Strong semistability/Arithmetic and geometric deformations/description

Let $C\rightarrow \operatorname {Spec} \,D$ be a smooth projective relative curve, where $D$ is a domain. A vector bundle ${\mathcal {S}}$ over $C$ induces on every fiber a vector bundle. If ${\mathcal {S}}$ is semistable in the generic fiber, then it is also semistable for the fibers over a non-empty open set of $\operatorname {Spec} \,D$. What can we say about strong semistability?

We have to distinguish two forms of deformations:

• Arithmetic deformations: $D$ contains $\mathbb {Z}$. Here the generic fiber is in characteristic zero, and suppose the bundle is semistable on the generic fiber. What can we say about strong semistability on the special fibers?

A problem of Miyaoka (85) asks in this setting (for $D$ of dimension one): does there exist infinitely many closed points ${\mathfrak {p}}$ in $\operatorname {Spec} \,D$ such that ${\mathcal {S}}_{\mathfrak {p}}$ is strongly semistable on $C_{\mathfrak {p}}$?

This is still open, but the stronger question whether for almost all closed points the restricition is strongly semistable (asked by Shepherd-Barron) has a negative answer.

• Geometric deformations: $D$ contains $\mathbb {Z} /(p)$. Here suppose that the bundle on the generic fiber is strongly semistable. Then $F^{e*}({\mathcal {S}})$ makes sense everywhere and it is semistable on the fibers over an open subset $U_{e}\subseteq \operatorname {Spec} \,D$, however this subset depends on $e$ and the intersection might become smaller and smaller.

In retrospective, Paul Monsky provided for both questions counter-examples. His results were formulated in terms of Hilbert-Kunz multiplicity, but this can be translated to strong semistability by work of Brenner and Trivedi.