Abel maps and limit linear series

TL;DR

This paper investigates the connection between limit linear series and Abel map fibers on nodal curves, providing explicit descriptions and properties of associated subschemes, and demonstrating compatibility with smoothings.

Contribution

It introduces a new association between certain limit linear series and subschemes of Abel map fibers, extending previous refined theories.

Findings

The associated subscheme is Cohen-Macaulay of pure dimension r.

Explicit computation of the Hilbert polynomial of the subscheme.

Compatibility of the construction with one-parameter smoothings.

Abstract

We explore the relationship between limit linear series and fibers of Abel maps in the case of curves with two smooth components glued at a single node. To an r-dimensional limit linear series satisfying a certain exactness property (weaker than the refinedness property of Eisenbud and Harris) we associate a closed subscheme of the appropriate fiber of the Abel map. We then describe this closed subscheme explicitly, computing its Hilbert polynomial and showing that it is Cohen-Macaulay of pure dimension r. We show that this construction is also compatible with one-parameter smoothings.