Restrictions on Seshadri constants on surfaces

TL;DR

This paper uses continued fractions and recent geometric results to derive effective restrictions on Seshadri constants on algebraic surfaces, supporting a conjecture for surfaces with Picard number 1.

Contribution

It introduces a novel approach combining continued fractions and Okounkov bodies to restrict Seshadri constants on surfaces.

Findings

Derived effective bounds on Seshadri constants

Provided evidence supporting the conjecture for Picard number 1 surfaces

Connected approximation techniques with geometric properties of line bundles

Abstract

Starting with the pioneering work of Ein and Lazarsfeld restrictions on values of Seshadri constants on algebraic surfaces have been studied by many authors. In the present note we show how approximation involving continued fractions combined with recent results of Kuronya and Lozovanu on Okounkov bodies of line bundles on surfaces lead to effective statements considerably restricting possible values of Seshadri constants. These results in turn provide strong aditional evidence to a conjecture governing the Seshadri constants on algebraic surfaces with Picard number 1.