The Waldschmidt constant of a standard k-configuration in P^2

Abstract

A k-configuration of type (d(1), ..., d(s)), where 1 <= d < ... < d(s) are integers, is a set of points in P2 that has a number of algebraic and geometric properties. For example, the graded Betti numbers and Hilbert functions of all k-configurations in P-2 are determined by the type (d(1),........., ds). However the Waldschmidt constant of a k-configuration in P-2 of the same type may vary. In this paper, we find that the Waldschmidt constant of a k-configuration in P-2 of type (d(1),..., d(s)) with d(1) >= s >= 1 is s. Then we deal with the Waldschmidt constants of standard k-configurations in P2 of type (a), (a, b), and (a, b, c) with a >= 1. In particular, we prove that the Waldschmidt constant of a standard k-configuration in P-2 of type (1, b, c) with c >= 2b+2 does not depend on c.