NumericalGodeaux -- Construction of numerical Godeaux surfaces

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

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

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

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 = (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 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 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

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

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

Recovering lines

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

We have precomputed parametrizations leading to the hyperelliptic locus in Q and special lines.

Precomputed Parametrization of Special Points and Lines

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

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

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

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

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

Authors

Version

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.

Exports

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