Beauville surface

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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[edit]

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[edit]

Hodge diamond:

1
0 0
0 2 0
0 0
1

References[edit]