On Buchsbaum bundles on quadric hypersurfaces

TL;DR

This paper classifies indecomposable rank two arithmetically Buchsbaum vector bundles on smooth quadric hypersurfaces, establishing that such bundles exist only up to dimension five and providing boundedness results for certain cases.

Contribution

It extends the classification of arithmetically Buchsbaum bundles from projective space to quadric hypersurfaces, identifying dimension bounds and providing new boundedness results.

Findings

Classification of arithmetically Buchsbaum bundles on quadrics for n ≤ 5

Existence only for n ≤ 5

Boundedness results for k-Buchsbaum bundles on Q_3

Abstract

Let $E$ be an indecomposable rank two vector bundle on the projective space $\PP^n, n \ge 3$, over an algebraically closed field of characteristic zero. It is well known that $E$ is arithmetically Buchsbaum if and only if $n=3$ and $E$ is a null-correlation bundle. In the present paper we establish an analogous result for rank two indecomposable arithmetically Buchsbaum vector bundles on the smooth quadric hypersurface $Q_n\subset\PP^{n+1}$, $n\ge 3$. We give in fact a full classification and prove that $n$ must be at most 5. As to $k$-Buchsbaum rank two vector bundles on $Q_3$, $k\ge2$, we prove two boundedness results.