A refined stable restriction theorem for vector bundles on quadric threefolds

TL;DR

This paper refines a restriction theorem for stable rank 2 vector bundles on smooth quadric threefolds, showing that most hyperplane restrictions remain stable and explicitly characterizing exceptions.

Contribution

It provides a more precise description of the hyperplanes where the restriction of a stable bundle fails to be stable, refining previous theorems by Ein, Sols, Coanda, and Barth.

Findings

Most hyperplanes preserve stability of the restricted bundle.

The subset of hyperplanes causing instability has codimension at least 2.

Explicit classification of bundles with non-stable restrictions.

Abstract

Let E be a stable rank 2 vector bundle on a smooth quadric threefold Q in the projective 4-space P. We show that the hyperplanes H in P for which the restriction of E to the hyperplane section of Q by H is not stable form, in general, a closed subset of codimension at least 2 of the dual projective 4-space, and we explicitly describe the bundles E which do not enjoy this property. This refines a restriction theorem of Ein and Sols [Nagoya Math. J. 96, 11-22 (1984)] in the same way the main result of Coanda [J. reine angew. Math. 428, 97-110 (1992)] refines the restriction theorem of Barth [Math. Ann. 226, 125-150 (1977)].