BRILL-NOETHER GEOMETRY ON MODULI SPACES OF SPIN CURVES

GAVRIL FARKAS

The aim of this paper is to initiate a study of geometric divisors of Brill-Noether type on the moduli space \overline{S}_{g} of spin curves of genus g. The moduli space \overline{S}_{g} is a compactification of the parameter space S_{g} of pairs [C, η], consisting of a smooth genus g curve C and a theta-characteristic η ∈ Pic^{g−1}(C), see [C]. The study of the birational properties of \overline{S}_{g} as well as other moduli spaces of curves with level structure has received an impetus in recent years, see [BV] [FL], [F2], [Lud], to name only a few results. Using syzygy divisors, it has been proved in [FL] that the Prym moduli space \overline{R}_{g} := \overline{M}_{g}(Bℤ_{2}) parameterizing curves together with a point of order 2 in the Jacobian variety, is a variety of general type for g ≥ 13 and g ≠ 15. The moduli space \overline{S}_{g}^{+} is known to be of general type for g > 8, uniruled for g < 8, see [F2], whereas the Kodaira dimension of \overline{S}_{8}^{+} is equal to zero, [FV]. This was the first example of a naturally defined moduli space of curves of genus g ≥ 2, having intermediate Kodaira dimension.

Virtually all attempts to show that a certain moduli space \overline{M}_{g,n} is of general type, rely on the calculation of certain effective divisors D ⊂ \overline{M}_{g,n} enjoying extremality properties in their effective cones Eff(\overline{M}_{g,n}), so that the canonical class K_{\overline{M}_{g,n}} lies in the cone spanned by [D], boundary classes δ_{i:S}, tautological classes λ, ψ_{1}, …, ψ_{n}, and possible other effective geometric classes. Examples of such a program being carried out, can be found in [EH2], [HM]-for the case of Brill-Noether divisors on \overline{M}_{g} consisting of curves with a 𝔤_{d}^{r} when ρ(g, r, d) = −1, [Log]-where pointed Brill-Noether divisors on \overline{M}_{g,n} are studied, and [F1]-for the case of Koszul divisors on \overline{M}_{g}, which provide counterexamples to the Slope Conjecture on \overline{M}_{g}. A natural question is what should be the analogous geometric divisors on the spin moduli space of curves \overline{S}_{g}?

In this paper we propose a construction and compute the class of spin Brill-Noether divisors on both \overline{S}_{g}^{+} and \overline{S}_{g}^{−}, defined in terms of difference varieties on Jacobians of curves. Precisely, we fix integers r, s ≥ 1 such that d := rs + r ≡ 0 mod 2, and then set g := rs + s. One can write d = 2i. By standard Brill-Noether theory, a general curve [C] ∈ M_{g} carries a finite number of (necessarily complete and base point free) linear series 𝔤_{d}^{r}. One considers the following loci of spin curves (both odd and even)

U_{g,d}^{r} := {[C, η] ∈ S_{g}^{∓} : ∃L ∈ W_{d}^{r}(C) such that η ⊗ L^{∨} ∈ C_{g−i−1} − C_{i}}.

Thus U_{g,d}^{r} consists of spin curves such that the embedded curve C ⟶^{|η⊗L|} P^{d−1} admits an i-secant (i − 2)-plane. We shall prove that for s ≥ 2, the locus U_{g,d}^{r} is always a divisor on S_{g}^{∓}, and we find a formula for the class of its compactification in \overline{S}_{g}^{∓}. For simplicity, we display this formula in the introduction only in the case r = 1, when g ≡ 2 mod 4:

Research partially supported by Sonderforschungsbereich ”Raum-Zeit-Materie”.

Theorem 0.1. We fix an integer a ≥ 1 and set g := 4a + 2. The locus

U_{4a+2,2a+2}^{1} := {[C, η] ∈ S_{4a+2}^{∓} : ∃L ∈ W_{2a+2}^{1}(C) such thatη ⊗ L^{∨} ∈ C_{3a} − C_{a+1}}

is an effective divisor and the class of its compactification in \overline{S}_{g}^{∓} is given by

\overline{U}_{4a+2,2a+2}^{1} ≡ \binom{4a}{a}\binom{4a + 2}{2a}\frac{a + 2}{8(2a + 1)(4a + 1)}((192a^{3} + 736a^{2} + 692a + 184)λ−

−(32a^{3} + 104a^{2} + 82a + 19)α_{0} − (64a^{3} + 176a^{2} + 148a + 36)β_{0} − ⋯) ∈ Pic(\overline{S}_{g}^{∓}).

To specialize further, we set a = 1 and find the expression of the class of spin curves [C, η] ∈ S_{6}^{∓} such that there exists a pencil L ∈ W_{4}^{1}(C) with the linear series

C ⟶^{|η⊗L|} P^{3} being not very ample:

\overline{U}_{6,4}^{1} ≡ 451λ − \frac{237}{4}α_{0} − 106β_{0} − ⋯ ∈ Pic(\overline{S}_{6}^{∓}).

The case s = 1 when necessarily L = K_{C} ∈ W_{2g−2}^{g−1}(C), produces a divisor only on

\overline{S}_{g}^{+}, and we recover the main calculation from [F2], used to prove that \overline{S}_{g}^{+} is a variety of general type for g > 8. We recall that Θ_{null} := {[C, η] ∈ S_{g}^{+} : H^{0}(C, η) ≠ 0} denotes the divisor of vanishing theta-nulls.

Theorem 0.2. Let π : \overline{S}_{g}^{+} → \overline{M}_{g} be the ramified covering which forgets the spin structure. For g ≥ 3, one has the following equality \overline{U}_{g,2g−2}^{g−1} = 2 · \overline{Θ}_{null} of codimension 1-cycles on the open subvariety π^{−1}(M_{g} ∪ Δ_{0}) of \overline{S}_{g}^{+}. Moreover, there is an equality of classes

\overline{U}_{g,2g−2}^{g−1} ≡ 2 · \overline{Θ}_{null} ≡ \frac{1}{2}λ − \frac{1}{8}α_{0} − 0 · β_{0} − ⋯ ∈ Pic(\overline{S}_{g}^{+}).

We remark once more, the low slope of the divisor \overline{Θ}_{null}. No similar divisor is known to exist on \overline{R}_{g}.

In the last section of the paper, we study the divisor \overline{Θ}_{g,1} on the universal curve \overline{M}_{g,1}, which consists of points in the support of odd theta-characteristics. This divisor, somewhat similar to the divisor W_{g} of Weierstrass points on \overline{M}_{g,1}, cf. [Cu], should be of importance in the study of the birational geometry of \overline{M}_{g,1}:

Theorem 0.3. The class of the compactification in \overline{M}_{g,1} of the effective divisor

Θ_{g,1} := {[C, q] ∈ M_{g,1} : q ∈ supp(η) for some [C, η] ∈ S_{g}^{−}}

is given by the following formula:

\overline{Θ}_{g,1} ≡ 2^{g−3}((2^{g} −1)(λ+2ψ)−2^{g−3}δ_{irr}−(2^{g} −2)δ_{1}−∑_{i=1}^{g−1}(2^{i}+1)(2^{g−i}−1)δ_{i}) ∈ Pic(\overline{M}_{g,1}).

When g = 2, the divisor Θ_{2} specializes to the divisor of Weierstrass points:

Θ_{2,1} = W_{2} := {[C, q] ∈ M_{2,1} : q ∈ C is a Weierstrass point}.

If we use Mumford’s formula λ = δ_{0}/10 + δ_{1}/5 ∈ Pic(\overline{M}_{2}), Theorem 0.3 reads

\overline{Θ}_{2,1} ≡ \frac{3}{2}λ + 3ψ − \frac{1}{4}δ_{irr} − \frac{3}{2}δ_{1} = −λ + 3ψ − δ_{1} ∈ Pic(\overline{M}_{2,1}),

that is, we recover the formula for the class of the Weierstrass divisor on \overline{M}_{2,1}, cf. [EH2]. The next case is g = 3: The condition [C, q] ∈ Θ_{3,1}, states that the point q ∈ C lies on one

of the 28 bitangent lines of the canonically embedded curve C ⟶^{|K_{C}|} P^{2}.

Corollary 0.4. The class of the compactification in \overline{M}_{3,1} of the bitangent locus Θ_{3,1} := {[C, q] ∈ M_{3,1} : q lies on a bitangent ofC}

is equal to \overline{Θ}_{3,1} ≡ 7λ + 14ψ − δ_{irr} − 9δ_{1} − 5δ_{2} ∈ Pic(\overline{M}_{3,1}).

If p : \overline{M}_{g,1} → \overline{M}_{g} is the map forgetting the marked point, we note the equality

\overline{D}_{3} ≡ p^{∗}(\overline{M}_{3,2}^{1}) + 2 · \overline{W}_{3} + 2ψ ∈ Pic(\overline{M}_{3,1}),

where \overline{W}_{3} ≡ −λ + 6ψ − 3δ_{1} − δ_{2} is the divisor of Weierstrass points on \overline{M}_{3,1}. Since the class ψ ∈ Pic(\overline{M}_{3,1}) is big and nef, it follows that \overline{Θ}_{3,1} (unlike the divisor \overline{Θ}_{2,1} ∈ Pic(\overline{M}_{2,1})), lies in the interior of the cone of effective divisors Eff(\overline{M}_{3,1}), or it other words, it is big. In particular, it cannot be contracted by a rational map \overline{M}_{3,1} −−→ X to any projective variety X. This phenomenon extends to all higher genera:

Corollary 0.5. For every g ≥ 3, the divisor \overline{Θ}_{g,1} ∈ Eff(\overline{M}_{g,1}) is big.

1. Generalities about \overline{S}_{g}

We recall basic facts about Cornalba’s stack of stable spin curves π : \overline{S}_{g} → \overline{M}_{g}, see [C], [F2] for details. If X is a nodal curve, a smooth rational component R ⊂ X is said to be exceptional if #(R ∩ \overline{X − R}) = 2. The curve X is said to be quasi-stable if #(R ∩ \overline{X − R}) ≥ 2 for any smooth rational component R ⊂ X, and moreover, any two exceptional components of X are disjoint. A quasi-stable curve is obtained from a stable curve by possibly inserting a rational curve at each of its nodes. We denote by [st(X)] ∈ \overline{M}_{g} the stable model of the quasi-stable curve X.

Definition 1.1. A spin curve of genus g consists of a triple (X, η, β), where X is a genus g quasi-stable curve, η ∈ Pic^{g−1}(X) is a line bundle of degree g − 1 such that η_{R} = O_{R}(1) for every exceptional component R ⊂ X, and β : η^{⊗2} → ω_{X} is a sheaf homomorphism which is generically non-zero along each non-exceptional component of X.

Stable spin curves of genus g form a smooth Deligne-Mumford stack \overline{S}_{g} which splits into two connected components \overline{S}_{g}^{+} and \overline{S}_{g}^{−}, according to the parity of h^{0}(X, η). Let f : C → \overline{S}_{g} be the universal family of spin curves of genus g. In particular, for every point [X, η, β] ∈ \overline{S}_{g}, there is an isomorphism between f^{−1}([X, η, β]) and the quasistable curve X. There exists a (universal) spin line bundle P ∈ Pic(C) of relative degree g − 1, as well as a morphism of O_{C}-modules B : P^{⊗2} → ω_{f} having the property that P_{|f^{−1}([X,η,β])} = η and B_{|f^{−1}([X,η,β])} = β : η^{⊗2} → ω_{X}, for all spin curves [X, η, β] ∈ \overline{S}_{g}.

1.1. The boundary divisors of \overline{S}_{g}.

We recall a few things about the boundary divisors of \overline{S}_{g}. We concentrate on the case of \overline{S}_{g}^{+}, the differences compared to the situation on \overline{S}_{g}^{−} being minor. We describe the pull-backs of the boundary divisors Δ_{i} ⊂ \overline{M}_{g} under the map π. First we fix an integer 1 ≤ i ≤ [g/2] and let [X, η, β] ∈ π^{−1}([C ∪_{y} D]), where [C, y] ∈ M_{i,1} and [D, y] ∈ M_{g−i,1}. For degree reasons, then X = C∪_{y_{1}} R∪_{y_{2}} D, where R is an exceptional (rational) component such that C ∩ R = {y_{1}} and D ∩ R = {y_{2}}. Moreover η = (η_{C}, η_{D}, η_{R} = O_{R}(1)) ∈ Pic^{g−1}(X), where η_{C}^{⊗2} = K_{C}, η_{D}^{⊗2} = K_{D}. The theta-characteristics η_{C} and η_{D} have the same parity in the case of \overline{S}_{g}^{+} (and opposite parities for \overline{S}_{g}^{−}). One denotes by A_{i} ⊂ \overline{S}_{g}^{+} the closure of the locus corresponding to pairs of pointed spin curves

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

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

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

If α := [A_{i}], β_{i} := [B_{i}] ∈ Pic(\overline{S}_{g}^{+}), we have the relation π^{∗}(δ_{i}) = α_{i} + β_{i}.

Next, we describe π^{∗}(δ_{0}) and pick a stable spin curve [X, η, β] such that st(X) = C_{yq} := C/y ∼ q, with [C, y, q] ∈ M_{g−1,2}. There are two possibilities depending on whether X possesses an exceptional component or not. If X = C_{yq} and η_{C} := ν^{∗}(η) where ν : C → X denotes the normalization map, then η_{C}^{⊗2} = K_{C}(y + q). For each choice of η_{C} ∈ Pic^{g−1}(C) as above, there is precisely one choice of gluing the fibres η_{C}(y) and η_{C}(q) such that h^{0}(X, η) ≡ 0 mod 2. We denote by A_{0} the closure in \overline{S}_{g}^{+} of the locus of points [C_{yq}, η_{C} ∈ \sqrt{K_{C}(y + q)}] as above.

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

[C ∪_{\{y,q\}} R, η_{C} ∈ \sqrt{K_{C}}, η_{R} = O_{R}(1)] ∈ \overline{S}_{g}^{+}.

A local analysis shows that B_{0} is the branch locus of π and the ramification is simple. If α_{0} = [A_{0}] ∈ Pic(\overline{S}_{g}^{+}) and β_{0} = [B_{0}] ∈ Pic(\overline{S}_{g}^{+}), we have the relation

(1)  π^{∗}(δ_{0}) = α_{0} + 2β_{0}.

2. Difference varieties and theta-characteristics

We describe a way of calculating the class of a series of effective divisors on both \overline{S}_{g}^{−} and \overline{S}_{g}^{+} defined in terms of the relative position of a theta-characteristic with respect to the divisorial difference varieties in the Jacobian of a curve. These loci, which should be thought of as divisors of Brill-Noether type on \overline{S}_{g}, inherit a determinantal description over the entire moduli stack of spin curves, via the interpretation of difference varieties in Pic^{g−2i−1}(C) as Raynaud theta-divisors for exterior powers of Lazarsfeld bundles provided in [FMP]. The determinantal description is then extended over a partial compactification S̃_{g} of S_{g}, using the explicit description of stable spin curves. The formulas we obtain for the class of these divisors are identical over both \overline{S}_{g}^{−} and \overline{S}_{g}^{+}, therefore we sometimes use the symbol \overline{S}_{g}^{∓} (or even \overline{S}_{g}), to denote one of the two spin moduli spaces.

We start with a curve [C] ∈ M_{g} and we denote as usual by Q_{C} := M_{K_{C}}^{∨} the associated Lazarsfeld bundle defined via the exact sequence

0 → M_{K_{C}} → H^{0}(C, K_{C}) ⊗ O_{C} →^{ev} K_{C} → 0.

Note that M_{K_{C}} is a semistable vector bundle on C (even stable, when the curve C is non-hyperelliptic), and μ(Q_{K_{C}}) = 2. For integers 0 ≤ i ≤ g −1, one defines the divisorial difference variety C_{g−i−1} − C_{i} ⊂ Pic^{g−2i−1}(C) as being the image of the difference map

ϕ : C_{g−i−1} × C_{i} → Pic^{g−2i−1}(C), ϕ(D, E) := O_{C}(D − E).

The main result from [FMP] provides a scheme-theoretic identification of divisors on the Jacobian variety

(2)  C_{g−i−1} − C_{i} = Θ_{∧^{i}Q_{C}} ⊂ Pic^{g−2i−1}(C),

where the right-hand-side denotes the Raynaud locus [R]

Θ_{∧^{i}Q_{C}} := {η ∈ Pic^{g−2i−1}(C) : H^{0}(C, ∧^{i}Q_{C} ⊗ η) ≠ 0}.

We fix integers r, s > 0 and set d := rs + r, g := rs + s, therefore the Brill-Noether number ρ(g, r, d) = 0. We assume moreover that d ≡ 0 mod 2, that is, either r is even or s is odd, and write d = 2i. We define the following locus in the spin moduli space S_{g}^{∓}:

U_{g,d}^{r} := {[C, η] ∈ S_{g}^{∓} : ∃L ∈ W_{d}^{r}(C) such that η ⊗ L^{∨} ∈ C_{g−i−1} − C_{i}}.

Using (2), the condition [C, η] ∈ U_{g,d}^{r} can be rewritten in a determinantal way as,

H^{0}(C, ∧^{i}M_{K_{C}} ⊗ η ⊗ L) ≠ 0.

Tensoring by η ⊗ L the exact sequence coming from the definition of M_{K_{C}}, namely

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

then taking global sections and finally using that M_{K_{C}} (hence all of its exterior powers) are semi-stable vector bundles, we find that [C, η] ∈ U_{g,d}^{r} if and only if the map

(3) ϕ(C, η, L) : ∧^{i}H^{0}(C, K_{C}) ⊗ H^{0}(C, η ⊗ L) → H^{0}(C, ∧^{i−1}M_{K_{C}} ⊗ K_{C} ⊗ η ⊗ L)

is not an isomorphism for a certain L ∈ W_{d}^{r}(C). Since μ(∧^{i−1}M_{K_{C}} ⊗K_{C} ⊗η⊗L) ≥ 2g−1 and ∧^{i−1}M_{K_{C}} is a semi-stable vector bundle on C, it follows that

h^{0}(C, ∧^{i−1}M_{K_{C}} ⊗ K_{C} ⊗ η ⊗ L) = χ(C, ∧^{i−1}M_{K_{C}} ⊗ K_{C} ⊗ η ⊗ L) = \binom{g}{i}d.

We assume that h^{1}(C, η ⊗ L) = 0. This condition is satisfied outside a locus of S_{g}^{∓} of codimension at least 2; if H^{1}(C, η ⊗ L) ≠ 0, then H^{1}(C, K_{C} ⊗ L^{⊗(−2)}) ≠ 0, in particular the Petri map μ_{0}(C, L) : H^{0}(C, L)⊗H^{0}(C, K_{C} ⊗L^{∨}) → H^{0}(C, K_{C}) is not injective. Then h^{0}(C, L ⊗ η) = d and we note that ϕ(C, η, L) is a map between vector spaces of the same rank. This obviously suggests a determinantal presentation of U_{g,d}^{r} as the (push-forward of) a degeneracy locus between vector bundles of the same rank. In what follows we extend this presentation over a partial compactification of \overline{S}_{g}^{∓}. We refer to [FL] Section 2 for a similar calculation over the Prym moduli stack \overline{R}_{g}.

We denote by M_{g}^{0} ⊂ M_{g} the open substack classifying curves [C] ∈ M_{g} such that W_{d−1}^{r}(C) = ∅, W_{d}^{r+1}(C) = ∅ and moreover H^{1}(C, L ⊗ η) = 0, for every L ∈ W_{d}^{r}(C) and each odd-theta characteristic η ∈ Pic^{g−1}(C). From general Brill-Noether theory one knows that codim(M_{g} − M_{g}^{0}, M_{g}) ≥ 2. Then we define Δ̃_{0} ⊂ Δ_{0} to be the open substack consisting of 1-nodal stable curves [C_{yq} := C/y ∼ q], where [C] ∈ M_{g−1} is a curve satisfying the Brill-Noether theorem and y, q ∈ C. We then set \overline{M}_{g}^{0} := M_{g}^{0} ∪ Δ̃_{0}, hence \overline{M}_{g}^{0} ⊂ M̃_{g} and then \overline{S}_{g}^{0} := π^{−1}(\overline{M}_{g}^{0}) = (\overline{S}_{g}^{0})^{+} ∪ (\overline{S}_{g}^{0})^{−}. Following [EH1], [F1], we consider the proper Deligne-Mumford stack

σ_{0} : 𝔊_{d}^{r} → \overline{M}_{g}^{0}

classifying pairs [C, L] with [C] ∈ \overline{M}_{g}^{0} and L ∈ W_{d}^{r}(C). For any curve [C] ∈ \overline{M}_{g}^{0} and L ∈ W_{d}^{r}(C) we have that h^{0}(C, L) = r + 1, that is, 𝔊_{d}^{r} parameterizes only complete linear series. For a point [C_{yq} := C/y ∼ q] ∈ Δ̃_{0}, we have the identification

σ_{0}^{−1}[C_{yq}] = {L ∈ W_{d}^{r}(C) : h^{0}(C, L ⊗ O_{C}(−y − q)) = r},

that is, we view linear series on singular curves as linear series on the normalization such that the divisor of the nodes imposes only one condition. We denote by f_{d}^{r} : ℭ_{g,d}^{r} := \overline{M}_{g,1}^{0} ×\overline{M}{0 \atop g} 𝔊_{d}^{r} → 𝔊_{d}^{r} the pull-back of the universal curve p : \overline{M}_{g,1}^{0} → \overline{M}_{g}^{0} to 𝔊_{d}^{r}. Once we have chosen a Poincaré bundle L on ℭ_{g,d}^{r} we can form the three codimension 1 tautological classes in A^{1}(𝔊_{d}^{r}): (4) 𝔞 := (f_{d}^{r})_{∗}(c_{1}(L)^{2}), 𝔟 := (f_{d}^{r})_{∗}(c_{1}(L) · c_{1}(ω_{f_{d}^{r}})), 𝔠 := (f_{d}^{r})_{∗}(c_{1}(ω_{f_{d}^{r}})^{2}) = (σ_{0})^{∗}((κ_{1})\overline{M}{0 \atop g}).

The dependence on 𝔞, 𝔟, 𝔠 on the choice of L is discussed in both [F2] and [FL]. We introduce the stack of 𝔤_{d}^{r}’s on spin curves σ : 𝔊_{d}^{r}(\overline{S}_{g}^{0}/\overline{M}_{g}^{0}) := \overline{S}_{g}^{0} ×\overline{M}{0 \atop g} 𝔊_{d}^{r} → \overline{S}_{g}^{0} and then the corresponding universal spin curve over the 𝔤_{d}^{r} parameter space

f′ : C_{d}^{r} := C ×\overline{S}{0 \atop g} 𝔊_{d}^{r}(\overline{S}_{g}^{0}/\overline{M}_{g}^{0}) → 𝔊_{d}^{r}(\overline{S}_{g}^{0}/\overline{M}_{g}^{0}).

We note that f′ is a family of quasi-stable curves carrying at the same time a spin structure as well as a 𝔤_{d}^{r}. Just like in [FL], the boundary divisors of 𝔊_{d}^{r}(\overline{S}_{g}^{0}/\overline{M}_{g}^{0}) are denoted by the same symbols, that is, one sets A′_{0} := σ^{∗}(A′_{0}) and B′_{0} := σ^{∗}(B′_{0}) and then α_{0} := [A′_{0}], β_{0} := [B′_{0}] ∈ A^{1}(𝔊_{d}^{r}(\overline{S}_{g}^{0}/\overline{M}_{g}^{0})). We observe that two tautological line bundles live on C_{d}^{r}, namely the pull-back of the universal spin bundle P_{d}^{r} ∈ Pic(C_{d}^{r}) and a Poincaré bundle L ∈ Pic(C_{d}^{r}) singling out the 𝔤_{d}^{r}’s, that is, L_{|f^{′−1}[X,η,β,L]} = L ∈ W_{d}^{r}(C), for each point

[X, η, β, L] ∈ 𝔊_{d}^{r}(\overline{S}_{g}^{0}/\overline{M}_{g}^{0}). Naturally, one also has the classes 𝔞, 𝔟, 𝔠 ∈ A^{1}(𝔊_{d}^{r}(\overline{S}_{g}^{0}/\overline{M}_{g}^{0})) defined by the formulas (4).

The following result is easy to prove and we skip details:

Proposition 2.1. We denote by f′ : C_{d}^{r} → 𝔊_{d}^{r}(\overline{S}_{g}^{0}/\overline{M}_{g}^{0}) the universal quasi-stable spin curve and by P_{d}^{r} ∈ Pic(C_{d}^{r}) the universal spin bundle of relative degree g − 1. One has the following formulas in A^{1}(𝔊_{d}^{r}(\overline{S}_{g}^{0}/\overline{M}_{g}^{0})):

(i) f′_{∗}(c_{1}(ω_{f′}) · c_{1}(P_{d}^{r})) = \frac{1}{2}𝔠. (ii) f′_{∗}(c_{1}(P_{d}^{r})^{2}) = \frac{1}{4}𝔠 − \frac{1}{2}β_{0}. (iii) f′_{∗}(c_{1}(L) · c_{1}(P_{d}^{r})) = \frac{1}{2}𝔟.

We determine the class of a compactification of U_{g,d}^{r} by pushing-forward a codi-

mension 1 degeneracy locus via the map σ : 𝔊_{d}^{r}(\overline{S}_{g}^{0}/\overline{M}_{g}^{0}) → \overline{S}_{g}^{0}. To that end, we define a

sequence of tautological vector bundles on 𝔊_{d}^{r}(\overline{S}_{g}^{0}/\overline{M}_{g}^{0}): First, for l ≥ 0 we set

A_{0,l} := f′_{∗}(L ⊗ ω_{f′}^{⊗l} ⊗ P_{d}^{r}).

It is easy to verify that R^{1}f′_{∗}(L⊗ω_{f′}^{⊗l} ⊗P_{d}^{r}) = 0, hence A_{0,l} is locally free over 𝔊_{d}^{r}(\overline{S}_{g}^{0}/\overline{M}_{g}^{0}) of rank equal to h^{0}(X, L ⊗ ω_{X}^{⊗l} ⊗ η) = l(2g − 2) + d. Next we introduce the global Lazarsfeld vector bundle M over C_{d}^{r} by the exact sequence

0 ⟶ M ⟶ (f′)^{∗}(f′_{∗}ω_{f′}) ⟶ ω_{f′} ⟶ 0,

and then for all integers a, j ≥ 1 we define the sheaf over 𝔊_{d}^{r}(\overline{S}_{g}^{0}/\overline{M}_{g}^{0})

A_{a,j} := f′_{∗}(∧^{a}M ⊗ ω_{f′}^{⊗j} ⊗ L ⊗ P_{d}^{r}).

In a way similar to [FL] Proposition 2.5 one shows that R^{1}f′_{∗}(∧^{a}M⊗ω_{f′}^{⊗(i−a)}⊗L⊗P_{d}^{r}) =

0, therefore by Grauert’s theorem A_{a,i−a} is a vector bundle over 𝔊_{d}^{r}(\overline{S}_{g}^{0}/\overline{M}_{g}^{0}) of rank

rk(A_{a,i−a}) = χ(X, ∧^{a}M_{ω_{X}} ⊗ ω_{X}^{⊗(i−a)} ⊗ L ⊗ η) = 2(i − a)g\binom{g − 1}{a}.

Furthermore, for all 1 ≤ a ≤ i − 1, the vector bundles A_{a,i−a} sit in exact sequences

(5)  0 ⟶ A_{a,i−a} ⟶ ∧^{a}f′_{∗}(ω_{f′}) ⊗ A_{0,i−a} ⟶ A_{a−1,i−a+1} ⟶ 0,

where the right exactness boils down to showing that H^{1}(X, ∧^{a}M_{ω_{X}} ⊗ω_{X}^{⊗(i−a)}⊗η⊗L) = 0 for all [X, η, β, L] ∈ 𝔊_{d}^{r}(\overline{S}_{g}^{0}/\overline{M}_{g}^{0}).

We denote as usual 𝔼 := f′_{∗}(ω_{f′}) the Hodge bundle over 𝔊_{d}^{r}(\overline{S}_{g}^{0}/\overline{M}_{g}^{0}) and then

note that there exists a vector bundle map

(6)  ϕ : ∧^{i}𝔼 ⊗ A_{0,0} → A_{i−1,1}

between vector bundles of the same rank over 𝔊_{d}^{r}(\overline{S}_{g}^{0}/\overline{M}_{g}^{0}). For [C, η, L] ∈ σ^{−1}(M_{g}^{0}) the fibre of this morphism is precisely the map ϕ(C, η, L) defined by (3).

Theorem 2.2. The vector bundle morphism ϕ : ∧^{i}𝔼 ⊗ A_{0,0} → A_{i−1,1} is generically nondegenerate over 𝔊_{d}^{r}(\overline{S}_{g}^{0}/\overline{M}_{g}^{0}). It follows that U_{g,d}^{r} is an effective divisor over S_{g}^{+} for all s ≥ 1, and over S_{g}^{−} as well for s ≥ 2.

Proof. We specialize C to a hyperelliptic curve, and denote by A ∈ W_{2}^{1}(C) the hyperelliptic involution. The Lazarsfeld bundle splits into a sum of line bundles Q_{C} ≅ A^{⊕(g−1)}, therefore the condition H^{0}(C, ∧^{i}M_{K_{C}} ⊗η⊗L) = 0 translates into H^{0}(C, η⊗A^{⊗i}⊗L^{∨}) = 0. Suppose that h^{0}(C, η ⊗ A^{⊗i} ⊗ L^{∨}) ≥ 1 for any L = A^{⊗r} ⊗ O_{C}(x_{1} + ⋯ + x_{d−2r}) ∈ W_{d}^{r}(C), where the x_{1}, …, x_{d−2r} ∈ C are arbitrarily chosen points. This implies that h^{0}(C, η ⊗ A^{⊗(i−r)}) ≥ d − 2r + 1. Any theta-characteristic on C is of the form

η = A^{⊗m} ⊗ O_{C}(p_{1} + ⋯ + p_{g−2m−1}),

where 1 ≤ m ≤ (g − 1)/2 and p_{1}, …, p_{g−2m−1} ∈ C are Weierstrass points. Choosing a theta-characteristic on C for which m ≤ i − r − 1 (which can be done in all cases except on S_{g}^{−} when i = r), we obtain that h^{0}(C, η ⊗ A^{⊗(i−r)}) ≤ d − 2r, a contradiction. □

Proof of Theorem 0.1. To compute the class of the degeneracy locus of ϕ we use repeatedly the exact sequence (5). We write the following identities in A^{1}(𝔊_{d}^{r}(\overline{S}_{g}^{0}/\overline{M}_{g}^{0})):

c_{1}(A_{i−1,1} − ∧^{i}𝔼 ⊗ A_{0,0}) = ∑_{l=0}^{i}(−1)^{l−1}c_{1}(∧^{i−l}𝔼 ⊗ A_{0,l}) =

= ∑_{l=0}^{i}(−1)^{l+1}((2l(g − 1) + d)\binom{g − 1}{i − l − 1}c_{1}(𝔼) + \binom{g}{i − l}c_{1}(A_{0,l})).

Using Proposition 2.1 one can show via the Grothendieck-Riemann-Roch formula applied to f′ : C_{d}^{r} → 𝔊_{d}^{r}(\overline{S}_{g}^{0}/\overline{M}_{g}^{0}) that one has that

c_{1}(A_{0,l}) = λ + (\frac{l^{2}}{2} − \frac{1}{8})𝔠 + \frac{1}{2}𝔞 + l𝔟 − \frac{1}{4}β_{0} ∈ A^{1}(𝔊_{d}^{r}(\overline{S}_{g}^{0}/\overline{M}_{g}^{0})).

To determine σ_{∗}(c_{1}(A_{i−1,1} − ∧^{i}𝔼)) ∈ A^{1}(\overline{S}_{g}) we use [F2]: If

N := deg(σ) = #(W_{d}^{r}(C))

denotes the number of 𝔤_{d}^{r}’s on a general curve [C] ∈ M_{g}, then there exists a precisely described choice of a Poincaré bundle on ℭ_{g,d}^{r} such that the push-forwards of the tau-

tological classes on 𝔊_{d}^{r}(\overline{S}_{g}^{0}/\overline{M}_{g}^{0}) are given as follows (cf. [F2], [Kh] and especially [FL] Section 2, for the similar argument in the Prym case):

σ_{∗}(𝔞) = \frac{dN}{(g − 1)(g − 2)}((gd − 2g^{2} + 8d − 8g + 4)λ + \frac{1}{6}(2g^{2} − gd + 3g − 4d − 2)(α_{0} + 2β_{0}))

and

σ_{∗}(𝔟) = \frac{dN}{2g − 2}(12λ − α_{0} − 2β_{0}) ∈ A^{1}(𝔊_{d}^{r}(\overline{S}_{g}^{0}/\overline{M}_{g}^{0})).

One notes that c_{1}(A_{i−1,1}−∧^{i}𝔼⊗A_{0,0}) ∈ A^{1}(𝔊_{d}^{r}(\overline{S}_{g}^{0}/\overline{M}_{g}^{0})) does not depend of the Poincaré bundle. Using the previous formulas, after some arithmetic, one computes the class of the partial compactification of U_{g,d}^{r} and finishes the proof. □

When s = 1 and r = g−1, then necessarily L = K_{C} ∈ W_{2g−2}^{g−1}(C) and the condition η − K_{C} ∈ −C_{g−1} is equivalent to H^{0}(C, η) ≠ 0, that is, on S_{g}^{+} we recover the theta-null divisor, or more precisely also taking into account multiplicities U_{g,2g−2}^{g−1} = 2 · Θ_{null}, cf. [?] (while on S_{g}^{+} one does not get a divisor at all). In particular,

\overline{U}_{g,2g−2}^{g−1} ≡ 2 · \overline{Θ}_{null} ≡ \frac{1}{2}λ − \frac{1}{8}α_{0} − 0 · β_{0} − ⋯ ∈ Pic(\overline{S}_{g}^{+}).

Another interesting case is when r = 2, hence g = 3s, L ∈ W_{2s+2}^{2}(C) and the condition

η ⊗ L^{∨} ∈ C_{2s−2} − C_{s+1} is equivalent to requiring that the embedded curve C ⟶^{|η⊗L|} P^{2s+1} has an (s + 1)-secant (s − 1)-plane:

Theorem 2.3. For g = 3s, d = 2s + 2, the class of the closure in \overline{S}_{g}^{∓} of the effective divisor

U_{g,d}^{2} := {[C, η] ∈ S_{3s}^{∓} : ∃L ∈ W_{2s+2}^{2}(C) such thatη ⊗ L^{∨} ∈ C_{2s−2} − C_{s+1}}

is given by the formula in Pic(\overline{S}_{g}^{∓}):

\overline{U}_{g,d}^{2} ≡ \binom{g}{s + 2}\binom{g}{s, s, s}\frac{1}{24g(g − 1)^{2}(g − 2)(s + 1)^{2}} (4(216s^{4}+513s^{3}−348s^{2}−387s+18)λ−

−(144s^{4} + 225s^{3} − 268s^{2} − 99s + 10)α_{0} − (288s^{4} + 288s^{3} + 320s^{2} + 32)β_{0} − ⋯).

For instance, for g = 9, we obtain the class of the closure of the locus spin curves [C, η] ∈ S_{9}^{∓}, for which there exists a net L ∈ W_{8}^{2}(C) such that η ⊗ L^{∨} ∈ C_{4} − C_{4}:

\overline{U}_{9,8}^{2} ≡ 235 · 35(\frac{36}{5}λ − α_{0} − \frac{428}{235}β_{0} − ⋯) ∈ Pic(\overline{S}_{9}^{∓}).

3. The divisor of points of odd theta-characteristics

In this section we compute the class of the divisor \overline{Θ}_{g,1}. We begin by recalling that p : \overline{M}_{g,1} → \overline{M}_{g} is the map frogetting the marked point. For i = 1, …, g − 1, the divisor Δ_{i} on \overline{M}_{g,1} is the closure of the locus of pointed curves [C ∪ D, q], where C and D are smooth curves of genus i and g − i respectively, and q ∈ C. Similarly, Δ_{irr} denotes the closure in \overline{M}_{g,1} of the locus of irreducible 1-pointed stable curves. We set δ_{i} := [Δ_{i}], δ_{irr} := [Δ_{irr}] ∈ Pic(\overline{M}_{g,1}), and recall that ψ ∈ Pic(\overline{M}_{g,1}) is the universal cotangent class. Clearly, p^{∗}(δ_{irr}) = δ_{irr} and p^{∗}(δ_{i}) = δ_{i} + δ_{g−i} ∈ Pic(\overline{M}_{g,1}). For g ≥ 3, the group Pic(\overline{M}_{g,1}) is freely generated by the classes λ, ψ, δ_{irr}, δ_{1}, …, δ_{g−1}, cf. [AC1]. For g = 2 the same classes generate Pic(\overline{M}_{2,1}) subject to the Mumford relation

λ = \frac{1}{10}δ_{irr} + \frac{1}{5}δ_{1},

expressing that λ is a boundary class. We expand the class [\overline{Θ}_{g,1}] in this basis of Pic(\overline{M}_{g,1}),

\overline{Θ}_{g,1} ≡ aλ + bψ − b_{irr}δ_{irr} − ∑_{i=1}^{g−1} b_{i}δ_{i} ∈ Pic(\overline{M}_{g,1}),

and determine the coefficients, by understanding the restriction of \overline{Θ}_{g,1} to sufficiently many geometric subvarieties of \overline{M}_{g,1}. To simplify calculations, we denote by

N_{g}^{−} := 2^{g−1}(2^{g} − 1) and N_{g}^{+} := 2^{g−1}(2^{g} + 1),

the number of odd (respectively even) theta-characteristic on a curve of genus g.

We define a few test-curves in the boundary of \overline{M}_{g,1}. For an integer 2 ≤ i ≤ g − 1, we choose general curves [C] ∈ M_{i} and [D, x, q] ∈ M_{g−i,2}. In particular, we may assume that x, q ∈ D do not appear in the support of any odd theta-characteristic η_{D}^{−} on D, and that h^{0}(D, η_{D}^{+}) = 0, for any even theta-characteristic η_{D}^{+}. By joining C and D at a variable point of x ∈ C, we obtain a family of 1-pointed stable curves

F_{g−i} := \{[C ∪_{x} D, q] : x ∈ C\} ⊂ Δ_{g−i} ⊂ \overline{M}_{g,1},

where the marked point q ∈ D is fixed. It is clear that F_{g−i} · δ_{g−i} = 2 − 2i, F_{g−i} · λ = F_{g−i} · ψ = 0. Moreover, F_{g−i} is disjoint from all the other boundary divisors of \overline{M}_{g,1}.

Proposition 3.1. For each 2 ≤ i ≤ g − 1, one has that b_{g−i} = N_{i}^{−} · N_{g−i}^{+}/2.

Proof. We observe that the curve F_{g−i} ×_{\overline{M}_{g,1}} \overline{S}_{g}^{−} splits into N_{i}^{+} · N_{g−i}^{−} + N_{i}^{−} · N_{g−i}^{+} irreducible components, each isomorphic to C, corresponding to a choice of a pair of theta-characteristics of opposite parities on C and D respectively. Let t ∈ F_{g−i} · \overline{Θ}_{g,1} be an arbitrary point in the intersection, with underlying stable curve C ∪_{x} D, and spin curves ([C, η_{C}], [D, η_{D}]) ∈ S_{i} × S_{g−i} on the two components.

Suppose first that η_{C} = η_{C}^{+} and η_{D} = η_{D}^{−}, that is, t corresponds to an even thetacharacteristic on C and an odd theta-characteristic on D. Then there exist non-zero sections σ_{C} ∈ H^{0}(C, η_{C}^{+} ⊗ O_{C}((g − i)x)) and σ_{D} ∈ H^{0}(D, η_{D}^{−} ⊗ O_{D}(ix)) such that

(7)  ord_{x}(σ_{C}) + ord_{x}(σ_{D}) ≥ g − 1, and σ_{D}(q) = 0.

In other words, σ_{C} and σ_{D} are the aspects of a limit 𝔤_{g−1}^{0} on C ∪_{x} D which vanishes at q ∈ D. Clearly, ord_{x}(σ_{C}) ≤ g − i − 1, hence div(σ_{D}) ≥ ix + p, that is, q ∈ supp(η_{D}^{−}), which is a contradiction because of the generality assumption on q ∈ D.

Thus, we are left to consider the case η_{C} = η_{C}^{−} and η_{D} = η_{D}^{+}. We denote again by σ_{C} ∈ H^{0}(C, η_{C}^{−} ⊗ O_{C}((g − i)x)) and σ_{D} ∈ H^{0}(D, η_{D}^{+} ⊗ O_{D}(ix)) the sections satisfying the compatibility relations (7). The condition h^{0}(D, η_{D}^{+} ⊗ O_{D}(x − q)) ≥ 1 defines a correspondence on D × D, cf. [DK], in particular, we can choose the points x, q ∈ D general enough such that H^{0}(D, η_{D}^{+} ⊗ O_{D}(x − q)) = 0. Then ord_{x}(σ_{D}) ≤ i − 2, thus ord_{x}(σ_{C}) ≥ g − i + 1. It follows that we must have equality ord_{x}(σ_{C}) = g − i + 1 and then, x ∈ supp(η_{C}^{−}). An argument similar to [EH3] Lemma 3.4, shows that each of these intersection points has to be counted with multiplicity 1, thus F_{g−i} · \overline{Θ}_{g,1} = #supp(η_{C}^{−}) · N_{i}^{−} · N_{g−i}^{+}. We conclude by noting that (2i − 2)b_{g−i} = F_{g−i} · \overline{Θ}_{g,1}. □

Proposition 3.2. The relation b = N_{g}^{−}/2 holds.

Proof. Having fixed a general curve [C] ∈ \overline{M}_{g}, by considering the fibre p^{∗}([C]) inside the universal curve, one writes the identity (2g − 2)b = p^{∗}([C]) · \overline{Θ}_{g,1} = (g − 1) · N_{g}^{−}. □

We describe the restriction of Θ_{g,1} over the locus of smooth pointed curves:

Proposition 3.3. One has the relation Θ_{g,1} ≡ N_{g}^{−}(ψ/2 + λ/4) ∈ Pic(M_{g,1}).

Proof. We consider the universal pointed spin curve pr : S_{g,1}^{−} := S_{g}^{−} ×_{M_{g}} M_{g,1} → M_{g,1}. As usual, P ∈ Pic(S_{g,1}^{−}) denotes the universal spin bundle, which over the stack S_{g,1}^{−}, is a root of the dualizing sheaf ω_{pr}, that is, 2c_{1}(P) = pr^{∗}(ψ). We introduce the divisor

Z := {[C, η, q] ∈ S_{g,1}^{−} : q ∈ supp(η)} ⊂ S_{g,1}^{−},

and clearly Θ_{g,1} := pr_{∗}(Z). We write [Z] = c_{1}(P) − c_{1}(pr^{∗}(pr_{∗}(P))), and we take into account that c_{1}(pr_{!}(P)) = 2c_{1}(pr_{∗}(P)) = −λ/2. The rest follows by applying the projection formula. □

In order to determine the remaining coefficients b_{0}, b_{1}, we study the pull-back of \overline{Θ}_{g,1} under the map ν : \overline{M}_{1,2} → \overline{M}_{g,1}, given by ν([E, x, q]) := [C ∪_{x} E, q] ∈ \overline{M}_{g,1}, where [C, x] ∈ M_{g−1,1} is a fixed general pointed curve.

On \overline{M}_{1,2} one has the following relations between divisors classes, see [AC2]:

ψ_{x} = ψ_{q}, λ = ψ_{x} − δ_{0:xq}, δ_{irr} = 12(ψ_{x} − δ_{0:xq}).

We describe the pull-back map ν^{∗} : Pic(\overline{M}_{g,1}) → Pic(\overline{M}_{1,2}) at the level of divisors:

ν^{∗}(λ) = λ, ν^{∗}(ψ) = ψ_{q}, ν^{∗}(δ_{irr}) = δ_{irr}, ν^{∗}(δ_{1}) = −ψ_{x}, ν^{∗}(δ_{g−1}) = δ_{0:xq}.

By direct calculation, we write ν^{∗}(\overline{Θ}_{g,1}) ≡ (a + b − 12b_{0} + b_{1})ψ_{x} − (a + b_{g−1} − 12b_{0})δ_{0:xq}. Next we describe ν^{∗}(\overline{Θ}_{g,1}) viewed as an explicit divisor on \overline{M}_{1,2}:

Proposition 3.4. One has the relation ν^{∗}(\overline{Θ}_{g,1}) ≡ N_{g−1}^{−} · \overline{𝔗}_{2} ∈ Pic(\overline{M}_{2,1}), where

𝔗_{2} := {[E, x, q] ∈ M_{1,2} : 2x ≡ 2q}.

Proof. We fix an arbitrary point t := [C ∪_{x} E, q] ∈ ν^{∗}(\overline{Θ}_{g,1}). Suppose first that E is a smooth elliptic curve, that is, j(E) ≠ ∞ and x ≠ q. Then there exist theta-characteristics of opposite parities η_{C}, η_{E} on C and E respectively, together with non-zero sections

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

such that σ_{E}(q) = 0 and ord_{x}(σ_{C}) + ord_{x}(σ_{E}) ≥ g − 1.

First we assume that η_{C} = η_{C}^{+} and η_{E} = η_{E}^{−}, thus, η_{E} = O_{E}. Since H^{0}(C, η_{C}^{+}) = 0, one obtains that ord_{x}(σ_{C}) = 0, that is ord_{x}(σ_{E}) = g −1, which is impossible, because σ_{E} must vanish at q as well. Thus, one is lead to study the remaining case, when η_{C} = η_{C}^{−} and η_{E} = η_{E}^{+}. Since x ∉ supp(η_{C}^{−}), we obtain ord_{x}(σ_{C}) ≤ 1, and then by compatibility, the last inequality becomes equality, while ord_{x}(σ_{E}) = g − 2, hence η_{E}^{+} = O_{E}(x − q), or equivalently, [E, x, q] ∈ 𝔗_{2}. The multiplicity N_{g−1}^{−} in the expression of ν^{∗}(\overline{Θ}_{g,1}) comes from the choices for the theta-characteristics η_{C}^{−}, responsible for the C-aspect of a limit 𝔤_{g−1}^{0} on C ∪_{x} E. It is an easy moduli count to show that the cases when j(E) = ∞, or [E, x, q] ∈ δ_{0:xq} (corresponding to the situation when x and q coalesce on E), do not occur generically on a component of ν^{∗}(\overline{Θ}_{g,1}). □

Proposition 3.5. \overline{𝔗}_{2} is an irreducible divisor on \overline{M}_{1,2} of class \overline{𝔗}_{2} ≡ 3ψ_{x} ∈ Pic(\overline{M}_{1,2}).

Proof. We write \overline{𝔗}_{2} ≡ αψ_{x}−βδ_{0:xq} ∈ Pic(\overline{M}_{1,2}), and we need to understand the intersection of \overline{𝔗}_{2} with two test curves in \overline{M}_{1,2}. First, we fix a general point [E, q] ∈ \overline{M}_{1,1} and consider the family E_{1} := \{[E, x, q] : x ∈ E\} ⊂ \overline{M}_{1,2}. Clearly, E_{1} · δ_{0:xq} = E_{1} · ψ_{x} = 1. On the other hand E_{1} · \overline{𝔗}_{2} is a 0-cycle simply supported at the points x ∈ E − \{q\} such that x − q ∈ Pic^{0}(E)[2], that is, E_{1} · \overline{𝔗}_{2} = 3. This yields the relation α − β = 3.

As a second test curve, we denote by [L, u, x, q] ∈ \overline{M}_{0,3} the rational 3-pointed

rational curve, and define the pencil R := \{[L ∪_{u} E_{λ}, x, q] : λ ∈ P^{1}\} ⊂ \overline{M}_{1,2}, where \{E_{λ}\} is a pencil of plane cubics. Then R ∩ \overline{𝔗}_{2} = ∅. Since R · λ = 1 and R · δ_{irr} = 12, we obtain the additional relation β = 0, which completes the proof. □

Putting together Proposition 3.1, 3.3 and 3.5, we obtain a system of equations

a + b_{g−1} − 12b_{irr} = 0, a − 12b_{irr} + b + b_{1} = 3N_{g−1}^{−}, a = \frac{1}{4}N_{g}^{−}, b = \frac{1}{2}N_{g}^{−}, b_{1} = \frac{3}{2}N_{g−1}^{−}.

Thus b_{irr} = 2^{2g−6} and b_{g−1} = 2^{g−3}(2^{g−1} + 1). This completes the proof of Theorem 0.3.

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

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