Cyclic branched coverings of surfaces with abelian quotient singularities

Artal Bartolo, Enrique (Universidad de Zaragoza) ; Cogolludo-Agustín, José Ignacio (Universidad de Zaragoza) ; Martín-Morales, Jorge (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 (X̃, C) in terms of a resolution of the ramification locus. Later, in [1] the first author applies 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: 10.48550/arXiv.1912.08670

Año: 2021

Publicado en: INDIANA UNIVERSITY MATHEMATICS JOURNAL 2 (2021), [27 pp.]
ISSN: 0022-2518

Originalmente disponible en: Texto completo de la revista

Factor impacto JCR: 1.059 (2021)

Categ. JCR: MATHEMATICS rank: 141 / 333 = 0.423 (2021) - Q2 - T2

Factor impacto CITESCORE: 2.0 - Mathematics (Q2)

Factor impacto SCIMAGO: 1.326 - Mathematics (miscellaneous) (Q1)

Financiación: info:eu-repo/grantAgreement/ES/DGA/E22-17R
Financiación: info:eu-repo/grantAgreement/ES/MINECO/MTM2016-76868-C2-2-P

Tipo y forma: Article (PrePrint)

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