The Frobenius map, rank 2 vector bundles and Kummer's quartic surface in characteristic 2 and 3

Abstract

Let X be a smooth projective curve of genus g⩾2 defined over an algebraically closed field k of characteristic p>0. Let M X ( r) be the moduli space of semi-stable rank r vector bundles with fixed trivial determinant. The relative Frobenius map F : X→X 1 induces by pull-back a rational map V : M X 1 (r)→ M X(r) . We determine the equations of V in the following two cases (1) ( g, r, p)=(2,2,2) and X nonordinary with Hasse–Witt invariant equal to 1 (see math.AG/0005044 for the case X ordinary), and (2) ( g, r, p)=(2,2,3). We also show the existence of base points of V, i.e., semi-stable bundles E such that F ∗E is not semi-stable, for any triple ( g, r, p).