Theta functions for singular curves

Abstract

Let $X$ be an irreducible singular Riemann surface, with desingularisation $\widetilde X$. The generalised Jacobian $J(X)$ of $X$ fibers over the Jacobian $J(\widetilde{X})$ of $\widetilde X$, and there is an Abel map $A$ of $\widetilde X$ to $J(X)$, lifting the Abel map to $J(\widetilde X)$. We build a theta function on a compactification of the generalised Jacobian $J(X)$ (giving a section of a suitable positive line bundle). The translation action on $J(X)$ then yields all line bundles of that degree, and the translates of the theta function, restricted to $A(\widetilde X)$, give a ``universal section'' of the line bundles of that degree over $X$. This extends to the singular case a classical result of Riemann.