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.

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Abstract. 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 Frobeniusdestabilized 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 [CZ]. In particular we prove a generalization of a result of Mochizuki to higher ranks.Dedicated to Vikram B. Mehta

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 The COVID-19 pandemic has negative impacts on most Sustainable Development Goals, which may subside in the medium and long terms. Key impeding factors causing the negative impacts include lockdowns, unemployment, and diluted focus on non-COVID-19-related issues. The COVID-19 pandemic has also opened a short-lived and narrow window of opportunity for sustainable transformation.