Seshadri constants on rational surfaces with anticanonical pencils

TL;DR

This paper investigates Seshadri constants on rational surfaces with anticanonical pencils, providing explicit formulas and classifications related to the anticanonical divisors and their degrees.

Contribution

It offers an explicit description of Seshadri constants on such surfaces and classifies certain log del Pezzo surfaces based on these constants.

Findings

Explicit formulas for Seshadri constants on rational surfaces with anticanonical pencils

Classification of log del Pezzo surfaces with specific anticanonical degrees

Identification of special cases where Seshadri constants reveal geometric properties

Abstract

We study a Seshadri constant at a general point on a rational surface whose anticanonical linear system contains a pencil. First, we describe a Seshadri constant of an ample line bundle on such a rational surface explicitly by the numerical data of the ample line bundle. Secondly, we classify log del Pezzo surfaces which are special in terms of the Seshadri constants of the anticanonical divisors when the anticanonical degree is between 4 and 9.