THE FERMAT CUBIC AND SPECIAL HURWITZ LOCI IN \overline{M}_{g}

GAVRIL FARKAS

Abstract: We compute the class of the compactified Hurwitz divisor \overline{𝔗ℜ}_{d} in \overline{M}_{2d−3} consisting of curves of genus g = 2d_{3} having a pencil 𝔤_{d}^{1} with two unspecified triple ramification points. This is the first explicit example of a geometric divisor on \overline{M}_{g} which is not pulled-back form the moduli space of pseudo-stable curves. We show that the intersection of \overline{𝔗ℜ}_{d} with the boundary divisor Δ_{1} in \overline{M}_{g} picks-up the locus of Fermat cubic tails.

1. Introduction

Hurwitz loci have played a basic role in the study of the moduli space of curves at least since 1872 when Clebsch, and later Hurwitz, proved that M_{g} is irreducible by showing that a certain Hurwitz space parameterizing coverings of P^{1} is connected (see [Hu], or [Fu2] for a modern proof). Hurwitz cycles on \overline{M}_{g} are essential in the work of Harris and Mumford [HM] on the Kodaira dimension of \overline{M}_{g} and are expected to govern the length of minimal affine stratifications of M_{g}. Faber and Pandharipande have proved that the class of any Hurwitz cycle on \overline{M}_{g,n} is tautological (cf. [FP]). Very few explicit formulas for the classes of such cycles are known.

We define a Hurwitz divisor in \overline{M}_{g} with n degrees of freedom as follows: We fix integers k_{1}, …, k_{n} ≥ 3 and positive integers d, g such that

k_{1} + k_{2} + ⋯ + k_{n} = 2d − g + n − 1.

Then H_{g: k_{1},...,k_{n}} is the locus of curves [C] ∈ M_{g} having a degree d morphism f : C → P^{1} together with n distinct points p_{1}, …, p_{n} ∈ C such that mult_{p_{i}}(f) ≥ k_{i} for i = 1, …, n. When n = 0 and g = 2d − 1, we recover the Brill-Noether divisor of d-gonal curves studied extensively in [HM]. For n = 1 we obtain Harris’ divisor H_{g: k} of curves having

a linear series C →^{d:1} P^{1} with a k = (2d − g + 1)-fold point, cf. [H]. If n = 1 and d = g − 1 then H_{g: g−1} specializes to S. Diaz’s divisor of curves [C] ∈ M_{g} having an exceptional Weierstrass point p ∈ C with h^{0}(C, O_{C}((g − 1)p)) ≥ 1 (cf. [Di]).

Since H_{g:k_{1},...,k_{n}} is the push-forward of a cycle of codimension n + 1 in M_{g,n}, as n increases the problem of calculating the class of \overline{H}_{g:k_{1},...,k_{n}} becomes more and more difficult. In this paper we carry out the first study of a Hurwitz locus having at least 2 degrees of freedom, and we treat the simplest non-trivial case, when n = 2, k_{1} = k_{2} = 3 and g = 2d − 3. Our main result is the calculation of the class of \overline{𝔗ℜ}_{d} := \overline{H}_{2d−3: 3,3}. As usual we denote by λ ∈ Pic(\overline{M}_{g}) the Hodge class and by δ_{0}, …, δ_{[g/2]} ∈ Pic(\overline{M}_{g}) the codimension 1 classes on the moduli stack corresponding to the boundary divisors of \overline{M}_{g}:

Research partially supported by an Alfred P. Sloan Fellowship and the NSF Grant DMS-0500747 .

Theorem 1.1. We fix d ≥ 3 and denote by 𝔗ℜ_{d} the locus of curves [C] ∈ M_{2d−3} having a covering C →^{d:1} P^{1} with two unspecified triple ramification points. Then 𝔗ℜ_{d} is an effective divisor on M_{2d−3} and the class of its compactification \overline{𝔗ℜ}_{d} inside \overline{M}_{2d−3} is given by the formula:

\overline{𝔗ℜ}_{d} ≡ 2\frac{(2d − 6)!}{d! (d − 3)!}(a λ − b_{0} δ_{0} − b_{1} δ_{1} − ⋯ − b_{d−2} δ_{d−2}) ∈ Pic(\overline{M}_{2d−3}),

where a = 24(36d^{4} − 36d^{3} − 640d^{2} + 1885 − 1475), b_{0} = 144d^{4} − 528d^{3} − 298d^{2} + 3049d − 2940,

and b_{i} = 12i(2d − 3 − i)(36d^{3} − 156d^{2} + 180d − 5), for 1 ≤ i ≤ d − 2.
The divisor \overline{𝔗ℜ}_{d} is also the first example of a geometric divisor in \overline{M}_{g} which

is not a pull-back of an effective divisor from the space \overline{M}_{g}^{ps} of pseudo-stable curves. Precisely, if we denote by R ⊂ \overline{M}_{g} the extremal ray obtained by attaching to a fixed pointed curve [C, q] of genus g − 1 a pencil of plane cubics, then R · λ = 1, R · δ_{0} = 12, R·δ_{1} = −1 and R·δ_{α} = 0 for α ≥ 2. If δ := δ_{0}+⋯+δ_{[g/2]} ∈ Pic(\overline{M}_{g}) is the total boundary, there exists a divisorial contraction of the extremal ray R ⊂ Δ_{1} ⊂ \overline{M}_{g} induced by the base point free linear system |11λ − δ| on \overline{M}_{g},

f : \overline{M}_{g} → \overline{M}_{g}^{ps}.

The image is isomorphic to the moduli space of pseudo-stable curves as defined by D. Schubert in [S]. A curve is pseudo-stable if it has only nodes and cusps as singularities, and each component of genus 1 (resp. 0) intersects the curve in at least 2 (resp. 3 points). The contraction f is the first step in carrying out the minimal model program for \overline{M}_{g}, see [HH]. One has an inclusion f^{∗}(Eff(\overline{M}_{g}^{ps})) ⊂ Eff(\overline{M}_{g}). All the geometric divisors on \overline{M}_{g} whose class has been computed (e.g. Brill-Noether or Gieseker-Petri divisors [EH], Koszul divisors [Fa1], [Fa2], or loci of curves with an abnormal Weierstrass point [Di]), lie in the subcone f^{∗}(Eff(\overline{M}_{g}^{ps})). The divisor \overline{𝔗ℜ}_{d} behaves quite differently: If i : Δ_{1} = \overline{M}_{1,1} × \overline{M}_{g−1,1} ↪ \overline{M}_{g} denotes the inclusion, then we have the relation

i^{∗}(\overline{𝔗ℜ}_{d}) = α · \{j = 0\} × \overline{M}_{g−1,1} + \overline{M}_{1,1} × D = α · \{Fermat cubic\} × \overline{M}_{g−1,1} + \overline{M}_{1,1} × D,

where α := \frac{3(2d−4)!}{d! (d−3)!} and D ⊂ \overline{M}_{g−1,1} is an explicitly described effective divisor. Hence when restricted to the boundary divisor Δ_{1} ⊂ \overline{M}_{g} of elliptic tails, \overline{𝔗ℜ}_{d} picks-up the locus of Fermat cubic tails!

The rich geometry of \overline{𝔗ℜ}_{d} can also be seen at the level of genus 2 curves. We denote by χ : \overline{M}_{2,1} → \overline{M}_{2d−3} be the map obtained by attaching a fixed tail [B, q] of genus 2d − 5 at the marked point of every curve of genus 2. Then the pull-back under χ of every known geometric divisor on \overline{M}_{2,1} is a multiple of the Weierstrass divisor \overline{W} of \overline{M}_{2,1} (cf. [HM], [EH], [Fa1]). In contrast, for \overline{𝔗ℜ}_{d} we have the following picture:

Theorem 1.2. If χ : \overline{M}_{2,1} → \overline{M}_{g} is as above, we have the following relation in Pic(\overline{M}_{2,1}):

χ^{∗}(\overline{𝔗ℜ}_{d}) = N_{1}(d) · \overline{W} + e(d, 2d − 5) · \overline{D}_{1} + a(d − 1, 2d − 5) · \overline{D}_{2} + a(d, 2d − 5) · \overline{D}_{3}, where W := {[C, p] ∈ M_{2,1} : p ∈ C is a Weierstrass point },

D_{1} := {[C, p] ∈ M_{2,1} : ∃x ∈ C − {p} such that3x ≡ 3p},

D_{2} := {[C, p] ∈ M_{2,1} : ∃l ∈ G_{3}^{1}(C), x ≠ y ∈ C−{p} witha_{1}^{l}(x) ≥ 3, a_{1}^{l}(y) ≥ 3, a_{1}^{l}(p) ≥ 2}, and

D_{3} := {[C, p] ∈ M_{2,1} : ∃l ∈ G_{4}^{1}(C), x ≠ y ∈ C−{p} witha_{1}^{l}(p) ≥ 4, a_{1}^{l}(x) ≥ 3, a_{1}^{l}(y) ≥ 3}.

The constants N_{1}(d), e(d, 2d − 5), a(d, 2d − 5), a(d − 1, 2d − 5) appearing in the statement are explicitly known and defined in Proposition 2.1. We used the notation a_{1}^{l}(p) := mult_{p}(l), for the multiplicity of a pencil l ∈ G_{d}^{1}(C) at a point p ∈ C. The classes of the divisors \overline{D}_{1}, \overline{D}_{2}, \overline{D}_{3} on \overline{M}_{2,1} are determined as well (The class of \overline{W} is of course well-known, see [EH]):

Theorem 1.3. One has the following formulas expressed in the basis \{ψ, λ, δ_{0}\} of Pic(\overline{M}_{2,1}):

\overline{D}_{1} ≡ 80ψ + 10δ_{0} − 120λ, \overline{D}_{2} ≡ 160ψ + 17δ_{0} − 200λ,
and \overline{D}_{3} ≡ 640ψ + 72δ_{0} − 860λ.

Acknowledgment: I have benefitted from discussions with R. Pandharipande (5 years ago!) on counting admissible coverings.

2. Admissible coverings with two triple points

We begin by recalling a few facts about admissible coverings in the context of points of triple ramification. Let H_{d}^{tr} be the Hurwitz space parameterizing degree d maps [f : C → P^{1}, q_{1}, q_{2}; p_{1}, …, p_{6d−12}], where [C] ∈ M_{2d−3}, q_{1}, q_{2}, p_{1}, …, p_{6d−12} are distinct points on P^{1} and f has one point of triple ramification over each of q_{1} and q_{2} and one point of simple ramification over p_{i} for 1 ≤ i ≤ 6d − 12. We denote by \overline{H}_{d}^{tr} the compactification of the Hurwitz space by means of Harris-Mumford admissible coverings (cf. [HM], [ACV] and [Di] Section 5; see also [BR] for a survey on Hurwitz schemes and their compactifications). Thus \overline{H}_{d}^{tr} is the parameter space of degree d maps

[f : X ⟶^{d:1} R, q_{1}, q_{2}; p_{1}, …, p_{6d−12}],

where [R, q_{1}, q_{2}; p_{1}, …, p_{6d−12}] is a nodal rational curve, X is a nodal curve of genus 2d − 3 and f is a finite map which satisfies the following conditions:

• f^{−1}(R_{reg}) = X_{reg} and f^{−1}(R_{sing}) = X_{sing}.

• f has a point of triple ramification over each of q_{1} and q_{2} and simple ramification over p_{1}, …, p_{6d−12}. Moreover f is étale over each point in R_{reg} − {q_{1}, q_{2}, p_{1}, …, p_{6d−12}}.

• If x ∈ X_{sing} and x ∈ X_{1} ∩ X_{2} where X_{1} and X_{2} are irreducible components of X, then f(X_{1}) and f(X_{2}) are distinct components of R and

mult_{x}{f_{|X_{1}} : X_{1} → f(X_{1})} = mult_{x}{f_{|X_{2}} : X_{2} → f(X_{2})}.

The group 𝔖_{2} × 𝔖_{6d−12} acts on \overline{H}_{d}^{tr} by permuting the triple and the ordinary ramification points of f respectively and we denote by ℌ_{d} := \overline{H}_{d}^{tr}/𝔖_{2} × 𝔖_{6d−12} for the quotient. There exists a stabilization morphism σ : ℌ_{d} → \overline{M}_{g} as well as a finite map β : ℌ_{d} → \overline{M}_{0,6d−10}. The description of the local rings of \overline{H}_{d}^{tr} can be found in [HM] pg. 61-62 or [BR] and will be used in the paper. In particular, the scheme \overline{H}_{d}^{tr} is smooth at points [f : X → R, q_{1}, q_{2}; p_{1}, …, p_{6d−12}] with the property that there are no automorphisms ϕ : X → X with f ∘ ϕ = f.

2.1. The enumerative geometry of pencils on the general curve. We shall determine the intersection multiplicities of \overline{𝔗ℜ}_{d} with standard test curves in \overline{M}_{g}. For this we need a variety of enumerative results concerning pencils on pointed curves which will be used throughout the paper. For a point p ∈ C and a linear series l ∈ G_{d}^{r}(C), we denote by

a^{l}(p) : (0 < a_{0}^{l}(p) < a_{1}^{l}(p) < … < a_{r}^{l}(p) ≤ d)

the vanishing sequence of l at p. If l ∈ G_{d}^{1}(C), we say that p ∈ C is an n-fold point if l(−np) ≠ ∅. We first recall the results from [HM] Theorem A and [H] Theorem 2.1.

Proposition 2.1. Let us fix a general curve [C, p] ∈ M_{g,1} and an integer d ≥ 2d − g − 1 ≥ 0.

• The number of pencils L ∈ W_{d}^{1}(C) satisfying h^{0}(L ⊗ O_{C}(−(2d − g − 1)p)) ≥ 1 equals

a(d, g) := (2d − g − 1)\frac{g!}{d! (g − d + 1)!}.

• The number of pairs (L, x) ∈ W_{d}^{1}(C) × C satisfying h^{0}(L ⊗ O_{C}(−(2d − g)x)) ≥ 2 equals

b(d, g) := (2d − g − 1)(2d − g)(2d − g + 1)\frac{g!}{d! (g − d)!}.

• Fix integers α, β ≥ 1 such that α + β = 2d − g. The number of pairs (L, x) ∈ W_{d}^{1}(C) × C satisfying h^{0}(L ⊗ O_{C}(−βp − γx)) ≥ 1 equals

c(d, g, γ) := (γ^{2}(2d − g) − γ)\binom{g}{d}.

• The number of pairs (L, x) ∈ W_{d}^{1}(C) × C satisfying the conditions

h^{0}(L ⊗ O_{C}(−(2d − g − 2)p)) ≥ 1 and h^{0}(L ⊗ O_{C}(−3x)) ≥ 1 equals

e(d, g) := 8\frac{g!}{(d − 3)! (g − d + 2)!} − 8\frac{g!}{d! (g − d − 1)!}.

We now prove more specialized results, adapted to our situation of counting pencils with two triple points:

Proposition 2.2. (1) We fix d ≥ 3 and a general 2-pointed curve [C, p, q] ∈ M_{2d−6}. The number of pencils l ∈ G_{d}^{1}(C) having triple points at both p and q equals

F (d) := (2d − 6)! (\frac{1}{(d − 3)!^{2}} − \frac{1}{d! (d − 6)!}).

(2) For a general curve [C] ∈ M_{2d−4}, the number of pencils l ∈ G_{d}^{1}(C) having triple ramification at unspecified distinct points x, y ∈ C, equals

N(d) := \frac{48(6d^{2} − 28d + 35) (2d − 4)!}{d! (d − 3)!} .

(3) We fix a general pointed curve [C, p] ∈ M_{2d−5,1}. The number of pencils L ∈ W_{d}^{1}(C) satisfying the conditions

h^{0}(L ⊗ O_{C}(−2p)) ≥ 1, h^{0}(L ⊗ O_{C}(−3x)) ≥ 1, h^{0}(L ⊗ O_{C}(−3y)) ≥ 1

for unspecified distinct points x, y ∈ C, is equal to

N_{1}(d) := 24(12d^{3} − 92d^{2} + 240d − 215)\frac{(2d − 4)!}{d! (d − 2)!}.

Remark 2.3. In the formulas for e(d, g) and F (d) we set 1/n! := 0 for n < 0.

Remark 2.4. As a check, for d = 3 Proposition 2.2 (2) reads N(3) = 80. Thus for a general curve [C] ∈ M_{2} there are 160 = 2 · 80 pairs of points (x, y) ∈ C × C, x ≠ y, such that 3x ≡ 3y. This can be seen directly by considering the map ψ : C × C → Pic^{0}(C) given by ψ(x, y) := O_{C}(3x − 3y). Then ψ^{∗}(0) = \frac{1}{2} ∫_{C×C} ψ^{∗}(ω ∧ ω) = 2 · 3^{2} · 3^{2} = 162, where ω is a differential form representing θ. To get the answer to our question we subtract from 162 the contribution of the diagonal Δ ⊆ C × C. This excess intersection contribution is equal to 2 (cf. [Di]), so in the end we get 160 = 162 − 2 pairs of distinct points (x, y) ∈ C × C with 3x ≡ 3y.

Proof. (1) This is a standard exercise in limit linear series and Schubert calculus in the spirit of [EH]. We let [C, p, q] ∈ M_{2d−6,2} degenerate to the stable 2-pointed curve [C_{0} := P^{1} ∪E_{1} ∪…∪E_{2d−6}, p_{0}, q_{0}], consisting of elliptic tails {E_{i}}_{i=1}^{2d−6} and a rational spine, such that {p_{i}} = E_{i} ∩ P^{1}, and the marked points p_{0}, q_{0} lie on the spine. We also assume that p_{1}, …, p_{2d−6}, p_{0}, q_{0} ∈ ℙ^{1} are general points, in particular p_{0}, q_{0} ∈ P^{1} − {p_{1}, …, p_{2d−6}}. Then F (d) is the number of limit 𝔤_{d}^{1}’s on C_{0} having triple ramification at both p_{0} and q_{0} and this is the same as the number of 𝔤_{d}^{1}’s on P^{1} having cusps at p_{1}, …, p_{2d−6} and triple ramification at p_{0} and q_{0}. This equals the intersection number of Schubert cycles σ_{(0,2)}^{2}σ_{(0,1)}^{2d−6} (computed in H^{top}(𝔾(1, d), ℤ)). The product can be computed using formula

(v) on page 273 in [Fu1] and one finds that

σ_{(0,2)}^{2} σ_{(0,1)}^{2d−6} = (2d − 6)! (\frac{1}{(d − 3)!^{2}} − \frac{1}{d! (d − 6)!}).

(2) This is more involved. We specialize [C] ∈ M_{2d−4} to [C_{0} := P^{1} ∪ E_{1} ∪ … ∪ E_{2d−4}], where E_{i} are general elliptic curves, {p_{i}} = P^{1} ∩ E_{i} and p_{1}, …, p_{2d−4} ∈ P^{1} are general points. Then N(d) is equal to the number of limit 𝔤_{d}^{1}’s on C_{0} with triple ramification at two distinct points x, y ∈ C_{0}. Let l be such a limit 𝔤_{d}^{1}. We can assume that both x and y are smooth points of C_{0} and by the additivity of the Brill-Noether number (see e.g.

[EH] pg. 365), we find that x, y must lie on the tails E_{i}. Since [E_{i}, p_{i}] ∈ M_{1,1} is general, we assume that j(E_{i}) ≠ 0 (that is, none of the E_{i}’s is the Fermat cubic). Then there can be no l_{i} ∈ G_{3}^{1}(E_{i}) carrying 3 triple ramification points. There are two cases we consider:

a) There are indices 1 ≤ i < j ≤ 2d − 4 such that x ∈ E_{i} and y ∈ E_{j}. Then a^{l_{E_{i}}} (p_{i}) = a^{l_{E_{j}}} (p_{j}) = (d − 3, d), hence 3x ≡ 3p_{i} on E_{i} and 3y ≡ 3p_{j} on E_{j}. There are 8 choices for x ∈ E_{i}, 8 choices for y ∈ E_{j} and \binom{2d−4}{2} choices for the tails E_{i} and E_{j} containing the triple points. On P^{1} we count 𝔤_{d}^{1}’s with cusps at {p_{1}, …, p_{2d−4}} − {p_{i}, p_{j}} and triple points at p_{i} and p_{j}. This number is again equal to σ_{(0,2)}^{2} σ_{(0,1)}^{2d−6} ∈ H^{top}(𝔾(1, d), ℤ) and we get a contribution of

(1)  64\binom{2d − 4}{2}σ_{(0,2)}^{2} σ_{(0,1)}^{2d−6} = 32(2d − 4)! (\frac{1}{(d − 3)!^{2}} − \frac{1}{d! (d − 6)!}) .

b) There is 1 ≤ i ≤ 2d − 4 such that x, y ∈ E_{i}. We distinguish between two subcases:

b_{1}) a^{l_{E_{i}}} (p_{i}) = (d − 3, d − 1). On ℙ^{1} we count 𝔤_{d−1}^{1}’s with cusps at p_{1}, …, p_{2d−4} and this number is σ_{(0,1)}^{2d−4} (in H^{top}(𝔾(1, d − 1), ℤ)). On E_{i} we compute the number of 𝔤_{3}^{1}’s having triple ramification at unspecified points x, y ∈ E_{i} − {p_{i}} and ordinary ramification at p_{i}. For simplicity we set [E_{i}, p_{i}] := [E, p]. If we regard p ∈ E as the origin of E, then the translation map (x, y) ↦ (y − x, −x) establishes a bijection between the set of pairs (x, y) ∈ E × E − Δ, x ≠ p ≠ y ≠ x, such that there is a 𝔤_{3}^{1} in which x, y, p appear with multiplicities 3, 3 and 2 respectively, and the set of pairs (u, v) ∈ E × E − Δ, with u ≠ p ≠ v ≠ u such that there is a 𝔤_{3}^{1} in which u, v, p appear with multiplicities 3, 2 and 3 respectively. The latter set has cardinality 16, hence the number of pencils 𝔤_{3}^{1} we are counting is 8 = 16/2. All in all, we find a contribution of

(2)  8(2d − 4) σ_{(0,1)}^{2d−4} = 16\binom{2d − 4}{d − 1} .

b_{2}) a^{l_{E_{i}}} (p_{i}) = (d − 4, d). This time, on P^{1} we look at 𝔤_{d}^{1}’s with cusps at {p_{1}, …, p_{2d−4}} − {p_{i}} and a 4-fold point at p_{1}. Their number is σ_{(0,3)} σ_{(0,1)}^{2d−5} ∈ H^{top}(𝔾(1, d), ℤ)). On E_{i} we compute the number of 𝔤_{4}^{1}’s for which there are distinct points x, y ∈ E_{i} −{p_{i}} such that p_{i}, x, y appear with multiplicities 4, 3 and 3 respectively. Again we set [E_{i}, p_{i}] := [E, p] and denote by Σ the closure in E × E of the locus

{(u, v) ∈ E × E − Δ : ∃l ∈ G_{4}^{1}(E) such that a_{1}^{l}(p) = 4, a_{1}^{l}(u) ≥ 3, a_{1}^{l}(v) ≥ 2}.

The class of the curve Σ can be computed easily. If F_{i} denotes the numerical equivalence class of a fibre of the projection π_{i} : E × E → E for i = 1, 2, then

(3)  Σ ≡ 10F_{1} + 5F_{2} − 2Δ.

The coefficients in this expression are determined by intersecting Σ with Δ and the fibres of π_{i}. First, one has that Σ ∩ Δ = {(x, x) ∈ E × E : x ≠ p, 4p ≡ 4x} and then Σ ∩ π_{2}^{−1}(p) = {(y, p) ∈ E × E : y ≠ p, 3p ≡ 3y}. These intersections are all transversal, hence Σ · Δ = 15, Σ · F_{2} = 8, whereas obviously Σ · F_{1} = 3. This proves (3).

The number of pencils l ⊆ |O_{E}(4p)| having two extra triple points will then be equal to 1/2 #(ramification points of π_{2} : Σ → E) = Σ^{2}/2 = 20. We have obtained in this case a contribution of

(4)  20(2d − 4) σ_{(0,3)} σ_{(0,1)}^{2d−5} = 80\binom{2d − 4}{d}.

Adding together (1),(2) and (4), we obtain the stated formula for N(d).

(3) We relate N_{1}(d) to N(d) by specializing the general curve from M_{2d−4} to [C ∪_{p} E] ∈ Δ_{1} ⊂ \overline{M}_{2d−4}, where [C, p] ∈ M_{2d−5,1} and [E, p] ∈ \overline{M}_{1,1}. Under this degeneration N(d)

becomes the number of admissible coverings f : X →^{d:1} R having as source a nodal curve X stably equivalent to C ∪_{p} E and as target a genus 0 nodal curve R. Moreover, f possesses distinct unspecified triple ramification points x, y ∈ X_{reg}. There are a number of cases depending on the position of x and y.

(3_{a}) x, y ∈ C − {p}. In this case deg(f_{C}) = d and because of the generality of [C, p], f_{C} has to be one of the finitely many 𝔤_{d}^{1}’s having two distinct triple points and a simple ramification point at p ∈ C. The number of such coverings is precisely N_{1}(d). By the compatibility condition on ramification indices at p, we find that deg(f_{E}) = 2 and the E-aspect of f is induced by |O_{E}(2p)|. The curve X is obtained from C ∪_{p} E by inserting d − 2 copies of P^{1} at the points in f_{C}^{−1}(f(p)) − {p}. We then map these rational curves isomorphically to f(E). This admissible cover has no automorphisms and it should be counted with multiplicity 1.

(3_{b}) x, y ∈ E − {p}. The curve [C] ∈ M_{2d−5} being Brill-Noether general, it carries no linear series 𝔤_{d−2}^{1}, hence deg(f_{C}) ≥ d − 1. We distinguish two subcases:

If deg(f_{C}) = d − 1, then f_{C} is one of the a(d − 1, 2d − 5) linear series 𝔤_{d−1}^{1} on C having p as an ordinary ramification point. Since C and E meet only at p, we have that deg(f_{E}) = 3, and f_{E} corresponds to a 𝔤_{3}^{1} on E having two unspecified triple points and a simple ramification point at p. There are 8 such 𝔤_{3}^{1}’s on E (see the proof of Proposition 2.2). To obtain a degree d admissible covering, we first attach a copy (P^{1})_{1} of P^{1} to E at the point q ∈ f_{E}^{−1}(f(p)) − {p}, then map (P^{1})_{1} and C map to the same component of R. Then we insert d − 2 copies of P^{1} at the points lying in the same fibre of f_{C} as p. All these rational curves map to the same copy of R as E. Each of these 8a(d − 1, 2d − 5) admissible coverings is counted with multiplicity 1.

If deg(f_{C}) = d, then f_{C} corresponds to one of the a(d, 2d − 5) linear series 𝔤_{d}^{1} with a 4-fold point at p. By compatibility, f_{E} corresponds to a 𝔤_{4}^{1} in which p and two unspecified points x, y ∈ E appear with multiplicities 4, 3 and 3 respectively. There are 20 such 𝔤_{4}^{1}’s on E, hence 20a(d, 2d − 5) admissible coverings.

(3_{c}) x ∈ E − {p}, y ∈ C − {p}. In this situation deg(f_{C}) = d and f_{C} corresponds to one of the e(d, 2d − 5) coverings 𝔤_{d}^{1} on C having a triple point at p and another unspecified triple point at y ∈ C. Then deg(f_{E}) = 3 and 3x ≡ 3p, that is, there are 8 choices of the E-aspect of f. We obtain X by attaching to C copies of P^{1} at the d − 3 points in f_{C}^{−1}(f(p)) − {p}, and mapping these curves isomorphically onto f(C).

By degeneration to [C ∪_{p} E], we have found the relation for [C, p] ∈ M_{2d−5,1}:
N(d) = N_{1}(d) + 20a(d, 2d − 5) + 8a(d − 1, 2d − 5) + 8e(d, 2d − 5).

This immediately leads to the claimed expression for N_{1}(d). □

3. The class of the divisor \overline{𝔗ℜ}_{d}

The strategy to compute the class [\overline{𝔗ℜ}_{d}] is similar to the one employed by Eisenbud and Harris in [EH] to determine the class of the Brill-Noether divisors [\overline{M}_{g,d}^{r}] of curves with a 𝔤_{d}^{r} in the case ρ(g, r, d) = −1: We determine the restrictions of \overline{𝔗ℜ}_{d} to \overline{M}_{0,g} and \overline{M}_{2,1} via obvious flag maps. However, because in the definition of \overline{𝔗ℜ}_{d} we allow 2 degrees of freedom for the triple ramification points, the calculations are much more intricate (and interesting) than in the case of Brill-Noether divisors.

Proposition 3.1. Consider the flag map j : \overline{M}_{0,g} → \overline{M}_{g} obtained by attaching g general elliptic tails at the g marked points. Then j^{∗}(\overline{𝔗ℜ}_{d}) = 0. If we have a linear relation

\overline{𝔗ℜ}_{d} ≡ a λ − ∑_{i=0}^{d−2} b_{i} δ_{i} ∈ Pic(\overline{M}_{g}), thenb_{i} = \frac{i(g − i)}{g − 1}b_{1}, for 1 ≤ i ≤ d − 2.

Proof. The second part of the statement is a consequence of the first: For an effective divisor D ≡ aλ − ∑_{i=0}^{d−2} b_{i}δ_{i} ∈ Pic(\overline{M}_{g}) satisfying the condition j^{∗}(D) = ∅, we have the relations among its coefficients: b_{i} = \frac{i(g−i)}{g−1}b_{1} for i ≥ 1 (cf. [EH] Theorem 3.1).

Suppose that [X := R ∪_{x_{1}} E_{1} ∪ … ∪_{x_{g}} E_{g}] ∈ j(\overline{M}_{0,g}) is a flag curve corresponding to a g-stable rational curve [R, x_{1}, …, x_{g}]. The elliptic tails {E_{i}}_{i=1}^{g} are general and we may assume that all the j-invariants are different from 0. In particular, none of the [E_{i}, x_{i}]’s carries a 𝔤_{3}^{1} with triple ramification points at x_{i} and at two unspecified points x, y ∈ E_{i} − \{x_{i}\}. Assuming that [X] ∈ \overline{𝔗ℜ}_{d}, there exists l ∈ \overline{G}_{d}^{1}(X) a limit 𝔤_{d}^{1}, together with distinct ramification points x ≠ y ∈ X, such that a_{1}^{l}(x) ≥ 3 and a_{1}^{l}(y) ≥ 3. By blowing-up if necessary the nodes x_{i} (that is, by inserting chains of P^{1}’s at the points x_{i}), we may assume that both x, y are smooth points of X.

We make use of the following facts: On R we have that the inequality

ρ(l_{R}, x_{1}, …, x_{g}, z_{1}, …, z_{t}) ≥ 0,

for any choice of distinct points z_{1}, …, z_{t} ∈ R − {x_{1}, …, x_{g}}. On the elliptic tails, we have that ρ(l_{E_{i}}, x_{i}, z) ≥ −1, for any point z ∈ E_{i} − {x_{i}}, with equality only if z − x_{i} ∈ Pic^{0}(E_{i}) is a torsion class. Using these remarks as well as and the additivity of the Brill-Noether number of l, since ρ(l, x, y) = −3 it follows that there must exist an index 1 ≤ i ≤ g such that x, y ∈ E_{i} − {x_{i}}, and ρ(l_{E_{i}}, x_{i}, x, y) = −3. This implies that a^{l_{E_{i}}} (x_{i}) = (d − 3, d) and that l_{E_{i}}(−(d − 3)x_{i}) ∈ G_{3}^{1}(E_{i}) has triple ramification points at distinct points x_{i}, x and y. This can happen only if E_{i} is isomorphic to the Fermat cubic, a contradiction. □

The next result highlights the difference between \overline{𝔗ℜ} and all the other geometric divisors in the literature, cf. [HM], [EH], [H], [Fa1], [Fa2]: \overline{𝔗ℜ} is the first example of a geometric divisor on \overline{M}_{g} not pulled-back from the space \overline{M}_{g}^{ps} of pseudo-stable curves.

Proposition 3.2. If \overline{𝔗ℜ}_{d} ≡ a λ − ∑_{i=0}^{d−2} b_{i} δ_{i} ∈ Pic(\overline{M}_{g}), then a − 12b_{0} + b_{1} = 4a(d, 2d − 4).

Proof. We use a standard test curve in \overline{M}_{g} obtained by attaching to the marked point of a general pointed curve [C, q] ∈ M_{2d−4,1} a pencil of plane cubics. If R ⊂ \overline{M}_{g} is the family induced by this pencils, then clearly R · λ = 1, R · δ_{0} = 12, R · δ_{1} = −1 and R · δ_{j} = 0 for j ≥ 2.

Set-theoretically, R∩\overline{𝔗ℜ}_{d} consists of the points corresponding to the elliptic curves [E, q] in the pencil, for which there exists l ∈ G_{3}^{1}(E) as well as two distinct points x, y ∈ E − {q} with a_{1}^{l}(q) = a_{1}^{l}(x) = a_{1}^{l}(y) = 3 (It is a standard limit linear series argument to show that the triple points of the limit 𝔤_{d}^{1} must specialize to the elliptic tail). Then E must be isomorphic to the Fermat cubic, (thus j(E) = 0, and this curve appears 12 times in the pencil. The pencil l ∈ G_{3}^{1}(E) is of course uniquely determined. Since Aut(E, q) = ℤ/6ℤ while a generic element from \overline{M}_{1,1} has automorphism group ℤ/2ℤ, each point of intersection will contribute 4 = 24/6 times in the intersection R ∩ \overline{𝔗ℜ}_{d}. On the side of the genus 2d − 4 component, we count pencils L ∈ W_{d}^{1}(C) with a_{1}^{L}(q) ≥ 3. Using Proposition 2.1 their number is finite and equal to a(d, 2d − 4), hence R · \overline{𝔗ℜ}_{d} = 4a(d, 2d − 4). □

Next we describe the restriction of \overline{𝔗ℜ}_{d} under the map χ : \overline{M}_{2,1} → \overline{M}_{2d−3} obtained by attaching a fixed tail B of genus 2d − 5 to each pointed curve [C, p] ∈ M_{2,1}. It is revealing to compare Theorem 1.2 to Propositions 4.1 and 5.5 in [EH]: When ρ(g, r, d) = −1, the pull-back of the Brill-Noether divisor χ^{∗}(\overline{M}_{g,d}^{r}) is irreducible and supported on \overline{W}. By contrast, \overline{𝔗ℜ}_{d} displays a much richer geometry.

Proof of Theorem 1.2. We fix a general pointed curve [B, p] ∈ M_{2d−5,1}. For each [C, p] ∈ M_{2,1}, we study degree d admissible coverings [f : X → R, q_{1}, q_{2}; p_{1}, …, p_{6d−12}] ∈ \overline{H}_{d}^{tr} with source curve X stably equivalent to C ∪_{p} B, and target R a nodal curve of genus 0. Moreover, f is assumed to have distinct points of triple ramification x, y ∈ X_{reg}, where f(x) = q_{1} and f(y) = q_{2}. It is easy to check that both x and y must lie either on C or on B (and not on rational components of X we may insert). Depending on their position we distinguish four cases:

(i) x, y ∈ B. A parameter count shows that deg(f_{B}) = d and p ∈ B must be a simple ramification point for f_{B}. By compatibility of ramification sequences at p, then f_{C} must also be simply ramified at p, that is, p ∈ C is a Weierstrass point and f_{C} is induced by |O_{C}(2p)|. There is a canonical way of completing {f_{C}, f_{B}} to an element in ℌ_{d}, by attaching rational curves to B at the points in f_{B}^{−1}(f(p)) − \{p\}. For a fixed [C, p] ∈ \overline{W}, the Hurwitz scheme is smooth at each of the points t ∈ \overline{H}_{d}^{tr} corresponding to an admissible coverings {f_{C}, f_{B}} of the type described above. Since t has no automorphisms permuting some of the branch points, it follows that ℌ_{d} = \overline{H}_{d}^{tr}/𝔖_{2} × 𝔖_{6d−12} is also smooth at each of the N_{1}(d) points in the fibre σ^{−1}([C ∪_{p} B]). This implies that N_{1}(d) · \overline{W} appears as an irreducible component in the pull-back divisor χ^{∗}(\overline{𝔗ℜ}_{d}).

(ii) x, y ∈ C, deg(f_{B}) = d. Clearly deg(f_{C}) ≥ 4 and the B-aspect of the covering must have a 4-fold point at p. There are a(d, 2d − 5) choices for f_{B}, whereas f_{C} corresponds to a linear series l_{C} ∈ G_{4}^{1}(C) with a_{1}^{l_{C}}(p) = 4 and which has two other points of triple ramification. To obtain the domain of an admissible covering, we attach to B rational curves at the (d − 4) points in f_{B}^{−1}(f(p)) − {p}. We map these curves isomorphically onto f_{C}(C). The divisor a(d, 2d − 5) · \overline{D}_{3} is an irreducible component of χ^{∗}(\overline{𝔗ℜ}_{d}).

(iii) x, y ∈ C, deg(f_{B}) = d − 1. In this case the B-aspect corresponds to one of the a(d − 1, 2d − 5) linear series l_{B} ∈ G_{d−1}^{1}(B) with simple ramification at p, while f_{C} is a degree 3 covering having two unspecified points of triple ramification and simple ramification at p ∈ C. To obtain a point in ℌ_{d}, we attach a rational curve T′ to C at the remaining point in f_{C}^{−1}(f(p) − {p}. We then map T′ isomorphically onto f_{B}(B). Next, we attach d − 3 rational curves to B at the points f_{B}^{−1}(f(p)) − {p}, which we map isomorphically onto f_{C}(C). Each resulting admissible covering has no automorphisms and is a smooth point of ℌ_{d}. Thus a(d − 1, 2d − 5) · \overline{D}_{2} is a component of χ^{∗}(\overline{𝔗ℜ}_{d}).

(iv) x ∈ C, y ∈ B. After a moment of reflection we conclude that deg(f_{B}) = d, that is, f_{B} corresponds to one of the e(d, 2d − 5) coverings l_{B} ∈ G_{d}^{1}(B) with a_{1}^{l_{B}}(p) = 3 and a_{1}^{l_{B}}(y) = 3 at some unspecified point y ∈ B − {p}. The C-aspect of f is determined by the choice of a point x ∈ C − \{p\} such that 3x ≡ 3p. Hence e(d, 2d − 5) · \overline{D}_{1} is the final irreducible component of χ^{∗}(\overline{𝔗ℜ}_{d}). □

As a consequence of Proposition 3.1 and Theorem 1.2 we are in a position to determine all the δ_{i}-coefficients (i ≥ 1) in the expansion of \overline{𝔗ℜ}_{d} in the basis of Pic(\overline{M}_{g}):

Theorem 3.3. If \overline{𝔗ℜ}_{d} ≡ a λ − ∑_{i=0}^{d−2} b_{i} δ_{i} ∈ Pic(\overline{M}_{g}), then we have that

b_{i} = \frac{(2d − 6)!}{2 d!(d − 3)!} i(2d − 3 − i)(36d^{3} − 156d^{2} + 180d − 5), for all1 ≤ i ≤ d − 2.

Proof. We use the obvious relations χ^{∗}(δ_{2}) = −ψ, χ^{∗}(λ) = λ, χ^{∗}(δ_{0}) = δ_{0}, χ^{∗}(δ_{1}) = δ_{1}. If for a class E ∈ Pic(\overline{M}_{2,1}) we denote by (E)_{ψ} the coefficient of ψ in its expansion in the basis \{ψ, λ, δ_{0}\} of Pic(\overline{M}_{2,1}) (see also the next section for details on the divisor theory of \overline{M}_{2,1}), then, using Proposition 3.2, we can write the following relation:

b_{2} = \frac{2(g − 2)}{g − 1}b_{1} = N_{1}(d)(\overline{W})_{ψ}+e(d, 2d−5)(\overline{D}_{1})_{ψ}+a(d−1, 2d−5)(\overline{D}_{2})_{ψ}+a(d, 2d−5)(\overline{D}_{3})_{ψ}.

We determine the coefficients (\overline{D}_{i})_{ψ} for 1 ≤ i ≤ 3 by intersecting each of these divisors with a general fibral curve F := \{[C, p]\}_{p∈C} ⊂ \overline{M}_{2,1} of the projection π : \overline{M}_{2,1} → \overline{M}_{2}. (Note that (\overline{W})_{ψ} = 3).

It is useful to recall that if [C, q] ∈ M_{2,1} is a fixed general pointed curve and a ≥ b ≥ 0 are integers, then the number of pairs (p, x) ∈ C × C, p ≠ x satisfying a linear equivalence relation a · x ≡ b · p + (a − b) · q in Pic^{a}(C), equals

(5)  r(a, b) := 2(a^{2}b^{2} − 1).

We start with \overline{D}_{1} and note that F · \overline{D}_{1} is the number of pairs (x, p) ∈ C × C with x ≠ p, such that 3x ≡ 3p, which is equal to r(3, 3) = 160 and then (\overline{D}_{1})_{ψ} = r(3, 3)/(2g−2) = 80. To compute F · \overline{D}_{2} we note that there are 80 = r(3, 3)/2 pencils L ∈ W_{3}^{1}(C) with two distinct triple ramification points. From the Hurwitz-Zeuthen formula, each such pencil has 4 more simple ramification points, thus (\overline{D}_{2})_{ψ} = 4 × 80/(2g − 2) = 160. Finally, F · \overline{D}_{3} = n_{0}/2, where by n_{0} we denote the number of pencils l ∈ W_{4}^{1}(C) having one unspecified point of total ramification and two further points of triple ramification, that is there exist mutually distinct points x, y, p ∈ C with a_{1}^{l}(p) = 4 and a_{1}^{l}(x) = a_{1}^{l}(y) = 3.

We compute n_{0} by letting C specialize to a curve of compact type [C_{0} := C_{1}∪_{q} C_{2}], where [C_{1}, q], [C_{2}, q] ∈ M_{1,1}. Then n_{0} is the number of admissible coverings f : X →^{4:1} R, where R is of genus 0 and X is stably equivalent to C_{0} and has a 4-fold ramification point p ∈ X_{reg} and triple ramification points x, y ∈ X_{reg}. We distinguish three cases:

(i) x, y ∈ C_{2} and p ∈ C_{1} (Or x, y ∈ C_{1} and p ∈ C_{2}). In this case deg(f_{C_{1}}) = deg(f_{C_{2}}) = 4 and we have the linear equivalence 4p ≡ 4q on C_{1}. This yields 15 choices for p ≠ q. On C_{2} we count 𝔤_{4}^{1}’s with total ramification at q, and two unspecified triple points. This number is equal to 20 (see the proof of Proposition 2.2). Reversing the role of C_{1} and C_{2} we double the number of coverings and we find 600 = 2 · 15 · 20 admissible 𝔤_{4}^{1}’s.

(ii) x, p ∈ C_{2} and y ∈ C_{1} (Or x, p ∈ C_{1} and y ∈ C_{2}). In this situation deg(f_{C_{1}}) = 3 and deg(f_{C_{2}}) = 4 and on C_{1} we have the linear equivalence 3y ≡ 3q, which gives 8 choices for y. On C_{2} we count l_{C_{2}} ∈ G_{4}^{1}(C_{2}) in which two unspecified points p, x ∈ C_{2} appear with multiplicities 4 and 3 respectively, while a_{1}^{l_{C_{2}}} (q) = 3. By translation, this is the same as the number of pairs of distinct points (u, v) ∈ C_{2} − {q} × C_{2} − {q} such that there exists l_{2} ∈ G_{4}^{1}(C_{2}) with a_{1}^{l_{2}}(q) = 4, a_{1}^{l_{2}}(x) = a_{1}^{l_{2}}(y) = 3. This number equals 40 (again, see the proof of Proposition 2.2). By reversing the role of C_{1} and C_{2} the total number of coverings in case (ii) is 640 = 2 · 8 · 40.

(iii) x, y, p ∈ C_{1} (or x, y, p ∈ C_{2}). A quick parameter count shows that deg(f_{C_{2}}) = 2 and mult_{q}(f_{C_{2}}) = mult_{q}(f_{C_{1}}) = 2. Hence f_{C_{2}} is induced by |O_{C_{2}}(2q)|. On C_{1} we count 𝔤_{4}^{1}’s in which the points p, x, y, q appear with multiplicities 4, 3, 3 and 2 respectively. The translation on C_{2} from p to q shows that we are yet again in the situation of Proposition 2.2 and this last number is 20. We interchange C_{1} and C_{2} and we find 40 admissible 𝔤_{4}^{1}’s on C_{1} ∪ C_{2} with all the non-ordinary ramification concentrated on a single component.

By adding (i), (ii) and (iii) together, we obtain n_{0} = 600 + 640 + 40 = 1280. This determines (\overline{D}_{3})_{ψ} = n_{0}/(2g − 2) = 640 and completes the proof. □

4. The divisor theory of \overline{M}_{2,1}

The remaining part of the calculation of [\overline{𝔗ℜ}_{d}] has been reduced to the problem of determining the divisor classes [\overline{D}_{i}] (i = 1, 2, 3) on \overline{M}_{2,1}. We recall some things about divisor theory on this space (see also [EH]). There are two boundary divisor classes:

• δ_{0}, whose generic point is an irreducible 1-pointed nodal curve of genus 2.

• δ_{1}, with generic point being a transversal union of two elliptic curves with the marked point lying on one of the components.

If π : \overline{M}_{2,1} → \overline{M}_{2} is the universal curve then ψ := c_{1}(ω_{π}) ∈ Pic(\overline{M}_{2,1}) denotes the tautological class and λ = π^{∗}(λ) ∈ Pic(\overline{M}_{2,1}) is the Hodge class. Unlike the case g ≥ 3, λ is a boundary class on \overline{M}_{2}, and we have Mumford’s genus 2 relation:

λ = \frac{1}{10}δ_{0} + \frac{1}{5}δ_{1}.

The classes ψ, λ and δ_{1} form a basis of Pic(\overline{M}_{2,1})⊗ℚ. The class of the Weierstrass divisor has been computed in [EH] Theorem 2:

(6)  \overline{W} ≡ 3ψ − λ − δ_{1}.

We start by determining the class of \overline{D}_{1} of 3-torsion points:

Proposition 4.1. The class of the closure in \overline{M}_{2,1} of the effective divisor

D_{1} = {[C, p] ∈ M_{2,1} : ∃x ∈ C − {p} such that3x ≡ 3p}

is given by [\overline{D}_{1}] = 80ψ + 10δ_{0} − 120λ ∈ Pic(\overline{M}_{2,1}).

Proof. We introduce the map χ : \overline{M}_{2,1} → \overline{M}_{4} given by χ([C, p]) := [B ∪_{p} C], where [B, p] is a general 1-pointed curve of genus 2. On \overline{M}_{4} we have the divisor of curves with an exceptional Weierstrass point 𝔇𝔦 := {[C] ∈ M_{4} : ∃x ∈ C such that h^{0}(C, 3x) ≥ 2}. Its class has been computed by Diaz [Di]: \overline{𝔇𝔦} ≡ 264λ − 30δ_{0} − 96δ_{1} − 128δ_{2} ∈ Pic(\overline{M}_{4}).

We claim that χ^{∗}(\overline{𝔇𝔦}) = \overline{D}_{1} + 16 · \overline{W}. Indeed, let [C, p] ∈ M_{2,1} be such that χ([C, p]) ∈ \overline{𝔇𝔦}. Then there is a limit 𝔤_{3}^{1} on X := B ∪_{p} C, say l = \{l_{B}, l_{C}\}, which has a point of total ramification at some x ∈ X_{reg}. There are two possibilities:

(i) If x ∈ C, then a^{l_{B}}(p) = (0, 3), hence l_{B} = |O_{B}(3p)|, while on C we have the linear equivalence 3p ≡ 3x, that is, [C, p] ∈ \overline{D}_{1}.

(ii) If x ∈ B, then a^{l_{C}}(p) = (1, 3), that is, p ∈ B is a Weierstrass point and moreover l_{C} = p + |O_{C}(2p)|. On B we have that a^{l_{B}}(p) = (0, 2) and a^{l_{B}}(x) = (0, 3), that is, 3x ≡ 2p + y for some y ∈ B − {p, y}. There are r(3, 1) = 16 such pairs (x, y).

Thus we have proved that χ^{∗}(\overline{𝔇𝔦}) = \overline{D}_{1} + 16 · \overline{W} (We would have obtained the same conclusion using admissible coverings instead of limit 𝔤_{3}^{1}’s). We find the formula for [\overline{D}_{1}] if we remember that χ^{∗}(δ_{0}) = δ_{0}, χ^{∗}(δ_{1}) = δ_{1}, χ^{∗}(δ_{2}) = −ψ and χ^{∗}(λ) = λ. □

4.1. The divisor \overline{𝔗ℜ}_{3} and the class of \overline{D}_{2}. We compute the class of the divisor \overline{D}_{2} on \overline{M}_{2,1} by determining directly the class of \overline{𝔗ℜ}_{3} in genus 3 (In this case \overline{D}_{3} = ∅). Much of the set-up we develop here is valid for arbitrary d ≥ 3 and will be used in the next section when we compute the class [\overline{𝔗ℜ}_{4}] on \overline{M}_{5}. We fix a general [C, p] ∈ M_{2d−4,1} and introduce the following enumerative invariant:

N_{2}(d) := #{l ∈ G_{d}^{1}(C) : ∃x ≠ y ∈ C − {p} such that l(−3x) ≠ ∅ and l(−p − 2y) ≠ ∅}.

For instance, N_{2}(3) is the number of pairs (x, y) ∈ C×C, x ≠ p ≠ y such that 3x ≡ p+2y, hence N_{2}(3) = r(3, 2) = 70 (cf. formula (5)).

For each d ≥ 4 we fix a general pointed curve [B, q] ∈ M_{2d−5,1} and define the invariant:

N_{3}(d) := #{l ∈ G_{d}^{1}(B) : ∃x ≠ y ∈ B − {q} such that l(−3x) ≠ ∅ and l(−2q − 2y) ≠ ∅}.

Theorem 4.2. The closure of the divisor 𝔗ℜ_{3} := {[C] ∈ M_{3} : ∃x ≠ p ∈ C with3x ≡ 3x} is linearly equivalent to the class

\overline{𝔗ℜ}_{3} ≡ 2912λ − 311δ_{0} − 824δ_{1} ∈ Pic(\overline{M}_{3}).

It follows that \overline{D}_{2} ≡ −200λ + 160ψ + 17δ_{0} ∈ Pic(\overline{M}_{2,1}).

Proof. For most of this proof we assume d ≥ 3 and we specialize to the case of \overline{M}_{3} only at the very end. We write \overline{𝔗ℜ}_{d} ≡ a λ − b_{0} δ_{0} − ⋯ − b_{d−2} δ_{d−2} ∈ Pic(\overline{M}_{g}) and we have already determined b_{1}, …, b_{d−2} (cf. Theorem 3.3) while we know that a − 12b_{0} + b_{1} = 4a(d, 2d − 4) (cf. Proposition 3.2). We need one more relation involving a, b_{0} and b_{1}, which we obtain by intersecting \overline{𝔗ℜ}_{d} with the test curve

C^{0} := \{\frac{C}{q ∼ p}\}_{p∈C} ⊂ Δ_{0} ⊂ \overline{M}_{g}

obtained from a general curve [C, q] ∈ M_{2d−4,1}. The number C^{0} · \overline{𝔗ℜ}_{d} counts (with appropriate multiplicities) admissible coverings

t := [f : X →^{d:1} R, q_{1}, q_{2} : p_{1}, …, p_{6d−12}] mod 𝔖_{2} × 𝔖_{6d−12} ∈ ℌ_{d},

where the source X is stably equivalent to the curve C ∪_{{p,q}} T (q ∈ C) obtained by ”blowing-up” \frac{C}{q∼p} at the node and inserting a rational curve T. These covers should possess two points of triple ramification x, y ∈ X_{reg} such that f(x) = q_{1}, f(y) = q_{2}. Suppose t ∈ C^{0} · \overline{𝔗ℜ} and again we distinguish a number of possibilities:

(i) x, y ∈ C. Then deg(f_{C}) = d and f_{C} corresponds to one of the N(d) linear series l ∈ G_{d}^{1}(C) with two points of triple ramification. The point q ∈ C is such that l(−p − q) ≠ ∅, which, after having fixed l, gives d − 1 choices. Clearly mult_{q}(f_{C}) = mult_{q}(f_{T}) = 1. This implies that deg(f_{T}) = 2 and f_{T} is given by |O_{T} (p + q)|. To obtain out of {f_{C}, f_{B}} a point t ∈ \overline{H}_{d}^{tr}, we attach rational curves to C at the points in f_{C}^{−1}(f(p)) − \{p, q\} and map these isomorphically onto the component f_{T} (T) of R. Each such cover has an automorphism ϕ : X → X of order 2 such that ϕ_{C} = id_{C}, ϕ_{T′} = id_{T′}, for every rational component T′ ≠ T of X, but ϕ_{T} interchanges the 2 branch points of T. Even though t ∈ \overline{H}_{d}^{tr} is a smooth point (because there is no automorphism of X preserving all the ramification points of f), if τ ∈ 𝔖_{6d−12} is the involution exchanging the marked points lying on f_{T} (T), then τ · t = t. Therefore \overline{H}_{d}^{tr}/𝔖_{2} → \overline{M}_{g} is simply ramified at t. In a general deformation [X → R] of [f : X → R] in \overline{H}_{d}^{tr} we blow-down T and obtain a rational double point, hence the image of R in \overline{M}_{g} meets Δ_{0} with multiplicity 2. Since \overline{H}_{d}^{tr}/𝔖_{2} → \overline{M}_{g} is ramified anyway, it follows that each of the (d − 1)N(d) admissible coverings found at this step is to be counted with multiplicity 1.

(ii) x ∈ C, y ∈ T. Since C has only finitely many 𝔤_{d−1}^{1}’s, all simply ramified and having no ramification in the fibre over q, we must have that deg(f_{C}) = d and deg(f_{T}) = 3. Moreover, C and T map via f onto the two components of the target R in such a way that f_{C}(p) = f_{C}(q) = f_{T} (p) = f_{C}(q). In particular, both f_{C} and f_{T} are simply ramified at either p or q. If f_{C} is ramified at q ∈ C, then f_{C} is induced by one of the e(d, 2d−4) linear series l ∈ G_{d}^{1}(C) with one unassigned point of triple ramification and one assigned point of simple ramification. Having fixed l, there are d − 2 choices for p ∈ C such that l(−2q − p) ≠ ∅. On T there is a unique 𝔤_{3}^{1} corresponding to a map f_{T} : T → P^{1} such that f_{T}^{∗}(0) = 2q + p and f_{T}^{∗}(∞) = 3y, for some y ∈ T − {q, p}. Finally, we attach d − 3 rational curves to C at the points in f_{C}^{−1}(f(q)) − {p, q} and we map these components isomorphically onto f_{T} (T).

The other possibility is that f_{C} is unramified at q and ramified at p. The number of such 𝔤_{d}^{1}’s is N_{2}(d). On the side of T, there is a unique way of choosing f_{T} : T →^{3:1} P^{1} such that f_{T}^{∗}(0) = q + 2p and f_{T}^{∗}(∞) = 3y. Because the map σ : ℌ_{d} → \overline{M}_{g} blows-down the component T, if [X → R] is a general deformation of [f : X → R] then σ(R) meets Δ_{0} with multiplicity 3 (see also [Di], pg. 47-52). Thus \overline{𝔗ℜ}_{d}·Δ_{0} has multiplicity 3 at the point [C/p ∼ q]. The admissible coverings constructed at this step have no automorphisms, hence they each must be counted with multiplicity 3. This yields a total contribution of 3(d − 2)e(d, 2d − 4) + 3N_{2}(d).

(iii) x, y ∈ T −{p, q}. Here there are two subcases. First, we assume that deg(f_{C}) = d−1, that is, f_{C} is induced by one of the \frac{(2d−4)!}{(d−1)!(d−2)!} linear series l ∈ G_{d−1}^{1}(C). For each such l, there are d − 2 possibilities for p such that l(−q − p) ≠ ∅. Clearly deg(f_{T}) = 3 and the admissible covering f is constructed as follows: Choose f_{T} : T → P^{1} such that f_{T}^{∗}(0) = 3x, f_{T}^{∗}(∞) = 3y and f_{T}^{∗}(1) = p + q + q′. We map C to the component of R other than f_{T} (T) by using l ∈ G_{d−1}^{1}(C) and f_{C}(p) = f_{T} (p) and f_{C}(q) = f_{T} (q). We attach to T a rational curve T′ at the point q′ and map T′ isomorphically onto f(C). Finally we attach d − 3 rational curves to C at the points in f_{C}^{−1}(f(q)) − \{q, p\}. Each of these \binom{2d−4}{d−1} elements of 𝔥_{d} is counted with multiplicity 2.

We finally deal with the case deg(f_{C}) = d. Since a 𝔤_{3}^{1} on P^{1} with two points of total ramification must be unramified everywhere else, it follows that deg(f_{T}) ≥ 4. The generality assumption on [C, q] implies that deg(f_{T}) = 4. The C-aspect of f is induced by l ∈ G_{d}^{1}(C) for which there are integers β, γ ≥ 1 with β + γ = 4 and a point p ∈ C such that l(−βp − γq) ≠ ∅. Proposition 2.1 gives the number c(d, 2d − 4, γ) of such l ∈ G_{d}^{1}(C). On the side of T, we choose f_{T} : T →^{4:1} P^{1} such that f_{T}^{∗}(0) = 3x, f_{T}^{∗}(∞) = 3y and f_{T}^{∗}(1) = βp + γq. When γ ∈ {1, 3}, up to isomorphism there is a unique such f_{T} having 3 triple ramification points. By direct computation we have the formula:

f_{T} : T → P^{1}, f_{T} (t) := \frac{2t^{3}(t − 2)}{2t − 1},

which has the properties that f_{T}^{(i)}(0) = f_{T}^{(i)}(∞) = f_{T}^{(i)}(1) = 0, for i = 1, 2. When γ = 2, there are two 𝔤_{4}^{1}’s with 2 points of triple ramification and 2 points of simple ramification lying in the same fibre. It is important to point out that f_{T} (and hence the admissible covering f as well), has an automorphism of order 2 which preserves the points of attachment p, q ∈ T but interchanges x and y (In coordinates, if x = 0, y = ∞ ∈ T, check that f_{T} (1/t) = 1/f_{T} (t)). This implies that \overline{H}_{d}^{tr} → \overline{M}_{d} is (simply) ramified at [X → R]. Furthermore, a calculation similar to [Di] pg. 47-50, shows that the image in \overline{M}_{g} of a generic deformation in \overline{H}_{d}^{tr} of [X → T] meets the divisor Δ_{0} with multiplicity 4 = β + γ. It follows that \overline{𝔗ℜ}_{d} · Δ_{0} has multiplicity 4/2 = 2 in a neighbourhood of [C/p ∼ q], that is, each covering found at this step gets counted with multiplicity 2 in the product C^{0} · \overline{𝔗ℜ}. Coverings of this type give a contribution of

2c(d, 2d − 4, 1) + 2c(d, 2d − 4, 3) + 4c(d, 2d − 4, 2) = 128\binom{2d − 4}{d}.

Thus we can write the following equation:

(7)  (2g − 2)b_{0} − b_{1} = C^{0} · \overline{𝔗ℜ}_{d} =

= (d − 1)N(d) + 3N_{2}(d) + 3(d − 2)e(d, 2d − 4) + 128\binom{2d − 4}{d} + 2\binom{2d − 4}{d − 1}.

For d = 3, when N_{2}(d) = 70, all terms in (7) are known and this finishes the proof. □

5. The divisor \overline{𝔗ℜ}_{5} and the class of \overline{D}_{3}

In this section we finish the computation of [\overline{𝔗ℜ}_{d}] (and implicitly compute [\overline{D}_{3}] ∈ Pic(\overline{M}_{2,1}) and determine N_{2}(d) for all d ≥ 3 as well). According to (7) it suffices to compute N_{2}(4) to determine [\overline{𝔗ℜ}_{4}] ∈ Pic(\overline{M}_{5}). Then applying Theorem 1.2 we obtain [\overline{D}_{3}] which will finish the calculation of [\overline{𝔗ℜ}_{d}] for g = 2d − 3. We summarize some of the enumerative results needed in this section:

Proposition 5.1. We fix a general 2-pointed elliptic curve [E, p, q] ∈ M_{1,2}.

(a) There are 11 pencils l ∈ G_{3}^{1}(E) such that there exist distinct points x, y ∈ E − {p, q} with a_{1}^{l}(x) = 3, a_{1}^{l}(q) = 2 and l(−p − 2y) ≠ ∅.

(b) There are 38 pencils l ∈ G_{4}^{1}(E) such that there exist distinct points x, y ∈ E − {p, q} with a_{1}^{l}(p) = 4, a_{1}^{l}(x) = 3 and l(−q − 2y) ≠ ∅.

Proof. (a) We denote by U the closure in E × E of the locus

{(u, v) ∈ E × E − Δ : ∃l ∈ G_{3}^{1}(E) such that a_{1}^{l}(q) = 3, a_{1}^{l}(u) ≥ 2, a_{1}^{l}(v) ≥ 2}

and denote by F_{i} the (numerical class of the) fibre of the projection π_{i} : E × E → E for i = 1, 2. Using that U ∩ Δ = {(u, u) : u ≠ q, 3u ≡ 3q} (this intersection is transversal!), it follows that U ≡ 4(F_{1} + F_{2}) − Δ. If q ∈ E is viewed as the origin of E, then the isomorphism E ×E ∋ (x, y) ↦ (−x, y −x) ∈ E ×E shows that the number of l ∈ G_{3}^{1}(E) we are computing, equals the intersection number U · V on E × E, where

V := {(u, v) ∈ E × E : 2v + u ≡ 4q − p}.

Since V ≡ 3F_{1} + 6F_{2} − 2Δ, we reach the stated answer by direct calculation.

(b) We specialize [E, p, q] ∈ M_{1,2} to the stable curve [E ∪_{r} T, p, q] ∈ \overline{M}_{1,2}, where [T, r, p, q] ∈ \overline{M}_{0,3}. We count admissible coverings [f : X ⟶^{4:1} R, p̃, q̃], where p̃, q̃ ∈ X_{reg}, R is a nodal curve of genus 0 and there exist points x, y ∈ X_{reg} with the property that the divisors 4p̃, 3x, q̃ + 2y on X all appear in distinct fibres of f. Moreover [X, p̃, q̃] is a pointed curve stably equivalent to [E ∪_{r} T, p, q]. There are three possibilities:

(1) x, y ∈ E. Then f_{T} : T →^{4:1} (P^{1})_{1} is uniquely determined by the properties f_{T}^{∗}(0) = 4p and f_{T}^{∗}(∞) = 3r + q, while f_{E} : E →^{3:1} (P^{1})_{2} is such that r and some point x ∈ E − {r} appear as points of total ramification. In particular, 3x ≡ 3r on E, which gives 8 choices for x. Each such f_{E} has 2 remaining points of simple ramification, say y_{1}, y_{2} ∈ E and we take a rational curve T′ which we attach to T at q and map isomorphically onto (P^{1})_{2}. Choose q̃ ∈ T′ with the property that f(q̃) = f_{E}(y_{i}) for i ∈ {1, 2} and obviously p̃ = p ∈ T. This procedure produces 16 = 8 · 2 admissible 𝔤_{4}^{1}’s.

(2) x ∈ T, y ∈ E. Now f_{T} : T →^{4:1} (P^{1})_{1} has the properties f_{T}^{∗}(0) = 4p, f_{T}^{∗}(1) ≥ 2r + q and f_{T}^{∗}(∞) ≥ 3x for some x ∈ T (Up to isomorphism, there are 2 choices for f_{T}). Then f_{E} : E →^{2:1} (P^{1})_{2} is ramified at r and at some point y ∈ E − {r} such that 2y ≡ 2r. This gives 3 choices for f_{E}. We attach two rational curve T′ and T′′ to T at the points q and q′ ∈ f_{T}^{−1}(f(q)) − {r, q} respectively. We then map T′ and T′′ isomorphically onto (P^{1})_{2}. Finally we choose p̃ = p ∈ T and q̃ ∈ T′ uniquely determined by the condition f_{T′}(q̃) = f_{E}(y). We have produced 6 = 2 · 3 coverings.

(3) x ∈ E, y ∈ T. Counting ramification points on T we quickly see that deg(f_{E}) = 3 and f_{E} : E → (P^{1})_{2} is such that f_{E}^{∗}(0) = 3x and f_{E}^{∗}(∞) = 3r, which gives 8 choices for f_{E}. Moreover f_{T} : T →^{4:1} (P^{1})_{1} must satisfy the properties f_{T}^{∗}(0) = 4p, f_{T}^{∗}(1) ≥ q + 2y and f_{T}^{∗}(∞) = 3r + r′ for some r′ ∈ T. If [T, p, q, r] = [P^{1}, 0, 1, ∞] ∈ \overline{M}_{0,3}, then

f_{T} (t) = \frac{t^{4}}{t − r′} , where r′ ∈ \{\frac{1 + \sqrt{−2}}{4}, \frac{1 − \sqrt{−2}}{4}\}.

Thus we obtain another 16 = 8·2 admissible 𝔤_{4}^{1}’s in this case. Adding (1), (2) and (3), we found 38 = 16+6+16 admissible coverings 𝔤_{4}^{1} on E ∪_{r} T and this finishes the proof. □

Proposition 5.2. We fix a general pointed curve [C, p] ∈ M_{3,1}. Then there are 210 pencils l = O_{C}(2p + 2x) ∈ G_{4}^{1}(C), x ∈ C, having an unspecified triple point.

Proof. We define the map ϕ : C × C → Pic^{1}(C) given by

ϕ(x, y) := O_{C}(2p + 2x − 3y).

A standard calculation shows that ϕ^{∗}(W_{1}(C)) = g(g − 1) · 2^{2} · 3^{2} = 216 (Use Poincaré’s formula [W_{1}(C)] = θ^{2}/2). Set-theoretically it is clear that ϕ^{∗}(W_{1}(C)) ∩ Δ = {(p, p)}. A local calculation similar to [Di] pg. 34-36, shows that the intersection multiplicity at the point (p, p) is equal to 6 = g(g − 1), hence the answer to our question. □

5.1. The invariant N_{2}(d). We have reached the final step of our calculation and we now compute N_{2}(d). We denote by \overline{A}_{d} the Hurwitz stack parameterizing admissible coverings of degree d

t := [f : (X, p) ⟶^{d:1} R, q_{0}; p_{0}; p_{1}, …, p_{6d−13}],

where [X, p] is a pointed nodal curve of genus 2d − 4, [R, q_{0}; p_{0} : p_{1}, …, p_{6d−13}] is a pointed nodal curve of genus 0, and f is an admissible covering in the sense of [HM] having a point of triple ramification x ∈ f^{−1}(q_{0}), a point of simple ramification y ∈ X − {p} such that f(y) = f(p) = p_{0} and points of simple ramification in the fibres over p_{1}, …, p_{6d−13}. The symmetric group 𝔖_{6d−13} acts on \overline{A}_{d} by permuting the branch points p_{1}, …, p_{6d−13} and the stabilization map

ϕ : \overline{A}_{d}/𝔖_{6d−13} → \overline{M}_{2d−4,1}, ϕ(t) := [X, p]

is generically finite of degree N_{2}(d).

We completely describe the fibre ϕ^{−1}([C ∪_{q} E, p]), where [C, q] ∈ M_{2d−5,1} and [E, q, p] ∈ M_{1,2} are general pointed curves. We count admissible covers f : (X, p̃) → R as above, where [X, p̃] is stably equivalent to [C ∪_{q} E, p]. Depending on the position of the ramification points x, y ∈ X we distinguish between the following cases:

(i) x ∈ C, y ∈ E. From Brill-Noether theory, we know that deg(f_{C}) ∈ {d − 1, d}. If deg(f_{C}) = d, then one possibility is that both f_{C} and f_{E} are triply ramified at q. In this case f_{C} is induced by one of the e(d, 2d − 5) linear series l ∈ G_{d}^{1}(C) with l(−3q) ≠ ∅ and l(−3x) ≠ ∅, for some x ∈ C − {q}. The covering f_{E} is of degree 3 and it induces a linear equivalence 3q ≡ 2y + p on E which has 4 solutions y ∈ E. To obtain X we attach to C rational curves at the d − 3 points in f_{C}^{−1}(f(q)) − {q}. We have exhibited in this way 4e(d, 2d − 5) automorphism-free points in ϕ^{−1}([C ∪_{q} E, p]) which are counted with multiplicity 1. Another possibility is that both f_{C} and f_{E} are simply ramified at q and the fibre f_{C}^{−1}(f(q)) contains a second point z ≠ q of simple ramification. The number of such l ∈ G_{d}^{1}(C) has been denoted by N_{3}(d). Having chosen f_{C}, then f_{E} : E →^{2:1} (P^{1})_{2} is induced by |O_{E}(2q)|. Then we attach a rational curve T to C at z, and we map T →^{2:1} (P^{1})_{2} using the linear system |O_{T} (2q)| in such a way that the remaining ramification point of f_{T} maps to f_{E}(p). We produce N_{3}(d) smooth points of \overline{A}_{d}/𝔖_{6d−13} via this construction. In both these cases p̃ = p ∈ C ∪ E.

(ii) x, y ∈ C. Now deg(f_{C}) = d − 1 and f_{C} is induced by one of the b(d − 1, 2d − 5) = e(d−1, 2d−5) linear series l ∈ G_{d−1}^{1}(C) with l(−3x) ≠ ∅ for some x ∈ C−{p}. Moreover, f_{C}(q) is not a branch point of f_{C} which implies that deg(f_{E}) = 2 and that f_{E} is induced by |O_{E}(p + q)|. Obviously, f_{C} and f_{E} map to different components of R. To obtain the source (X, p̃) of our covering, we first attach d − 2 rational curves to C at all the points in f_{C}^{−1}(f(q)) − {q} and map these curves 1 : 1 onto f_{E}(E). Then we attach a curve T′ ≅ P^{1}, this time to E at the point q and map T′ isomorphically onto f_{C}(C). The point q̃ ∈ X lies on the tail T′ and is characterized by the property f_{T′}(p̃) = f_{C}(y), where y ∈ C is one of the 6d − 16 simple ramification points of l. This procedure produces (6d − 16)b(d − 1, 2d − 5) admissible coverings in ϕ^{−1}([C ∪_{q} E, p]).

(iii) x ∈ E, y ∈ E. If deg(f_{C}) = d, then deg(f_{E}) ≥ 4 and f_{C} is given by one of the a(d, 2d − 5) linear series l ∈ G_{d}^{1}(C) such that l(−4q) ≠ ∅. Then f_{E} : E →^{4:1} P^{1} has the properties that (up to an automorphism of the base) f_{E}^{∗}(0) = 4q, f_{E}^{∗}(1) ≥ p + 2y and f^{∗}(∞) ≥ 3x, for some points x, y ∈ E − {p, q}. The number of such 𝔤_{4}^{1}’s has been computed in Proposition 5.1 (b) and it is equal to 38. Therefore this case produces 38a(d, 2d − 5) coverings. If on the contrary, deg(f_{C}) = d − 1, then f_{C} is induced by one of the a(d−1, 2d−5) linear series l ∈ G_{d−1}^{1}(C) such that l(−2q) ≠ ∅, while f_{E} : E →^{3:1} P^{1} is such that (up to an automorphism of the base) f_{E}^{∗}(0) ≥ 2q, f_{E}^{∗}(1) = p + 2y, f_{E}^{∗}(∞) = 3x for some x, y ∈ E − {p, q}. After making these choices, we attach d − 3 rational curves to C at the point {q′} = f_{C}^{−1}(f(q)) − {q} and we map these isomorphically onto f_{E}(E). Furthermore, we attach a rational curve T′ to E at the point {q′} = f_{E}^{−1}(f(q)) − {q} and map T′ isomorphically onto f_{C}(C). Using Proposition 5.1 (a), we obtain 11a(d−1, 2d−5) admissible coverings. Altogether part (iii) provides 38a(d−1, 2d−5)+11a(d−1, 2d−5) points in \overline{A}_{d}/𝔖_{6d−13}.

(iv) x ∈ E, y ∈ C. In this case, since p and y lie in different components, we know that we have to “blow-up” the point p and insert a rational curve which is mapped to the component f_{C}(C) of R. Thus deg(f_{C}) ≤ d − 1, and by Brill-Noether theory it follows that deg(f_{C}) = d − 1. Precisely, f_{C} is induced by one of the a(d − 1, 2d − 5) linear series l ∈ G_{d−1}^{1}(C) such that l(−2q) ≠ ∅. Furthermore, f_{E} : E →^{3:1} P^{1} can be chosen such that f_{E}^{∗}(0) = p + 2q and f_{E}^{∗}(∞) = 3x for some x ∈ E. This gives the linear equivalence 3x ≡ p + 2q on E which has 9 solutions. We attach d − 3 rational curves at the points in f_{C}^{−1}(f(q)) − {q} and map these 1 : 1 onto f_{E}(E). Finally, we attach a rational curve T′ to E at the point p and map T′ such that f(T′) = f(C). We pick p̃ ∈ T′ with the property that f_{T′}(p̃) = f_{C}(y), where y ∈ C is one of the 6d − 15 ramification points of f_{C}. We have obtained 9(6d − 15)a(d − 1, 2d − 5) admissible coverings in this way.

We have completely described ϕ^{−1}([C ∪_{q} E, p]) and it is easy to check that all these coverings have no automorphisms, hence they give rise to smooth points in \overline{A}_{d} and that the map ϕ is unramified at each of these points. Thus

N_{2}(d) = deg(ϕ) = 4e(d, 2d − 5) + (6d − 16)b(d − 1, 2d − 5) + 38a(d, 2d − 5)+
+11a(d − 1, 2d − 5) + 9(6d − 15) a(d − 1, 2d − 5) + N_{3}(d).

For d = 4, we know that N_{3}(4) = 210 (cf. Proposition 5.2), which determines N_{2}(4) and the class [\overline{D}_{3}]. We record these results:

Theorem 5.3. The locus D_{3} of pointed curves [C, p] ∈ M_{2,1} with a pencil l ∈ G_{4}^{1}(C) totally ramified at p and having two points of triple ramification, is a divisor on M_{2,1}. The class of its compactification in \overline{M}_{2,1} is given by the formula:

\overline{D}_{3} ≡ 640ψ − 860λ + 72δ_{0} ∈ Pic(\overline{M}_{2,1}).

Theorem 5.4. For a general pointed curve [C, p] ∈ M_{2d−4,1} the number of pencils L ∈ W_{d}^{1}(C) satisfying the conditions

h^{0}(L ⊗ O_{C}(−3x)) ≥ 1 and h^{0}(L ⊗ O_{C}(−p − 2y)) ≥ 1

for some points x, y ∈ C − {p}, is equal to

N_{2}(d) = \frac{6(40d^{2} − 179d + 212) (2d − 4)!}{d! (d − 3)!} .

Remark 5.5. As a check, for d = 3, the number N_{2}(3) computes the number of pairs (x, y) ∈ C × C such that p ≠ x ≠ y ≠ p and 3x ≡ p + 2y. This number is equal to r(3, 2) = 70 which matches Theorem 5.4.

Theorem 5.6. We fix an integer d ≥ 4. For a general pointed curve [C, p] ∈ M_{2d−5,1}, the number of pencils L ∈ W_{d}^{1}(C) satisfying the conditions

h^{0}(L ⊗ O_{C}(−3x)) ≥ 1 and h^{0}(L ⊗ O_{C}(−2p − 2y)) ≥ 1

for some points x, y ∈ C − {p}, is equal to

N_{3}(d) = \frac{84(d − 3)(2d^{2} − 10d + 13) (2d − 4)!}{d! (d − 2)!} .

Remark 5.7. For d = 4, Theorem 5.6 specializes to Proposition 5.2 and we find again that N_{3}(4) = 210.

References

[ACV] D. Abramovich, A. Corti and A. Vistoli, Twisted bundles and admissible coverings, Communications in Algebra 8 (2003), 3547-3618.

[BR] J. Bertin and M. Romagny, Champs de Hurwitz, math.AG/0701680.

[Di] S. Diaz, Exceptional Weierstrass points and the divisor on moduli space that they define, Memoirs of the American Mathematical Society 327 (1985).

[Fa1] G. Farkas, Koszul divisors on moduli spaces of curves, math.AG/0607475, to appear in American Journal of Mathematics (2008).

[Fa2] G. Farkas, Syzygies of curves and the effective cone of \overline{M}_{g}, Duke Mathematical Journal 135 (2006), 53-98.

[Fu1] W. Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer Verlag, second edition 1997.

[Fu2] W. Fulton, Hurwitz schemes and the irreducibility of the moduli of algebraic curves, Annals of Mathematics 90 (1969), 542575.

[FP] C. Faber and R. Pandharipande, Relative maps and tautological classes, Journal of the European Mathematical Society 7 (2005), 13-49.

[H] J. Harris, The Kodaira dimension of the moduli space of curves, II: The even genus case, Inventiones Mathematicae 75 (1984), 437-466.

[Hu] A. Hurwitz, A, Über Riemann’sche Flächen mit gegebenen Verzweigungspunkten, Mathematische Annalen 39 (1891), 1- 61.

[EH] D. Eisenbud and J. Harris, The Kodaira dimension of the moduli space of curves of genus ≥ 23, Inventiones Mathematicae 90 (1987), 359-387.

[HM] J. Harris and D. Mumford, On the Kodaira dimension of the moduli space of curves, Inventiones Mathematicae 67 (1982), 23-86.

[HH] B. Hassett and D. Hyeon, Log canonical models for the moduli space of curves: First divisorial contraction, math.AG/0607477.

[S] D. Schubert, A new compactification of the moduli space of curves, Compositio Math. 78 (1991), 297-313.

Humboldt Universität zu Berlin, Institut für Mathematik, 10099 Berlin

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