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NumericalGodeaux :: NumericalGodeaux

NumericalGodeaux -- Construction of numerical Godeaux surfaces


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 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 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 3 ×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 3 ×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

  • modelInP3xP3xP3xP3 -- compute the model of the Fano variety F(Q) in P3xP3xP3xP3
  • collapsingOneCStar -- compute the hypersurface of bidegree (4,6) in P3xP5
  • furtherCollapsing -- computes the 5-dimensional anti-canonical hypersurface in the cox ring of a toric variety

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



This documentation describes version 1.1 of NumericalGodeaux.

Source code

The source code from which this documentation is derived is in the file NumericalGodeaux.m2.