Beauville surface

In mathematics, a Beauville surface is one of the surfaces of general type introduced by Beauville (1996, exercise X.13 (4)). They are examples of "fake quadrics", with the same Betti numbers as quadric surfaces.

Construction

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 nontrival 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.

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.

Invariants

Hodge diamond:

		1
	0		0
0		2		0
	0		0
		1

References

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. 4, Springer-Verlag, Berlin, ISBN 978-3-540-00832-3, MR 2030225
Beauville, Arnaud (1996), Complex algebraic surfaces, London Mathematical Society Student Texts 34 (2nd ed.), Cambridge University Press, ISBN 978-0-521-49510-3, MR 1406314