Limit linear series in positive characteristic and Frobenius-unstable vector bundles on curves

Abstract

(cont.) yield a new proof of a result of Mochizuki yield a new proof of a result of Mochizuki Frobenius-unstable bundles for C general, and hence obtaining a self-contained proof of the resulting formula for the degree of V₂.

Using limit linear series and a result controlling degeneration from separable maps to inseparable maps, we give a formula for the number of self-maps of P¹ with ramification to order e[sub]i at general points P[sub]i the case that all e[sub]i are less than the characteristic. We also develop a new, more functorial construction for the basic theory of limit linear series, which works transparently in positive and mixed characteristics, yielding a result on lifting linear series from characteristic p to characteristic 0, and even showing promise for generalization to higher-dimensional varieties. Now, let C be a curve of genus 2 over a field k of positive characteristic, and V₂ the Verschiebung rational map induced by pullback under Frobenius on moduli spaces of semistable vector bundles of rank two and trivial determinant. We show that if the Frobenius-unstable vector bundles are deformation-free in a suitable sense, then they are precisely the undefined points of V₂, and may each be resolved by a single blow-up; in this setting, we are able to calculate the degree of V₂ in terms of the number of Frobenius-unstable bundles, and describe the image of the exceptional divisors. We finally examine the Frobenius-unstable bundles on C by studying connections with vanishing p-curvature on certain unstable bundles on C. Using explicit formulas for p-curvature, we completely describe the Frobenius-unstable bundles in characteristics 3, 5, 7. We classify logarithmic connections with vanishing p-curvature on vector bundles of rank 2 on P¹ in terms of self-maps of P¹ with prescribed ramification. Using our knowledge of such maps, we then glue the connections to a nodal curve and deform to a smooth curve to