Maximally Frobenius-destabilized vector bundles over smooth algebraic curves

Abstract

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 ...

On Maximally Frobenius Destabilised Vector bundles

“…If r > p, Zhao [11,Proposition 2.8] showed that any maximally Frobenius destabilised rank-r vector bundle over an arbitrary smooth projective curve of genus g ≥ 2 in characteristic p > 0 is not semistable. If r = p, then F X * (L ) is a maximally Frobenius destabilised rank-p stable vector bundle for any line bundle L on an arbitrary smooth projective curve X of genus g ≥ 2 in characteristic p > 0 (see [5] and [11]).…”

“…If r = p, then F X * (L ) is a maximally Frobenius destabilised rank-p stable vector bundle for any line bundle L on an arbitrary smooth projective curve X of genus g ≥ 2 in characteristic p > 0 (see [5] and [11]). If r < p, Zhao [11,Proposition 2.14] showed that for any given natural numbers p > 0, g ≥ 2 and r > 0 with r < p and p g − 1, there exists some maximally Frobenius destabilised rank-r stable vector bundle over some smooth projective curve of genus g ≥ 2 in characteristic p. Under the assumption p > r(r − 1)(r − 2)(g − 1), Joshi and Pauly [3] gave a correspondence between maximally Frobenius destabilised L. Li [2] stable vector bundles of degree 0 and dormant operatic loci, and proved the existence of Frobenius destabilised stable vector bundles of rank r and degree 0. Further results about Frobenius destabilised stable vector bundles can be found in [4,6,7] and [8].…”