The minimal resolution property for points on general curves

TL;DR

This paper fully solves the Minimal Resolution Conjecture for general points on general curves and proves related stability conjectures for syzygy bundles across all characteristics.

Contribution

It provides a complete solution to the Minimal Resolution Conjecture for general points on general curves and establishes stability results for syzygy bundles in arbitrary characteristic.

Findings

Complete resolution shape for general points on curves

Proof of strong Butler's Conjecture on syzygy bundle stability

Frobenius semistability of syzygy bundles in positive characteristic

Abstract

We present an essentially complete solution to the Minimal Resolution Conjecture for general curves, determining the shape of the minimal resolution of general sets of points on a general curve C of degree d>2r-1 in P^r. Our methods also provide a proof (valid in arbitrary characteristic) of the strong version of Butler's Conjecture on the stability of syzygy bundles on a general curve of every genus at least 3, as well as of the Frobenius semistability in positive characteristic of the syzygy bundle of a general curve in the range d>2r-1.