Del Pezzo Surfaces, Rigid Line Configurations and Hirzebruch-Kummer Coverings

Ingrid Bauer, Fabrizio Catanese

Published 2018-03-08Version 1

We prove the equisingular rigidity of the singular Hirzebruch-Kummer coverings X(n, \mathcal{L}) of the projective plane branched on line configurations \mathcal{L}, satisfying some technical condition. In the case, \mathcal{L} = the complete quadrangle, we give explicit equations of the Hirzebruch-Kummer covering S_n (=the minimal desingularisation of X(n, \mathcal{L})) in a product of four Fermat curves of degree n. Since S_n is the (\mathbb{Z}/n)^5 covering of the Del Pezzo surface Y_5 of degree 5 branched on the 10 lines, these equations are derived from explicit equations of the image of Y_5 in (\mathbb{P}^1)^4.