Description
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 $$\ell \subset Q \subset \mathbb{P}^{11}. $$ The variety $ Q \subset \mathbb{P}^{11} $ 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 $\ell \subset Q$ from the first step. For a general line $\ell$ one has a $\mathbb{P}^3$ 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 $\mathbb{Z}/3$ and $\mathbb{Z}/5$. 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 $\mathbb{Z}/2-$Godeaux surfaces.
Random construction over finite fields and the rational numbers
Setup for the construction
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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 $\mathbb{P}^3 \times \mathbb{P}^3 \times \mathbb{P}^3 \times \mathbb{P}^3$. The additional grading is coming from the $G = (G_m)^3$-action, and a quotient of this action is computed in two steps. First we collapse one $ G_m$-action and obtain a hypersurface $H_{4,6}$ of bidegree (4,6) in a $\mathbb{P}^3 \times \mathbb{P}^5$.The final result is a model of the quotient $F(Q)//G$ realized as a hypersurface $Y$ in a toric variety. The hypersurface $H_{4,6}$ 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
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randomPoint -- compute a rational point in a variety
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randomLine -- compute a line through a given point which is completely contained in the Pfaffian variety
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randomSection -- choose a point in the solution space defined by the linear relations
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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