Ulrich and aCM bundles from invariant theory

TL;DR

This paper constructs special arithmetically Cohen-Macaulay (aCM) and Ulrich vector bundles on hypersurfaces using invariant theory, revealing new bundles on general cubic hypersurfaces and fourfolds.

Contribution

It introduces a method using prehomogeneous representations to construct aCM and Ulrich bundles on hypersurfaces, including new examples on cubic fourfolds and sevenfolds.

Findings

A general cubic hypersurface of dimension seven admits an indecomposable Ulrich bundle of rank nine.

A general cubic fourfold admits an unsplit aCM bundle of rank six.

The method links invariant theory with vector bundle construction on hypersurfaces.

Abstract

We use certain special prehomogeneous representations of algebraic groups in order to construct aCM vector bundles, possibly Ulrich, on certain families of hypersurfaces. Among other results, we show that a general cubic hypersurface of dimension seven admitsan indecomposable Ulrich bundle of rank nine, and that a general cubic fourfold admits an unsplit aCM bundle of rank six.