The Severi bound on sections of rank two semistable bundles on a Riemann surface

Let E be a semistable, rank two vector bundle of degree d on a Riemann surface C of genus g > 1, i.e. such that the minimal degree s of a tensor product of E with a line bundle having a nonzero section is nonnegative. We give an analogue of Clifford's lemma by showing that E has at most (ds)/2 + 6 independent sections, where 6 is 2 or 1 according to whether the Krawtchouk polynomial Kr(n, N) is zero or not at r = (ds)/2 + 1, n = g, N = 2g-s (the analogous bound for nonsemistable rank two bundles being stronger but easier to prove). This gives an answer to the problem posed by Severi asking for the minimal degree of a directrix of a ruled surface. In some cases, namely if s has maximal value s = g, or if s > gonality(C) 2, or if E is general among those of the same Segre invariant s, or also if the genus is a power of two, we prove the bound holds with 3 = 1. The theory of Krawtchouk polynomials investigates which triples (g, s, d) provide zeros of Kr(n,N). Then, they generate invariants which one may expect to be associated to a Severi bundle, i.e., to a rank two semistable bundle reaching the bound with 3 = 2. According to this theory, there are only a finite number of such triples (g, s, d) for each value of d s, with the exception that there are infinitely many triples with ds = 2 or 4. We then find all the Severi bundles corresponding to those two exceptional values of ds.