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

• 成果类型: Article
• 署名作者: Cilleruelo, J; Sols, I
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.2307/3062146
• 发表日期: 2001
• 页码: 739-758
• 关键词: diophantine equations; maximal subbundles; special divisors; vector-bundles; CURVES

摘要

Let E be a semistable, rank two vector bundle of degree d on a Riemann surface C of genus g greater than or equal to 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 6 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 8 has maximal value s = g, or if s greater than or equal to 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, 8, 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 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.