RANK-2 VECTOR BUNDLES WITH MANY SECTIONS AND LOW c 2 ON A SURFACE
Recently ([2], , , , ) there was much interest in the classification of rank-n vector bundles on a projective variety V, dim(V) = n, with very low top Chern class and many sections, e.g. ample and generated by global sections. Here we shall give a quantitative version of 'how much and how spread over V' are the sections of a vector bundle.
We work over an algebraically closed field ~: with char0:) = 0. Let V be a projective variety (always reduced and irreducible), dim(V)= n, and E a rank-n vector bundle on V with t:= c.(E)> 0; we assume that E is generated by global sections (we shall say also that E is spanned). Since char 0:) = 0, there is a non-empty open dense subset U of HΒ°(V, E) such that for every s ~ U, the locus of zeros (s)0 is formed by t smooth points of V. We obtain a morphism 9: U ~ St(V) (symmetric power). Set s(E):= dim(Im( )). Since E is spanned, we have n <~ s(E) <~ nc,(E). We think that s(E) is a useful invariant which in extremal cases seems to determine the geometry of (V, E) or of V. In this paper we give a few cases in which this happens.
In Section 4 we describe the pairs (V, E), dim(V) = n, E spanned, c,(E) > 1, with s(E) = n. This description is a complete classification if n = 2 (4.3). If E is ample and n > 2, this description would be a complete classification, modulo a proof of the conjecture stated in , without the smoothness assumption of [16-1.
In Section 1 we discuss a few elementary properties of pairs (V, E) with s(E) >t nc,(E) -1. In particular, if s(E) = nc,(E), V is rational.
From now on, S is a smooth projective surface, and E a rank-2 vector bundle on S. In Section 2 (see 2.1) we classify all the pairs (S,E