Abel maps of Gorenstein curves

TL;DR

This paper constructs Abel maps for Gorenstein curves, showing their properties and connections to moduli schemes, thereby extending classical Abel map theory to singular curves.

Contribution

It introduces Abel maps for Gorenstein curves and links them to Seshadri's moduli scheme, providing new insights into their geometric and moduli-theoretic properties.

Findings

Abel maps are embeddings away from rational subcurves meeting separating nodes.

The images of Abel maps have the same arithmetic genus as the original curve.

A connection between Abel maps and Seshadri's moduli scheme U_X(1) is established.

Abstract

For a Gorenstein curve X and a nonsingular point P of X, we construct Abel maps A from X to J_X^1 and A_P from X to J_X^0, where J_X^i is the moduli scheme for simple, torsion-free, rank-1 sheaves on X of degree i. The image curves of A and A_P are shown to have the same arithmetic genus of X. Also, A and A_P are shown to be embeddings away from rational subcurves L of X meeting the closure of X-L in separating nodes. Finally, we establish a connection with Seshadri's moduli scheme U_X(1) for semistable, torsion-free, rank-1 sheaves on X, obtaining an embedding of A(X) into U_X(1).