ACM bundles on cubic surfaces

TL;DR

This paper proves the existence of a rich family of ACM and Ulrich vector bundles on cubic surfaces, revealing new geometric structures and extremal properties related to their cohomology and generators.

Contribution

It establishes the nonemptiness of moduli spaces of stable Ulrich bundles of arbitrary rank on cubic surfaces and characterizes ACM bundles within these spaces.

Findings

Existence of nonempty smooth open subsets of ACM bundles in moduli spaces.

Construction of indecomposable Ulrich bundles of arbitrarily high rank.

Identification of extremal bundles with minimal generator numbers.

Abstract

In this paper we prove that, for every $r \geq 2$, the moduli space $M^s_X(r;c_1,c_2)$ of rank $r$ stable vector bundles with Chern classes $c_1=rH$ and $c_2=(3r^2-r)/2$ on a nonsingular cubic surface $X \subset \mathbb{P}^3$ contains a nonempty smooth open subset formed by ACM bundles, i.e. vector bundles with no intermediate cohomology. The bundles we consider for this study are extremal for the number of generators of the corresponding module (these are known as Ulrich bundles), so we also prove the existence of indecomposable Ulrich bundles of arbitrarily high rank on $X$.