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

TL;DR

This paper extends the understanding of the stratification of Jacobians of cyclic $C_{rs}$ curves using sigma functions, providing new vanishing theorems related to the Weierstrass semigroup and connections to Sato's Grassmannian.

Contribution

It introduces new vanishing results for derivatives of sigma functions on specific strata of the Jacobian of $C_{rs}$ curves, linking algebraic geometry and integrable systems.

Findings

Vanishing theorems for sigma derivatives on Jacobian strata

Connection between Weierstrass semigroup and sigma function behavior

Applicability of results to Sato's Grassmannian stratification

Abstract

Previous work by the authors (this journal, \vol{60} (2008), 1009-1044) produced equations that hold on certain loci of the Jacobian of a cyclic $C_{rs}$ curve. A curve of this type generalizes elliptic curves, and the equations in question are given in terms of (Klein's) generalization of Weierstrass' $\sigma$-function. The key tool is a matrix with entries that are polynomial in the coordinates of the affine plane model of the curve, thus can be expressed in terms of $\sigma$ and its derivatives. The key geometric loci on the Jacobian of the curve give a stratification of Brill-Noether type. The results are of the type of Riemann-Kempf singularity theorem, the methods are germane to those used by J.D. Fay, who gave vanishing tables for Riemann's $\theta$-function and its derivatives. The main objects we use were developed by several contemporary authors, aside from the classical definitions: meromorphic differentials were expressed in terms of the coordinates mainly by V.M. Buchstaber, J.C. Eilbeck, V.Z. Enolski, D.V. Leykin, and Taylor expansions for $\sigma$ in terms of Schur polynomials also contributed by A. Nakayashiki, in terms of Sato's $\tau$-function. Within this framework, following specific results for $\sigma$-derivatives given by Y. \^Onishi, we arrive at our main results, namely statements on the vanishing on given strata of the partial derivatives of $\sigma$ indexed by Young-diagrams subsets that can be worked out in terms of the Weierstrass semigroup of the curve at its point at infinity. The combinatorial statements hold not only for Jacobians but for the stratification of Sato's infinite-dimensional Grassmann manifold as well.