A generalization of Riemann's theta functions for singular curves

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.