randomStandardResolution -- compute a random standard resolution of an S-module R

Using our construction method, this procedure computes a standard resolution of an S = k[x₀,x₁,y₀,...,y₃]-module R

with a skew-symmetric map d₂: F₁^*(-17) →F₁ .

The main steps of the construction are the choice of a (random) line in complete intersection Q ⊂ℙ¹¹ and a (random) point in a linear solution space (depending on the chosen line). The procedure computes only modules R which satisfy the ring condition. Such an module R leads to a numerical Godeaux surface if the surface Proj(R) has only canonical singularities. However, this condition is not checked. If an additional string s is given, the procedure computes standard resolutions which lead to surfaces 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.

This procedure works mainly over finite fields. Over the rational numbers the procedure may not terminate.