On the Poincare formula and the Riemann singularity theorem over nodal curves

Abstract

The symmetric powers of a smooth curve determine effective cycles in the Jacobian of the curve. The classical Poincare formula expresses these cycles in terms of the powers of the theta divisor of the Jacobian. Here we prove an analogue of this well-known Poincare formula for the desingularisation of the compactified Jacobian of an irreducible nodal curve with arbitrary number of nodes. We also prove an analogue of the Riemann singularity theorem and show that these effective cycles are normal.