lineConditionsTorsZ5 -- compute a list of possible loci for Z/5Z-Godeaux surfaces

The order of the torsion group does only depend on the choice of the line in the complete intersection of the quadratic relations $Q$ in $\mathbb{P}^{11}$. A numerical Godeaux surface $X$ with torsion group $\mathbb{Z}/5$ has two special reducible bicanonical curves of the form $D_i+D_{5-i}$, where $D_i \in |K_x + t_i|$ with a torsion element $t_i$ of order $i =1,\ldots,4$. The rank of the $e$-matrix drops from three to two at the corresponding two (different) points in $\mathbb{P}^1$. Thus, the associated line in $Q$ must intersect the loci given by the 3x3-minors of the $e$-matrix in two different points. We choose two different $\mathbb{P}^3s$ in this loci and evaluate the condition that a line through two general points is completely contained in the variety $Q$. The resulting zero loci decomposes in a union of several surfaces of type $\mathbb{P}^{1} \times \ \mathbb{P}^{1} \subset \ \mathbb{P}^{3} \times \ \mathbb{P}^{3}$ and $\mathbb{P}^{2} \times \ \mathbb{P}^{0} \ \subset \ \mathbb{P}^{3} \times \ \mathbb{P}^{3}$ or $\mathbb{P}^{0} \times \ \mathbb{P}^{2} \ \subset \ \mathbb{P}^{3} \times \ \mathbb{P}^{3}$. The last two types do not lead to numerical Godeaux surfaces. Picking a point in one of the $\mathbb{P}^{1} \times \mathbb{P}^{1}$- components gives a line which generically leads to a Godeaux surface with torsion group $\mathbb{Z}/5$.