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 $$(\mathbb {P}^1)^4$$ .We describe more generally determinantal equations for all Del Pezzo surfaces of degree $$9-k \le 6$$ as subvarieties of the k-fold product of the projective line.