On coherent systems of type (n,d,n+1) on Petri curves

TL;DR

This paper analyzes the geometry and non-emptiness of moduli spaces of coherent systems of type (n,d,n+1) on Petri curves, providing new results on stability, critical values, and applications to Brill-Noether theory.

Contribution

It describes the moduli space structure for large and small parameters, determines critical values, and proves Butler's conjecture in new cases.

Findings

Identified the top critical value of lpha and its positive codimension flip.

Established non-emptiness conditions for various lpha ranges.

Proved Butler's conjecture in specific cases for low genus.

Abstract

We study coherent systems of type $(n,d,n+1)$ on a Petri curve $X$ of genus $g\ge2$. We describe the geometry of the moduli space of such coherent systems for large values of the parameter $\alpha$. We determine the top critical value of $\alpha$ and show that the corresponding ``flip'' has positive codimension. We investigate also the non-emptiness of the moduli space for smaller values of $\alpha$, proving in many cases that the condition for non-emptiness is the same as for large $\alpha$. We give some detailed results for $g\le5$ and applications to higher rank Brill-Noether theory and the stability of kernels of evaluation maps, thus proving Butler's conjecture in some cases in which it was not previously known.