Linked determinantal loci and limit linear series

TL;DR

This paper extends the concept of linked determinantal loci, proving their Cohen-Macaulay property and applying this to analyze the geometry of moduli spaces of limit linear series and Brill-Noether loci.

Contribution

It generalizes linked determinantal loci, establishes their Cohen-Macaulayness, and compares different scheme structures of limit linear series.

Findings

Linked determinantal loci are Cohen-Macaulay when of expected codimension.

Cohen-Macaulayness and flatness of moduli spaces of limit linear series are proven.

A crucial comparison between scheme structures of Eisenbud-Harris and recent limit linear series is established.

Abstract

We study (a generalization of) the notion of linked determinantal loci recently introduced by the second author, showing that as with classical determinantal loci, they are Cohen-Macaulay whenever they have the expected codimension. We apply this to prove Cohen-Macaulayness and flatness for moduli spaces of limit linear series, and to prove a comparison result between the scheme structures of Eisenbud-Harris limit linear series and the spaces of limit linear series recently constructed by the second author. This comparison result is crucial in order to study the geometry of Brill-Noether loci via degenerations.