On the behaviour of strong semistability in geometric deformations

TL;DR

This paper constructs examples of vector bundles over families of curves in positive characteristic that are generically semistable but not strongly semistable on special fibers, illuminating complex behaviors of stability and Hilbert-Kunz multiplicities.

Contribution

It provides explicit examples of vector bundles exhibiting specific stability behaviors in geometric deformations, advancing understanding of semistability in positive characteristic.

Findings

Examples of vector bundles that are generically semistable but not strongly semistable.

Illustrations of the behavior of Hilbert-Kunz multiplicities in families.

Insights into stability phenomena in algebraic geometry over positive characteristic.

Abstract

Let $Y \to B$ be a relative smooth projective curve over an affine integral base scheme $B$ of positive characteristic. We provide for all prime characteristics example classes of vector bundles $\mathcal{S}$ over $Y$ such that $\mathcal{S}$ is generically strongly semistable and semistable but not strongly semistable for some special fibre. This also provides new examples of the behaviour of Hilbert-Kunz multiplicities in geometric families.