Orbifold splice quotients and log covers of surface pairs

Walter D. Neumann and Jonathan Wahl

Journal of Singularities
volume 23 (2021), 151-169

Received: 4 December 2020. Received in revised form: 5 July 2021.

DOI: 10.5427/jsing.2021.23i

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Abstract:

A three-dimensional orbifold (Σ, γ_i, n_i)$, where Σ 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 QHS 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, proving the existence of the universal abelian log cover of a pair. A 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.

2000 Mathematical Subject Classification:

32S50, 14J17, 57M25, 57N10

Key words and phrases:

surface singularity, splice quotient singularity, orbifold homology, rational homology sphere, singular pair, abelian cover

Author(s) information:

Walter D. Neumann
Department of Mathematics
Barnard College, Columbia University
New York, NY 10027
email: neumann@math.columbia.edu

Jonathan Wahl
Department of Mathematics
The University of North Carolina
Chapel Hill, NC 27599-3250
email: jmwahl@email.unc.edu