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

TL;DR

This paper establishes the rigidity of certain Hirzebruch-Kummer coverings of the projective plane branched on line configurations and provides explicit equations for special cases involving Del Pezzo surfaces and Fermat curves.

Contribution

It proves equisingular rigidity of Hirzebruch-Kummer coverings and derives explicit equations for these coverings in special configurations, expanding understanding of Del Pezzo surfaces.

Findings

Proved rigidity of specific Hirzebruch-Kummer coverings.

Derived explicit equations for coverings in special line configurations.

Extended results to determinantal equations for Del Pezzo surfaces.

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.