Bounds for Seshadri Constants

Introduction In this paper we present an alternative approach to the boundedness of Seshadri constants of nef and big line bundles at a general point of a complex–projective variety. Seshadri constants ε(L, x), which have been introduced by Demailly [De92], measure the local positivity of a nef line bundle L at a point x ∈ X of a complex–projective variety X, and can be defined as ε(L, x) := inf C∋x L · C mult x (C) , where the infimum is taken over all reduced irreducible curves C ⊂ X passing through x (cf. also §1 below, [De92] or [EKL] for further characterizations and properties of Seshadri constants). Over the last years there has been quite some activity in studying Seshadri constants, starting with the somehow surprising result by Ein and Lazarsfeld (cf. [EL]) that the Seshadri constant of an ample line bundle on a smooth surface is bounded below by 1 for all except perhaps countably many points. On the other hand examples by Miranda (cf. [EKL, 1.5]) show that for any integral n ≥ 2 and real δ > 0 there is a smooth n−dimensional variety X, an ample line bundle L on X and a point x ∈ X with ε(L, x) < δ; in other words, there does not exist a universal lower bound for Seshadri constants valid for all X and ample L at every point x ∈ X. Then it was proven by Ein-Küchle-Lazarsfeld [EKL] that, for a nef and big line bundle L on an n−dimensional projective variety, the Seshadri constant at very general points 1991 Mathematics Subject Classification. 14C20. 1 2 (i.e. outside a countable union of proper subvarieties) is bounded below by 1 n , and that this implies the existence of a lower bound at general points depending only on n. Finally we want to mention the recent papers by Nakamaye [Na] and Steffens [St] dealing with the problem of " maximality " of Seshadri constants on abelian varieties, as well as variants due to Küchle [Kü] and Paoletti [Pa] concerning Seshadri constants along several points and higher dimensional subvarieties respectively. Here we first reprove the existence of lower bounds for Seshadri constants of nef and big line bundles at general points of a projective variety. Although the bound obtained does, in general, not improve the one in [EKL], the method at hand may give better results in certain …