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Let X be a smooth projective curve of genus g ≥ 2 defined over an algebraically closed field k of characteristic p > 0 . For p > r ( r − 1 ) ( r − 2 ) ( g − 1 ) we construct an atlas for the locus of all Frobenius-destabilized bundles of rank r (i.e. we construct all Frobenius-destabilized bundles of rank r and degree zero up to isomorphism). This is done by exhibiting a surjective morphism from a certain Quot-scheme onto the locus of stable Frobenius-destabilized bundles. Further we show that there is a bijective correspondence between the set of stable vector bundles E over X such that the pull-back F ⁎ ( E ) under the Frobenius morphism of X has maximal Harder–Narasimhan polygon and the set of opers having zero p -curvature. We also show that, after fixing the determinant, these sets are finite, which enables us to derive the dimension of certain Quot-schemes and certain loci of stable Frobenius-destabilized vector bundles over X . The finiteness is proved by studying the properties of the Hitchin–Mochizuki morphism; an alternative approach to finiteness has been realized in [3] . In particular we prove a generalization of a result of Mochizuki to higher ranks. [ABSTRACT FROM AUTHOR]

We study two natural questions about subvarieties of moduli spaces. In the first section, we study the locus of curves equipped with F-nilpotent bundles and its relationship to the p-rank zero locus of the moduli space of curves of genus g. In the second section, we study subvarieties of moduli spaces of vector bundles on curves. We prove an analogue of a result of F. Oort about proper subvarieties of moduli of abelian varieties. [ABSTRACT FROM AUTHOR]