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Beauville surface

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 numbers as quadric surfaces.

Construction

Let C<sub>1</sub> and C<sub>2</sub> be smooth curves with genera g<sub>1</sub> and g<sub>2</sub>. Let G be a finite group acting on C<sub>1</sub> and C<sub>2</sub> such that

  • G has order (g<sub>1</sub> &minus; 1)(g<sub>2</sub> &minus; 1)
  • No nontrivial element of G has a fixed point on both C<sub>1</sub> and C<sub>2</sub>
  • C<sub>1</sub>/G and C<sub>2</sub>/G are both rational.

Then the quotient (C<sub>1</sub> × C<sub>2</sub>)/G is a Beauville surface.

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

Invariants

Hodge diamond:

References