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

T. Szemberg proposed in 2001 a generalization to arbitrary varieties of M. Nagataʹs 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 Nagataʹs original conjecture implies Szembergʹs 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 where 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=dimX.