Del Pezzo surfaces, rigid line configurations and Hirzebruch-Kummer coverings. Del Pezzo surfaces, rigid line configurations and Hirzebruch-Kummer covering. In beloved memory of Paolo (de Bartolomeis)
In this paper the authors study certain Galois covers \(X_{\mathcal L}(n)\) of the projective plane with group \((\mathbb Z/n)^r\) branched in a configuration \(\mathcal L\) of \(r+1\) lines. The minimal resolution of \(X_{\mathcal L}(n)\) is known as the Hirzebruch-Kummer cover \(HK(n,\mathcal L)\). The first interesting example occurs for the complete quadrangle \(\mathcal C \mathcal Q\) and \(n=5\). The surface \(HK(5,\mathcal C \mathcal Q)\) as shown by Hirzebruch, is a ball quotient, in particular a rigid surface admitting a Hermitian metric with strongly negative curvature. In a previous work, the authors could prove that the surface \(HK(n,\mathcal C \mathcal Q)\) is rigid if and only if \(n \geq 4\). Since the rigidity of the cover implies the rigidity of the line configuration, it is natural to ask, if \(HK(n,\mathcal L)\) is a rigid surface for any rigid configuration \(\mathcal L\) and \(n\) sufficiently large. In the first part of the paper the rigidity of the equisingular deformations of \(X_{\mathcal L}(n)\) is established under the assumption that the configuration is singularly saturated, meaning that it has at least four singular points, each line in the configuration contains at least two singular points and the line defined by any two singular points belongs to the configuration. About the question of the existence of a metric with negative curvature very few is known. In fact it is only settled for \(\mathcal C \mathcal Q\) in the special case where \(5\) divides \(n\), thanks to the work of \textit{F. Zheng} [Commun. Anal. Geom. 7, No. 4, 755--786 (1999; Zbl 0937.53036)]. Motivated by this question, the authors establish in the second part of the paper explicit equations for \(HK(n,\mathcal C \mathcal Q)\) as a submanifold of the product of four Fermat curves of degree \(n\). Their idea is to consider \(HK(n,\mathcal C \mathcal Q)\) as a cover of the blowup of the plane in the four points of the complete quadrangle, which is the Del Pezzo surface of degree \(5\). They first derive determinantal equations for the Del Pezzo surfaces of degree \(9-k \leq 6\) as submanifolds of \((\mathbb P^1)^k\). Then they use the equations for the Del Pezzo of degree \(5\), together with the natural \((\mathbb Z/n)^2\) cover from the Fermat curve of degree \(n\) to the projective line, to derive also equations for \(HK(n,\mathcal C \mathcal Q)\).
scientific article; zbMATH DE number 7074501
scientific article