On semistable vector bundles over curves (Q950131)

Let \(X\) be a geometrically irreducible, smooth projective curve defined over a field \(k\), and let \(E\) be a vector bundle on \(X\). Then \(E\) is semistable if and only if there exists a vector bundle \(F\) on \(X\) such that \(H^i(X,F \otimes E)=0\) for \(i=0,1\). This result is due to \textit{G. Faltings} [J. Algebr. Geom. 2, 507--568 (1993; Zbl 0790.14019)] for the case where \(k\) is algebraically closed and due to the first two authors [Generalization of a criterion for semistable vector bundles, \url{arXiv:0804.4120}] for the case where \(k\) is perfect. Furthermore, there exists a bound for the rank of such a vector bundle \(F\) [\textit{M. Popa}, Duke Math. J. 107, 469--495 (2001; Zbl 1064.14032)]. The purpose of this note is twofold: Firstly, the perfectness assumption is removed, and secondly, a better bound for the rank of \(F\) is given. This bound depends only on the rank and degree of \(E\) if \(k\) is infinite, and additionally on the genus of \(X\) if \(k\) is finite. The proof makes use of the geometry of the moduli space of semistable vector bundles on \(X\) of fixed determinant and rank, showing that there must exist a point that parametrizes a vector bundle \(F\) with the required properties.