Abstract
Let $E$ be a semistable rank two vector bundle of degree $d$ on a Riemann surface $C$ of genus $g\ge 1$, 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 $(d-s)/2+\delta$ independent sections, where $\delta$ is $2$ or $1$ according to whether the Krawtchouk polynomial $K_r(n,N)$ is zero or not at $r = (d-s)/2+1$, $n = g$, $N = 2g-s$ (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 $s\ge \mathrm{gonality}(C)-2$, 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 $\delta =1$.
The theory of Krawtchouk polynomials investigates which triples $(g,s,d)$ provide zeros of $K_r(n,N)$. 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 $\delta =2$. 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 $d-s = 2$ or $4$. We then find all the Severi bundles corresponding to those two exceptional values of $d-s$.
