Rank 3 arithmetically Cohen-Macaulay bundles on hypersurfaces

TL;DR

This paper characterizes when rank 3 arithmetically Cohen-Macaulay bundles on high-dimensional hypersurfaces split into line bundles, confirming a conjecture for this case through cohomological vanishing conditions.

Contribution

It proves a splitting criterion for rank 3 ACM bundles on hypersurfaces and confirms a conjecture by Buchweitz, Greuel, and Schreyer.

Findings

Rank 3 ACM bundles split as sums of line bundles if certain cohomology groups vanish.

The paper proves a conjecture for rank 3 ACM bundles on hypersurfaces.

Provides a cohomological criterion for bundle splitting.

Abstract

Let $X$ be a smooth projective hypersurface of dimension $\geq 5$ and let $E$ be an arithmetically Cohen-Macaulay bundle on $X$ of any rank. We prove that $E$ splits as a direct sum of line bundles if and only if $H^i_*(X, \wedge^2 E) = 0$ for $i = 1,2,3,4$. As a corollary this result proves a conjecture of Buchweitz, Greuel and Schreyer for the case of rank 3 arithmetically Cohen-Macaulay bundles.