On Seshadri constants of adjoint divisors on surfaces and threefolds in arbitrary characteristic

TL;DR

This paper introduces a new method for establishing lower bounds on Seshadri constants of ample adjoint divisors on smooth projective varieties in any characteristic, with specific bounds for surfaces and threefolds.

Contribution

It develops a novel approach to bounding Seshadri constants and provides explicit bounds for surfaces and threefolds, extending known results to arbitrary characteristic.

Findings

For surfaces, established lower bounds such as /4 for /4 of the Seshadri constant.

For threefolds, proved bounds involving the intersection number and multiplicity, approaching 1/(27/2) minus an arbitrarily small .

Identified conditions under which the Seshadri constant is rational and achieved by a specific curve.

Abstract

We develop a new approach towards obtaining lower bounds of the Seshadri constants of ample adjoint divisors on smooth projective varieties $X$ in arbitrary characteristic. Let $x\in X$ be a closed point and $A$ an ample divisor on $X$. If $X$ is a surface, we recover some known lower bounds by proving, e.g., that $\varepsilon(K_X+4A;x)\geq 3/4$. If $X$ is a threefold, we prove that for all $\delta>0$ and all but finitely many curves $C$ through $x$, we have $\frac{(K_X+6A).C}{\operatorname{mult}_x C}\geq\frac{1}{2\sqrt{2}}-\delta$. In particular, if $\varepsilon(K_X+6A;x)<1/(2\sqrt{2})$, then $\varepsilon(K_X+6A;x)$ is a rational number, attained by a Seshadri curve $C$.