randomGodeauxSurface -- compute a birational model of a numerical Godeaux surface

Using the procedure randomStandardResolution we compute first a (standard resolution of an) $S$-module $R$ satisfying the ring condition. Such an module $R$ leads to a numerical Godeaux surface if the surface $Proj(R)$ has only canonical singularities. Verifying this is computationally involved and thus this condition is not checked. A sufficient and easier criterion to check is whether the obtained surface $Y$ in $\mathbb{P}(2^2,3^4)$ is has only canonical singularities or is smooth. If an additional string $s$ is given, the procedure computes a surface with the desired configuration of base points of the bicanonical system. If no string $s$ is indicated, the procedure uses the default setting "1111". Moreover, an additional number $n$ indicates the order of the torsion group of the resulting surface. Thus, $n$ must be an integer between 1 and 5. Note that any numerical Godeaux surface with a torsion group of an odd order cannot have double points. Hence, if the configuration of base points and the order of the torsion group are not compatible, an error message is printed. For the integers $n=2,\ldots,5$ we have a unirational parametrization of the space of all possible lines. If the option preComputedParametrization is true, we use this parametrization to choose a line instead of computing one randomly. The default value option is preComputedParametrization => true. In case of Certify=>true we check whether the surface $Y$ is smooth ( and hence $Proj(R)$). The message "singular surface" means that we have to repeat the construction to get a smooth surface. We do this at most k times if Attempts=>k. If we fail every time we return null. The default values of the options are Certify=>false and Attempts=>1.

As an example we consider first a torsion-free Godeaux surface:

The birational model in $\mathbb{P}^3$ is a hypersurface of degree 9:

The following is an example of a Godeaux surface with torsion group $\mathbb{Z}/4$ over the rational numbers.

The birational model in $\mathbb{P}^3$ is a hypersurface of degree 8:

In the cases $n=2,\ldots,5$, this procedure works over any finite field (of characteristic $\geq 5$) and over the rational numbers, as we have a unirational description of the space of possible lines in these cases. For torsion-free numerical Godeaux surfaces, such a unirational parametrization is so far not known to us and the procedure works only over finite fields.