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

Abstract

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 L 2 is a square. More generally, we prove the inequality ε n−1(L,r)⩾ε n−1(L,1)ε n−1 O P n (1),r , where ε n−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.