A note on Seshadri constants on general $K3$ surfaces

TL;DR

This paper establishes a lower bound for Seshadri constants on certain K3 surfaces and characterizes when the constant equals a specific value based on the square of the line bundle.

Contribution

It provides a new lower bound for Seshadri constants on K3 surfaces with Picard group generated by a single line bundle and characterizes cases where the constant equals the square root of the line bundle's self-intersection.

Findings

Established a lower bound for Seshadri constants on K3 surfaces.

Characterized when the Seshadri constant equals the square root of the line bundle's self-intersection.

Connected the Seshadri constant value to the geometry of the surface and line bundle.

Abstract

We prove a lower bound on the Seshadri constant $\epsilon (L)$ on a $K3$ surface $S$ with $\Pic S \simeq \ZZ[L]$. In particular, we obtain that $\epsilon (L)=\alpha$ if $L^2=\alpha^2$ for an integer $\alpha$.