Constructing torsion-free numerical Godeaux surfaces
Numerical Godeaux surfaces provide the first case in the geography of minimal surfaces of general type. It is known that the torsion group of such a surface is cyclic of order m ≤ 5 and a full classification has been given for m = 3, 4, 5 by Reid, Miyaoka. In my talk I will discuss a homological approach to construct a numerical Godeaux surface X based on a former project of Frank-Olaf Schreyer. The main idea is to study the syzygies of the canonical ring R(X) considered as a module over some (weighted) graded polynomial ring. We focus on the case Tors(X) = 0 and pay particular emphasis to the genus-4 fibration given by the bicanonical system.