Seshadri constants over fields of characteristic zero

Abstract

Let X be a smooth projective variety defined over a field k of characteristic 0 and let L be a nef line bundle defined over k. We prove that if x∈X is a k-rational point then the Seshadri constant ε(X,L,x) over k‾ is the same as that over k. We show, by constructing families of examples, that there are varieties whose global Seshadri constant ε(X) is zero. We also prove a result on the existence of a Seshadri curve with a natural (and necessary) hypothesis.