Cyclic branched coverings of surfaces with Abelian quotient singularities

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