Limit Linear Series in Positive Characteristic and Frobenius-Unstable Vector Bundles on Curves

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 ei at general points Pi, in the case that all ei 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 PV, 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 pcurvature, 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 1P1 in terms of self-maps of 1P1 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 yield a new proof of a result of Mochizuki giving the number of Frobenius-unstable bundles for C general, and hence obtaining a self-contained proof of the resulting formula for the degree of V2. Thesis Supervisor: Aise Johan de Jong Title: Professor