Beauville surface

{{No footnotes|date=July 2020}} In mathematics, a '''Beauville surface''' is one of the surfaces of general type introduced by {{harvs|txt|last=Beauville|first=Arnaud|authorlink=Arnaud Beauville|year=1996|loc=exercise X.13 (4)}}. They are examples of "fake quadrics", with the same Betti numbers as quadric surfaces.

==Construction== Let ''C''₁ and ''C''₂ be smooth curves with genera ''g''₁ and ''g''₂. Let ''G'' be a finite group acting on ''C''₁ and ''C''₂ such that *''G'' has order (''g''₁ − 1)(''g''₂ − 1) *No nontrivial element of ''G'' has a fixed point on both ''C''₁ and ''C''₂ *''C''₁/''G'' and ''C''₂/''G'' are both rational.

Then the quotient (''C''₁ × ''C''₂)/''G'' is a Beauville surface. The corresponding group ''G'' is called a Beauville group.

One example is to take ''C''₁ and ''C''₂ both copies of the genus 6 quintic ''X''⁵ + ''Y''⁵ + ''Z''⁵ =0, and ''G'' to be an elementary abelian group of order 25, with suitable actions on the two curves.

==Invariants== '''Hodge diamond:''' {{Hodge diamond|style=font-weight:bold | 1 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 1 }}

==References== *{{Citation | last1=Barth | first1=Wolf P. |author-link1=Wolf Barth | last2=Hulek | first2=Klaus |author-link2=Klaus Hulek | last3=Peters | first3=Chris A.M. | last4=Van de Ven | first4=Antonius | title=Compact Complex Surfaces | publisher= Springer-Verlag, Berlin | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. | isbn=978-3-540-00832-3 | mr=2030225 | year=2004 | volume=4}} *{{Citation | last=Beauville | first=Arnaud | authorlink=Arnaud Beauville| title=Complex algebraic surfaces | publisher=Cambridge University Press | edition=2nd | series=London Mathematical Society Student Texts | isbn=978-0-521-49510-3 | mr=1406314 | year=1996 | volume=34}}

Category:Algebraic surfaces Category:Complex surfaces

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