Cyclic branched coverings of surfaces with abelian quotient singularities

TL;DR

This paper extends the theory of cyclic branched coverings to surfaces with quotient singularities, providing new formulas and conditions to analyze their irregularity, with applications to distinguishing non-homeomorphic curves on weighted projective planes.

Contribution

It generalizes the classical theory to singular surfaces, introduces simplified proofs, and develops criteria for irregularity of coverings with partial resolutions.

Findings

Extended the theory to surfaces with quotient singularities.

Provided conditions for irregularity of cyclic branched coverings.

Constructed examples of non-homeomorphic curves with identical singularities.

Abstract

Esnault-Viehweg developed the theory of cyclic branched coverings $\tilde X\to X$ of smooth surfaces providing a very explicit formula for the decomposition of $H^1(\tilde X,\mathbb{C})$ in terms of a resolution of the ramification locus. Later, the first author applies this to the particular case of coverings of $\mathbb{P}^2$ reducing the problem to a combination of global and local conditions on projective curves. In this paper we extend the above results in three directions: first, the theory is extended to surfaces with quotient singularities, second the ramification locus can be partially resolved and need not be reduced, and finally global and local conditions are given to describe the irregularity of cyclic branched coverings of the weighted projective plane. The techniques required for these results are conceptually different and provide simpler proofs for the classical results. For instance, the local contribution comes from certain modules that have the flavor of quasi-adjunction and multiplier ideals on singular surfaces. As an application, a Zariski pair of curves on a singular surface is described. In particular, we prove the existence of two cuspidal curves of degree 12 in the weighted projective plane $\mathbb{P}^2_{(1,1,3)}$ with the same singularities but non-homeomorphic embeddings. This is shown by proving that the cyclic covers of $\mathbb{P}^2_{(1,1,3)}$ of order 12 ramified along the curves have different irregularity. In the process, only a partial resolution of singularities is required.