On the exceptional locus of the birational projections of normal surface singularity into a plane

TL;DR

This paper studies the local geometry of exceptional divisors in birational projections of normal surface singularities into a plane, providing new insights into their structure and characterizations of minimal singularities.

Contribution

It offers a new characterization of minimal singularities based on the geometry of exceptional divisors in birational projections.

Findings

Dimension of tangent space equals the number of exceptional components meeting at Q

Derived conditions for contracting a prescribed number of irreducible curves

Provided new criteria for identifying minimal singularities

Abstract

Given a normal surface singularity $(X, Q)$ and a birational morphism to a non- singular surface $\pi : X \to S$, we investigate the local geometry of the exceptional divisor $L$ of $\pi$. We prove that the dimension of the tangent space to $L$ at $Q$ equals the number of exceptional components meeting at $Q$. Consequences relative to the existence of such birational projections contracting a prescribed number of irreducible curves are deduced. A new characterization of minimal singularities is obtained in these terms.