An experimental approach to numerical Godeaux surfaces
by
Frank-Olaf Schreyer
Universität des Saarlandes

A (numerical) Godeaux surface is a minimal surface X of general type with K²=1 and p_g=0, hence also q=0 and H₁(X, Q)=0. So in some sense these are the surfaces of general type with smallest possible invariants. Godeaux constructed a family of such surfaces as quotients of a quintic hypersurface by fix point free action of Z₅. By the work of Miyaoka it is known, that torsion group T=H¹(X, Z) is a cyclic group of order at most 5. The surfaces with T=Z_d for d=3, 4, 5 have a moduli space which in each case consists of one 8-dimensional component by work of Reid and Miyaoka. For T=Z₂ or T=0 much less is known. Existence of such surface was proved by Rebecca Barlow using a complicated quotient construction.

Traditionally there are two approaches to construct numerical Godeaux surfaces: Either via a Godeaux approach as quotient of a simpler surface by a possibly non free group action, or via a Campedelli approach as a double plane branched along a curve with a specific configuration of singularities.

In this talk I present a third approach based on homological algebra.

Date received: March 23, 2005