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 ⊂ℙ^{11}. ## Random construction over finite fields and the rational numbers

## Setup for the construction

* F(Q)* in *ℙ*^{3} ×ℙ^{3} ×ℙ^{3} ×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 *ℙ*^{3} ×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

*Q* and special lines.## Precomputed Parametrization of Special Points and Lines

## Steps of the construction

## From free resolutions to different models of Godeaux surfaces

## Calculation of the Unirational Parametrization of the Loci of Special Points and Lines

The variety * Q ⊂ℙ ^{11} * is a complete intersection of four quadrics and plays, together with the corresponding Fano variety of lines

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

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.

- randomStandardResolution -- compute a random standard resolution of an S-module R
- randomGodeauxSurface -- compute a birational model of a numerical Godeaux surface

- setupGodeaux -- summarize the single steps for the general set-up of the construction

- 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

- precomputedModelInP3xP3xP3xP3 -- load the precomputed ideal of the model of F(Q) in P3xP3xP3xP3
- precomputedModelInP3xP5 -- load the precomputed model
- precomputedCoxModel -- load the equation of the 5-dimensional hypersurface in a Cox ring of a toric variety

- findPointInP3xP5 -- find a point on the model in P3xP5
- pointOnARationalCodim1Hypersurface -- choose a QQ-rational point on a codimension 1 rational subvariety of the model in P3xP5
- fromPointInP3xP5ToPointInP3xP3xP3xP3 -- compute a point in the model in P3xP3xP3xP3
- fromPointInP3xP3xP3xP3ToLine -- compute a line in Q from a point in the model in P3xP3xP3xP3
- fromLineToGodeauxSurface -- compute a birational model of a numerical Godeaux surface from a given line

- precomputedHyperellipticLocus -- get the ideal of the hyperelliptic locus
- precomputedHyperellipticPoint -- compute a point in the hyperelliptic locus using the unirational parametrization
- precomputedTorsZ2Line -- compute a line leading generically to a Z/2-Godeaux surface using a unirational parametrization
- precomputedTorsZ3Line -- compute a line leading generically to a Z/3-Godeaux surface using a unirational parametrization
- precomputedTorsZ4Line -- compute a line leading generically to a Z/4-Godeaux surface using a unirational parametrization
- precomputedTorsZ5Line -- compute a line leading generically to a Z/5-Godeaux surface using a unirational parametrization

- 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

- surfaceInWeightedP5 -- compute the surface in P(2,2,3,3,3,3)
- bihomogeneousModel -- compute a birational model of a numerical Godeaux surface in P1xP3
- tricanonicalModelInP3 -- computes the tricanonical model of a numerical Godeaux surface in P3
- canonicalRing -- computes the canonical ring of a numerical Godeaux surface

- calculationOfTheUnirationalParametrizationOfTorsZ5Lines -- describe the unirational parametrization of the locus of Z/5-lines
- verifyThmHypLocus -- print commands which verify the assertions on the hyperelliptic locus
- computeParametrizationOfHypLocus -- print commands which compute the parametrization

- Functions and commands
- allLoci -- compute all exceptional loci at which the dimension of the solution space may rise
- allLociTors0 -- compute all exceptional loci for torsion-free numerical Godeaux surfaces
- associatedLineInP11 -- compute the associated line in the P11 of a-variables
- bihomogeneousModel -- compute a birational model of a numerical Godeaux surface in P1xP3
- calculationOfTheUnirationalParametrizationOfTorsZ5Lines -- describe the unirational parametrization of the locus of Z/5-lines
- canonicalRing -- computes the canonical ring of a numerical Godeaux surface
- collapsingOneCStar -- compute the hypersurface of bidegree (4,6) in P3xP5
- complexModuloRegularSequence -- set-up for minimal free resolution modulo x0,x1
- computeParametrizationOfHypLocus -- print commands which compute the parametrization
- findPointInP3xP5 -- find a point on the model in P3xP5
- fromLineToGodeauxSurface -- compute a birational model of a numerical Godeaux surface from a given line
- fromLineToStandardResolution -- compute a standard resolution F of an S-module R from a given line
- fromPointInP3xP3xP3xP3ToLine -- compute a line in Q from a point in the model in P3xP3xP3xP3
- fromPointInP3xP5ToPointInP3xP3xP3xP3 -- compute a point in the model in P3xP3xP3xP3
- furtherCollapsing -- computes the 5-dimensional anti-canonical hypersurface in the cox ring of a toric variety
- getAMatrix -- compute the a-matrix of a given matrix
- getChainComplexes -- resolve the two linear submatrices of the solution matrices over the coordinate ring of the Pfaffians
- getEMatrix -- compute the e-matrix of a given matrix
- getP11 -- the polynomial ring which depends only on the a-variables
- getRelationsAndNormalForm -- compute a minimal set of the relations and a normal form for d1' and d2
- globalVariables -- introduce the main variables for the construction
- homologyLocus -- compute the homology of the two chain complexes C1 and C2
- isSmoothBihomModel -- check whether the model in P1xP3 is smooth or not
- isSmoothModelInP5 -- check whether the model in the weighted P5 is smooth or not
- jacobianQ -- compute the Jacobian matrix of the quadratic relations
- lineConditionsTorsZ2 -- compute a list of possible loci for Z/2Z-Godeaux surfaces
- lineConditionsTorsZ4 -- compute a list of possible loci for Z/4Z-Godeaux surfaces
- lineConditionsTorsZ5 -- compute a list of possible loci for Z/5Z-Godeaux surfaces
- lowerRankLociA -- compute the loci at which the rank of the a-matrix drops
- lowerRankLociE -- compute the loci at which the rank of the e-matrix drops
- modelInP1BundleOverP2xP5 -- compute the projection from a double point of H_{4,6}
- modelInP3xP3xP3xP3 -- compute the model of the Fano variety F(Q) in P3xP3xP3xP3
- modelInP13 -- compute the image of a variety in P(2,2,3,3,3,3) under a embedding to P13
- normalBundleLineInQ -- compute the normal bundle of a line in Q
- pointOnARationalCodim1Hypersurface -- choose a QQ-rational point on a codimension 1 rational subvariety of the model in P3xP5
- precomputedCoxModel -- load the equation of the 5-dimensional hypersurface in a Cox ring of a toric variety
- precomputedHyperellipticLocus -- get the ideal of the hyperelliptic locus
- precomputedHyperellipticPoint -- compute a point in the hyperelliptic locus using the unirational parametrization
- precomputedModelInP3xP3xP3xP3 -- load the precomputed ideal of the model of F(Q) in P3xP3xP3xP3
- precomputedModelInP3xP5 -- load the precomputed model
- precomputedTorsZ2Line -- compute a line leading generically to a Z/2-Godeaux surface using a unirational parametrization
- precomputedTorsZ3Line -- compute a line leading generically to a Z/3-Godeaux surface using a unirational parametrization
- precomputedTorsZ4Line -- compute a line leading generically to a Z/4-Godeaux surface using a unirational parametrization
- precomputedTorsZ5Line -- compute a line leading generically to a Z/5-Godeaux surface using a unirational parametrization
- randomGodeauxSurface -- compute a birational model of a numerical Godeaux surface
- randomLine -- compute a line through a given point which is completely contained in the Pfaffian variety
- randomLineTors0 -- compute a line for a torsion-free numerical Godeaux surface
- randomLineTorsZ2 -- compute a line for a numerical Godeaux surface with a cyclic torsion group of order 2
- randomLineTorsZ3 -- compute a line for a numerical Godeaux surface with a cyclic torsion group of order 3
- randomLineTorsZ4 -- compute a line for a numerical Godeaux surface with a cyclic torsion group of order 4
- randomLineTorsZ5 -- compute a line for a numerical Godeaux surface with a cyclic torsion group of order 5
- randomPoint -- compute a rational point in a variety
- randomSection -- choose a point in the solution space defined by the linear relations
- randomStandardResolution -- compute a random standard resolution of an S-module R
- setupGeneralMatrices -- compute the general set-up for the construction
- setupGodeaux -- summarize the single steps for the general set-up of the construction
- setupSkewMatrices -- compute four skew-symmetric matrices whose Pfaffians are among the quadratic relations
- singleSolutionMatricesLine -- evaluate the single solution matrices at a line
- singleSolutionMatricesOverP11 -- display the single solution matrices over the P^n of a-variables
- singularLocusQ -- compute the minimal primes of the singular locus of the Pfaffian relations
- solutionMatrix -- display the relations linear in the c- and o-variables as a matrix
- standardResolution -- compute a standard resolution of an S-module R obtained from the given input
- surfaceInWeightedP5 -- compute the surface in P(2,2,3,3,3,3)
- tangentSpacePoint -- compute the complete intersection of quadrics in the tangent space at a given point
- tricanonicalModelInP3 -- computes the tricanonical model of a numerical Godeaux surface in P3
- verifyAssertions -- verify the ring condition
- verifyThmHypLocus -- print commands which verify the assertions on the hyperelliptic locus

- Symbols
- Attempts -- optional argument in randomGodeauxSurface
- Certify -- optional argument in randomGodeauxSurface
- PrecomputedParametrization -- optional argument for using a precomputed unirational parametrization