Fabrizio Catanese, Roberto Pignatelli
On simply connected Godeaux surfaces
Preprint series: Mathematica Gottingensis
14J25 Special surfaces, {For Hilbert modular surfaces, See 14G35}
14J10 Families, moduli, classification: algebraic theory
Abstract: \begin{abstract}
In this paper we provide a first step towards the
classification of the numerical Godeaux surfaces
in the still unknown open cases where $Tors(S) = 0$, or
$Tors(S) = \Z/2\Z$.

Our method works in both cases, but in this paper, after some results
which we establish in a greater generality, we mostly restrict ourselves to the
case where the Torsion group is zero.

The bicanonical system yields, on a
suitable blow up $\tilde{S}$ of the minimal model $S$, a fibration
$f:\tilde{S} \rightarrow {\mathbb P}^1$ in curves of genus $2\leq g \leq 4$,
and the invariants of this fibration determine the equations of the
image of $S$ under a map which, in the general case, is the
product of the tricanonical and of the bicanonical map.

This allows us to subdivide our surfaces into
four classes, according to the behaviour of the
bicanonical system. For each of these classes we have a complete
description but the existence questions are not yet solved.

Keywords: Surfaces, Godeaux,fibrations, Hyperelliptic fibres,