Sigma Function as A Tau Function

TL;DR

This paper explores the sigma function associated with (n,s)-curves, expressing it via tau functions within the Grassmannian framework, and uses this to prove properties of the sigma function's series expansion.

Contribution

It provides a new expression of the tau function as a multivariate sigma function and proves fundamental properties of the sigma function's series expansion.

Findings

Expressed the tau function as a multivariate sigma function.

Proved fundamental properties of the sigma function's series expansion.

Connected the affine ring of (n,s)-curves with the universal Grassmann manifold.

Abstract

The tau function corresponding to the affine ring of a certain plane algebraic curve, called (n,s)-curve, embedded in the universal Grassmann manifold is studied. It is neatly expressed by the multivariate sigma function. This expression is in turn used to prove fundamental properties on the series expansion of the sigma function established in a previous paper in a different method.