On vector bundles destabilized by Frobenius pull-back

Abstract

Let X be a smooth projective curve of genus $g > 1$ over an algebraically closed field of positive characteristic. This paper is a study of a natural stratification, defined by the absolute Frobenius morphism of X, on the moduli space of vector bundles. In characteristic two, there is a complete classification of semi-stable bundles of rank 2 which are destabilized by Frobenius pull-back. We also show that these strata are irreducible and obtain their respective dimensions. In particular, the dimension of the locus of bundles of rank two which are destabilized by Frobenius is $3g-4$. These Frobenius destabilized bundles also exist in characteristics two, three and five with ranks 4, 3 and 5, respectively. Finally, there is a connection between (pre)-opers and Frobenius destabilized bundles. This allows an interpretation of some of the above results in terms of pre-opers and provides a mechanism for constructing Frobenius destabilized bundles in large characteristics.