On a problem of Miyaoka

Abstract

We give an example of a vector bundle E on a relative curve C --> Spec Z such that the restriction to the generic fiber in characteristic zero is semistable but such that the restriction to positive characteristic p is not strongly semistable for infinitely many prime numbers. Moreover, under the hypothesis that there exist infinitely many Sophie Germain primes, there are also examples such that the density of primes with non strongly semistable reduction is arbitrarily high.