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

Abstract

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 $(\mathbb{P}^1)^4$. Version2: We added a new section, describing more generally determinantal equations for all Del Pezzo surfaces of degree $9-k \leq 6$ as subvarieties of the k-fold product of the projective line.