Prym enumerative geometry and a Hurwitz divisor in $\overline{\mathcal{R}}_{2i}$

Abstract

For i ≥ 2, we compute the first coefficients of the class [D(µ; 3)] in the rational Picard group of the moduli of Prym curves R 2i , where D(µ; 3) is the divisor parametrizing pairs [C, η] for which there exists a degree 2i map π : C → P 1 having ramification profile (2, . . . , 2) above two points q 1 , q 2 , a triple ramification somewhere else and satisfying O C ( π * (q 1 )−π * (q 2 ) 2 ) ∼ = η. Furthermore, we provide several new Prym enumerative results related to this situation.