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 ℙ¹¹. A numerical Godeaux surface X with torsion group ℤ/5 has two special reducible bicanonical curves of the form D_i+D_5-i, where D_i ∈|K_x + t_i| with a torsion element t_i of order i =1,...,4. The rank of the e-matrix drops from three to two at the corresponding two (different) points in ℙ¹. 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 ℙ³s 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 ℙ¹ × ℙ¹ ⊂ ℙ³ × ℙ³ and ℙ² × ℙ⁰  ⊂ ℙ³ × ℙ³ or ℙ⁰ × ℙ²  ⊂ ℙ³ × ℙ³. The last two types do not lead to numerical Godeaux surfaces. Picking a point in one of the ℙ¹ ×ℙ¹- components gives a line which generically leads to a Godeaux surface with torsion group ℤ/5.