On positivity of line bundles on Enriques surfaces

Abstract

We study linear systems on Enriques surfaces. We prove rationality of Seshadri constants of ample line bundles on Enriques surfaces and provide lower bounds on these numbers. We show the nonexistence of k-very ample line bundles on Enriques surfaces of degree 4k + 4 for k > 1, thus answering an old question of Ballico and Sommese. INTRODUCTION In this note we answer a question posed by Ballico and Sommese [1] (see also Terakawa [14]) concerning the existence of k-very ample line bundles of degree 4k + 4 on Enriques surfaces. We show that such line bundles exist if and only if k = 0 and that in fact the degree of a k-very ample line bundle on an Enriques surface is subject to a much more restrictive bound (Theorem 2.4). This has strong consequences for the local positivity of line bundles on Enriques surfaces. We address this questions in section 3. In particular we show that Seshadri constants of ample line bundles on Enriques surfaces are always rational. 1. AUXILIARY MATERIAL AND NOTATION The basic reference for this paper is the monograph [6] by Cossec and Dolgachev. Here we briefly recall properties of Enriques surfaces needed for our considerations. First of all, an Enriques surface is a surface Y of Kodaira dimension zero with no differential forms. In particular, ho(Ky) 0 and the canonical bundle Ky is a 2torsion element in Pico(Y), i.e. 2Ky Oy. To alleviate notation it is convenient to use the following Convention. In the sequel we make frequent use of vanishing theorems and statements on adjoint line bundles. Since the canonical divisor Ky is numerically trivial, the adjoint line bundle Ky + L has the same numerical properties as L. In particular, if L is nef and big (ample) then so is Ky + L (by the Nakai-Moishezon criterion). So in order to get vanishing statements for L we apply vanishing theorems to Ky + L and use the fact that 2Ky is trivial. In the sequel we do so without comment, and hope to cause no confusion this way. Received by the editors June 1, 2000. 2000 Mathematics Subject Classification. Primary 14J28; Secondary 14C20, 14E25.