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)
type: Item
instance of: scholarly article
title: 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)
zbMATH Open document ID: 1432.14004
author: Fabrizio Catanese
author: Ingrid Bauer
published in: Bollettino dell'Unione Matematica Italiana
publication date: 2019-06-27T00:00:00Z
review text: 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 surfac…
reviewed by: Christian Gleißner
Mathematics Subject Classification ID: 14B12
Mathematics Subject Classification ID: 14J15
Mathematics Subject Classification ID: 14J29
Mathematics Subject Classification ID: 14J45
Mathematics Subject Classification ID: 14E20
Mathematics Subject Classification ID: 14F17
Mathematics Subject Classification ID: 32G05
Mathematics Subject Classification ID: 32S15
Mathematics Subject Classification ID: 32J15
Mathematics Subject Classification ID: 52C35
zbMATH DE Number: 7074501
zbMATH Keywords: rigid complex manifolds and varieties
zbMATH Keywords: branched coverings
zbMATH Keywords: Hirzebruch Kummer coverings
zbMATH Keywords: deformation theory
zbMATH Keywords: configurations of lines
zbMATH Keywords: del Pezzo surfaces
MaRDI profile type: Publication
full work available at URL: s40574-018-0169-x
OpenAlex ID: W2963258354
cites work: Q3788124
cites work: A volume maximizing canonical surface in 3-space
cites work: Burniat surfaces. III: Deformations of automorphisms and extended Burniat surfaces
cites work: On rigid compact complex surfaces and manifolds
cites work: What makes a complex exact?
cites work: Q3807349
cites work: Kodaira fibrations and beyond: methods for moduli theory
cites work: The direct image of the relative dualizing sheaf needs not be semiample
cites work: Answer to a question by Fujita on Variation of Hodge Structures
cites work: Vector bundles on curves coming from variation of Hodge structures
cites work: On Stability of Compact Submanifolds of Complex Manifolds
cites work: A compact Kähler surface of negative curvature not covered by the ball
cites work: Complex surfaces with CAT(0) metrics
cites work: Q4100345
cites work: Hirzebruch-Kato surfaces, Deligne-Mostow's construction, and new examples of negatively curved compact Kähler surfaces
Wikidata QID: Q129158302
DOI: 10.1007/S40574-018-0169-X
description: scientific article; zbMATH DE number 7074501