Orbifold splice quotients and log covers of surface pairs

TL;DR

This paper explores the structure of orbifold covers and log covers of surface pairs, providing methods to compute orbifold homology and constructing universal abelian log covers under specific conditions.

Contribution

It introduces the concept of orbifold splice quotients and develops a framework for their construction and analysis in the context of surface pairs with singularities.

Findings

Computed orbifold homology from resolutions of pairs.

Defined universal abelian log covers for surface pairs.

Constructed orbifold splice quotients under certain conditions.

Abstract

A three-dimensional orbifold $(\Sigma, \gamma_i, n_i)$, where $\Sigma$ is a rational homology sphere, has a universal abelian orbifold covering, whose covering group is the first orbifold homology. A singular pair $(X,C)$, where $X$ is a normal surface singularity with $\mathbb Q$HS link and $C$ is a Weil divisor, gives rise on its boundary to an orbifold. One studies the preceding orbifold notions in the algebro-geometric setting, in particular defining the universal abelian log cover of a pair. A first key theorem computes the orbifold homology from an appropriate resolution of the pair. In analogy with the case where $C$ is empty and one considers the universal abelian cover, under certain conditions on a resolution graph one can construct pairs and their universal abelian log covers. Such pairs are called orbifold splice quotients.