Beauville surface

Let C₁ and C₂ be smooth curves with genera g₁ and g₂. Let G be a finite group acting on C₁ and C₂ such that

• G has order (g₁ − 1)(g₂ − 1)
• No nontrivial element of G has a fixed point on both C₁ and C₂
• C₁/G and C₂/G are both rational.

Then the quotient (C₁ × C₂)/G is a Beauville surface. The corresponding group G is called a Beauville group.

One example is to take C₁ and C₂ both copies of the genus 6 quintic X⁵ + Y⁵ + Z⁵ =0, and G to be an elementary abelian group of order 25, with suitable actions on the two curves.

Hodge diamond:

• Barth, Wolf P.; Hulek, Klaus; Peters, Chris A.M.; Van de Ven, Antonius (2004), Compact Complex Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 4, Springer-Verlag, Berlin, ISBN 978-3-540-00832-3, MR 2030225
• Beauville, Arnaud (1996), Complex algebraic surfaces, London Mathematical Society Student Texts, vol. 34 (2nd ed.), Cambridge University Press, ISBN 978-0-521-49510-3, MR 1406314

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