Seshadri constants on surfaces of general type

TL;DR

This paper investigates the possible values of Seshadri constants of the canonical bundle on minimal surfaces of general type, revealing their specific form when small and linking their behavior to the surface's geometry.

Contribution

It characterizes the values of Seshadri constants between 0 and 1 and relates small constants to the geometric properties of the surface.

Findings

Seshadri constants between 0 and 1 are of the form (m-1)/m for m ≥ 2.

Small Seshadri constants are influenced by the geometry of the surface.

The paper provides a detailed analysis for very general points on the surface.

Abstract

We study Seshadri constants of the canonical bundle on minimal surfaces of general type. First, we prove that if the Seshadri constant $\eps(K_X,x)$ is between 0 and 1, then it is of the form $(m-1)/m$ for some integer $m\ge 2$. Secondly, we study values of $\eps(K_X,x)$ for a very general point $x$ and show that small values of the Seshadri constant are accounted for by the geometry of $X$.