In mathematics, a Beauville surface is one of the surfaces of general type introduced by . They are examples of "fake quadrics", with the same Betti number

Betti number

In algebraic topology, a mathematical discipline, the Betti numbers can be used to distinguish topological spaces. Intuitively, the first Betti number of a space counts the maximum number of cuts that can be made without dividing the space into two pieces....

s as quadric surfaces.

Construction

Let C₁ and C₂ be smooth curves with genuses 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