The Severi Bound on Sections of Rank Two Semistable Bundles on a Riemann Surface

Abstract

Let C be a compact Riemann surface of genus g≥1 . For a line bundle L on C of degree d with 0≤d≤2g−2 , the classical theorem of Clifford states that the dimension h 0 (L) of the space of global sections of L satisfies h 0 (L)≤d 2 +1 . It is also quite easy to list the cases which give equality (apart from the trivial bundle and the canonical bundle, these exist only on hyperelliptic curves). This theorem can be extended to a semistable bundle E on C of rank n and degree d with 0≤d≤n(2g−2) , when it states that h 0 (E)≤d 2 +n . Now let E be a semistable bundle of rank 2 . In this case the authors obtain a more refined result depending on the Segre invariant s(E) , which is defined as the minimum value of degE/L−degL for L a line subbundle of E . (In particular, the semistability condition becomes s(E)≥0 .) The main theorem is that, if degE=d and s≤d≤4g−4−s , then h 0 (E)≤d−s 2 +δ , where δ=1 or δ=2 . (Note that d−s is always even.) In a previous paper, E. Arrondo and the second author conjectured this inequality with δ=1 (with a small number of exceptions similar to those for rank 1 ). However, this is not correct and δ=2 is sometimes necessary; the authors call the bundles which attain this bound Severi bundles. The number δ is obtained as follows. We define polynomials K r (n,N) (the Krawtchouk polynomials) by means of the generating function ∑ r K r (n,N)z r =(1−z) n (1+z) N−n . Let r=d−s 2 +1 ; then we can take δ=1 if K r (g,2g−s)≠0 and δ=2 otherwise. The connection with Krawtchouk polynomials is intriguing, since these are used in coding theory and recent work of T. Johnsen has established links between coding theory and semistable bundles of rank 2 on Riemann surfaces; these links make essential use of the Segre invariant s . The key to the proof of the inequality is the following lemma. Let E be a bundle of rank 2 and degree d on a curve of genus g having at least r+1≤g independent sections. If K r (g,2g+2r−2−d)≠0, then there is a nonzero section of E vanishing at r points. This is proved using a computation of Chern classes, formal properties of K r (n,N) and the Porteous formula. The authors give examples to show that, for any allowable values of g,s,d , the bound δ=1 can be attained (in fact on a hyperelliptic curve). A good deal is known about the vanishing of K r (n,N) , thus providing a restricted list of candidates for the invariants of Severi bundles, but these values do not all arise from actual bundles. The Severi bundles with d−s=0 are all known, and the authors compute those with d−s=2 and d−s=4 . In particular the bundles obtained by I. Grzegorczyk [Ulam Quart. 3 (1996), no. 2, 41 ff., approx. 3 pp. (electronic) ] are Severi bundles with g=5 , s=4 , d=8 . The authors also obtain sharp bounds for the numbers of independent sections of non-semistable bundles.