A Hurwitz divisor on the moduli of Prym curves

Abstract

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