On the behaviour of strong semistability in geometric deformations

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.

arXiv comment12 pages, v2: extended results and fixed several typos, v3: fixed typos, improved exposition, 1 new example, v4: Added some references, improvements in exposition