A relation between one-point and multi-point Seshadri constants

T. Szemberg proposed in 2001 a generalization to arbitrary varieties of M. Nagata's 1959 open conjecture, which claims that the Seshadri constant of r ≥ 9 very general points of the projective plane is maximal. Here we prove that Nagata's original conjecture implies Szemberg's for all smooth surfaces X with an ample divisor L generating NS(X) and such that L2 is a square. More generally, we prove the inequality. En-1 (L, r) ≥ En-1 (L, 1)En-1 (O ℙn (1), r), where En-1 (L, r) stands for the (n - 1)-dimensional Seshadri constant of the ample divisor L at r very general points of a normal projective variety X, and n = dim X. © 2004 Elsevier Inc. All rights reserved.