ACM sheaves on the double plane

TL;DR

This paper investigates arithmetically Cohen-Macaulay (aCM) and Ulrich sheaves on non-integral projective varieties, showing splitting results for rank two bundles on the double plane and high-dimensional quadrics, and classifying low-rank aCM sheaves on the double plane.

Contribution

It provides the first study of aCM and Ulrich sheaves on non-integral varieties, proving splitting theorems and classifying low-rank aCM sheaves on the double plane.

Findings

Rank two aCM vector bundles on the double plane split into line bundles.

High-dimensional quadrics also exhibit splitting of rank two aCM bundles.

Existence of non-splitting aCM and Ulrich bundles on multiple hyperplanes.

Abstract

The goal of this paper is to start a study of aCM and Ulrich sheaves on non-integral projective varieties. We show that any aCM vector bundle of rank two on the double plane is a direct sum of line bundles. As a by-product, any aCM vector bundle of rank two on a sufficiently high dimensional quadric hypersurface also splits. We consider aCM and Ulrich vector bundles on a multiple hyperplanes and prove the existence of such bundles that do not split, if the multiple hyperplane is linearly embedded into a sufficiently high dimensional projective space. Then we restrict our attention to the double plane and give a classification of aCM sheaves of rank at most $3/2$ on the double plane and describe the family of isomorphism classes of them.