Abstract

For even genus g=2ige 4 and the length g-1 partition mu = (4,2,ldots ,2,-2,ldots ,-2) of 0, we compute the first coefficients of the class of overline{D}(mu ) in mathrm {Pic}_{mathbb {Q}}(overline{{mathcal {R}}}_g), where D(mu ) is the divisor consisting of pairs [C,eta ]in {mathcal {R}}_g with eta cong {mathcal {O}}_C(2x_1+x_2+cdots + x_{i-1}-x_i-cdots -x_{2i-1}) for some points x_1,ldots , x_{2i-1} on C. We further provide several enumerative results that will be used for this computation.

Highlights

  • The moduli space Rg parametrizing pairs [C, η] consisting of a curve C of genus g and a 2-torsion line bundle η on C received considerable attention following the influential papers [5,24]

  • The description of Rg in [5] as a coarse moduli space of a stack, together with the algebraic theory of Prym curves developed by Mumford in [24] brought this topic to the attention of algebraic geometers

  • Let A be the test curve in Mg consisting of a generic genus g − 1 curve C glued at a generic point x to a pencil of elliptic curves along a base point

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Summary

Introduction

The moduli space Rg parametrizing pairs [C, η] consisting of a curve C of genus g and a 2-torsion line bundle η on C received considerable attention following the influential papers [5,24]. We consider the basis of PicQ(Rg) consisting of the classes λ, δ0, δ0 , δ0ram together with δi , δg−i , δi:g−i for 1 ≤ i ≤ [g/2] In this basis, we compute the first coefficients of our divisor D(μ) and we obtain: Theorem 1.1 Let g = 2i ≥ 4 and μ the length g − 1 partition We intersect our divisor with some classical test curves and compute the number of admissible covers above this intersection, along with their multiplicities We get in this way a system of 8 equations with 7 unknowns which will be compatible and will conclude Theorem 1.1

Admissible covers and enumerative geometry
Various enumerative results
A Hurwitz space over the elliptic curves I
A Hurwitz space over the elliptic curves II
Test curves
Test curve A
Test curve B
Test curves Ci
Intersection numbers
Intersection with test curves of type A
Intersection with test curves of type Cg−1
Intersection with test curves of type B
A divisor in R2i+1
Full Text
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