Catanese surface

In mathematics, a Catanese surface is one of the surfaces of general type introduced by Fabrizio Catanese (1981).

Construction

The construction starts with a quintic V with 20 double points. Let W be the surface obtained by blowing up the 20 double points. Suppose that W has a double cover X branched over the 20 exceptional −2-curves. Let Y be obtained from X by blowing down the 20 −1-curves in X. If there is a group of order 5 acting freely on all these surfaces, then the quotient Z of Y by this group of order 5 is a Catanese surface. Catanese found a 4-dimensional family of curves constructed like this.

Invariants

The Catanese surface is a numerical Campedelli surface and hence has Hodge diamond

and canonical degree [math]\displaystyle{ K^2 = 2 }[/math]. The fundamental group of the Catanese surface is [math]\displaystyle{ \mathbf{Z}/5\mathbf{Z} }[/math], as can be seen from its quotient construction.

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., 4, Springer-Verlag, Berlin, ISBN 978-3-540-00832-3 
• Catanese, Fabrizio (1981), "Babbage's conjecture, contact of surfaces, symmetric determinantal varieties and applications", Inventiones Mathematicae 63 (3): 433–465, doi:10.1007/BF01389064, ISSN 0020-9910