NumericalGodeaux : Table of Contents

• NumericalGodeaux -- Construction of numerical Godeaux surfaces
• 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
• Attempts -- optional argument in randomGodeauxSurface
• 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
• Certify -- optional argument in randomGodeauxSurface
• collapsingOneCStar -- compute the hypersurface of bidegree (4,6) in P3xP5
• complexModuloRegularSequence -- set-up for minimal free resolution modulo x0,x1
• complexModuloRegularSequence4 -- set-up the minimal free resolution modulo $x_0,x_1$ in the case of a fat base point
• complexModuloRegularSequence22 -- set-up the minimal free resolution modulo $x_0,x_1$ in the case of 2 double base points
• complexModuloRegularSequence211 -- set-up the minimal free resolution modulo $x_0,x_1$ for the configuration "211" of base points
• complexModuloRegularSequence1111 -- set-up the minimal free resolution modulo $x_0,x_1$ in the case of 4 distinct base points
• 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
• globalVariables4 -- introduce the main variables in the case of a fat base point
• globalVariables22 -- introduce the main variables in the case of 2 double base points
• globalVariables211 -- introduce the main variables for the configuration "211" of the base points
• globalVariables1111 -- introduce the main variables in the case of 4 distinct base points
• 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
• PrecomputedParametrization -- optional argument for using a precomputed 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
• randomGodeauxSurface -- compute a birational model of a numerical Godeaux surface
• randomGodeauxSurface(...,Height=>...) -- optional argument for the random computations
• 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