Del Pezzo surfaces, rigid line configurations and Hirzebruch–Kummer coverings

In beloved memory of Paolo (De Bartolomeis)
  • Ingrid BauerEmail author
  • Fabrizio Catanese


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.


Rigid complex manifolds and varieties Branched coverings Hirzebruch Kummer coverings Deformation theory Configurations of lines Del Pezzo surfaces 

Mathematics Subject Classification

14B12 14J15 14J29 14J45 14E20 14F17 32G05 32S15 32J15 52C35 


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Copyright information

© Unione Matematica Italiana 2018

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität BayreuthBayreuthGermany

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