On Algebraic Theta Divisors and Rational Solutions of the KP Equation

TL;DR

This paper classifies singular curves with algebraic theta divisors, relates these divisors to tau functions of the KP hierarchy, and determines their degrees based on curve singularities.

Contribution

It provides a classification of singular curves with algebraic theta divisors and establishes a connection to KP hierarchy tau functions, including degree calculations.

Findings

Classification of singular curves with algebraic theta divisors

Explicit relation between algebraic theta functions and KP tau functions

Degree determination of algebraic theta divisors based on curve singularities

Abstract

In this paper we classify the singular curves whose theta divisors in their generalized Jacobians are algebraic, meaning that they are cut out by polynomial analogs of theta functions. We also determine the degree of an algebraic theta divisor in terms of the singularities of the curve. Furthermore, we show a precise relation between such algebraic theta functions and the corresponding tau functions for the KP hierarchy.