On the ample vector bundles over curves in positive characteristic (Q1886970)

Here the authors prove the following extension to positive characteristic of a result of \textit{D. Barlet}, \textit{T. Peternell} and \textit{M. Schneider} [Math. Ann. 286, 13--25 (1990; Zbl 0716.32011)]. Let \(E\) be an ample vector bundle over a smooth projective curve defined over an algebraically closed field of positive characteristic. The authors construct a family of curves in the total space of \(E\), parametrized by an affine space, that surjects onto the total space of \(E\) and gives a deformation of a (nonreduced) zero section of \(E\). The main tool is a strong theorem concerning the Harder-Narasimhan filtration of the Frobenius pull-backs of \(E\) proved by \textit{A. Langer} [Ann. Math. (2) 159, 251--276 (2004; Zbl 1080.14014)].