Beauville surface
# Beauville surface

In mathematics, a Beauville surface is one of the surfaces of general type introduced by Arnaud Beauville (1996, 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.

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:

| | | 1 | | |
| - | - | - | - | - |
| | 0 | | 0 | |
| 0 | | 2 | | 0 |
| | 0 | | 0 | |
| | | 1 | | |
