The Gieseker-Petri theorem and imposed ramification

TL;DR

This paper proves a smoothness result for linear series with prescribed ramification on elliptic curves and uses it to provide a new proof and generalization of the Gieseker-Petri theorem in characteristic 0.

Contribution

It introduces a new approach to the Gieseker-Petri theorem using limit linear series and intersection theory on Grassmannians.

Findings

Established smoothness of certain linear series spaces on elliptic curves.

Provided a new proof of the Gieseker-Petri theorem in characteristic 0.

Generalized the theorem to include prescribed ramification at up to two points.

Abstract

We prove a smoothness result for spaces of linear series with prescribed ramification on twice-marked elliptic curves. In characteristic 0, we then apply the Eisenbud-Harris theory of limit linear series to deduce a new proof of the Gieseker-Petri theorem, along with a generalization to spaces of linear series with prescribed ramification at up to two points. Our main calculation involves the intersection of two Schubert cycles in a Grassmannian associated to almost-transverse flags.