Catanese Surface

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

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 $K^2 = 2$. The fundamental group of the Catanese surface is $\mathbf/5\mathbf$, as can be seen from its quotient construction.

References

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*{{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

Algebraic surfaces
Complex surfaces