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 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 nontrivial 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
00
020
00
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., vol. 4, Springer-Verlag, Berlin, ISBN 978-3-540-00832-3, MR 2030225
  • Beauville, Arnaud (1996), Complex algebraic surfaces, London Mathematical Society Student Texts, vol. 34 (2nd ed.), Cambridge University Press, ISBN 978-0-521-49510-3, MR 1406314


This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.