Cyclic branched coverings of surfaces with Abelian quotient singularities

Artal Bartolo, E. (Universidad de Zaragoza) ; Cogolludo-Agustín, J. I. (Universidad de Zaragoza) ; Martín-Morales, J. (Universidad de Zaragoza)

Resumen: In [9], Esnault-Viehweg developed the theory of cyclic branched coverings X̃ → X of smooth surfaces providing a very explicit formula for the decomposition of H 1 (X̃, C) in terms of a resolution of the ramification locus. Later, in [1] the first author applied this to the particular case of coverings of P² 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 abelian 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 P²(1,1,3) with the same singularities but non-homeomorphic embeddings. This is shown by proving that the cyclic covers of P²(1,1,3) of order 12 ramified along the curves have different irregularity. In the process, only a partial resolution of singularities is required.

Idioma: Inglés

DOI: 10.1512/iumj.2022.71.8768

Año: 2022

Publicado en: INDIANA UNIVERSITY MATHEMATICS JOURNAL 71, 1 (2022), 213-249
ISSN: 0022-2518

Factor impacto JCR: 1.1 (2022)

Categ. JCR: MATHEMATICS rank: 101 / 329 = 0.307 (2022) - Q2 - T1

Factor impacto CITESCORE: 1.7 - Mathematics (Q3)

Factor impacto SCIMAGO: 1.223 - Mathematics (miscellaneous) (Q1)

Financiación: info:eu-repo/grantAgreement/ES/MICINN/MTM2016-76868-C2-2-P

Tipo y forma: Artículo (PostPrint)

Área (Departamento): Área Geometría y Topología (Dpto. Matemáticas)