An experimental approach to numerical Godeaux surfaces
Wednesday, October 4, 2006 - 11:15am - 12:15pm
A (numerical) Godeaux surface is a minimal surface X of general type with K_2=1 and p_g=0, hence also q=0 and H_1(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 a fixed point free action of Z_5. By the work of Miyaoka it is known that the torsion group T=H_1(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 consists of one 8 dimensional component by work of Reid and Miyaoka. For T=Z_2 or T=0 much less is known. Existence of such surfaces was proved by Rebecca Barlow, by a complicated quotient construction. Traditionally there are two approaches to construct numerical Godeaux surfaces: Either by a Godeaux approach as a quotient of a simpler surface by a possibly non free group action, or by 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 and computer algebra.