Minimal resolutions, Chow forms and Ulrich bundles on K3 surfaces

TL;DR

This paper proves the Minimal Resolution Conjecture for general points on a curve with general moduli, and shows that K3 surfaces admit Ulrich bundles of all ranks, leading to new descriptions of Chow forms.

Contribution

It establishes the MRC for general points on curves with general moduli and constructs Ulrich bundles of all ranks on K3 surfaces, connecting to Chow forms.

Findings

MRC holds for general points on curves with degree d > 2r - 1.

Full solution to the Ideal Generation Conjecture for such curves.

Existence of Ulrich bundles of every rank on K3 surfaces.

Abstract

The Minimal Resolution Conjecture (MRC) for points on a projective variety X predicts that the Betti numbers of general sets of points in X are as small as the geometry (Hilbert function) of X allows. To a large extent, we settle this conjecture for a curve C with general moduli. We show that, independently of the genus, MRC holds for a general linear system of degree d and dimension r on C if and only if d>2r-1. We then proceed to find a full solution to the Ideal Generation Conjecture for curves with general moduli. In a different direction, we prove that K3 surfaces admit Ulrich bundles of every rank. We apply this to describe a pfaffian equation for the Chow form of a K3 surface.