A generalization of Riemann's theta functions for singular curves

TL;DR

This paper extends Riemann's theta functions to singular curves, providing a generalized framework that parallels classical results for smooth Riemann surfaces, thus broadening the scope of the Jacobi inversion problem.

Contribution

It introduces a new generalization of Riemann's theta functions and constants applicable to singular curves, extending classical theory to more general algebraic curves.

Findings

Defined generalized Riemann's theta functions for singular curves

Established analogous results to classical Riemann theory for these generalized functions

Provided explicit solutions to the Jacobi inversion problem in the singular case

Abstract

Let $X$ be a compact Riemann surface of genus $g$. Jacobi's inversion theorem states that the Abel-Jacobi map $\varphi : X^{(g)} \longrightarrow J(X)$ is surjective, where $X^{(g)}$ is the symmetric product of $X$ of degree $g$ and $J(X)$ is the Jacobi variety of $X$. Riemann obtained the explicit solution of the Jacobi inversion problem introducing Riemann's theta functions. We study such a problem for singular curves. We define a generalization of Riemann's theta functions and Riemann's constants. We obtain similar results for singular curves.