philosophy to prove that, in certain cases, the moduli spaces of curves with torsion bundles—the modular varieties of the title—are also of general type. These modular varieties are natural generalizations, in higher genus, of the much studied modular curves X_{1}(ℓ) := H/Γ_{1}(ℓ) classifying elliptic curves together with an ℓ-torsion point in their Jacobian.

To explain what we mean by “extra” syzygies, consider a finitely generated graded module M over a polynomial ring S = ℂ[x_{0}, …, x_{n}]. Such a module has a minimal free resolution of the form

0 ← M ← F_{0} ← ⋯ ← F_{i} ← ⋯,

where F_{i} = ∑_{j} S(−i − j)^{b_{i,j}}. The numbers b_{i,j} = b_{i,j}(M), called the graded Betti numbers of M, are uniquely defined; in fact b_{i,j} is the dimension of the degree i + j component of Tor_{i}^{S}(M, ℂ). We say that the resolution of M is natural if, for each j, the number b_{i,j−i}(M) is nonzero for at most one value of i, that is, at most one Betti number on each diagonal of the Betti diagram of M is non-zero. In an irreducible flat family of modules M_{λ} the b_{i,j}(M_{λ}) are semicontinuous, and simultaneously take on minimum values on an open set. We say that M_{λ} has “extra” syzygies when one of the values b_{i,j}(M_{λ}) is larger than this minimum. If the resolution of some M_{λ} is natural in the sense above, then its Betti numbers have the minimum value.

For example, given a curve C, a line bundle L ∈ Pic(C) and a sheaf F on C, we consider the S := Sym H^{0}(C, L)-module

Γ_{C}(F, L) := ⨁_{q∈ℤ} H^{0}(C, F ⊗ L^{⊗q}).

Following notation introduced by Mark Green, the vector space Tor_{i}^{S}(Γ_{C}(F, L), ℂ)_{i+j} is often written K_{i,j}(C; F, L) and one sets K_{i,j}(C, L) := K_{i,j}(C; O_{C}, L). Green’s Conjecture [G] for generic curves of genus g (proved in [V1] and [V2]), asserts that the resolution of the canonical ring Γ_{C}(O_{C}, K_{C}), is natural. For odd genus, it follows that the resolution is pure, which in Green’s notation says

(1)  K_{\frac{g−1}{2} ,1}(C, K_{C}) = 0.

The locus of curves whose canonical ring has extra syzygies is a divisor supported on the Hurwitz locus M_{g, \frac{g+1}{2}}^{1} used in [HM] to prove that these moduli spaces have general type.

Generalizing the case above, we study modules of the form Γ_{C}(ξ, K_{C} ⊗ η) where η and ξ are line bundles of degree 0 on a smooth curve C. As a first step, we prove:

Theorem 0.1. Let C be a general curve of genus g, η ∈ Pic^{0}(C)[ℓ] a torsion bundle of order ℓ ≥ 2 and ξ ∈ Pic^{0}(C) a general line bundle of degree degree 0. Then the module Γ_{C}(ξ, K_{C} ⊗η) has natural resolution.

We shall use a refinement of this result to compute the Kodaira dimension of the moduli spaces R_{g,ℓ} parametrizing level ℓ curves [C, η], where C is a smooth curve of genus g and η ∈ Pic^{0}(C) is a torsion line bundle of order ℓ. In particular, we shall prove:

Theorem 0.2. R_{g,3} is a variety of general type for g ≥ 12. Furthermore, the Kodaira dimension of R_{11,3} is at least 19.

It is known that R_{g,3} is rational for g ≤ 4, see [BC], [BV]. The level ℓ modular variety that has received most attention so far is the moduli space R_{g,2} classifying Prym varieties of dimension g − 1. It is shown in [FL] that R_{g,2} is a variety of general type for g > 13, whereas R_{g,2} is unirational for g ≤ 7, see [FV2] and references therein.

We now explain the statements concerning the naturality of the resolution of certain modules one associates to a level curve [C, η] ∈ R_{g,ℓ}. Such predictions can be made for any g and, just like for Green’s Conjecture, in about half of the cases they amount to saying that the resolution is actually pure. In such case, the locus of points [C, η] where the corresponding resolution is not pure is a virtual divisor on R_{g,ℓ}, that is, the degeneracy locus of a morphism between vector bundles of the same rank over the stack R_{g,ℓ} which is coarsely represented by R_{g,ℓ}. Proving the corresponding syzygy conjecture amounts to showing that the respective degeneracy locus is a genuine divisor on R_{g,ℓ}.

A. Prym-Green Conjecture. For a general level ℓ curve [C, η] ∈ R_{g,ℓ} of genus g ≥ 6, the homogeneous coordinate ring of the paracanonical curve ϕ_{K_{C}⊗η} : C ↪ P^{g−2} has a natural resolution. Equivalently, in even genus, the resolution is pure and the paracanonical curve satisfies property (N_{\frac{g}{2} −3}), that is,

(2)  K_{\frac{g}{2} −2,1}(C, K_{C} ⊗ η) = 0,

whereas in odd genus

(3)  K_{\frac{g−3}{2} ,1}(C, K_{C} ⊗ η) = 0 and K_{\frac{g−7}{2} ,2}(C, K_{C} ⊗ η) = 0.

The name of the conjecture is justified by the analogy with Green’s Conjecture. In Section 4, we verify the Prym-Green Conjecture for all values of g ≤ 18 and small ℓ, with the exception of 2-torsion in genus g = 8 and g = 16. Our findings suggest that very likely, for 2-torsion and g a power of 2 (or perhaps multiple of 8), the Prym-Green Conjecture is actually false. (With our methods, an experiment with g = 24 would take about 6500 years.) Our verification of Conjecture A is computational via the use of Macaulay2. We verify condition (2) for g-nodal rational curves over a finite field (see Section 4 for details). For g := 2i + 6, we denote by

Z_{g,ℓ} := {[C, η] ∈ R_{g,ℓ} : K_{i+1,1}(C, K_{C} ⊗ η) ≠ 0},

the failure locus of Conjecture A. This is a virtual divisor on R_{g,ℓ}.

The second conjecture we address concerns the resolution of torsion bundles.

B. Torsion Bundle Conjecture. Let [C, η] ∈ R_{g,ℓ} be a general level ℓ curve of even genus g ≥ 4. For each 1 ≤ k ≤ ℓ − 2, the module Γ_{C}(η^{⊗k}, K_{C} ⊗ η) has natural resolution, unless

η^{⊗(2k+1)} = O_{C} andg ≡ 2 mod 4 and \binom{g − 3}{\frac{g}{2} − 1} ≡ 1 mod 2.

Equivalently, one has the vanishing statement

(4)  K_{\frac{g}{2} −1,1}(C; η^{⊗k}, K_{C} ⊗ η) = 0.

In the exceptional cases there is precisely one extra syzygy.

Theorem 0.1 can be viewed as a weak form of Conjecture B. The first exceptional genera in the Conjecture B are g = 6, 10, 18, 34 or 66. For levels ℓ ≥ 3, we set g := 2i + 2 and denote the corresponding virtual divisor by

D_{g,ℓ} := {[C, η] ∈ R_{g,ℓ} : K_{i,1}(C; η^{⊗(ℓ−2)}, K_{C} ⊗ η) ≠ 0}.

Conjecture B can be reformulated in the spirit of the Minimal Resolution Conjecture [FMP] for points on paracanonical curves as follows. We view a divisor Z ∈ |K_{C} ⊗ η^{⊗(1−k)}| as a 0-dimensional subscheme of ϕ_{L}(C) ⊂ P^{g−2}. Then using [FMP] Proposition 1.6, Conjecture B is equivalent to the statement b_{\frac{g}{2} ,1}(Z) = b_{\frac{g}{2} −1,2}(Z) = 0, which amounts to the minimality of the number of syzygies of Z ⊂ P^{g−2}.

The exceptions to Conjecture B can be explained by (surprising) symmetries in the Koszul differentials (see Section 4). We verify Conjecture B for genus g ≤ 16 and small level. In any of these cases D_{g,ℓ} is a divisor on R_{g,ℓ}.

A prediction about the naturality of the resolution of the module Γ_{C}(η, K_{C}) can also be made. This time we expect no exceptions and this turns out to be the case:

Theorem 0.3. Let [C, η] ∈ R_{g,ℓ} be a general level ℓ curve of genus g ≥ 3. Then the resolution of Γ_{C}(η, K_{C}) is natural (respectively pure for odd g), that is,

(5)  K_{⌊ \frac{g−1}{2} ⌋,1}(C; η, K_{C}) = 0.

The proof of Theorem 0.3 is by specialization to hyperelliptic curves. Via the following equality of cycles in the Jacobian of any curve of odd genus, see [FMP],

C_{\frac{g−1}{2}} − C_{\frac{g−1}{2}} = \{ξ ∈ Pic^{0}(C) : K_{\frac{g−1}{2} ,1}(C; ξ, K_{C}) ≠ 0\},

where the left hand side denotes the top difference variety of C, Theorem 0.3 admits the following geometrically transparent reformulation:

Corollary 0.4. For a general curve C of odd genus, the top difference variety C_{\frac{g−1}{2}} − C_{\frac{g−1}{2}} contains no non-trivial ℓ-torsion points, for any ℓ ≥ 2.

To pass from syzygy statements to the birational structure of R_{g,ℓ}, one calculates the cohomology classes of the failure loci of the conditions (2), (4) and (5) respectively on a compact moduli of level ℓ curves. The space R_{g,ℓ} admits a compactification \overline{R}_{g,ℓ}, which is the coarse moduli space associated to the smooth proper Deligne-Mumford stack \overline{R}_{g,ℓ} of level twisted curves, that is, triples [C, η, ϕ], where C is a genus g twisted curve, η is a faithful line bundle on the stack C and ϕ : η^{⊗ℓ} → O_{C} is an isomorphism, see [CF] and Section 1 of this paper. We denote by f : \overline{R}_{g,ℓ} → \overline{M}_{g} the forgetful map.

An essential ingredient in the proof of Theorem 0.2 is one of the main results of [CF]. It is shown that the non log-canonical singularities of \overline{R}_{g,3} do not impose adjunction conditions, that is, if ϵ : R̂_{g,3} → \overline{R}_{g,3} denotes a resolution of singularities, then for each n ≥ 1, there exists an isomorphism at the level of spaces of global sections

ϵ^{∗} : H^{0}(\overline{R}_{g,3}^{reg}, K_{\overline{R}_{g,3}}^{⊗n}) ⟶^{≅} H^{0}(R̂_{g,3}, K⊗n ; R̂_{g,3}).

A similar extension result has been proved in [HM] for level one curves, that is, for the moduli space \overline{M}_{g} itself, and in [FL] in the case of \overline{R}_{g,2}. We refer to [CF] for partial generalizations of this extension result for higher levels. Using [CF], we thus conclude that the Kodaira dimension of \overline{R}_{g,3} is equal to the Kodaira-Iitaka dimension of the canonical bundle K_{\overline{R}_{g,3}}. In order to prove that for a given g, the moduli space \overline{R}_{g,3} is of general type, it suffices to express K_{\overline{R}_{g,3}} ∈ Pic(\overline{R}_{g,3}) as a positive combination of the Hodge class λ ∈ Pic(\overline{R}_{g,3}), the class [\overline{𝔇}] of the closure of a certain effective divisor 𝔇 on R_{g,3} and boundary divisor classes corresponding to singular level curves. In our case, the divisor 𝔇 is a jumping locus for Koszul cohomology groups of paracanonically embedded curves, defines by one of the conditions (2), (4) or (5).

We denote by M̃_{g} the open subvariety of \overline{M}_{g} classifying irreducible stable curves of genus g and set R̃_{g,ℓ} := f^{−1}(M̃_{g}). The boundary R̃_{g,ℓ} −f^{−1}(M_{g}) is the union of three proper and closed subloci Δ0^{′} ∪ Δ0^{′′} ∪ Δ_{0}^{ram}, where Δ0^{′′} and Δ_{0}^{ram} can be characterized as follows (see Section 1, §1.4 for further details). The locus Δ_{0}^{ram} is the ramification divisor of f. The locus Δ0^{′′} is the locus of order-ℓ analogues of Wirtinger covers. These boundary subloci are not irreducible in general and we refer to [CF, §1.4.3-4] for a decomposition into irreducible components. When ℓ = 2 or 3 however, the loci Δ0^{′}, Δ0^{′′} and Δ_{0}^{ram} are irreducible, whereas for a prime level ℓ > 3, both Δ′′_{0} and Δ_{0}^{ram} are reducible and decompose into ⌊\frac{ℓ}{2}⌋ irreducible components. For Δ_{0}^{ram}, we have a decomposition Δ_{0}^{ram} = ⋃_{a=1}^{⌊ℓ/2⌋} Δ_{0}^{(a)}, where the irreducible components are determined by a local index a at the node of the generic level ℓ curve (see Definitions 1.1 and 1.2) and yielding ℚ-divisors δ_{0}^{(a)} = [Δ_{0}^{(a)}]_{ℚ} at the level of the moduli stack, fitting in the formula

f^{∗}(δ_{0}) = δ′_{0} + δ′′_{0} + ℓ ∑_{a=1}^{⌊ℓ/2⌋} δ_{0}^{(a)}.

Here δ_{0} is the ℚ-divisor attached to Δ_{0} = M̃_{g} − M_{g}. As shown in Section 1, §1.3 and §1.4, the above identity extends word for word for possibly composite levels ℓ ≥ 2.

For any ℓ ≥ 2, following the established practice of passing to lower case symbols for the divisor classes on the moduli stack, we have the following results:

Theorem 0.5. Write g = 2i + 1 and ℓ ≥ 2. The class of the closure in R̃_{g,ℓ} of the effective divisor U_{g,ℓ} := {[C, η] ∈ R_{g,ℓ} : K_{i,1}(C; η, K_{C}) ≠ 0} is equal to

[\overline{U}_{g,ℓ}] = \frac{1}{2i − 1}\binom{2i}{i}((3i+1)λ−\frac{i}{2}(δ0^{′}+δ0^{′′})−∑_{a=1}^{⌊ \frac{ℓ}{2} ⌋} \frac{1}{2ℓ}(iℓ^{2}+2a^{2}i−2aℓi−a^{2}+aℓ) δ_{0}^{(a)}) ∈ Pic(R̃_{g,ℓ}).

Keeping g = 2i + 1 and setting ℓ = 3, in order to form an effective representative of K_{\overline{R}_{g,3}}, we use Theorem 0.5 together with the formula of the slope s(\overline{M}_{g,i+1}^{1}) = \frac{6(i+2)}{i+1} of the class of the closure in \overline{M}_{g} of the Hurwitz divisor [HM], [EH]

M_{g,i+1}^{1} := {[C] ∈ M_{g} : W_{i+1}^{1}(C) ≠ ∅},

in order to obtain that, for suitable rational constants α, β > 0, the ℚ-divisor class

(6)  α · [\overline{U}_{g,3}] + β · [f^{∗}(\overline{M}_{g,i+1}^{1})] = \frac{6(2i + 3)}{i + 1}λ − 2(δ0^{′} + δ0^{′′}) − 4δ_{0}^{(1)} ∈ Pic(R̃_{g,3})

is effective. Comparing this formula against that of the canonical class K_{\overline{R}_{g,3}} (see Section 1), we note that whenever the following inequality

\frac{6(2i + 3)}{i + 1} < 13 ⇔ i > 5,

holds, the canonical class K_{\overline{R}_{g,3}} is big. Using the extension result of [CF], we conclude that \overline{R}_{g,3} is of general type for odd genus g ≥ 13. This argument also shows that when g = 11 the class given in (6) is an effective representative for the canonical class; furthermore, one has the following inequalities

κ(\overline{R}_{11,3}, K_{\overline{R}_{11,3}}) ≥ κ(\overline{R}_{11,3}, f^{∗}(\overline{M}_{11,6}^{1})) ≥ κ(\overline{M}_{11}, \overline{M}_{11,6}^{1}) = 19,

where the last equality has been proved in [FP]. It is an interesting open question whether the equality κ(\overline{R}_{11,3}) = 19 holds. It is known [FV1] that the universal Picard variety over the moduli space \overline{M}_{11} has Kodaira dimension equal to 19 as well.

We compute the (virtual) class of the failure loci given by Conjectures A and B.

Theorem 0.6. Set g := 2i + 6 with i ≥ 0 and ℓ ≥ 2. The virtual class of the closure in R̃_{g,ℓ} of the locus Z_{g,ℓ} of level ℓ curves [C, η] ∈ R_{g,ℓ} with K_{i+1,1}(C, K_{C} ⊗ η) ≠ 0 is equal to

[\overline{Z}_{g,ℓ}]^{virt} = \binom{2i + 2}{i}(\frac{3(2i + 7)}{i + 3}λ − (δ′_{0} + δ0^{′′}) − ∑_{a=1}^{⌊ \frac{ℓ}{2} ⌋} \frac{a^{2} − aℓ + ℓ^{2}}{2} δ_{0}^{(a)}) ∈ Pic(R̃_{g,ℓ}).

We explain the meaning of this result. In Section 3 we construct tautological vector bundles A and B over the stack R̃_{g,ℓ} with rk(A) = rk(B), as well as a vector bundle morphism φ : A → B, such that over the open part R_{g,ℓ} ⊂ \overline{R}_{g,ℓ}, the degeneracy locus of φ equals the scheme Z_{g,ℓ}. Accordingly, we define [\overline{Z}_{g,ℓ}]^{virt} := c_{1}(B − A). Whenever the Prym-Green Conjecture holds, that is, φ is generically non-degenerate and \overline{Z}_{g,ℓ} is a divisor, we have that [\overline{Z}_{g,ℓ}]^{virt} − [\overline{Z}_{g,ℓ}] is a (possibly empty) effective class entirely supported on the boundary of R̃_{g,ℓ}. In particular, the class computed in Theorem 0.6 is effective. Next we describe the universal failure locus of Conjecture B.

Theorem 0.7. Set g := 2i + 2 ≥ 4 and ℓ ≥ 3 such that i ≡ 1 mod 2 or \binom{2i−1}{i} ≡ 0 mod 2. The virtual class of the closure in R̃_{g,ℓ} of the locus D_{g,ℓ} of level ℓ curves [C, η] ∈ R_{g,ℓ} such that K_{i,1}(C; η^{⊗(ℓ−2)}, K_{C} ⊗ η) ≠ 0 is equal to

[\overline{D}_{g,ℓ}]^{virt} = \frac{1}{i − 1}\binom{2i − 2}{i}((6i+1)λ−i(δ0^{′}+δ0^{′′})− \frac{1}{ℓ} ∑_{a=1}^{⌊ \frac{ℓ}{2} ⌋}(iℓ^{2}+5a^{2}i−5aiℓ−2a^{2}+2aℓ)δ_{0}^{(a)}).

To establish Theorem 0.2 in even genus, when g ≥ 14 but g ≠ 16, we can use the class [\overline{Z}_{g,3}] to show that K_{\overline{R}_{g,3}} is big. In genus g = 16, when the Prym-Green Conjecture appears to fail, we use the class [\overline{D}_{g,3}] instead. Interestingly, for g = 12, the Prym-Green divisor \overline{Z}_{12,3} ≡ 13λ − 2(δ0^{′} + δ0^{′′}) − \frac{14}{3}δ_{0}^{(1)} has slope equal to that of the canonical divisor K_{\overline{R}_{12,3}}. However, one can form the effective linear combination

\frac{31}{36}[\overline{Z}_{12,3}] + \frac{1}{36 · 7}[\overline{D}_{12,3}] = (13 − \frac{1}{12})λ − 2(δ0^{′} + δ0^{′′}) − 4δ_{0}^{(1)} = KR̃_{12,3} − \frac{1}{12}λ ∈ Eff(R̃_{12,3}),

thus showing that \overline{R}_{12,3} is of general type as well.

Section 5 is devoted to the highly surprising failure of the Prym-Green Conjecture for g = 8 and ℓ = 2. We give a probabilistic proof of the fact that for a general genus 8 Prym canonical curve ϕ_{K_{C}⊗η} : C ↪ P^{6}, the multiplication map

I_{2}(C, K_{C} ⊗ η) ⊗ H^{0}(C, K_{C} ⊗ η) → I_{3}(C, K_{C} ⊗ η),

has a non-trivial 1-dimensional kernel, corresponding to a syzygy of rank 6.

A curve ϕ_{L} : C ↪ P^{6} corresponding to a general element [C, L] of the universal Jacobian variety 𝔓𝔦𝔠_{8}^{14} → M_{8} is linked via five quadrics to a genus 14 curve with general moduli C′ ⊂ P^{6}, embedded such that K_{C′}(−1) ∈ W_{8}^{1}(C′) is a pencil of minimal degree.

The universal Koszul locus 𝔎𝔬𝔰𝔷 := {[C, L] ∈ 𝔓𝔦𝔠_{8}^{14} : K_{1,2}(C, L) ≠ 0} is a divisor that has at least two components 𝔎𝔬𝔰𝔷_{6} and 𝔎𝔬𝔰𝔷_{7}, distinguished by whether the extra syzygy has rank 6 or 7. The generic curve in 𝔎𝔬𝔰𝔷_{6} corresponds via linkage to a curve in the Petri divisor GP_{14,8}^{1} on M_{14}, while the generic curve in 𝔎𝔬𝔰𝔷_{7} links to a 7-gonal curve of genus 14 such that K_{C′}(−1) has a base point. Using a structure theorem for rank 6 syzygies in P^{6}, our findings show that R_{8,2} lies in the component 𝔎𝔬𝔰𝔷_{6}, that is, the extra syzygy of a Prym-canonical curve is never of maximal rank.

Structure of the paper. The first section is dedicated to the geometry of the stack of twisted level ℓ curves. In Section 2 we describe the syzygy formalism needed to formulate Conjectures A and B and then we prove Theorems 0.1 and 0.3. In Section 3 proofs of Theorems 0.5, 0.6 and 0.7 are provided, whereas in Section 4 we explain how to verify with Macaulay2 the syzygy conjectures formulated in the Introduction. The last section is devoted entirely to genus 8 and we explain the unexpected failure of the Prym-Green Conjecture on R_{8,2}.

Disclaimer. This paper is a collaborative effort marrying techniques ranging from computer algebra to stacks. The work of the second author is mainly reflected in sections 4 and 5, while the work of the first author is mainly reflected in sections 1–3.

Acknowledgments. We are grateful to Anand Patel for pointing out an error in an earlier version of this paper, to Burcin Eröcal and Florian Geiss for help with the computational aspects, and to Hans-Christian Graf von Bothmer and Alessandro Verra, who made suggestions that allowed us to improve the results of Section 5.

1. Level ℓ curves

This section contains background material concerning the moduli space \overline{R}_{g,ℓ} and complements the paper [CF]. We begin by setting terminology. For a Deligne-Mumford stack X be denote by X its associated coarse moduli space. The morphism π_{X} : X → X is universal with respect to morphisms from X to algebraic spaces. The coarsening of a morphism f : X → Y between Deligne-Mumford stacks is the induced morphism f : X → Y between coarse moduli spaces. For a Deligne-Mumford stack X we denote by Pic(X) the Picard group of the stack with rational coefficients.

We fix two integers g and ℓ ≥ 2, the genus and the level. Throughout §1.2 we assume that ℓ is prime, but we generalize all statements to any, possibly composite, level ℓ ≥ 2 in §1.3.

1.1. The geometric points of the moduli space of level ℓ curves. The geometric points of the moduli space of level ℓ curves can be interpreted in two relatively simple ways, as quasi-stable ℓth roots and twisted ℓth roots respectively. We are recall their definitions.

A quasi-stable curve is a nodal curve X such that (i) for each smooth rational component E ⊂ X the inequality k_{E} := |E ∩ \overline{X − E}| ≥ 2 holds, and (ii) if E, E′ are rational components with k_{E} = k_{E′} = 2, then E ∩E′ = ∅. Rational components of X meeting the rest of the curve in two points are called exceptional. If X is a quasi-stable curve, there exists a stabilization morphism st : X → C, obtained by collapsing all rational curves E ⊂ X with k_{E} = 2. Abusing terminology, we say that X is a blow-up of the curve C.

Definition 1.1. A quasi-stable level ℓ curve consists of a triple (X, η, ϕ), where X is a quasi-stable curve, η ∈ Pic^{0}(X) is a locally free sheaf of total degree 0 and ϕ : η^{⊗ℓ} → O_{X} is a sheaf homomorphism satisfying the following properties:

(i) η_{E} = O_{E}(1), for every exceptional component E ⊂ X; (ii) ϕ is an isomorphism along each non-exceptional component of X;

(iii) if E is an exceptional component and \{p, q\} := E ∩ \overline{X − E}, then
ord_{p}(ϕ) + ord_{q}(ϕ) = ℓ.

The last condition refers to the vanishing orders of the section 0 ≠ ϕ ∈ H^{0}(X, η^{⊗(−ℓ)}).

We recall that a balanced twisted curve is a Deligne-Mumford stack C whose coarse moduli space C is a stable curve, and such that locally at a node, the stack C comes endowed with an action of ℤ_{ℓ} of determinant 1 (this is equivalent to impose the condition that C be smoothable).

Definition 1.2. A twisted ℓth root (C → T, η, ϕ), is a balanced twisted curve C of genus g over a base T, a faithful line bundle η on C (i.e. a representable morphism η : C → Bℤ_{ℓ}) and an isomorphism ϕ : η^{⊗ℓ} → O_{C}. If η has order ℓ in Pic(C), then (C → T, η, ϕ) is a level ℓ curve.

There exist two (3g − 3)-dimensional projective schemes Root_{g,ℓ} (see [J1, CCC]) and \overline{M}_{g}(Bℤ_{ℓ}) (see [AV, ACV]) whose (geometric) points represent isomorphism classes of quasi-stable ℓth roots and twisted ℓth roots, respectively. The moduli space \overline{R}_{g,ℓ} of level ℓ curves arises as a connected component of \overline{M}_{g}(Bℤ_{ℓ}). As we illustrate in detail below, Root_{g,ℓ} and \overline{M}_{g}(Bℤ_{ℓ}) are not isomorphic unless ℓ = 2 or 3, but there exists a natural morphism nor : \overline{M}_{g}(Bℤ_{ℓ}) → Root_{g,ℓ}, which sets a bijection on the sets of geometric points, and may be regarded, scheme-theoretically, as a normalization morphism (for ℓ > 3 the singularities of Root_{g,ℓ} are not normal). The aim of §1.2 is to describe the twisted ℓth root (C → T, η, ϕ) and the quasi-stable ℓth root (X, η, ϕ) that correspond to each other under the map nor. In fact, following [C2], we lift this correspondence to a correspondence between a universal twisted ℓth root and a universal quasi-stable ℓth root both defined on the moduli stack of level ℓ curves. This correspondence allows us to study the enumerative geometry of the moduli of level curves both in schemetheoretic and stack-theoretic terms (see Remark 1.4).

1.2. The moduli stack of level ℓ curves (when ℓ is prime). We consider the categories of quasi-stable ℓth roots and of twisted ℓ roots. For the sake of clarity, in this section, we assume that ℓ is prime.

A family of quasi-stable ℓth roots consists of a triple (f, η, ϕ), where f : X → T is a flat family of quasi-stable curves, η is a line bundle on X and ϕ : η^{⊗ℓ} → O_{X} is a morphism of sheaves such that for each geometric point t ∈ T, the restriction

(X_{t} := f^{−1}(t), η_{t} := η_{|X_{t}}, ϕ_{|X_{t}} : η_{t}^{⊗ℓ} → O_{X_{t}})

is a quasi-stable ℓth root as defined above. The category of level ℓ curves gives rise to a proper Deligne-Mumford stack Root_{g,ℓ} with associated coarse moduli space Root_{g,ℓ}, see [CCC, J1]. Unfortunately, as already mentioned, this stack is singular as soon as ℓ > 3 (see (8)). To obtain a smooth stack whose coarse moduli space is the normalization of Root_{g,ℓ} one can use twisted ℓ roots [AV, ACV].

Indeed, the category \overline{M}_{g}(Bℤ_{ℓ}) of twisted ℓth roots forms a smooth and proper Deligne-Mumford stack, whose coarse moduli space is the normalization of Root_{g,ℓ}. The connected component parametrizing order-ℓ line bundles is the Deligne-Mumford stack \overline{R}_{g,ℓ}; the coarse space \overline{R}_{g,ℓ} is the (3g − 3)-dimensional projective variety studied in this paper. The following diagram is commutative

(7)  \xymatrix{\overline{M}_{g}(Bℤ_{ℓ}) \ar@{-->}[dr]_{f} \ar[rr]^{nor} &  & Root_{g,ℓ} \ar@{-->}[dl]^{h} \\  & _{\overline{M}_{g}}}

and endows \overline{M}_{g}(Bℤ_{ℓ}) with a universal twisted ℓth root and a universal quasi-stable ℓth root. The universal twisted ℓth root

(u_{g,ℓ}^{C} : C_{g,ℓ} → \overline{M}_{g}(Bℤ_{ℓ}), η_{g,ℓ}^{C} ∈ Pic(C_{g,ℓ}), ϕ_{g,ℓ}^{C} : (η_{g,ℓ}^{C})^{⊗ℓ} ⟶^{∼} O)

consists of a universal balanced twisted curve, a line bundle and an isomorphism. The universal quasi-stable level ℓ curve pulled back via nor^{∗}

(u_{g,ℓ}^{X} : X_{g,ℓ} → \overline{M}_{g}(Bℤ_{ℓ}), η_{g,ℓ}^{X} ∈ Pic(X_{g,ℓ}), ϕ_{g,ℓ}^{X} : (η_{g,ℓ}^{X})^{⊗ℓ} → O)

consists of a family of quasi-stable curves, endowed with a line bundle and a homomorphism of line bundles. For any morphisms T → \overline{M}_{g}(Bℤ_{ℓ}) from a scheme T, we consider the pullbacks (X_{g,ℓ})_{T} and (C_{g,ℓ})_{T}; the stabilization of the quasi-stable curve (X_{g,ℓ})_{T} and the coarsening of the balanced twisted curve (C_{g,ℓ})_{T} coincide and yield a representable morphism C_{g,ℓ} → \overline{M}_{g}(Bℤ_{ℓ}) which may be also regarded as the universal stable curve pulled back from \overline{M}_{g} via f. On C_{g,ℓ}, we have the following identity of sheaves (see [J2] and, in particular, (1-3) after Lemma 3.3.8, the proof of Theorem 3.3.9, and Figure 1; see also [C2, Lem. 2.2.5] for a more complete statement).

Proposition 1.3. Over \overline{M}_{g}(Bℤ_{ℓ}), consider the stabilization morphism st_{g,ℓ} : X_{g,ℓ} → C_{g,ℓ} and the coarsening morphism π_{g,ℓ} : C_{g,ℓ} → C_{g,ℓ}. On the universal stable curve C_{g,ℓ}, we have the following identity between coherent sheaves (π_{g,ℓ})_{∗}(η_{g,ℓ}^{C}) = (st_{g,ℓ})_{∗}(η_{g,ℓ}^{X}). □

Let us fix a closed point \overline{τ} := [C, η, ϕ] representing a twisted ℓth root and its image \overline{t} := [X, η, ϕ] representing a quasi-stable ℓth root. By the above proposition, the coarsening C of C equals the stabilization of X. Consider the exceptional components E^{1}, …, E^{k} ⊂ X and write \{p_{i}, q_{i}\} := E^{i} ∩ \overline{X − E^{i}} and a_{i} := ord_{p_{i}}(ϕ) and b_{i} := ord_{q_{i}}(ϕ), for i = 1, …, k. Then a_{i} + b_{i} = ℓ and we have gcd(a_{i}, b_{i}) = 1 (because ℓ is prime). The only nontrivial stabilizers of C occur at the nodes n_{1},…,n_{k} of C mapping to the nodes st(E_{1}), …, st(E_{1}) of C. The local picture at n_{i} is the spectrum of ℂ[x̃_{i}, ỹ_{i}]/(x̃_{i}ỹ_{i} = 0) and the local coordinates x_{i} and y_{i} of \overline{X − E^{i}} at p_{i} and at q_{i} are related to x̃_{i} and to ỹ_{i} by x_{i}^{ℓ} = x_{i} and y_{i}^{ℓ} = y_{i}. The action of ℤ_{ℓ} on the local picture Spec R[t] → Spec R of η → C at n_{i} is given by (x̃_{i}, ỹ_{i}) ↦ (ξ_{ℓ}x̃_{i}, ξ_{ℓ}^{−1}ỹ_{i}) and t_{i} ↦ ξ_{ℓ}^{a_{i}}t_{i}. Following [CF], we refer to the pair (a_{i}, b_{i}) as the multiplicity of the quasi-stable ℓ-root [X, η, ϕ] along the exceptional component E^{i} and, equivalently, of the twisted ℓth root at n_{i}.

We further describe the local picture of the universal quasi-stable ℓth root over Root_{g,ℓ} along one exceptional component E = E^{i} meeting the rest of X at p and q and having multiplicity (a, b) := (a_{i}, b_{i}). It is proved in [CCC, §3.1], that if one denotes by C_{a,b} ⊂ 𝔸_{w,z}^{2} the affine plane curve given by the equation w^{a} = z^{b} and the following surface by

(8) S_{a,b} := {((x, y, z, w), [s_{0} : s_{1}]) ∈ 𝔸_{x,y,z,w}^{4} × P^{1} : xs_{0} = ws_{1}, ys_{1} = zs_{0}, w^{a} = z^{b}},

then the covering f_{a,b} : S_{a,b} → C_{a,b} given by f_{a,b} : ((x, y, w, z), [s_{0} : s_{1}]) ↦ (w, z) is a local model for the simultaneous smoothing of the nodes p and q within the universal quasi-stable ℓth root over Root_{g,ℓ}. In particular, as soon as a, b > 1, the space C_{a,b} and hence the moduli space Root_{g,ℓ} are not normal. The normalization f̃_{a,b} : S̃_{a,b} → 𝔸_{τ}^{1} of f_{a,b} is constructed by setting

(9)  S̃_{a,b} := {((x, y, τ), [s_{0} : s_{1}]) ∈ 𝔸_{x,y,τ}^{3} × P^{1} : xs_{0} = τ^{b}s_{1}, ys_{1} = τ^{a}s_{0}},

and mapping 𝔸_{τ}^{1} → C_{a,b} via τ ↦ (τ^{b}, τ^{a}). At the level of the normalization, the point q ∈ X corresponds to the A_{a−1}-singularity ((0, 0, 0), [1 : 0]) ∈ S̃_{a,b}, whereas the A_{b−1}- singularity ((0, 0, 0), [0 : 1]) ∈ S̃_{a,b} corresponds to the point p ∈ X. Globalizing this description, one obtains a local picture of the diagram (7)at the points \overline{τ} and \overline{t}. There exist local coordinates such that

(10) Ô_{Root_{g,ℓ}, \overline{t}} = ℂ[[w_{1}, z_{1}, …, w_{k}, z_{k}, t_{k+1}, …, t_{3g−3}]]/(w_{i}^{a_{i}} = z_{i}^{b_{i}}), for i = 1, …, k,
(11) Ô_{\overline{M}_{g},st(X)} = ℂ[[t_{1}, …, t_{3g−3}]],
(12) Ô_{\overline{M}_{g}(Bℤ_{ℓ}), \overline{τ}} = ℂ[[τ_{1}, …, τ_{k}, t_{k+1}, …, t_{3g−3}]].

Locally, the morphisms f, nor and h are given by

f : t_{i} ↦ \begin{cases}τ_{i}^{ℓ} & i ≤ k \\ t_{i} & i > k \end{cases}  nor : \begin{cases}w_{i} ↦ τ_{i}^{b_{i}} & i ≤ k \\ z_{i} ↦ τ_{i}^{a_{i}} & i ≤ k \\ t_{i} ↦ t_{i} & i > k \end{cases}  h : t_{i} ↦ \begin{cases}w_{i}z_{i} & i ≤ k \\ t_{i} & i > k. \end{cases} (13)

We regard X as a fiber of a universal quasi-stable curve on \overline{M}_{g}(Bℤ_{ℓ}) over \overline{τ}. Set τ := τ_{i}, (a, b) := (a_{i}, b_{i}). Then, at p, we obtain an A_{b−1}-singularity of equation x(s_{0}/s_{1}) = τ^{b} along the locus where the node p persists. At q, we obtain an A_{a−1}-singularity of equation y(s_{1}/s_{0}) = τ^{a} along the locus where the node q persists. In view of intersection theory computations, following [C2], we desingularize such singularities. For any étale morphism T → \overline{M}_{g}(Bℤ_{ℓ}) the desingularization of (X_{g,ℓ})_{T} yields a semi-stable curve, globally on \overline{M}_{g}(Bℤ_{ℓ}),

X′_{g,ℓ} → X_{g,ℓ} → \overline{M}_{g}(Bℤ_{ℓ})

equipped with a line bundle P := η_{g,ℓ}^{X′} and a homomorphism Φ = ϕ_{g,ℓ}^{X′} from the ℓth tensor power of P to O: the pull-backs of η_{g,ℓ}^{X} and ϕ_{g,ℓ}^{X}. The curve X′ over \overline{τ} ∈ \overline{M}_{g}(Bℤ_{ℓ}) is obtained by iterated blow-ups; i.e. for a fibre of X_{g,ℓ} of the form C′ ∪ E with E exceptional and E ∪C′ = {p, q}. One inserts a chain of ℓ−a−1 rational curves E_{1}, …, E_{ℓ−a−1} at p and a chain of a − 1 rational curves E_{ℓ−a+1}, …, E_{ℓ−1} at q. Setting E_{ℓ−a} := E, we obtain the nodal semi-stable curve C′ ∪ E_{1} ∪ … ∪ E_{ℓ−1} of Figure 1, where C′ = \overline{(X − E)}.

By iterating this procedure at all exceptional curves we get X′, whose stable model is C = st(X), obtained by contracting the chains of the form E_{1} ∪ … ∪ E_{ℓ−1}: this explains that the singular locus C_{g,ℓ} → \overline{M}_{g}(Bℤ_{ℓ}) is formed by A_{ℓ−1}-singularities (since C_{g,ℓ} is the pullback via f of the universal stable curve of \overline{M}_{g}, this may be regarded as a consequence of the local description of f of (13)). The restriction of P = η_{g,ℓ}^{X′} to X′ is a line bundle η′ satisfying η′_{E_{i}} = O_{E_{i}} for i ≠ ℓ − a and η′_{E_{ℓ−a}} = O_{E_{ℓ−a}}(1).

Figure 1. The curve X′.

Remark 1.4. There is an isomorphism H^{0}(X′, ω_{X′} ⊗ η′) ≅ H^{0}(X, ω_{X} ⊗ η) and we often identify the two spaces. This may be regarded as a consequence of the fact that X′_{g,ℓ} → X_{g,ℓ} is crepant (the relative canonical bundles match under pullback) and of Proposition 1.3. In practice this means that any cohomological question involving kernel bundles on a twisted ℓth root can be settled by working, equivalently, over \overline{M}_{g}(Bℤ_{ℓ}), with the twisted ℓth root (C_{g,ℓ}, η_{g,ℓ}^{C}, ϕ_{g,ℓ}^{C}), with the quasi-stable level ℓ curve (X_{g,ℓ}, η_{g,ℓ}^{X}, ϕ_{g,ℓ}^{X}), or with the pullback P := η_{g,ℓ}^{X′} of η_{g,ℓ}^{X} on the semi-stable curve X′_{g,ℓ}.

1.3. The moduli stack of level ℓ curves (for any level ℓ). Let us assume only ℓ ∈ ℤ_{≥2}, without any condition on ℓ being prime. The local picture of Root_{g,ℓ} (10)is still valid; in particular the local model for the curve smoothing the nodes p and q of an exceptional component E is still S_{a,b} where (a, b) are the multiplicites of the quasistable ℓth root at the nodes. However, since a and b are no longer necessarily coprime, S_{a,b} has d := gcd(a, b) irreducible components, each one isomorphic to S_{\frac{a}{d} , \frac{b}{d}}. Then the local picture of f, nor and h of (7)is

f : t_{i} ↦ \begin{cases}τ_{i}^{ℓ/d_{i}} & i ≤ m \\ t_{i} & i > m \end{cases}  ñor : \begin{cases}w_{i} ↦ τ_{i}^{ℓ−a_{i}/d_{i}} & i ≤ k \\ z_{i} ↦ τ_{i}^{a_{i}/d_{i}} & i ≤ k \\ t_{i} ↦ t_{i} & i > k \end{cases}  h : t_{i} ↦ \begin{cases}w_{i}z_{i} & i ≤ k \\ t_{i} & i > k, \end{cases} (14)

for d_{i} = gcd(a_{i}, b_{i}). This happens because the stabilizers of a node of C attached to an exceptional component whose multiplicities satisfy d_{i} = gcd(a_{i}, b_{i}) have order ℓ/d_{i}. Notice that the normalization morphism nor induces a surjection at the level of closed points, but does not induce an injection as soon as ℓ is composite. Furthermore the desingularization of the pullback of the universal quasi-stable curve on \overline{M}_{g}(Bℤ_{ℓ}) is a semistable curve with chain of rational curves of length ℓ/d_{i} − 1 over the node corresponding to the the exceptional component E^{i}. Fortunately, there is a simple way to avoid these exceptions.

In [C1] the first author introduced a series of variants of smooth modifications M̃_{g}(Bℤ_{ℓ}) of \overline{M}_{g}(Bℤ_{ℓ}) allowing us to reproduce, even when ℓ is composite, the same configuration with chains of ℓ − 1 rational curves illustrated in Figure 1. The stack M̃_{g}(Bℤ_{ℓ}) parametrizes triples (C, η → C, ϕ : η^{⊗ℓ} → O), where the stabilizer G_{n} at a node n is trivial if and only if the restriction η_{|n} is trivial and, otherwise, satisfies G_{n} ≅ ℤ_{ℓ}. In this way, by removing the condition of faithfulness and by imposing stabilizers of order ℓ on n whenever η_{|n} is nontrivial, the local picture of C at each node n_{1}, …, n_{m} where η has nontrivial restriction is given by the spectrum of R = ℂ[x̃_{i}, ỹ_{i}]/(x̃_{i}ỹ_{i} = 0) with ℤ_{ℓ} operating as (x̃_{i}, ỹ_{i}) ↦ (ξ_{ℓ}x̃_{i}, ξ_{ℓ}^{−1}ỹ_{i}). There, the line bundle η → C has local picture Spec R[t_{i}] → Spec R with ℤ_{ℓ} operating as t_{i} ↦ ξ_{ℓ}^{a_{i}}t_{i} with a_{i} ∈ {1, …, ℓ − 1}, not necessarily prime to ℓ, for i = 1, …, m. As a consequence, there exist local coordinates at the closed point τ̃ ∈ M̃_{g}(Bℤ_{ℓ}) representing (C, η, ϕ)

Ô_{M̃_{g}(Bℤ_{ℓ}), τ̃} = ℂ[[τ̃_{1}, …, τ̃_{m}, t_{m+1}, …, t_{3g−3}]].

and a morphism e : M̃_{g}(Bℤ_{ℓ}) → \overline{M}_{g}(Bℤ_{ℓ}) fitting in

(15)  \xymatrix{M̃_{g}(Bℤ_{ℓ}) \ar[r]^{e} \ar[drr]_{f̃} & \overline{M}_{g}(Bℤ_{ℓ}) \ar[dr]^{f} \ar[rr]^{nor} &  & Root_{g,ℓ} \ar[dl]^{h} \\  &  & _{\overline{M}_{g}}}

given (at the level of local rings) by e : τ_{i} ↦ τ_{i}^{d_{i}} for i ≤ m and by f̃ : t_{i} ↦ τ_{i}^{ℓ} for i ≤ m.

The morphism e yields an isomorphism at the level of coarse spaces. Therefore, as in [C2], when ℓ is not prime, we can work throughout the rest of the paper with the substack \overline{R}_{g,ℓ} arising as the connected component of M̃_{g}(Bℤ_{ℓ}) of triples (C, η, ϕ) where η has order ℓ. This is a smooth Deligne-Mumford stack whose coarse space is a projective (3g − 3)-dimensional variety \overline{R}_{g,ℓ}. With a slight abuse of notation, let us denote by X_{g,ℓ} the pullback of the universal quasi-stable curve on \overline{R}_{g,ℓ} via nor ∘ e. By construction, the singularities arising at the nodes p_{i} and q_{i} of an exceptional curve E_{i} are of type A_{ℓ−a_{i}−1} and A_{a_{i}−1} and the desingularization X′_{g,ℓ} is fibred in semi-stable curves obtained by inserting chains of ℓ − a_{i} − 1 rational curves at p_{i} and chains of a_{i} − 1 rational curves at q_{i} as in Figure 1. Remark 1.4 generalizes word for word. From now on we shall work with \overline{R}_{g,ℓ} ⊂ M̃_{g}(Bℤ_{ℓ}) equipped with the universal semistable curve X′_{g,ℓ} and with the universal line bundle P := η_{g,ℓ}^{X′} pulled back from Root_{g,ℓ}.

1.4. The boundary divisors of \overline{R}_{g,ℓ}. We briefly discuss the geometry of the boundary divisors of \overline{R}_{g,ℓ} by describing the structure of the fibre f^{−1}([C]) corresponding to a general point of each boundary of the boundary divisors Δ_{0}, …, Δ_{⌊ \frac{g}{2} ⌋} of \overline{M}_{g}. Let us assume first that C := C_{1} ∪_{p} C_{2} is a transverse union of two smooth curves C_{1} and C_{2} of genus i and g − i respectively. If [X, η, ϕ] ∈ f^{−1}([C]), then necessarily X = C and η is uniquely determined by the data of two line bundles η_{C_{1}} ∈ Pic^{0}(C_{1})[ℓ] and η_{C_{2}} ∈ Pic^{0}(C_{2})[ℓ]. Depending on which of these line bundles is trivial, we define the boundary divisors Δ_{i}, Δ_{g−i} and Δ_{i:g−i} respectively. For 1 ≤ i ≤ g − 1, the general point of Δ_{i} corresponds to a level curve of compact type

[C_{1} ∪_{p} C_{2}, η_{C_{1}} of order ℓ , η_{C_{2}} ≅ O_{C_{2}}] ∈ \overline{R}_{g,ℓ}.

Finally, we denote by Δ_{i:g−i} the closure in \overline{R}_{g,ℓ} of the locus of twisted level curves on C_{1} ∪ C_{2} such that η_{C_{1}} ≅ O̸_{C_{1}} and η_{C_{2}} ≅ O̸_{C_{2}}. Denoting by δ_{i} := [Δ_{i}]_{ℚ}, δ_{g−i} := [Δ_{g−i}]_{ℚ}, δ_{i:g−i} := [Δ_{i:g−i}]_{ℚ} the corresponding classes in Pic(\overline{R}_{g,ℓ}), we have the following relation, showing that the morphism of stacks f : \overline{R}_{g,ℓ} → \overline{M}_{g} is étale over Δ_{i} where i ≥ 1.

(16)  f^{∗}(δ_{i}) = δ_{i} + δ_{g−i} + δ_{i:g−i}.

For i = 1 and ℓ ≥ 3, observe that at the level of coarse moduli spaces the formula f^{∗}(Δ_{1}) = 2(Δ_{1} + Δ_{1:g−1}) + Δ_{g−1} holds, in particular f is ramified along Δ_{1} and Δ_{1:g−1}.

Suppose now that [C] ∈ Δ_{0} is a general irreducible 1-nodal curve of genus g with normalization nor : C′ → C, and let p, q ∈ C′ be such that nor(p) = nor(q) ∈ Sing(C). Assume that [X, η, ϕ] ∈ π^{−1}([C]). We write down the exact sequence

(17)  1 ⟶ ℤ_{ℓ} ⟶ Pic^{0}(C)[ℓ] ⟶^{nor^{∗}} Pic^{0}(C′)[ℓ] ⟶ 0.

If X = C, we set η_{C′} := nor^{∗}(η) ∈ Pic^{0}(C′)[ℓ]. We denote by Δ0^{′} the closure in \overline{R}_{g,ℓ} of the locus of level curves [C, η, ϕ] as above, where η_{C′} ≅ O̸_{C′}. An ℓ-torsion line bundle on C is determined by the choice of η_{C′} and the choice of a ℤ_{ℓ}-gluing of the fibres η_{C′}(p) and η_{C′}(q). We observe that, when ℓ is prime, deg(Δ0^{′}/Δ_{0}) = ℓ(ℓ^{2g−2} − 1) (see [CF] for the general statement).

We denote by Δ0^{′′} the closure in \overline{R}_{g,ℓ} of the locus of level curves [C, η, ϕ] such that η_{C′} ≅ O_{C′} (we referred to these curves as Wirtinger covers in the introduction; these arise as the preimage of a constant section via the morphism η → O between total spaces). Using (17), an order-ℓ line bundle η ∈ Pic^{0}(C)[ℓ] with nor^{∗}(η) = O_{C′} is determined by a ℤ_{ℓ}^{∗}-gluing of the fibres η_{C′}(p) and η_{C′}(q), that is, a root of unity ξ_{ℓ}^{a} ∈ ℤ_{ℓ}^{∗}, with 1 ≤ a ≤ ℓ − 1 prime to ℓ, such that sections σ ∈ η_{C′} that descend to C are characterized by the equation σ(p) = ξ_{ℓ}^{a}σ(q). Since by reversing the role of p and q we interchange a and ℓ − a, we obtain the a decomposition into irreducible ⌊l/2⌋ components, each of them of order 2 over Δ_{0}. For any ℓ ∈ ℤ_{≥2} we get deg(Δ0^{′′}/Δ_{0}) = ℓ − 1.

We now consider the case X = C′ ∪_{{p,q}} E, where E is an exceptional component. Then by definition η_{E} = O_{E}(1), therefore deg(η_{C′}) = −1. Furthermore, there exists an integer 1 ≤ a ≤ ℓ − 1 such that η_{C′}^{⊗(−ℓ)} = O_{C′}(a · p + (ℓ − a) · q). Let us denote by Δ_{0}^{(a)} the closure in \overline{R}_{g,ℓ} of the locus of such points. By switching the role of p and q we can obviously restrict ourselves to the case 1 ≤ a ≤ ⌊\frac{ℓ}{2}⌋. Since the choice of η_{C′} ∈ Pic^{−1}(C′) as above uniquely determines the level curve [C′ ∪ E, η, ϕ], it follows that, when ℓ is prime, deg(Δ_{0}^{(a)}/Δ_{0}) = 2ℓ^{2g−2} (the factor 2 accounts for the possibility of interchanging p and q). We refer to [CF] for the treatment of the general case ℓ ∈ ℤ_{≥2} and the decomposition into irreducible components.

We follow again the convention of passing to lower case symbols for the divisor classes in the moduli stack \overline{R}_{g,ℓ}: δ′_{0} = [Δ′_{0}]_{ℚ}, δ′′_{0} = [Δ′′_{0}]_{ℚ} and δ_{0}^{(a)} = [Δ_{0}^{(a)}]_{ℚ}. As a direct consequence of the local description (13)the morphism f is étale over Δ′_{0} and Δ′′_{0} and ramified with order ℓ at Δ_{0}^{(a)}. (When ℓ is not prime the order of ramification at Δ_{0}^{(a)} is still ℓ because \overline{R}_{g,ℓ} as been defined as a substack of the covering M̃_{g}(Bℤ_{ℓ}) of \overline{M}_{g}(Bℤ_{ℓ})). Summarizing we have the following relation in Pic(\overline{R}_{g,ℓ})

f^{∗}(δ_{0}) = δ0^{′} + δ0^{′′} + ℓ ∑_{a=1}^{⌊ \frac{ℓ}{2} ⌋} δ_{0}^{(a)}.

Proposition 1.5. For ℓ ≥ 3, the canonical class of the coarse moduli space \overline{R}_{g,ℓ} is equal to

K_{\overline{R}_{g,ℓ}} = 13λ − 2(δ0^{′} + δ0^{′′}) − (ℓ + 1) ∑_{a=1}^{⌊ \frac{ℓ}{2} ⌋} δ_{0}^{(a)} − 2 ∑_{i=1}^{⌊ \frac{g}{2} ⌋}(δ_{i} + δ_{g−i} + δ_{i:g−i}) − δ_{g−1} ∈ Pic(\overline{R}_{g,ℓ}).

Proof. It follows from the Hurwitz formula K_{\overline{R}_{g,ℓ}} = f^{∗}K_{\overline{M}_{g}}+δ_{1}+δ_{1:g−1}+(ℓ−1) ∑_{a=1}^{⌊ \frac{ℓ}{2} ⌋} δ_{0}^{(a)}, coupled with the expression for K_{\overline{M}_{g}}, cf. [HM] Theorem 2 and formula (16). □

1.5. The geometry of the universal semi-stable level ℓ curve. Consider the universal semi-stable curve u : X′_{g,ℓ} → \overline{R}_{g,ℓ} introduced in §1.2, endowed with the tautological bundle P := η_{g,ℓ}^{X′} and the homomorphism of line bundles

Φ : P^{⊗ℓ} → O_{X′_{g,ℓ}}.

For sake of clarity and in preparation of the enumerative geometry studied in this paper, we detail the proof of a preliminary result on the self-intersection of c_{1}(P). First, let us fix an integer 1 ≤ a ≤ ⌊\frac{ℓ}{2}⌋ and observe that the the pull-back of the boundary divisor Δ_{0}^{(a)} in \overline{R}_{g,ℓ} splits into irreducible components

u^{∗}(Δ_{0}^{(a)}) = E_{1}^{(a)} + … + E_{ℓ−1}^{(a)},

where E_{i}^{(a)} parametrizes the closure of the locus of points that lie on the ith rational component of the chain E_{1} ∪ ⋯ ∪ E_{ℓ−1} of ℓ − 1 rational lines over a general point of Δ_{0}^{(a)}.

Proposition 1.6. The following relations hold in Pic(\overline{R}_{g,ℓ}):

(1) u_{∗}([E_{i}^{(a)}] · [E_{i+1}^{(a)}]) = δ_{0}^{(a)} and u_{∗}([E_{i}^{(a)}]^{2}) = −2δ_{0}^{(a)}, for each 1 ≤ i ≤ ℓ − 1.

(2) u_{∗}(c_{1}(P) · c_{1}(ω_{u})) = 0.

(3) u_{∗}(c_{1}^{2}(P)) = − ∑_{1≤a≤⌊ \frac{ℓ}{2} ⌋} \frac{a(ℓ−a)}{ℓ}δ_{0}^{(a)}.

Proof. The first two statements being immediate, we proceed to the last one (see also [C2, Lem. 3.1.4]). For each 1 ≤ a ≤ ⌊\frac{ℓ}{2}⌋, we fix a general quasi-stable ℓth root (X^{a}, η, ϕ) over Δ_{0}^{(a)}. Here X^{a} := C′∪_{{p,q}}E is a quasi-stable curve and η_{C′}^{⊗ℓ} = O_{C′}(−a·p−(ℓ−a)·q). The corresponding semi-stable fibre of X′_{g,ℓ} mapping to X^{a} is a semi-stable curve X′ obtained by setting E_{ℓ−a}^{(a)} := E, and gluing to the points p, q ∈ C smooth rational curves E_{1}^{(a)}, …, E_{ℓ−a−1}^{(a)}, E_{ℓ−a}^{(a)}, E_{ℓ−a+1}^{(a)}, …, E_{ℓ−1}^{(a)} forming a chain. The line bundle η′ on X′ is the restriction P_{|X′} isomorphic to η_{C′} on C′, to O on E_{i}^{(a)} for i ≠ ℓ − a, and to O(1) on E_{ℓ−a}^{(a)}. By construction we have that X′ ∩ E_{i}^{(a)} = E_{i}^{(a)}.

We claim that the vanishing locus of the section Φ ∈ H^{0}(X′_{g,ℓ}, P^{⊗(−ℓ)}) is precisely the divisor

(18)  ∑_{a=1}^{⌊ \frac{ℓ}{2} ⌋}(a ∑_{i=1}^{ℓ−a} i E_{i}^{(a)} + (ℓ − a) ∑_{i=1}^{a−1}(a − i)E_{ℓ−a+i}^{(a)}).

Indeed, we already know that D := supp(Z(Φ)) can be expressed as a linear combination of exceptional divisors ∑_{a,i} c_{i}^{(a)}E_{i}^{(a)}, where 1 ≤ i ≤ ℓ − 1 and 1 ≤ a ≤ ⌊\frac{ℓ}{2}⌋, which is furthermore characterized by two sets of conditions:

(i) For each 1 ≤ a ≤ ⌊\frac{ℓ}{2}⌋, one has that c_{1}^{(a)} = a and c_{ℓ−1}^{(a)} = ℓ − a.
(ii) For each i ≠ ℓ − a, one has deg(D_{|E_{i}^{(a)}}) = 0.

The first condition expressed the fact that the vanishing order of Φ_{|C} at p (respectively q) is equal to a (respectively ℓ − a). The second condition expresses the fact that the restriction of Φ on E_{i}^{(a)} is the trivial morphism along each exceptional divisor E_{i}^{(a)} with i ≠ ℓ − a, inserted when passing from X_{g,ℓ} to X′_{g,ℓ}. Taking into account that O(E_{i}^{(a)}) =

O(−2) on E_{i}^{(a)}, the two conditions lead to a linear system of equation in the coefficients c_{i}^{(a)}, which we then solve to obtain the claimed formula (18), which also computed −ℓc_{1}(P). Squaring this formula, pushing it forward while using formulas (1), (2), we obtain

ℓ^{2}u_{∗}(c_{1}^{2}(P)) = −2(∑_{i=1}^{ℓ−a} a^{2}i^{2} + ∑_{i=1}^{a−1}(ℓ − a)^{2}i^{2} + ∑_{i=1}^{ℓ−a−1} a^{2}i(i + 1) + ∑_{i=1}^{a−1}(ℓ − a)^{2}i(i + 1))δ_{0}^{(a)},

which after manipulations leads to the claimed formula. □

2. Syzygy jumping loci over \overline{R}_{g,ℓ}

Following a principle already explained in [F1], one can construct tautological divisors on moduli spaces of curves defined in terms of syzygies of the parametrized objects. Such loci have a determinantal structure over an open subset of R_{g,ℓ}, hence one can speak of their expected (co)dimension. When the expected codimension is 1, one has virtual divisors on R_{g,ℓ}.

We begin by setting notation. For a smooth curve C, a line bundle L and a sheaf F on C, we define the Koszul cohomology group K_{p,q}(C; F, L) as the cohomology of the complex

⋀^{p+1} H^{0}(C, L) ⊗ H^{0}(C, F ⊗ L^{⊗(q−1)}) ⟶^{d_{p+1,q−1}} ⋀^{p} H^{0}(C, L) ⊗ H^{0}(C, F ⊗ L^{⊗q}) ⟶^{d_{p,q}}

⟶^{d_{p,q}} ⋀^{p−1} H^{0}(C, L) ⊗ H^{0}(C, F ⊗ L^{⊗(q+1)}).

When F = O_{C}, one writes K_{p,q}(C, L) := K_{p,q}(C; O_{C}, L). When F is a line bundle, these Koszul cohomology groups can be interpreted in terms of syzygies of certain 0dimensional subschemes of C.

Example 2.1. Suppose [C, η] ∈ R_{g,ℓ} and η ∉ C_{2} − C_{2}. Let L := K_{C} ⊗ η ∈ Pic^{2g−2}(C) be the very ample paracanonical line bundle inducing an embedding ϕ_{L} : C → P^{g−2}. For an integer 1 ≤ k ≤ ℓ − 1, since H^{0}(C, η^{⊗k}) = 0 one has that

K_{p,1}(C; η^{⊗k}, L) = Ker{⋀^{p} H^{0}(L) ⊗ H^{0}(L ⊗ η^{⊗k}) → ⋀^{p−1} H^{0}(L) ⊗ H^{0}(L^{⊗2} ⊗ η^{⊗k})}.

This group can be viewed as parametrizing the syzygies of a 0-dimensional scheme Z ⊂ C ⊂ P^{g−2}, where Z ∈ |K_{C} ⊗ η^{⊗(1−k)}|, see [FMP] Proposition 1.6. Precisely,

(19)  b_{p+1,1}(Z) = dim K_{p,1}(C; η^{⊗k}, L).

We explain Conjecture B and Theorems 0.1 and 0.3 formulated in the introduction. We fix line bundles η, ξ ∈ Pic^{0}(C), with ξ ≠ O_{C} and η ⊗ ξ ≠ O_{C}. To the paracanonical linear series L := K_{C} ⊗ η we associate the kernel bundle defined via the exact sequence

0 ⟶ M_{L} ⟶ H^{0}(C, L) ⊗ O_{C} ⟶ L ⟶ 0.

Using standard arguments, see e.g. [FMP] Proposition 1.6, one has the identifications

(20)  K_{i,1}(C; ξ, K_{C} ⊗ η) = H^{0}(C, ⋀^{i} M_{L} ⊗ L ⊗ ξ),

K_{i−1,2}(C; ξ, K_{C} ⊗ η) = H^{1}(C, ⋀^{i} M_{L} ⊗ L ⊗ ξ).

The difference of the dimensions of the two cohomology groups is the Euler-Poincaré characteristic of a vector bundle on C, which in the case η ≠ O_{C} is equal to

(21) dim K_{i,1}(C; ξ, K_{C} ⊗ η) − dim K_{i−1,2}(C; ξ, K_{C} ⊗ η) = (g − 1)\binom{g − 2}{i}(1 − \frac{2i}{g − 2}).

The naturality of the resolution of Γ_{C}(ξ, L) as a Sym H^{0}(C, L)-module is equivalent to the vanishing for all i of one of the groups K_{i,1}(C; ξ, L) or K_{i−1,2}(C; ξ, L), depending on the sign computed by (21). The resolution being minimal, it suffices to verify this for the values of i when the expression (21), viewed as a function of i, changes sign. The resolution is pure, if the corresponding cohomology group vanishes for i such that the expression in (21) is equal to zero.

Lemma 2.2. Let C be a curve of genus g and η, ξ ∈ Pic^{0}(C) − {O_{C}} such that η ⊗ ξ ≠ O_{C}.

(1) The resolution of Γ_{C}(ξ, K_{C} ⊗ η) as a SymH^{0}(C, K_{C} ⊗ η)-module is natural if and only if

K_{⌊ \frac{g−1}{2} ⌋,1}(C; ξ, K_{C} ⊗ η) = 0 and K_{⌊ \frac{g−1}{2} ⌋,1}(C; (η ⊗ ξ)^{∨}, K_{C} ⊗ η) = 0.

(2) The resolution of Γ_{C}(ξ, K_{C}) as a SymH^{0}(C, K_{C})-module is natural if and only if

K_{⌊ \frac{g}{2} ⌋,1}(C; ξ, K_{C}) = 0 and K_{⌊ \frac{g}{2} ⌋,1}(C; ξ^{∨}, K_{C}) = 0.

Proof. Assume that g is odd and write g = 2i+1. From (21), the resolution of the module Γ_{C}(ξ, K_{C} ⊗ η) is natural if and only if K_{i,1}(C; ξ, K_{C} ⊗ η) = K_{i−2,2}(C; ξ, K_{C} ⊗ η) = 0. By the duality theorem [G], the dual of the last group is canonically isomorphic to

K_{i,0}(C; K_{C} ⊗ ξ^{∨}, K_{C} ⊗ η) = K_{i,1}(C; (η ⊗ ξ)^{∨}, K_{C} ⊗ η).

The case of the resolution of Γ_{C}(ξ, K_{C}) (that is, η = O_{C}) is similar and we skip details. □

We now describe the structure of the locus U_{g,ℓ} mentioned in Theorem 0.3. For g = 2i + 1, a result from [FMP] provides an identification of cycles valid for each curve:

{η ∈ Pic^{0}(C) : h^{0}(C, ⋀^{i} M_{K_{C}} ⊗ K_{C} ⊗ η) ≥ 1} = C_{i} − C_{i} ⊂ Pic^{0}(C).

The right hand side denotes the i-th difference variety consisting of line bundles of the form O_{C}(D − E), where D, E ∈ C_{i}. This establishes the following equivalence

K_{i,1}(C; η, K_{C}) ≠ 0 ⇔ η ∈ C_{i} − C_{i}.

After tensoring with K_{C} ⊗ η and taking cohomology in the short exact sequence

0 ⟶ ⋀^{i} M_{K_{C}} ⟶ ⋀^{i} H^{0}(C, K_{C}) ⊗ O_{C} ⟶ ⋀^{i−1} M_{K_{C}} ⊗ K_{C} ⟶ 0,

we reformulate the last condition as follows: [C, η] ∈ U_{g,ℓ} if and only if the map

(22)  χ([C, η]) : ⋀^{i} H^{0}(K_{C}) ⊗ H^{0}(K_{C} ⊗ η) → H^{0}(C, ⋀^{i−1} M_{K_{C}} ⊗ K_{C}^{⊗2} ⊗ η)

is an isomorphism. Since h^{0}(⋀^{i−1} M_{K_{C}} ⊗ K_{C}^{⊗2} ⊗ η) = χ(⋀^{i−1} M_{K_{C}} ⊗ K_{C}^{⊗2} ⊗ η) = \binom{2i}{i−1}(4i + 2), note that the locus U_{g,ℓ} can be defined as the degeneracy locus of a morphism between vector bundles of the same rank over R_{g,ℓ}, whose fibre over a point [C, η] is precisely the map χ([C, η] defined above. Thus U_{g,ℓ} is a virtual divisor over R_{g,ℓ}.

We show that that χ([C, η]) is an isomorphism for a general level curve, thus establishing Theorem 0.3. In fact, we prove a more precise result valid for all genera:

Theorem 2.3. Let [C, p] ∈ M_{g,1} be a general hyperelliptic curve of genus g ≥ 2 with a Weierstrass point and η ∈ Pic^{0}(C)[ℓ] − {O_{C}}. One has that H^{0}(C, η((g − 1)p)) = 0.

First we explain how Theorem 2.3 implies Theorem 0.3. Let C be hyperelliptic, A = O_{C}(2p) the hyperelliptic line bundle and i := ⌊\frac{g}{2}⌋. Then M_{K_{C}}^{∨} = A^{⊕(g−1)}, therefore

⋀^{i} M_{K_{C}}^{∨} = (A^{⊗i})^{⊕\binom{g−1}{i}}.

It follows from Lemma (2.2) that K_{i,1}(C; η, K_{C}) = 0 if and only if H^{0}(C, η((g −1)p)) = 0 when g is odd, respectively H^{0}(C, η((g − 2)p)) = 0, when g is even. Clearly, the statement of Theorem 2.3 implies the naturality of the resolution Γ_{C}(η, K_{C}) for all genera.

Proof of Theorem 2.3. Inductively we assume that [C′, η′] ∈ R_{g−1,ℓ} is a hyperelliptic level ℓ curve with a point p′ ∈ C′ such that h^{0}(C′, O_{C′}(2p′)) = 2 and H^{0}(C′, η′((g − 2)p′))= 0. We consider a pointed elliptic curve [E, p′] ∈ M_{1,1}, which we attach to C′ at the point p′, then choose p ∈ E − \{p′\} such that 2(p − p′) ≡ 0. Then [C := C ∪_{p′} E, p] ∈ \overline{M}_{g,1} is a degenerate hyperelliptic curve and p ∈ C is a hyperelliptic Weierstrass point. This follows by exhibiting an admissible double covering

φ : C → (P^{1})_{1} ∪_{t} (P^{1})_{2},

where φ_{C′} : C′ → (P^{1})_{1} is the hyperelliptic cover and φ_{E} : E → (P^{1})_{2} is the double cover ramified at p and p′. Note that φ(p′) = t ∈ (P^{1})_{1} ∩ (P^{1})_{2}.

We choose an ℓ-torsion point η_{E} ∈ Pic^{0}(E)[ℓ] such that η_{E} ≠ O_{C}((g − 1)(p′ − p)). Such a choice is possible since there at least two points of order ℓ inside Pic^{0}(E). Then

[C′ ∪_{p′} E, η_{C′}, η_{E}] ∈ \overline{R}_{g,ℓ},

and we claim that this curve does not lie in the closure of the locus of hyperelliptic level ℓ curves [X_{t}, η_{t}] ∈ R_{g,ℓ} with a point p_{t} ∈ X_{t} such that h^{0}(X_{t}, O_{X_{t}}(2p_{t})) = 2 and h^{0}(X_{t}, η_{t}((g − 1)p_{t})) ≥ 1. Indeed, assuming this not to be true, applying a limit linear series argument we find non-zero sections

σ_{C′} ∈ H^{0}(C′, η_{C′} ⊗ O_{C′}((g − 1)p′)) and σ_{E} ∈ H^{0}(E, η_{E} ⊗ O_{E}((g − 1)p)),

such that ord_{p′}(σ_{C′}) + ord_{p′}(σ_{E}) ≥ g − 1. Our assumption implies ord_{p′}(σ_{E}) ≤ g − 2, therefore ord_{p′}(σ_{C′}) ≥ 1, that is, H^{0}(C′, η_{C′} ⊗O_{C′}((g −2)p′)) ≠ 0. This is a contradiction and completes the proof. □

2.1. The syzygies of the twisted paracanonical module. We now show that for a general point [C, η] ∈ R_{g,ℓ}, the module Γ_{C}(ξ, K_{C} ⊗ η) has a pure resolution for a general choice of ξ ∈ Pic^{0}(C). The argument relies on specialization to bielliptic curves.

Suppose E is an elliptic curve, C a curve of genus g and f : C → E a double cover ramified along the divisor R ∈ C_{2g−2} and branched along the divisor B ∈ E_{2g−2}. Let δ ∈ Pic^{g−1}(E) denote the line bundle determining f, that is, δ^{⊗2} = O_{E}(B). In particular,

f_{∗}O_{C} = O_{E} ⊕ δ^{∨}, f^{∗}(δ) = O_{C}(R).

Proposition 2.4. Let ϵ ∈ Pic^{0}(E)[ℓ] an ℓ-torsion point and η := f^{∗}(ϵ) ∈ Pic^{0}(C)[ℓ]. Then

M_{K_{C}⊗η} = f^{∗}M_{δ⊗ϵ}.

Proof. From adjunction K_{C} = f^{∗}(δ), therefore from the push-pull formula

H^{0}(C, K_{C} ⊗ η) = f^{∗}H^{0}(E, δ ⊗ ϵ) ⊕ f^{∗}H^{0}(E, ϵ) ≅ H^{0}(E, δ ⊗ ϵ).

In particular, pulling-back via f the exact sequence defining the kernel bundle on E

0 ⟶ M_{δ⊗ϵ} ⟶ H^{0}(E, δ ⊗ ϵ) ⟶ δ ⊗ ϵ ⟶ 0,

we retrieve the sequence defining the kernel bundle on C, that is, M_{K_{C}⊗η} = f^{∗}(M_{δ⊗ϵ}). □

Lemma 2.5. The push-forward f_{∗}ξ of a general line bundle ξ ∈ Pic^{0}(C) is stable (respectively semistable) if g is even (respectively odd).

Proof. We treat only the case of even genus g = 2i + 2, the odd genus case being similar. The push-forward f_{∗}ξ is a rank two vector bundle on E with det f_{∗}(ξ) = Nm_{f}(ξ) ⊗ δ^{∨}, in particular deg(f_{∗}ξ) = 1 − g. Assume that f_{∗}ξ is not semistable. Then there exists a line subbundle M ↪ f_{∗}ξ with deg(M) ≥ deg(f_{∗}ξ)/2, that is, deg(M) ≥ −i. Then H^{0}(C, ξ ⊗ f^{∗}M^{∨}) ≠ 0, hence we can write ξ = f^{∗}(M)(D), where D is an effective divisor on C, with deg(D) ≤ 2i. Counting parameters, line bundles on C having this type depend on at most 2i+1 = g −1 parameters, hence they do not fill-up Pic^{0}(C). □

Thus one obtains a rational map f_{∗} : Pic^{0}(C) −−→ U_{E}(2, 1 − g). By describing the differential of this map, it is easy to show that this map is dominant, see also [B].

We complete the proof of Theorem 0.1. More precisely we prove the following:

Theorem 2.6. Let f : C → E be a bielliptic curve and η ∈ Pic^{0}(C)[ℓ] as above. Then

K_{⌊ \frac{g−1}{2} ⌋,1}(C; ξ, K_{C} ⊗ η) = 0,

for a general ξ ∈ Pic^{0}(C). In particular, the resolution of Γ_{C}(ξ, K_{C} ⊗ η) is natural.

Proof. We treat only the case g = 2i + 2, the odd genus case being quite similar. Using identification (20) we write

K_{i,1}(C; ξ, K_{C} ⊗ η) = H^{0}(C, ⋀^{i} M_{K_{C}⊗η} ⊗ K_{C} ⊗ η ⊗ ξ) = H^{0}(E, ⋀^{i} M_{δ⊗ϵ} ⊗ δ ⊗ ϵ ⊗ f_{∗}ξ).

It is well-known that the vector bundle M_{δ⊗ϵ} is stable, hence using also Lemma 2.5, the vector bundle F := ⋀^{i} M_{δ⊗ϵ} ⊗ ϵ ⊗ δ ⊗ f_{∗}ξ is semistable. Futhermore, the choice of a general ξ ∈ Pic^{0}(C) corresponds to the choice of a general F ∈ U_{E}(\binom{g−2}{i}, 0). Since on an elliptic curve, there exists precisely one semistable bundle of prescribed rank, degree zero and with a section, we find that H^{0}(E, F) = 0, which finishes the proof. □

Theorem 2.6 admits the following reformulation in the case of even genus.

Corollary 2.7. For a general curve [C, η] ∈ R_{g,ℓ} of even genus, the vector bundle ⋀^{\frac{g−2}{2}} M_{K_{C}⊗η} admits a theta divisor.

Since μ(⋀^{\frac{g−2}{2}} M_{K_{C}⊗η}) = g − 1 ∈ ℤ it makes sense to ask whether the vector bundle in question admits a theta divisor. Conjecture B is a more refined statement, predicting which line bundles K_{C} ⊗ η^{⊗k} belong to the theta divisor of ⋀^{\frac{g−2}{2}} M_{K_{C}⊗η}. Note that it is proved in [FMP] that all powers ⋀^{i} M_{K_{C}} admit a theta divisor for every smooth curve C.

3. Intersection theory on \overline{R}_{g,ℓ}

The aim of this section is to describe the characteristic classes of tautological bundles on \overline{R}_{g,ℓ} that are used to calculate the classes [\overline{Z}_{g,ℓ}]^{virt}, [\overline{D}_{g,ℓ}]^{virt} and [\overline{U}_{g,ℓ}] respectively. For our purposes it will suffice to consider only level ℓ curves whose underlying stable model is irreducible. Let R̃_{g,ℓ} be the open substack of level ℓ curves whose underlying stable model is a 1-nodal irreducible curves of arithmetic genus g. We denote by u : X̃_{g,ℓ} → R̃_{g,ℓ} the restriction of the universal level ℓ curve, by P ∈ Pic(X̃_{g,ℓ}) the tautological ℓth root bundle and by Φ : P^{⊗ℓ} → OX̃_{g,ℓ} the universal sheaf homomorphism. We shall use the Hodge bundle 𝔼 := u_{∗}(ω_{u}) respectively the Prym-Hodge bundle 𝔼′ := u_{∗}(ω_{u} ⊗ P). These are locally free sheaves over R̃_{g,ℓ} of ranks g and g − 1 respectively.

The following technical statement will be used to show that various tautological sheaves on R̃_{g,ℓ} are locally free.

Proposition 3.1. For a level curve [X, η, ϕ] ∈ R̃_{g,ℓ} and integers b ≥ 2 and 0 ≤ j ≤ \frac{g}{2}, the following vanishing statements hold:

(i) H^{1}(X, ⋀^{j} M_{ω_{X}} ⊗ ω_{X}^{⊗b} ⊗ η) = 0.
(ii) H^{1}(X, ⋀^{j} M_{ω_{X}⊗η} ⊗ ω_{X}^{⊗b} ⊗ η^{⊗(b−2)}) = 0.

Proof. As pointed out in Section 1, such a question can be studied at the level of root curves. To ease notation we identify nor([X, η, ϕ]) ∈ Root_{g,ℓ} and [X, η, ϕ]. If X is smooth, since μ(∧^{j}M_{ω_{X}} ⊗ ω_{X}^{⊗b} ⊗ η) = b(2g − 2) − 2j ≥ 2g − 1, the statements are a consequence of the semistability of M_{ω_{X}} and that of M_{ω_{X}⊗η} respectively.

Assume now that X is 1-nodal and η ∈ Pic(X) is locally free, that is, [X, η, ϕ] ∈ Δ0^{′} ∪ Δ0^{′′}. Let nor : C → X be the normalization map, with nor^{−1}(X_{sing}) = {p, q} and set η_{C} := nor^{∗}(η). Via the exact sequence 0 → O_{X} → nor_{∗}(O_{C}) → nor_{∗}(O_{C})/O_{X} → 0, to show that (ii) holds it suffices to show that

H^{1}(C, ⋀^{j} M_{K_{C}(p+q)⊗η_{C}} ⊗ K_{C}^{⊗b} ⊗ O_{C}(bp + bq) ⊗ η_{C}) = 0,

and

H^{1}(C, ⋀^{j} M_{K_{C}(p+q)⊗η_{C}} ⊗ K_{C}^{⊗b} ⊗ O_{C}((b − 1)p + (b − 1)q) ⊗ η_{C}) = 0.

This again is a consequence of the stability of the vector bundle M_{K_{C}(p+q)⊗η_{C}}, which in turn follows from [FL] Proposition 2.4.

Assume now that [X, η, ϕ] ∈ Δ_{0}^{ram} and X = C ∪_{{p,q}} E, where E ≅ P^{1} and [C, p, q] ∈ M_{g−1,2}. Furthermore η_{E} = O_{E}(1) and deg(η_{C}) = −1. The kernel bundles M_{ω_{X}} and M_{ω_{X}⊗η} have the following restrictions to the components of X:

M_{ω_{X}|C} = M_{K_{C}(p+q)} and M_{ω_{X}|E} = O_{E}^{⊕(g−1)},

respectively

M_{ω_{X}⊗η|C} = M_{K_{C}(p+q)⊗η_{C}} and M_{ω_{X}⊗η|E} = O_{E}(−1) ⊕ O_{E}^{⊕(g−3)}.

Via the Mayer-Vietoris sequence on X, a sufficient condition for (i) to hold is given by the following vanishing statements:

H^{1}(⋀^{j} M_{ω_{X}|C} ⊗K_{C}^{⊗b}(b(p+q))⊗η_{C}) = 0, H^{1}(⋀^{j} M_{ω_{X}|C} ⊗K_{C}^{⊗b}((b−1)(p+q))⊗η_{C}) = 0,

as well as H^{1}(E, ⋀^{j} M_{ω_{X}|E} ⊗ η_{E}) = 0 (note that ω_{X|E} = O_{E}). The vanishing on E is immediate, whereas that on C follows again by using that M_{K_{C}(p+q)} and M_{K_{C}(p+q)⊗η_{C}} are semistable and computing the slopes of the corresponding vector bundles whose first cohomology group is supposed to vanish. Statement (ii) is entirely similar and we skip details. □

To be able to define tautological sheaves over R̃_{g,ℓ}, we consider the global kernel bundle defined via the exact sequence over the universal curve X̃_{g,ℓ}

0 ⟶ M_{u} ⟶ u^{∗}(𝔼) ⟶ ω_{u} ⟶ 0,

respectively the global Prym kernel bundle

0 ⟶ M′_{u} ⟶ u^{∗}(𝔼′) ⟶ ω_{u} ⊗ P ⟶ 0.

3.1. Tautological sheaves. We introduce tautological vector bundles over R̃_{g,ℓ} whose fibres are various Koszul cohomology groups. For integers 0 ≤ j ≤ \frac{g}{2} and b ≥ 2, as well as for (j, b) = (0, 1), we define the sheaves

𝔼_{j,b} := u_{∗}(⋀^{j} M_{u} ⊗ ω_{u}^{⊗b} ⊗ P) and 𝔽_{j,b} := u_{∗}(⋀^{j} M′_{u} ⊗ ω_{u}^{⊗b} ⊗ P^{⊗(b−2)}).

Grauert’s theorem used via Proposition 3.1 imply that 𝔼_{j,b} and 𝔽_{j,b} are locally free. Note that 𝔼_{0,1} = 𝔼′. One can now carry a Grothendieck-Riemann-Roch calculation over the universal level ℓ curve u : X̃_{g,ℓ} → R̃_{g,ℓ} and prove the following:

Proposition 3.2. For each integer b ≥ 1, the following formula holds in Pic(R̃_{g,ℓ}):

(i) c_{1}(𝔼_{0,b}) = λ + \binom{b}{2}κ_{1} − \frac{1}{2ℓ} ∑_{a=1}^{⌊ \frac{ℓ}{2} ⌋} a(ℓ − a)δ_{0}^{(a)}.

(ii) c_{1}(𝔽_{0,b}) = λ + \binom{b}{2}κ_{1} − \frac{(b−2)^{2}}{2ℓ} ∑_{a=1}^{⌊ \frac{ℓ}{2} ⌋} a(ℓ − a)δ_{0}^{(a)}.

3.2. The divisor \overline{U}_{g,ℓ}. The divisor U_{g,ℓ} has a scheme theoretic characterization via condition (22). We extend this description over the boundary of R̃_{g,ℓ} and compute the class of the resulting degeneracy locus, thus completing the proof of Theorem 0.5.

By taking exterior powers in the sequence defining the sheaf M_{u} and then using Proposition 3.1, we find that for each 1 ≤ j ≤ i − 1, one has the following exact sequences, which can be used to compute inductively the Chern classes of the vector bundles 𝔼_{j,b}, starting from level j = 0:

(23)  0 ⟶ 𝔼_{j,i+1−j} ⟶ ⋀^{j} 𝔼 ⊗ 𝔼_{0,i+1−j} ⟶ 𝔼_{j−1,i+2−j} ⟶ 0.

With these ingredients, we can prove Theorem 0.5 and calculate [\overline{U}_{g,ℓ}]:

Proof of Theorem 0.5. We consider the morphism χ : ⋀^{i} 𝔼 ⊗ 𝔼_{0,1} → 𝔼_{i−1,2} of vector bundles over the stack R̃_{g,ℓ}, which at the level of fibres is given by the map

χ([X, η, ϕ]) : ⋀^{i} H^{0}(X, ω_{X}) ⊗ H^{0}(X, ω_{X} ⊗ η) → H^{0}(X, ⋀^{i−1} M_{ω_{X}} ⊗ ω_{X}^{⊗2} ⊗ η).

The intersection of the degeneracy locus of χ with R_{g,ℓ} is precisely the divisor U_{g,ℓ}, therefore the difference c_{1}(𝔼_{i−1,2} − ⋀^{i} 𝔼 ⊗ 𝔼_{0,1}) − [\overline{U}_{g,ℓ}] ∈ Pic(R̃_{g,ℓ}) is a (possibly empty) effective class supported only on the boundary classes of R̃_{g,ℓ}. In particular, the class c_{1}(𝔼_{g−1,2} − ⋀^{i} 𝔼 ⊗ 𝔼_{0,1}) is effective. In order to compute it, we use (23) and write that:

c_{1}(𝔼_{i−1,2} − ⋀^{i} 𝔼 ⊗ 𝔼_{0,1}) = ∑_{b=0}^{i}(−1)^{b+1}c_{1}(⋀^{i−b} 𝔼 ⊗ 𝔼_{0,b+1}) =

∑_{b=0}^{i}(−1)^{b+1}[\binom{g}{i − b}(λ + \binom{b + 1}{2}κ_{1} − ∑_{a=1}^{⌊ \frac{ℓ}{2} ⌋} \frac{a(ℓ − a)}{2ℓ}δ_{0}^{(a)}) + (2b + 1)(g − 1)\binom{g − 1}{i − b − 1}λ],

which after routine calculations leads to Theorem 0.5. □

3.3. The virtual divisor \overline{D}_{g,ℓ}. We prove Theorem 0.7. Since the proof resembles the previous calculation, we succintly explain the main points. We set g := 2i + 2 and recall that [C, η] ∈ D_{g,ℓ} if and only if the map

⋀^{i} H^{0}(K_{C} ⊗ η) ⊗ H^{0}(C, K_{C} ⊗ η^{∨}) → H^{0}(C, ⋀^{i−1} M_{K_{C}⊗η} ⊗ K_{C}^{⊗2})

is not an isomorphism. Equivalently, D_{g,ℓ} is the restriction to R_{g,ℓ} of the degeneracy locus of the vector bundle morphism ⋀^{i} 𝔼′ ⊗ 𝔽_{0,1} → 𝔽_{i−1,2}.

Proof of Theorem 0.7. For each integer 1 ≤ j ≤ i − 1 we have the exact sequence on R̃_{g,ℓ}

0 ⟶ 𝔽_{i−j,j+1} ⟶ ⋀^{i−j} 𝔼′ ⊗ 𝔽_{0,j+1} ⟶ 𝔽_{i−j−1,j+2} ⟶ 0,

thus one writes that

[\overline{D}_{g,ℓ}]^{virt} = ∑_{j=0}^{i}(−1)^{j+1}c_{1}(⋀^{i−j} 𝔼′ ⊗ 𝔽_{0,j+1}) =

∑_{j=0}^{i}(−1)^{j+1}[(g − 1)(2j + 1)\binom{g − 2}{i − j − 1}c_{1}(𝔼′) + \binom{g − 1}{i − j}c_{1}(𝔽_{0,j+1})].

Using Proposition 3.2 and that 𝔼′ = G_{0,1}, we finish the proof after some calculations. □

3.4. The Prym-Green divisor. Recall that 𝔼′ is the Prym-Hodge bundle with fibres 𝔼′([X, η, ϕ]) = H^{0}(X, ω_{X} ⊗ η). For b ≥ 1, we set ℍ_{0,b} := Sym^{b}𝔼′. Then for a ≥ 1, we define inductively the sheaves ℍ_{a,b} via the following exact sequences over R̃_{g,ℓ}:

(24)  0 ⟶ ℍ_{a,b} ⟶ ⋀^{a} 𝔼′ ⊗ ℍ_{0,b} ⟶ ℍ_{a−1,b+1} ⟶ 0.

Note that ℍ_{a,b} is locally free and its fibre over a point [X, η, ϕ] inducing a paracanonical map ϕ_{L} : X → P^{g−2}, where L := ω_{X} ⊗ η is canonically identified with the space H^{0}(P^{g−2}, Ω_{P^{g−2}}^{1}(a + b)).

For each b ≥ 1, we also define the sheaf 𝔾_{0,b} := u_{∗}(ω_{u}^{⊗b} ⊗ P^{⊗b}). Note that 𝔾_{0,1} = 𝔼′ and there exist sheaf homomorphisms φ_{0,b} : ℍ_{0,b} → 𝔾_{0,b}, which fibrewise over [X, η, ϕ] ∈ \overline{R}_{g,ℓ} correspond to the multiplication maps of global sections

Sym^{b}H^{0}(X, ω_{X} ⊗ η) → H^{0}(X, ω_{X}^{⊗b} ⊗ η^{⊗b}).

Inductively, for each a ≥ 1, we define sheaves 𝔾_{a,b} over R̃_{g,ℓ} via the exact sequences

(25)  0 ⟶ 𝔾_{a,b} → ⋀^{a} 𝔼′ ⊗ 𝔾_{0,b} → 𝔾_{a−1,b+1} ⟶ 0,

where the right exactness of the sequence (25) is to be soon justified. Inductively, one also constructs sheaf homomorphisms

φ_{a,b} : ℍ_{a,b} → 𝔾_{a,b},

fibrewise given by restriction of twisted forms on projective space

φ_{a,b}([X, η, ϕ]) : H^{0}(P^{g−2}, ⋀^{a} M_{P}_{g−2}(b)) → H^{0}(X, ⋀^{a} M_{L} ⊗ L^{⊗b}).

The right exactness of the sequence (25) is established via Grauert’s theorem by the following:

Proposition 3.3. Let [X, η, β] ∈ R̃_{g,ℓ} a twisted level curve. Then for all integers a ≥ 0, b ≥ 2, the vanishing H^{1}(X, ⋀^{a} M_{L} ⊗ L^{⊗b}) = 0 holds.

Proof. It follows closely [FL] Proposition 3.2, where the corresponding statement when ℓ = 2 is established. □

We now compute the Chern classes of the above defined tautological bundles (this matches the main theorem of [C2] for s = bℓ via \frac{1}{2}B_{2}(\frac{s}{ℓ}) = \binom{b}{2} + 1/6; we refer to Section 3.2, (44) for the explicit example in degree 1).

Proposition 3.4. For b ≥ 1 we have the following formula in Pic(R̃_{g}):

c_{1}(𝔾_{0,b}) = λ + \binom{b}{2}κ_{1} − \frac{b^{2}}{2} ∑_{a=1}^{⌊ \frac{ℓ}{2} ⌋} \frac{a(ℓ − a)}{ℓ}δ_{0}^{(a)}.

Proof. We apply Grothendieck-Riemann-Roch for the universal curve u : X̃_{g,ℓ} → M̃_{g} and the sheaf 𝔾_{0,b}. Noting that R^{1}u_{∗}(ω_{u}^{⊗b} ⊗ P^{⊗b}) = 0, we write:

ch(𝔾_{0,b}) = u_{∗}[(1+b c_{1}(ω_{u}⊗P)+ \frac{b^{2}}{2} c_{1}^{2}(ω_{u}⊗P)+⋯)·(1− \frac{c_{1}(ω_{u})}{2} + \frac{c_{1}^{2}(ω_{u}) + [Sing(u)]}{12})],

where Sing(u) ⊂ X̃_{g,ℓ} denotes the codimension 2 singular locus of the u, and clearly u_{∗}([Sing(u)]) = f^{∗}(δ_{0}), where δ_{0} ∈ Pic(M̃_{g}). Using Mumford’s formula [HM] u_{∗}(c_{1}^{2}(ω_{u})) = 12λ − δ as well as Proposition 1.6, we obtained the claimed formula by evaluating the push-forward under u of the quadratic terms. □

We now compute the virtual class of the Prym-Green divisor.

Proof of Theorem 0.6. We set g = 2i + 6 and observe that the locus Z_{g,ℓ} of smooth level curves [C, η] ∈ R_{g,ℓ} is the degeneracy locus of the morphism φ_{i,2} : ℍ_{i,2|R_{g,ℓ}} → 𝔾_{i,2|R_{g,ℓ}}. Whenever Conjecture A is true, the class [\overline{Z}_{g,ℓ}]^{virt} is effective, and differs from the class of the closure \overline{Z}_{g,ℓ} by a (possibly empty) effective combination of boundary divisors. We now compute [\overline{Z}_{g,ℓ}]^{virt} := c_{1}(𝔾_{i,2} − ℍ_{i,2}) as follows:

c_{1}(ℍ_{i,2}) = ∑_{j=0}^{i}(−1)^{j}c_{1}(⋀^{i−j} ℍ_{0,1} ⊗ Sym^{j+2}(ℍ_{0,1})),

and

c_{1}(𝔾_{i,2}) = ∑_{j=0}^{i}(−1)^{j}\binom{g − 1}{i − j}c_{1}(𝔾_{0,i+2}) + ∑_{j=0}^{i}(−1)^{j}(g − 1)(2j + 3)\binom{g − 2}{i − j − 1}c_{1}(𝔼′).

Using Proposition 3.4 we can finish the proof. □

Remark 3.5. In the course of the proof of Theorem 0.2 it is important to note that for any ℓ ≥ 2 and g ≤ 23, in order to establish that the class K_{\overline{R}_{g,ℓ}} is effective (respectively big), it suffices to prove the seemingly weaker statement that the restriction KR̃_{g,ℓ} is effective (respectively big). Precisely, if D is an effective divisor on R_{g,ℓ} such that

[\overline{D}] = sλ − 2(δ0^{′} + δ0^{′′}) − ∑_{a=1}^{⌊ \frac{ℓ}{2} ⌋} b_{0}^{(a)}δ_{0}^{(a)} − ∑_{i=1}^{⌊ \frac{g}{2} ⌋}(b_{i} δ_{i} + b_{g−i} δ_{g−i} + b_{i:g−i}δ_{i:g−i}) ∈ Pic(\overline{R}_{g,ℓ}),

with s ≤ 13 and b_{0}^{(a)} ≥ ℓ + 1 for a = 1, …, ⌊\frac{ℓ}{2}⌋, then K_{\overline{R}_{g,ℓ}} − [\overline{D}] ∈ Eff(\overline{R}_{g,ℓ}) is an effective class, that is b_{g−1} ≥ 3 and b_{i}, b_{i:g−i} ≥ 2, for all i = 1, …, g − 1. The proof uses pencils of level curves on K3 surfaces and is similar to [FL] Proposition 1.2. For 1 ≤ i ≤ min\{\frac{g}{2}, 11\}, we fix a general curve [C_{2}, p] ∈ M_{g−i,1}, an ℓ-torsion point η_{C_{2}} ∈ Pic^{0}(C_{2}) and a moduli curve B_{i} ⊂ \overline{M}_{i,1}, induced by a Lefschetz pencil \{(C_{t}, p)\}_{t∈P^{1}} of pointed curves on genus i on a fixed K3 surface S ⊂ P^{i}, the marked point p being one of the base points of the pencil. We construct three 1-dimensional families fillingup the divisors Δ_{g−i}, Δ_{i} and Δ_{i:g−i} as follows. Firstly, A_{g−i} ⊂ Δ_{g−i} consists of level curves \{[C_{t} ∪_{p} C_{2}, η_{C_{t}} = O_{C_{t}}, η_{C_{2}}]\}_{t∈P^{1}} ⊂ \overline{R}_{g,ℓ}. Then A_{i} ⊂ Δ_{i} parametrizes level curves

\{[C_{t} ∪_{p} C_{2}, η_{C_{t}} ∈ \overline{Pic}^{0}(C_{t})[ℓ], O_{C_{2}}]\}_{t∈P^{1}}. Finally, A_{i:g−i} ⊂ Δ_{i:g−i} consists of level curves

\{[C_{t} ∪_{p} C_{2}, η_{C_{t}} ∈ \overline{Pic}^{0}(C_{t})[ℓ], η_{C_{2}}]\}_{t∈P^{1}}. For i ≠ 10, the curves A_{i}, A_{g−i} and A_{i:g−i} fillup the respective boundary divisors in \overline{R}_{g,ℓ}. By imposing the conditions A_{i} · \overline{D} ≥ 0, A_{g−i} · \overline{D} ≥ 0 and A_{i:g−i} · \overline{D} ≥ 0 and computing the actual intersection numbers via [FP], we obtain the desired bounds on the boundary coefficients of [\overline{D}]. The case g = 10 is a little special but can be handled in a similar way, cf. [FP] Theorem 1.1 (b).

4. Syzygy computations with nodal curves

In this section we explain how to verify computationally Conjectures A and B for small g and bounded level ℓ using nodal curves.

Let C be a rational g-nodal curve with normalization nor : P^{1} → C and denote by {P_{j}, Q_{j}}_{j=1}^{g} the preimages of the nodes of C. A line bundle L ∈ Pic^{d}(C) is given by an isomorphism nor^{∗}(L) ≅ O_{P}_{1}(d) and gluing data between residue class fields

a_{j} : O_{P}_{1}(d) ⊗ κ(P_{j})≅O_{P}_{1}(d) ⊗ κ(Q_{j}).

In particular, Pic^{0}(C) ≅ G_{m} × … × G_{m} is a g-dimensional torus.

Let K[x_{0}, x_{1}] be the homogeneous coordinate ring of P^{1}, and let z = \frac{x_{1}}{x_{0}} be the affine coordinate on the chart U_{0} ≅ 𝔸^{1}. We assume that all preimage points of the nodes are contained in U_{0}, say P_{j} = (1 : p_{j}) and Q_{j} = (1 : q_{j}). Then

H^{0}(C, L) ≅ {f ∈ K[z]_{≤d} : a_{j}f(p_{j}) = f(q_{j}) for i = 1, …, g}

can be identified with the space of polynomials f of degree at most d whose values in p_{j} and q_{j} differ by the factor a_{j} ∈ K^{∗}. In terms of coefficients of polynomials, the space H^{0}(C, L) ⊂ H^{0}(P^{1}, O_{P}_{1}(d)) is the solution space of a homogeneous system of equations.

Example 4.1. A basis of sections of the dualizing sheaf ω_{C} is given by

ω_{j} = \frac{dz}{(z − p_{j})(z − q_{j})}, for j = 1, …, g,

and the canonical map is induced by

ϕ_{ω_{C}} : P^{1} → P^{g−1}, z ↦ (∏_{i≠1}(z − p_{i})(p − q_{i}) : …: ∏_{i≠g}(z − p_{i})(z − q_{i})).

Thus nor^{∗}(ω_{C})^{=}O_{P}_{1}(2g−2) and the the canonical multipliers are a_{j}^{can} = ∏_{i≠j} \frac{(q_{j}−p_{i})(q_{j}−q_{i})}{(p_{j}−p_{i})(p_{j}−q_{i})}.

Example 4.2. Suppose A, B ∈ Pic(C) correspond to the pairs (O_{P}_{1}(d), (a_{1}, …, a_{g})) and (O_{P}_{1}(e), (b_{1}, …, b_{g})) respectively, then A ⊗ B is given by (O_{P}_{1}(d + e), (a_{1}b_{1}, …, a_{g}b_{g})).

Example 4.3. Let η ∈ Pic^{0}(C) be a line bundle corresponding to (O_{P}_{1}, (ζ_{1}, …, ζ_{g})). Then η is a torsion bundle, if and only if all multipliers ζ_{j} are roots of unity. In particular, one obtains a level ℓ paracanonical curve by altering some or all of the multipliers a_{j}^{can} by a primitive ℓth root of unity.

We verify the Prym-Green Conjecture following the following steps:

(i) If ℓ = 2, we take r = 1 and choose a prime p of moderate size, for instance 10^{4} < p < 3 · 10^{4}. In case ℓ > 2 we choose an integer r and a prime p such that r represents a primitive ℓth root of unity in K = 𝔽_{p}. So p is one of prime factors of r^{ℓ} − 1.

(ii) Randomly pick 2g points P_{1}, Q_{1}, …, P_{g}, Q_{g} ∈ P^{1}(K) and compute the canonical

multipliers a_{j}^{can} for j = 1, …, g. (iii) Alter all (or some) multiplier a_{j}^{η} = ra_{j}^{can} and compute a basis f_{0}, …, f_{g−2} of the

Prym canonical system H^{0}(C, ω_{C}⊗η) ⊂ H^{0}(P^{1}, O_{P}_{1}(2g−2)). By Riemann-Roch, this space is (g − 1)-dimensional.

(iv) Compute the kernel I_{C} of the map K[y_{0}, …, y_{g−2}] → K[x_{0}, x_{1}] given by y_{j} ↦ f_{j} and its free resolution up to the appropriate step.

A Macaulay2 code which does this job can be found in the package NodalCurves.m2 available at http://www.math.uni-sb.de/ag/schreyer/home/computeralgebra.htm.

Proposition 4.4. The Prym-Green Conjecture holds for all even genera with g ≤ 18 and ℓ ≤ 5 over a field K of characteristic 0 with the possible exception of the cases (g, ℓ) = (8, 2) or (16, 2).

Proof. By the Macaulay2 computation documented above, the statement holds true for some g-nodal rational curve defined over a finite field 𝔽_{p}. This computation can be viewed as the reduction mod p of the same computation of an example defined over ℤ. By semicontinuity of Betti numbers in families, then there exists a g-nodal rational curve defined over ℚ with a pure resolution. Applying again semicontinuity, a general pair [C, η] ∈ R_{g,ℓ} has a pure resolution of its paracanonical embedding. □

In more concrete terms, Proposition 4.4 says that a general level ℓ paracanonical curve of genus 10 has the following syzygies

         0  1  2   3   4   5  6  7
 total: 1 18 42 126 210 162 63 10
      0: 1  .  .   .   .   .  .  .
      1: . 18 42   .   .   .  .  .
      2: .  .  . 126 210 162 63 10

for all 2 ≤ ℓ ≤ 5. The relevant computation in this example takes about 1.80 seconds. Using 2 minutes of cpu we can extend this result for g = 10 to all levels ℓ ≤ 30.

For larger genus the approach has to to modified to be still computationally feasible. Recall [E] that for a graded Cohen-Macaulay module M of dimension d over a standard graded polynomial ring S, if x_{1}, …, x_{d} ∈ S_{1} is an M-sequence, the minimal free resolution of the artinian quotient A = M/⟨x_{1}, …, x_{d}⟩M over the polynomial ring T = S/⟨x_{1}, …, x_{d}⟩ is obtained from the minimal free resolution of M as an S-module by tensoring with T. In particular the graded Betti numbers

b_{ij}(M) = dim_{K} Tor_{i}^{S}(M, K)_{i+j} = dim_{K} Tor_{i}^{T} (A, K)_{i+j} = b_{ij}(A)

coincide. We also can use duality [E, Theorem 21.15] : Let ω_{S} ≅ S(− dim S) denote the dualizing module of S, and ω_{M} := Ext_{S}^{codim M}(M, ω_{S}). If M is Cohen-Macaulay, then the minimal free resolution of ω_{M} is obtained by applying Hom_{S}(−, ω_{S})[− codim M] to the minimal free resolution of M. In particular,

dim_{K} Tor_{i}^{S}(M, K)_{j} = dim_{K} Tor_{codim M−i}^{S}(ω_{M}, K)_{dim S−j}.

Finally, we can use the Koszul complex, that is, the minimal free resolution of K as an S-module in order to compute Tor^{S}(M, K). We use all three options.

(i) Compute a basis s_{0}, …, s_{g−2} of H^{0}(C, K_{C} ⊗ η) ⊂ K[x_{0}, x_{1}]_{2g−2} and ω_{1}, …, ω_{g} of H^{0}(C, K_{C}) ⊂ K[x_{0}, x_{1}]_{2g−2}.

(ii) Check that s_{g−3}, s_{g−2} have no common zero in P^{1}, i.e. they correspond to a

R-sequence for R = K[y_{0}, …, y_{g−2}]/I_{C} ≅ Γ_{C}(O_{C}, K_{C} ⊗ η). (iii) Compute representatives in K[x_{0}, x_{1}] of a K-basis for the artinian reduction

A = ω_{R}/⟨(y_{g−3}, y_{g−2}⟩ω_{R}. Note that A is a graded artinian T = K[y_{0}, … y_{g−4}]- module with Hilbert series H_{A}(t) = ∑_{d} dim A_{d}t^{d} = g + (g − 3)t + t^{2}.

(iv) Set m = \frac{g}{2}. Substitute s_{0}, … s_{g−4} into the m^{th} Koszul matrix on y_{0}, …, y_{g−4} and compute the tensor product with the 1 × g matrix (ω_{1}, …, ω_{g})^{t}. The result is a \binom{g−3}{\frac{g}{2} −1} × g\binom{g−3}{\frac{g}{2}} matrix with entries in H^{0}(C, ω_{C}^{2} ⊗ η) ⊂ K[x_{0}, x_{1}]_{4g−4}.

(v) Reduce the entries module the ideal ⟨s_{g−3}, s_{g−2}⟩⟨ω_{1}, …, ω_{g}⟩ ⊂ K[x_{0}, x_{1}]. The result is now a matrix M_{pol} of polynomials with entries in the (g−3)-dimensional space of representatives of A_{2}.

(vi) Compute the (g − 3)\binom{g−3}{\frac{g}{2} −1} × g\binom{g−3}{\frac{g}{2}} coefficient matrix of M_{field} with values in K. The kernel of M_{field} is isomorphic to

Tor_{S}^{m}(ω_{R}, K)_{m} ≅ Tor_{S}^{g−3−m}(R, K)_{g−1−m} ≅ K_{g−3−m,2}(C, K_{C} ⊗ η),

whose dimension equals dim K_{\frac{g}{2} −2,1}(C, K_{C} ⊗ η) since g − 3 − m + 1 = \frac{g}{2} − 2. (vii) Use the finite field linear algebra subroutines/packages FFLAS [DGP] to com-

pute the rank of M_{field} over the finite field K = 𝔽_{p}.

Our approach to the torsion bundle conjecture (Conjecture B) for bounded g and ℓ is similar. Again we work with a g-nodal curve C and carry out the following steps:

(i) Compute bases of s_{0}, …, s_{g−2} of H^{0}(C, K_{C} ⊗η) ⊂ K[x_{0}, x_{1}]_{2g−2} and t_{0}, …, t_{g−3} of H^{0}(C, K_{C} ⊗ η^{k}) ⊂ K[x_{0}, x_{1}]_{2g−2} and the kernel of the multiplication map

μ : H^{0}(C, K_{C} ⊗ η) ⊗ H^{0}(C, K_{C} ⊗ η^{⊗k}) → H^{0}(C, K_{C}^{2} ⊗ η^{⊗(k+1)}).

Since the map is surjective, Ker(μ) is (g − 1)^{2} − (3g − 3) = (g − 1)(g − 4) dimensional.

(ii) Ker(μ) can be reinterpreted as the linear presentation matrix

ϕ : S^{g^{2}−5g+4}(−2) → S^{g−1}(−1)

of M = Γ_{C}(η^{⊗k}, K_{C} ⊗η) as an S = K[y_{0}, …, y_{g−2}] module. We wish to compute Tor_{\frac{g}{2} −1}^{S}(M, K)_{\frac{g}{2}} ≅ K_{\frac{g}{2} −1,1}(η^{⊗k}, K_{C} ⊗ η).

(iii) Check that s_{g−3}, s_{g−2} have no common zero in P^{1} and compute basis for the artinian reduction B = M/⟨y_{g−3}, y_{g−2}⟩M. This time B is a graded artinian T = K[y_{0}, … y_{g−4}]-module with Hilbert series H_{B}(t) = (g − 1)t + (g − 1)t^{2}. (iv) Reinterpret the tensor product of the (\frac{g}{2} − 1)^{th} Koszul matrix on y_{0}, …, y_{g−4} with B as a (g − 1)\binom{g−3}{\frac{g}{2}} square matrix with entries in the ground field, and compute its rank. The dimension of the kernel is the desired Betti number.

Proposition 4.5. Conjecture B hold over a field of characteristic zero for all 6 ≤ g ≤ 18 and all primitive ℓ-torsion bundle with 3 ≤ ℓ ≤ 6.

In more concrete terms the result says for g = 10 that a general pair [C, η] ∈ \overline{R}_{g,ℓ} the module Γ_{C}(C; η^{⊗k}, K_{C} ⊗ η) for 1 ≤ k ≤ ℓ − 2 has syzygies

        0  1   2   3   4   5  6 7
total: 9 54 126 126 126 126 54 9
     0: 9 54 126 126   .   .  . .
     1: .  .   .   . 126 126 54 9

unless ℓ = 2k + 1, in which case we have instead

        0  1   2   3   4   5  6 7
total: 9 54 126 126 126 126 54 9
     0: 9 54 126 126   1   .  . .
     1: .  .   .   1 126 126 54 9

Proof. By our computation, Conjecture B holds for an g-nodal example over a finite field. For documentation of the computation based on the Macaulay2 package NodalCurves.m2 see http://www.math.uni-sb.de/ag/schreyer/home/computeralgebra.htm. To complete the proof with a semi-continuity argument, we have to show that in the exceptional cases the Betti number cannot become zero.

The extra syzygy are explained as follows. The resolution of η^{⊗k} is self-dual if and only if

\begin{aligned}η^{⊗k} & ≅ & Ext_{O}^{g−3}(η^{⊗k}, O(−g)) & ≅ & Ext_{O}^{g−3}(η^{⊗k}, ω_{P}_{g−2})(−1) \\ & ≅ & Hom(η^{⊗k}, K_{C}) ⊗ K_{C}^{−1} ⊗ η^{−1} & ≅ & η^{⊗(−k−1)} \end{aligned}

that is, η^{⊗(2k+1)} = O_{C}. In this case the artinian reduction, B = B_{1} ⊕ B_{2} is selfdual as well, that is, B_{2} = B_{1}^{∨} and the multiplication map V ⊗ B_{1} → B_{2} with V = K[y_{0}, …, y_{g−4}]_{1} corresponds to a symmetric (g − 1) square matrix with entires in V^{∨}. The relevant Koszul cohomology map for B

κ_{B} : Λ^{(g−2)/2}V ⊗ B_{1} → Λ^{(g−4)/2}V ⊗ B_{2}

is obtained from the Koszul map

κ : Λ^{(g−2)/2}V → Λ^{(g−4)/2}V ⊗ V

by tensor product with μ′ : B_{1} → B_{2} ⊗ V^{∨} and contraction V^{∨} ⊗ V → K. We now apply the following well-known fact:

Lemma 4.6. The middle Koszul matrix

κ′ : Λ^{m+1}V ⊗ T → Λ^{m}V ⊗ T (1)

of a polynomial ring T = Sym V in 2m+1 variables is symmetric if m ≡ 0 mod 2. Otherwise κ′ is skew-symmetric.

Thus in case ℓ = 2k + 1 the matrix κ_{B} is as tensor product of matrices with symmetry properties, skew-symmetric precisely in case \frac{g−4}{2} ≡ 1 mod 2 i.e. g ≡ 2 mod 4. If in addition \binom{g−3}{\frac{g}{2} −1} is odd, then κ_{B} is a skew-symmetric matrix of odd size, and κ_{B} cannot have maximal rank. This completes the argument and justifies the exceptions in the statement of Conjecture B. □

Proof of Lemma 4.6. Under the pairing Λ^{m}V × Λ^{m+1}V → Λ^{2m+1}V ≅ K the matrix κ′ corresponds to the composition

ψ : Λ^{m+1}V ⊗ Λ^{m+1}V → V ⊗ Λ^{m}V ⊗ Λ^{m+1}V → V ⊗ Λ^{2m+1}V.

Hence for basis elements y_{I} = y_{i_{0}} ∧ … ∧ y_{i_{m}}, y_{J} = y_{j_{0}} ∧ … ∧ y_{j_{m}} ∈ Λ^{m+1}V for I, J ⊂ {0, …, 2m} we have that ψ(y_{I} ⊗ y_{J}) = 0 unless I ∪ J = {0, …, 2m}, which means that I ∩ J consists of precisely one element. Furthermore for disjoint sets I′ = {i_{1}, …, i_{m}} and J′ = {j_{1}, …, j_{m}} we compute that

ψ(y_{i_{0}} ∧ y_{I′} ⊗ y_{i_{0}} ∧ y_{J′}) = y_{i_{0}} ⊗ y_{I′} ∧ y_{i_{0}} ∧ y_{J′} = (−1)^{m}y_{i_{0}} ⊗ y_{i_{0}} ∧ y_{I′} ∧ y_{J′}

= y_{i_{0}} ⊗ y_{i_{0}} ∧ y_{J′} ∧ y_{I′} = (−1)^{m}y_{i_{0}} ⊗ y_{J′} ∧ y_{i_{0}} ∧ y_{I′} = (−1)^{m}ψ(y_{i_{0}} ∧ y_{J′} ⊗ y_{i_{0}} ∧ y_{I′})

as claimed. □

Remark 4.7. Notice that for a field of characteristic 2 the matrix ψ is symmetric with zeroes on the diagonal. Thus for ℓ = 2k + 1 and \binom{g−3}{\frac{g}{2} −1} ≡ 1 mod 2 a torsion bundle has an extra syzygy independent of the parity of \frac{g}{2}.

5. The failure of the Prym-Green Conjecture on R_{8}

We argue that the Prym-Green Conjecture A fails on R_{8}. Our findings suggest that for a general Prym canonical curve ϕ_{K_{C}⊗η} : C → P^{6} of genus 8, the non-vanishing K_{2,1}(C, K_{C} ⊗ η) ≠ 0 holds and the corresponding Betti table is the following:

         0 1  2  3  4 5
 total: 1 8 36 56 35 8
      0: 1 .  .  .  . .
      1: . 7  1  .  . .
      2: . 1 35 56 35 8

Moreover, in all cases the extra syzygy in K_{2,1}(C, K_{C} ⊗ η) is never of maximal rank 7. We analyze this situation geometrically.

Throughout this section ℓ = 2 and we set R_{g} := R_{g,2}. We recall that GP_{g,d}^{r} is the closure in M_{g} of the locus of curves with a base point free linear series L ∈ W_{d}^{r}(C) such that the Petri map μ_{0}(L) : H^{0}(C, L) ⊗ H^{0}(C, K_{C} ⊗ L^{∨}) → H^{0}(C, K_{C}) is not injective. The syzygy locus Z_{8,2} ⊂ R_{8} considered in Theorem 0.6 can be extended to the level of the universal genus 8 Jacobian 𝔓𝔦𝔠_{8}^{14} → M_{8} and we introduce the Koszul locus

𝔎𝔬𝔰𝔷 := {[C, L] ∈ 𝔓𝔦𝔠_{8}^{14} : K_{2,1}(C, L) ≠ 0}.

Note that dim K_{2,1}(C, L) = dim K_{1,2}(C, L), for each [C, L] ∈ 𝔓𝔦𝔠_{8}^{14}. It is proved in [Ve] that 𝔎𝔬𝔰𝔷 is indeed a divisor on 𝔓𝔦𝔠_{8}^{14}, that is, 𝔎𝔬𝔰𝔷 ≠ 𝔓𝔦𝔠_{8}^{14}. Via the map ι : R_{8} ↪ 𝔓𝔦𝔠_{8}^{14} given by ι([C, η]) := [C, K_{C} ⊗ η], one has that ι^{∗}(𝔎𝔬𝔰𝔷) = Z_{8,2}.

Let us consider the parameter space Σ of triples (C, L, V), where [C, L] ∈ 𝔓𝔦𝔠_{8}^{14} induces a map ϕ_{L} : C → P^{6} and V ∈ G(5, H^{0}(P^{6}, I_{C/P^{6}}(2))). The variety Σ is birational to a Grassmann bundle over an open subset of 𝔓𝔦𝔠_{8}^{14}. Denoting by X := Bs|V | ⊂ P^{6} the base locus of the system of quadrics in V, one can write that X = C + C′, where the linked curve C′ ⊂ P^{6} has genus 14, degree deg(C′) = 18 and h^{1}(C′, O_{C′}(1)) = 2. For a general (C, L, V) ∈ Σ, we have an isomorphism

(26)  H^{0}(P^{6}, I_{C/P^{6}}(2))/V ⟶^{≅} H^{0}(C′, K_{C′}(−1)).

This linkage construction induces a dominant rational map

φ : Σ −−→ M_{14}, (C, L, Σ) ↦ [C′].

The image of 𝔎𝔬𝔰𝔷 under this map has a transparent geometric interpretation. It is proved in [Ve] Lemma 4.4 that the Petri map for the genus 14 curve, that is,

μ_{0}(O_{C′}(1)) : H^{0}(C′, O_{C′}(1)) ⊗ H^{0}(C′, K_{C′}(−1)) → H^{0}(C′, K_{C′})

is an isomorphism if and only if K_{2,1}(C, L) = 0. We summarize the situation as follows:

Proposition 5.1. A paracanonical genus 8 curve C ⊂ P^{6} belongs to the Koszul divisor if and only if the residual genus 14 curve C′ ⊂ P^{6} is Petri special.

The locus of such Petri special curves C′ ⊂ P^{6} splits into two components depending on whether K_{C′}(−1) ∈ W_{8}^{1}(C′) is base point free, and then [C′] ∈ GP_{14,8}^{1}, or else, it has a base point, in which case C′ is 7-gonal.

We fix a point [C, L] ∈ 𝔎𝔬𝔰𝔷 corresponding to an embedded curve ϕ_{L} : C ↪ P^{6} of degree 14 having Betti diagram as above, where S denote the homogeneous coordinate ring of P^{6}. Consider the second syzygy matrix S^{7}(−2) ⊕ S(−3) ← S(−3) ⊕ S^{35}(−4) of S/I_{C}. The entries of the submatrix S^{7}(−2) ← S(−3) are given by linear forms (ℓ_{0}, …, ℓ_{6}) corresponding to a syzygy

0 ≠ γ := ∑_{i=0}^{6} ℓ_{i} ⊗ q_{i} ∈ Ker{H^{0}(P^{6}, O_{P}_{6}(1)) ⊗ H^{0}(P^{6}, I_{C/P^{6}}(2)) → H^{0}(P^{6}, I_{C/P^{6}}(3))}.

Definition 5.2. The syzygy scheme Syz(γ) of γ ∈ K_{2,1}(C, L) is the largest subscheme Y ⊂ P^{6} such that γ ∈ H^{0}(P^{6}, O_{P}_{6}(1)) ⊗ I_{Y} (2). The rank of γ is the dimension of the linear subspace ⟨ℓ_{0}, …, ℓ_{6}⟩ ⊂ H^{0}(P^{6}, O_{P}_{6}(1)).

We refer to [AN] and [vB] for general background on syzygy schemes. In case of rank 6 first order syzygies among quadrics, say ℓ_{0} = 0, there exists a 6 × 6 skewsymmetric matrix of linear forms (a_{ij})_{i,j=1}^{6} such that q_{i} = ∑_{j=1}^{6} ℓ_{j}a_{ji} for i = 1, …, 6 by [S, Lemma 4.3]. In this case the syzygy scheme is C′ = P^{6} ∩ X_{6}, where X_{6} ⊂ P^{20} := P_{ℓ_{i},a_{ij}} is the universal rank 6 syzygy scheme, given by equations {q_{i} = 0}_{i=1}^{6} and Pfaff((a_{ij})) = 0.

We consider first the case when γ ∈ K_{2,1}(C, L) is a rank 7 syzygy, which we can view as a section of the vector bundle Ω_{P^{6}}^{1}(3). By direct calculation using the Euler sequence, one finds c_{6}(Ω_{P^{6}}^{1}(3)) = 43, which is the number of zeroes of a general section of Ω_{P^{6}}^{1}(3). Since γ ∈ H^{0}(P^{6}, Ω_{P^{6}}^{1}(3) ⊗ I_{C}), to account for the contribution of C we use the excess intersection formula and from c_{6}(Ω_{P^{6}}^{1}(3)) we subtract

c_{1}(Ω_{P^{6}}^{1}(3)_{|C} − N_{C/P^{6}}) = deg(M_{L} ⊗ L^{⊗2}) − deg(N_{C/P^{6}}) = 11 · 14 − 8 · 14 = 42.

This shows that Z(γ) = C ∪ {p}, where p ∈ P^{6} − C. Choose now a 5-dimensional subspace V ⊂ H^{0}(P^{6}, I_{C/P^{6}}(2)) and let C′ ⊂ P^{6} be the corresponding linked curve. Since all the quadrics in H^{0}(P^{6}, I_{C/P^{6}}(2)) vanish at p, we find via the isomorphism (26) that the pencil K_{C′}(−1) ∈ W_{8}^{1}(C′) has a base point at C′. In conclusion, via linkage, to rank 7 syzygies γ ∈ K_{2,1}(C, L) correspond 7-gonal curves C′ of genus 14.

Assume now that γ is a rank 6 syzygy. Let p ∈ P^{6} denote the point defined by the 6 forms. The syzygy γ may be interpreted as a section in H^{0}(P^{6}, F(3)), where F is the first syzygy sheaf of the ideal sheaf I_{p/P^{6}}:

0 → F → O_{P^{6}}^{⊕6}(−1) → I_{p/P^{6}} → 0.

Since rk(F) = 5, the scheme Syz(γ) is expected to be 1-dimensional away from p. Indeed, the zero locus of a general global section of F(3) vanishes in p and a smooth half canonically embedded curve C′ of degree 21 and genus 22, see [ELMS, Theorem 4.4]. We refer to [ES] for more details.

In our case C′ is reducible, because C is a component of C′. In fact

C′ = C ∪ E,

where E is an elliptic normal curve of degree 7, intersecting C in 14 points. Moreover,

C · E ∈ |O_{E}(2)| and C · E ∈ |L^{⊗2} ⊗ K_{C}^{∨}|.

Observe that [C, L] ∈ R_{8} if and only if E · C ∈ |K_{C}|. Reversing this construction, we obtain the following result.

Theorem 5.3. The divisor 𝔎𝔬𝔰𝔷 ⊂ 𝔓𝔦𝔠_{8}^{14} is reducible and has a unirational component 𝔎𝔬𝔰𝔷_{6} whose general point is a paracanonical curve with extra syzygies of rank 6, as well as a component 𝔎𝔬𝔰𝔷_{7} whose general point is a curve with syzygies of rank 7. The syzygy scheme corresponding to a general point [C, L] ∈ 𝔎𝔬𝔰𝔷_{6} is the disjoint union of a point p ∈ P^{6} and a nodal half-canonical reducible curve C ∪ E of genus 22, where E is a smooth elliptic normal curve of degree 7 which intersects C in 14 distinct points.

Proof. In order to construct 𝔎𝔬𝔰𝔷_{6}, we reverse the construction and start with an elliptic normal curve E ⊂ P^{6} and a point p ∈ P^{6} − E. The minimal free resolution of E has the following Betti table:

              0  1  2  3  4 5
      total: 1 14 35 35 14 1
          0: 1  .  .  .  . .
          1: . 14 35 35 14 .
          2: .  .  .  .  . 1

The group K_{1,2}(E, O_{E}(1)) is 35-dimensional and there is a 22-dimensional vector space H^{0}(P^{6}, Ω_{P^{6}}^{1}(3) ⊗ I_{E∪{p}}) of second syzygies, whose order ideal vanishes at the point p. The zero locus of such a syzygy consists of p together with a half-canonical curve, having E as one of the components. Its residual curve is the desired point [C, O_{C}(1)] ∈ 𝔎𝔬𝔰𝔷. To show that such curves fill-up a component of 𝔎𝔬𝔰𝔷, we count dimensions. Modulo the action of P GL(7), septic elliptic curves E ⊂ P^{6} depend on one parameter. Indeed, if H_{7} ⊂ SL_{7}(ℂ) denotes the level 7 Heisenberg group, there is a 1 : 1 correspondence between elliptic curves with a level 7 structure and H_{7}-equivariantly embedded normal elliptic curves E ⊂ P^{6}. The choice of p ∈ P^{6} gives 6 dimensions. Having chosen E and p as above, the choice [γ] ∈ P(H^{0}(P^{6}, Ω_{P^{6}}^{1}(3) ⊗ I_{E∪{p}})) gives another 21 dimensions. Since the construction depends on 1 + 6 + 21 = dim(𝔓𝔦𝔠_{8}^{14}) − 1 parameters, curves obtained in this way fill-up a component 𝔎𝔬𝔰𝔷_{6} of 𝔎𝔬𝔰𝔷. To complete the proof of the unirationality of 𝔎𝔬𝔰𝔷_{6}, it remains to show that the construction leads to a smooth curve for general choices of parameters. By semi-continuity this can checked with Macaulay2 over a finite field, for details see the function unirationaliyOfD1 in our Macaulay2 package KoszulDivisorOnPic14M8.

By running the function getCurveOnKoszulDivisor contained in the Macaulay2 package KoszulDivisorOnPic14M8, we observe that there exist points [C, L] ∈ 𝔎𝔬𝔰𝔷 with syzygies 0 ≠ γ ∈ K_{1,2}(C, L) having full rank 7. They cannot lie in the closure of 𝔎𝔬𝔰𝔷_{6} for the rank of a syzygy is upper semi-continuous, therefore 𝔎𝔬𝔰𝔷_{6} ⊊ 𝔎𝔬𝔰𝔷. □

Remark 5.4. Since 𝔓𝔦𝔠_{8}^{14} is unirational [M], [Ve], over a finite field 𝔽_{p} one can find points on absolutely irreducible components of 𝔎𝔬𝔰𝔷 with probability approximately equal to \frac{1}{p}. By our experiment1 of the package KoszulDivisorOnPic14M8, we observe that the rate of points [C, L] ∈ 𝔎𝔬𝔰𝔷 corresponding to syzygies of rank 6 respectively 7 is approximately the same. We conclude that most likely, 𝔎𝔬𝔰𝔷 has precisely two components, namely 𝔎𝔬𝔰𝔷_{6} and 𝔎𝔬𝔰𝔷_{7}.

The remaining part of this section is devoted to obtain strong evidence for the inclusion R_{8} ⊂ 𝔎𝔬𝔰𝔷. Since we do not have a unirational parametrization of R_{8} (however see [FV3] for results in this direction), we are only able to perform Macaulay2 experiments with smooth curves from lower dimensional subvarieties of \overline{R}_{8}, for instance, we tested the curves constructed by the function getCanonicalCurveOfGenus8With2Torsion of our package PrymCanonicalCurves. All these calculations lead to curves in the component 𝔎𝔬𝔰𝔷_{6}, having an extra syzygy of rank 6. We first observe that in order to show the inclusion R_{8} ⊂ 𝔎𝔬𝔰𝔷, it suffices to show that general 1-nodal curves from the boundary divisor Δ0^{′} ⊂ \overline{R}_{8} lie in the locus \overline{Z}_{8,2}.

Proposition 5.5. Assuming that the inclusion Δ0^{′} ⊂ \overline{Z}_{8,2} holds, then \overline{Z}_{8,2} = \overline{R}_{8} and the Prym-Green Conjecture fails on R_{8}.

Proof. We proceed by contradiction and assume that \overline{Z}_{8,2} is a divisor on \overline{R}_{8} whose class is computed in Theorem 0.6. Precisely, we have the relation

[\overline{Z}_{8,2}]^{virt} − [\overline{Z}_{8,2}] = 27λ − 4(δ0^{′} + δ0^{′′}) − 6δ_{0}^{(1)} − [\overline{Z}_{8,2}] ∈ ℚ_{≥0}⟨δ0^{′}, δ0^{′′}, δ_{0}^{(1)}⟩.

First we observe that the class [\overline{Z}_{8,2}]^{virt} − 16δ0^{′′} is effective, that is, the morphism of vector bundles φ_{1,2} : ℍ_{1,2} → 𝔾_{1,2} over the stack R̃_{8,2} is degenerate with multiplicity at least 16 along the boundary divisor Δ0^{′′}. Indeed, assume that [X, η, ϕ] ∈ Δ0^{′′} is a general point corresponding to a normalization nor : C → X, where [C] ∈ M_{7}. Since nor^{∗}(η) = O_{C}, it quickly follows that K_{1,2}(X, ω_{X} ⊗η) = K_{1,2}(C, K_{C}) and the latter space is 16-dimensional. This shows that the class c_{1}(𝔾_{1,2} − ℍ_{1,2}) − 16δ0^{′′} is effective.

Assume now that Δ0^{′} ⊂ \overline{Z}_{8,2}, therefore [\overline{Z}_{8,2}]^{virt}−δ0^{′}−16δ0^{′′} = 27λ−5δ0^{′}−20δ0^{′′}−6δ_{0}^{(1)} is still effective. Combining this class with that of the pull-back of the Brill-Noether divisor [\overline{M}_{8,7}^{2}] = 22λ − 3δ_{0} ∈ Eff(M̃_{8}), see [EH], after routine manipulations we can form an effective representative of the canonical class K_{\overline{R}_{8}}. This is a contradiction, since \overline{R}_{8} is uniruled, see [FV3]. □

Even though we do not know whether Δ0^{′} is unirational, we quote from [FV3]:

Theorem 5.6. [FV3] The Prym moduli space R_{7} is unirational.

Sketch of proof. A general Prym canonical curve C ⊂ P^{5} of genus 7 has the Betti table

              0  1  2  3 4
      total: 1 11 27 24 7
          0: 1  .  .  . .
          1: .  3  .  . .
          2: .  8 27 24 7

The quadrics in H^{0}(P^{5}, I_{C/P^{5}}(2)) intersect in a Nikulin surface Y ⊂ P^{5}, that is, a smooth K3 surface containing 8 disjoint lines L_{1}, …, L_{8} with L_{i}^{2} = −2 and C · L_{i} = 0, for i = 1, …, 8. Furthermore, 2(C − H) ≡ L_{1} + ⋯ + L_{8}. The linear system

|O_{Y} (C − L_{1} − … − L_{7})|

is zero dimensional and consists of a single rational normal quintic curve R_{5} ⊂ P^{5}. The unirational parametrization of R_{7} reverses this construction. One starts with a fixed rational normal curve R_{5} ⊂ P^{5} together with seven general secant lines L_{1}, …, L_{8}. The union R_{5} ∪ L_{1} ∪ … ∪ L_{7} is a 14-nodal stable curve which lies on three quadrics intersecting in a smooth Nikulin surface Y. A general C ∈ |O_{Y} (R_{5} + L_{1} + … + L_{7})| is a general curve in R_{7}. This establishes a dominant rational map from a P^{7}-bundle over \overline{M}_{0,14} onto R_{7}. In the function randomPrymCanonicalCurveOfGenus7 of our package PrymCanonicalCurves you find an ”implementation” of the unirational parametrization.

□

Based on Theorem 5.6 we have strong evidence that the inclusion Δ′_{0} ⊂ R_{8} ∩ Z_{1} holds, therefore also for the equality \overline{Z}_{8,2} = \overline{R}_{8}. We start with a randomly chosen curve in C ∈ R_{7}(𝔽_{p}) for a prime p with 10^{4} < p < 3 · 10^{4}. By the Hasse-Weil theorem, one has the following estimate for the number of rational points of a curve C over 𝔽_{p}:

p + 1 − 2g\sqrt{p} ≤ |C(𝔽_{p})| ≤ p + 1 + 2g\sqrt{p}.

Rational points are easy to find: We consider a plane model C_{pl} ⊂ P^{2} of C defined over 𝔽_{p}. At least about 50% of the lines L ∈ (P^{2})^{∨}(𝔽_{p}) intersect C in at least one 𝔽_{p}-rational point. Indeed there are p^{2} + O(p) lines. The set

{(L, P) ∈ (P^{2})^{∨}(𝔽_{p}) × C(𝔽_{p}) | P ∈ L}

has about p^{2} + O(p^{3/2}) points, since there are p + 1 lines through a given point. The secant line spanned by two points of C(𝔽_{p}) are counted twice in this set. Thus at least half the lines, i.e. p^{2}− \frac{1}{2}p^{2}+O(p^{3/2}) intersect C in at least one 𝔽_{p}-rational point. (A better estimate should be 1 − \frac{1}{2} + \frac{1}{3!} − \frac{1}{4!} + … = 1 − \frac{1}{e} ≈ 63% for the fraction of 𝔽_{p}-rational lines which intersect C in at least one 𝔽_{p}-rational point.)

We find the 𝔽_{p}-rational intersection points by computing the primary decomposition of I_{C_{pl}}+I_{L} ⊂ 𝔽_{p}[x, y, z]. A probabilistic algorithm gives us two points P, Q ∈ C(𝔽_{p}). Consider now the curve [C′ = \frac{C}{P ∼Q}] ∈ Δ_{0} ⊂ \overline{M}_{8} obtained by identifying P and Q, and let nor : C → C′ be the normalization map. We now apply the following:

Proposition 5.7. Let nor : C → C′ be the partial normalization of a single node of a irreducible nodal curve C′ of genus g defined over a field 𝔽 and let P, Q ∈ C(𝔽) be the preimages of the node of C′. Let η = O_{C}(D_{1} − D_{2}) ∈ Pic^{0}C[2] be a 2-torsion line bundle on C curve with D_{1}, D_{2} being effective divisors defined over 𝔽 with support disjoint from {P, Q}. Then there exists a two torsion line bundle η′ ∈ Pic^{0}C[2] defined over 𝔽 with nor^{∗}η′ ≅ η if and only if \frac{f(P)}{f(Q)} ∈ (𝔽^{∗})^{2} where f ∈ 𝔽(C) is the rational function with (f) = 2D_{1} − 2D_{2}.

Proof. Consider an embedding C ⊂ P^{r} defined over 𝔽, e.g. a (pluri)canonical model. Suppose \frac{f(P)}{f(Q)} = α^{2}. Consider an arbitrary rational function g = \frac{g_{0}}{g_{1}} ∈ 𝔽(C), where g_{0}, g_{1} are linear forms on P^{r}. We choose a matrix

(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ SL(2, 𝔽)

such that the rational function h = \frac{ag_{0}+bg_{1}}{cg_{0}+dg_{1}} takes values h(P) = 1 and h(Q) = α. Let

(h) = (h)_{0} − (h)_{∞} be the principal divisor of h on C. The divisor E_{1} − E_{2} with E_{1} = D_{1}+(h)_{0} and E_{2} = D_{2}+(h)_{∞} is 2-torsion on C′ since \frac{(fh)(P)}{(fh)(Q)} = 1 and nor^{∗}O_{C′}(E_{1}−E_{2}) = O_{C}(D_{1} − D_{2}), because (E_{1} − E_{2}) − (D_{1} − D_{2}) is a principal divisor on C. This proves that the condition is sufficient. To establish necessity, we note that 1 ∈ (𝔽^{∗})^{2} □

Thus for a finite field of char(𝔽_{p}) ≠ 2, in about 50% of the cases we can descend the 2-torsion bundle from C to C′. In this way we find many 𝔽_{p}-rational points on Δ′_{0}. Computing the syzygies of the resulting Prym canonical curves we always land in the component 𝔎𝔬𝔰𝔷_{6}, see randomOneNodalPrymCanonicalCurveOfGenus8 in our package PrymCanonicalCurves. This does not establish the inclusion Δ′_{0} ⊂ 𝔎𝔬𝔰𝔷_{6} (hence the equality Z_{g,2} = R_{8}), since for any fixed prime p, we only have finitely many points in Δ′_{0}(𝔽_{p}). However, the possibility that Δ′_{0} ⊄ 𝔎𝔬𝔰𝔷_{6} appears to be exceedingly unlikely.

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[ES] F. Eusen and F.-O. Schreyer, A remark on a conjecture of Paranjape and Ramanan, preprint available at http://www.math.uni-sb.de/˜ agschreyer/PS/eus.ps, to appear in Geometry and Arithmetic Schiermonnikoog 2010 (C. Faber, G. Farkas, R. de Jong eds), EMS 2012.

[F1] G. Farkas, Koszul divisors on moduli spaces of curves, American Journal of Math. 131 (2009), 819-869.

[FL] G. Farkas and K. Ludwig, The Kodaira dimension of the moduli space of Prym varieties, Journal of the European Math. Society 12 (2010), 755-795.

[FMP] G. Farkas, M. Mustaţă and M. Popa, Divisors on M_{g,g+1} and the Minimal Resolution Conjecture for points on canonical curves, Annales Scientifique École Normale Supérieure 36 (2003), 553-581.

[FP] G. Farkas and M. Popa, Effective divisors on \overline{M}_{g}, curves on K3 surfaces and the Slope Conjecture, Journal of Algebraic Geometry 14 (2005), 151-174.

[FV1] G. Farkas and A. Verra, The classification of the universal Jacobian over the moduli space of curves, arXiv:1005.5354, to appear in Commentarii Math. Helvetici.

[FV2] G. Farkas and A. Verra, Moduli of theta characteristics via Nikulin surfaces, arXiv:1104.0273, to appear in Mathematische Annalen.

[FV3] G. Farkas and A. Verra, R_{8} is uniruled, preprint.

[GS] D. Grayson and M. Stillman, Macaulay2, a software system for research in algebraic geometry, http://www.math.uiuc.edu/Macaulay2/

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[J1] T. Jarvis, Geometry of the moduli of higher spin curves, International J. of Math., 11 (2000), 637-663.

[J2] T. Jarvis, The Picard group of the moduli of higher spin curves, New York J. of Math. 7 (2001), 23-47.

[M] S. Mukai, Curves and Grassmannians, in: Algebraic Geometry and Related Topics, Inchon 1992, (J.-H. Yang, Y. Namikawa, K. Ueno, eds), 1992, 19-40, International Press.

[S] F.-O. Schreyer, A standard basis approach to syzygies of canonical curves, J. Reine Angew. Math. 421 (1991), 83-123.

[vB] H.-C. Graf v. Bothmer, Generic syzygy schemes, J. Pure Appl. Algebra 208 (2007), 867-876.

[Ve] A. Verra, The unirationality of the moduli space of curves of genus 14 or lower, Compositio Math. 141 (2005), 1425-1444.

[V1] C. Voisin, Green’s canonical syzygy conjecture for generic curves of odd genus, Compositio Math. 141 (2005), 1163-1190.

[V2] C. Voisin, Green’s generic syzygy conjecture for curves of even genus lying on a K3 surface, Journal of European Mathematical Society 4 (2002), 363-404.

Université de Grenoble, U.M.R. CNRS 5582, Institut Fourier,

BP 74, 38402 Saint Martin d’Hres, France

      E-mail address: chiodo@ujf-grenoble.fr

University of California, Department of Mathematics, Berkeley, CA 94720, USA

      E-mail address: eisenbud@math.berkeley.edu

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

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

Uversität des Saarlandes, Campus E2 4, D-66123 Saarbrücken, Germany

      E-mail address: schreyer@math.uni-sb.de