Numerical properties of exceptional divisors of birational morphisms of smooth surfaces

TL;DR

This paper provides a detailed analysis of the numerical properties of exceptional divisors in birational morphisms of smooth surfaces and extends Miyaoka's inequality to a broader class of surfaces.

Contribution

It offers a comprehensive study of effective divisors in the exceptional locus and improves existing bounds on canonical singularities for non-minimal surfaces.

Findings

Extended Miyaoka's inequality to non-minimal surfaces

Improved bounds on the number of canonical singularities

Detailed characterization of numerical properties of exceptional divisors

Abstract

We make a very detailed analysis of the numerical properties of effective divisors whose support is contained in the exceptional locus of a birational morphism of smooth projective surfaces. As an application we extend Miyaoka's inequality on the number of canonical singularities on a projective normal surface with non-negative Kodaira dimension to the non minimal case, obtaining a slightly better result than known extensions by Megyesi and Langer.