This package contains the implementation of our construction method for numerical Godeaux surfaces from
An 8-dimensional family of simply connected Godeaux surfaces and
Marked Godeaux surfaces with special bicanonical fibers. The articles focus on numerical Godeaux surfaces whose bicanonical systems have 4 distinct base points. The main construction has two big steps. The first step consists of choosing a line
l⊂Q ⊂ℙ11.
The variety Q ⊂ℙ11 is a complete intersection of four quadrics and plays, together with the corresponding Fano variety of lines F(Q) a crucial part in our construction.
The second step of our construction consists of solving a linear system of equations which depends on the line
l⊂Q from the first step. For a general line
l one has a
ℙ3 of solutions. A general line together with a general solution specify a simply connected Godeaux surface. We call this the dominant component. Special lines lead to different components. We describe the family of lines in
Q leading to numerical Godeaux surfaces with torsion group
ℤ/3 and
ℤ/5. Another highlight is the parametrization of the hyperelliptic locus in
Q which leads to torsion-free numerical Godeaux surfaces with hyperelliptic bicanonical fibers.
An important result of the package not covered by the preprints above, is the construction of a 8-dimensional locally complete unirational family of
ℤ/2-Godeaux surfaces.
Random construction over finite fields and the rational numbers
Setup for the construction
- setupGodeaux -- summarize the single steps for the general set-up of the construction
We compute a model of the variety of lines
F(Q) in
ℙ3 ×ℙ3 ×ℙ3 ×ℙ3. The additional grading is coming from the
G = (Gm)3-action, and a quotient of this action is computed in two steps. First we collapse one
Gm-action and obtain a hypersurface
H4,6 of bidegree (4,6) in a
ℙ3 ×ℙ5.The final result is a model of the quotient
F(Q)//G realized as a hypersurface
Y in a toric variety. The hypersurface
H4,6 contains some codimension 1 rational subvariety
Z and given a randomly chosen point in
Z, functions are added which recover a line in the corresponding
G-orbit of lines in
F(Q).
Models of F(Q) and its quotients
Precomputed models
Recovering lines
We have precomputed parametrizations leading to the hyperelliptic locus in
Q and special lines.
Precomputed Parametrization of Special Points and Lines
Steps of the construction
- randomPoint -- compute a rational point in a variety
- randomLine -- compute a line through a given point which is completely contained in the Pfaffian variety
- randomSection -- choose a point in the solution space defined by the linear relations
- standardResolution -- compute a standard resolution of an S-module R obtained from the given input
From free resolutions to different models of Godeaux surfaces
Calculation of the Unirational Parametrization of the Loci of Special Points and Lines