Catanese surface

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.

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

{{Hodge diamond|style=font-weight:bold

| 1

| 0 | 0

| 0 | 8 | 0

| 0 | 0

| 1

}}

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

• {{Citation | last1=Barth | first1=Wolf P. | last2=Hulek | first2=Klaus | last3=Peters | first3=Chris A.M. | last4=Van de Ven | first4=Antonius | title=Compact Complex Surfaces | publisher= Springer-Verlag, Berlin | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. | isbn=978-3-540-00832-3 | mr=2030225 | year=2004 | volume=4}}
• {{Citation | last1=Catanese | first=Fabrizio| authorlink=Fabrizio Catanese | title=Babbage's conjecture, contact of surfaces, symmetric determinantal varieties and applications | doi=10.1007/BF01389064 | mr=620679 | year=1981 | journal=Inventiones Mathematicae | issn=0020-9910 | volume=63 | issue=3 | pages=433–465}}

Category:Algebraic surfaces

Category:Complex surfaces