Lower semi-continuity of the Waldschmidt constants

Abstract

In this paper, we study the Waldschmidt constant of a generalized fa point subscheme Z = m(1)p(1) +...+ m(r)p(r) of P-2, where p1, ... p(r) are essenally distinct points on P-2, satisfying the proximity inequalities. Furthermore, we prove its lower semi-continuity for r <= 8. Using thisrty, we also calculate the Waldschmidt constants of the fat point subschemes Z = p(1) + ... + p(5) giving weak del Pezzo surfaces of degree 4.