Seshadri Constants Over Fields Of Characteristic Zero

TL;DR

This paper investigates Seshadri constants on smooth projective varieties over characteristic zero fields, establishing their invariance over algebraic closures for rational points and exploring conditions for the existence of Seshadri curves.

Contribution

It proves the invariance of Seshadri constants over algebraic closures for rational points and constructs examples with zero global Seshadri constants, advancing understanding of their behavior.

Findings

Seshadri constants over algebraic closure equal those over the base field for rational points

Existence of varieties with zero global Seshadri constant demonstrated

Conditions for the existence of Seshadri curves established

Abstract

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