Abstract. Vector bundles in positive characteristics have a tendency to be destabilized after pulling back by the Frobenius morphism. In this paper, we closely examine vector bundles over curves that are, in an appropriate sense, maximally destabilized by the Frobenius morphism. Then we prove that such bundles of rank 2 exist over any curve in characteristic 3, and are unique up to twisting by a line bundle. We also give an application of such bundles to the study of ample vector bundles, which is valid in all characteristics. IntroductionGiven a normal projective variety X over an algebraically closed field k of characteristic p = 0, with a fixed ample divisor, it can happen that pulling back by an inseparable morphism f : Y → X destroys semistability of vector bundles over X. In the simplest case, where X is a smooth curve and f = F rel is the (relative) Frobenius morphism, semistable vector bundles whose pullback under f fail to be semistable are called Frobenius-destabilized vector bundles. They are closely related to the study of the generalized Verschiebung.In this paper, we are primarily interested in rank-r vector bundles E over a curve X of genus g ≥ 2 with the following property: ( * ) the Harder-Narasimhan filtration of F * rel E admits line bundle quotientsIf themselves semistable, such bundles are the "most destabilized" ones possible. We will call a semistable vector bundle E with property ( * ) a maximally Frobenius-destabilized vector bundle (cf. Def. 2.11). These vector bundles appear in the works of many authors. Notably, over a genus-2 curve, all rank-2 Frobenius-destabilized vector bundles are maximally Prop. 3.3]); the work of S. Mochizuki [17] gave a precise formula counting the number of such bundles with trivial determinant over a general curve in arbitrary characteristic, and B. Osserman [18] counted them over an arbitrary curve in small characteristics. In another direction, K. Joshi et al. [9] gave a relation between certain Frobeniusdestabilized bundles and pre-opers, a concept originated from the geometric Langlands program; their observation was later used by X. Sun [20] to prove that stability is preserved under Frobenius-pushforward.We prove that vector bundles with property ( * ) exhibit an interesting trichotomy: Here L max and L min are measures of maximal slope of subbundles and minimal slope of quotient bundles respectively, taken over all finite pullbacks. We split Shepherd-Barron's inequality intoas well as giving an equality criterion for both. As we will see, the improved inequalities (0.1) have many independent applications as well. The result (ii) shows that maximally Frobenius-destabilized vector In characteristic p = 3, B. Osserman [18] has constructed rank-2 maximally Frobenius-destabilized vector bundles with trivial determinant over an arbitrary curve of genus 2, and showed that there are exactly #Pic(X)[2] = 16 of them. We derive from the previous theorem that, for arbitrary genus, such bundles are precisely the rank-2 vector bundles E satisfyingwhere S 2 (E) is ...

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Let X be a smooth, geometrically connected curve over a perfect field k. Given a connected, reductive group G, we prove that central extensions of G by the sheaf K 2 on the big Zariski site of X, studied in Brylinski-Deligne [BD01], are equivalent to factorization line bundles on the Beilinson-Drinfeld affine Grassmannian Gr G . Our result affirms a conjecture of Gaitsgory-Lysenko [GL16] and classifies factorization line bundles on Gr G . the loop group by the work of R. Reich [Re12]. Finally, one obtains (0.2) by taking the trace of Frobenius. 0.2.5. The association of covering groups to Brylinski-Deligne data is thus seen to factor as the following composition of functors: Brylinski-Deligne data ΦG − − → factorization line bundles on Gr G Kummer −−−−−→ factorization gerbes on Gr G Reich −−−→ multiplicative gerbes on the loop group Tr(Frob) −−−−−→ covering groups.

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