Intersection Theory on Abelian-Quotient $V$-Surfaces and $\mathbf{Q}$-Resolutions

TL;DR

This paper develops intersection theory for surfaces with abelian quotient singularities, applying it to construct and analyze weighted blow-ups and $Q$-resolutions of singularities in algebraic surfaces.

Contribution

It introduces a framework for intersection theory on abelian-quotient $V$-surfaces and applies it to construct $Q$-resolutions of surface singularities.

Findings

Derived properties of quotients of weighted projective planes.

Constructed embedded $Q$-resolutions of plane curve singularities.

Provided methods for abstract $Q$-resolutions of surfaces.

Abstract

In this paper we study the intersection theory on surfaces with abelian quotient singularities and we derive properties of quotients of weighted projective planes. We also use this theory to study weighted blow-ups in order to construct embedded $\mathbf{Q}$-resolutions of plane curve singularities and abstract $\mathbf{Q}$-resolutions of surfaces.