Theta divisors and the Frobenius morphism

We introduce theta divisors for vector bundles and relate them to the ordinariness of curves in characteristic p > 0. We prove, following M. Raynaud, that the sheaf of locally exact differentials in characteristic p > 0 has a theta divisor, and that the generic curve in (any) genus g ≥ 2 and (any) characteristic p > 0 has a cover that is not ordinary (and which we explicitely construct). 1 Theta divisors for vector bundles Let k be an algebraically closed field and X a smooth proper connected curve over Spec k having genus g. We assume throughout that g ≥ 2. If E is a vector bundle (i.e. a locally free invertible sheaf) of rank r and degree d over X, we define its slope to be λ = d/r. The Riemann-Roch formula gives the Euler-Poincaré characteristic of E: χ(X, E) = h(X, E)− h(X,E) = r(λ− (g − 1)) In particular for λ = g − 1 (the critical slope) we have χ(X,E) = 0; moreover, it is still true for any invertible sheaf L of degree 0 over X that χ(X, E⊗L) = 0, in other words, h(X, E⊗L) = h(X, E⊗L). Under those circumstances, it is natural to ask the following question: Question 1.1. Suppose E has critical slope. Then for which invertible sheaves L of degree 0 (if any) is it true that h(X, E ⊗ L) = 0 (and consequently also h(X, E ⊗ L) = 0)? Is this true for some L, for many L, or for none? We start with a necessary condition. Suppose there were some subbundle F 1⁄2 E having slope λ(F ) > λ(E) = g − 1. Then we would have