Jacobi inversion on strata of the Jacobian of the $C_{rs}$ curve $y^r = f(x)$

Abstract

By using the generalized sigma function of a $C_{rs}$ curve $y^r = f(x)$ , we give a solution to the Jacobi inversion problem over the stratification in the Jacobian given by the Abel image of the symmetric products of the curve. We show that determinants consisting of algebraic functions on the curve, whose zeros give the Abelian pre-image of the strata, are written by ratios of certain derivatives of the sigma function on the strata. We also discuss the order of vanishing of abelian functions on the strata in terms of intersection theory.