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

2018/03/08 by Ingrid Bauer, Fabrizio Catanese, Bauer, Ingrid +1
Mathematics · #14B12 #14E20 #14F17 #14J15 #14J29 #14J45 #32G05 #32J15 #32S15 #52C35 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #math.AG #msc:14B12 #msc:14E20 #msc:14F17 #msc:14J15 #msc:14J29 #msc:14J45 #msc:32G05 #msc:32J15 #msc:32S15 #msc:52C35

paper · pdf · doi:10.48550/arxiv.1803.02984

21 pages

openalex publication_date 2018/03/08 · arxiv created 2018/05/02 · arxiv updated 2018/05/03 · openalex created_date 2022/09/28 · openalex updated_date 2026/07/28

Abstract

We prove the equisingular rigidity of the singular Hirzebruch-Kummer coverings X(n, L) of the projective plane branched on line configurations L, satisfying some technical condition. In the case, L = the complete quadrangle, we give explicit equations of the Hirzebruch-Kummer covering S_n (=the minimal desingularisation of X(n, L)) in a product of four Fermat curves of degree n. Since S_n is the (ℤ/n)⁵ covering of the Del Pezzo surface Y₅ of degree 5 branched on the 10 lines, these equations are derived from explicit equations of the image of Y₅ in (ℙ¹)⁴. Version2: We added a new section, describing more generally determinantal equations for all Del Pezzo surfaces of degree 9-k ≤ 6 as subvarieties of the k-fold product of the projective line.

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