MultiprojectiveVarieties : Table of Contents

• MultiprojectiveVarieties -- Multi-projective varieties and multi-rational maps

  • MultiprojectiveVariety -- the class of all multi-projective varieties

    • projectiveVariety -- the closed multi-projective subvariety defined by a multi-homogeneous ideal
    • projectiveVariety(List,Ring) -- product of projective spaces
    • projectiveVariety(List,List,Ring) -- the Segre-Veronese variety
    • projectiveVariety(MultidimensionalMatrix) -- the multi-projective variety defined by a multi-dimensional matrix
  • MultirationalMap -- the class of all multi-rational maps

    • multirationalMap -- the multi-rational map defined by a list of rational maps
    • rationalMap(List,MultiprojectiveVariety) -- the multi-rational map defined by a list of rational maps
• ∏ -- product of multi-projective varieties
• ⋂ -- intersection of multi-projective varieties
• ⋃ -- union of multi-projective varieties
• ambient(MultiprojectiveVariety) -- the ambient multi-projective space of the variety
• ambientVariety -- the ambient variety of a projective subvariety
• baseLocus -- the base locus of a multi-rational map
• check(MultirationalMap) -- check that a multi-rational map is well-defined
• chowForm(EmbeddedProjectiveVariety) -- chow forms of a projective variety
• clean(MultirationalMap) -- clean the internal information of a multi-rational map
• codim(MultiprojectiveVariety) -- the codimension of the variety
• coefficientRing(MultiprojectiveVariety) -- the coefficient ring of the variety
• coefficientRing(MultirationalMap) -- the coefficient ring of a multi-rational map
• coneOfLines -- cone of lines on a subvariety passing through a point
• conormalVariety(EmbeddedProjectiveVariety) -- the conormal variety of a projective variety
• cycleClass -- determine the expression of the class of a cycle as a linear combination of Schubert classes
• decompose(MultiprojectiveVariety) -- irreducible components of a variety
• degree(MultiprojectiveVariety) -- the degree of the variety
• degree(MultirationalMap) -- degree of a multi-rational map
• degree(MultirationalMap,Option) -- degree of a multi-rational map using a probabilistic approach
• degrees(MultiprojectiveVariety) -- degrees for the minimal generators
• degreeSequence -- the (multi)-degree sequence of a (multi)-rational map
• describe(MultiprojectiveVariety) -- describe a multi-projective variety
• describe(MultirationalMap) -- describe a multi-rational map
• dim(MultiprojectiveVariety) -- the dimension of the variety
• dual(EmbeddedProjectiveVariety) -- the variety projectively dual to an embedded projective variety
• EmbeddedProjectiveVariety -- the class of all embedded projective varieties
• EmbeddedProjectiveVariety ! -- print a more detailed description of an embedded projective variety
• EmbeddedProjectiveVariety ++ EmbeddedProjectiveVariety -- join of projective varieties
• EmbeddedProjectiveVariety ===> EmbeddedProjectiveVariety -- try to find an isomorphism between two projective varieties
• entries(MultirationalMap) -- list the defining polynomials of a rational map
• euler(MultiprojectiveVariety) -- topological Euler characteristic of a (smooth) multi-projective variety
• factor(MultirationalMap) -- the list of rational maps defining a multi-rational map
• Fano(ZZ,EmbeddedProjectiveVariety) -- Fano scheme of a projective variety
• fiberProduct -- fiber product of multi-projective varieties
• forceImage(MultirationalMap,MultiprojectiveVariety) -- declare which is the image of a multi-rational map
• GG -- the Grassmannian of k-dimensional linear subspaces of an n-dimensional projective space
• GG(ZZ,MultirationalMap) -- induced automorphism of the Grassmannian
• graph(MultirationalMap) -- the graph of a multi-rational map
• GrassmannianVariety -- the class of all Grassmannians of linear subspaces of projective spaces
• hilbertPolynomial(EmbeddedProjectiveVariety) -- the Hilbert polynomial of the variety
• Hom(MultiprojectiveVariety,MultiprojectiveVariety) -- get the hom-set of rational maps between two multi-projective varieties
• ideal(MultiprojectiveVariety) -- the defining ideal of the variety
• image(MultirationalMap) -- image of a multi-rational map
• inverse(MultirationalMap) -- inverse of a birational map
• inverse2 -- inverse of a birational map using a faster algorithm for a special class of maps
• isIsomorphism(MultirationalMap) -- whether a birational map is an isomorphism
• isMember(MultirationalMap,RAT) -- test membership in a hom-set of rational maps
• isMorphism(MultirationalMap) -- whether a multi-rational map is a morphism
• isSubset(MultiprojectiveVariety,MultiprojectiveVariety) -- whether one variety is a subvariety of another
• isWellDefined(MultirationalMap) -- whether a multi-rational map is well-defined
• linearlyNormalEmbedding -- get the linearly normal embedding
• linearSpan -- the linear span of an embedded projective variety
• multidegree(MultiprojectiveVariety) -- the multidegree of the variety
• multidegree(MultirationalMap) -- projective degrees of a multi-rational map
• multidegree(ZZ,MultirationalMap) -- i-th projective degree of a multi-rational map using a probabilistic approach
• MultiprojectiveVariety % MultiprojectiveVariety -- subvariety of a projective variety
• MultiprojectiveVariety * MultiprojectiveVariety -- intersection of two multi-projective varieties
• MultiprojectiveVariety ** MultiprojectiveVariety -- product of two multi-projective varieties
• MultiprojectiveVariety ** Ring -- change the coefficient ring of a multi-projective variety
• MultiprojectiveVariety + MultiprojectiveVariety -- union of two multi-projective varieties
• MultiprojectiveVariety == MultiprojectiveVariety -- equality of multi-projective varieties
• MultiprojectiveVariety \ MultiprojectiveVariety -- difference of multi-projective varieties
• MultiprojectiveVariety \\ MultiprojectiveVariety -- difference of multi-projective varieties
• MultiprojectiveVariety ^ ZZ -- power of a multi-projective variety
• MultirationalMap * MultirationalMap -- composition of multi-rational maps
• MultirationalMap ** Ring -- change the coefficient ring of a multi-rational map
• MultirationalMap << MultiprojectiveVariety -- force the change of the target in a multi-rational map
• MultirationalMap <==> MultirationalMap -- equality of multi-rational maps with checks on internal data
• MultirationalMap == MultirationalMap -- equality of multi-rational maps
• MultirationalMap ^** MultiprojectiveVariety -- inverse image via a multi-rational map
• MultirationalMap | MultiprojectiveVariety -- restriction of a multi-rational map
• MultirationalMap | MultirationalMap -- product of multi-rational maps
• MultirationalMap || MultiprojectiveVariety -- restriction of a multi-rational map
• MultirationalMap || MultirationalMap -- product of multi-rational maps
• MultirationalMap MultiprojectiveVariety -- direct image via a multi-rational map
• multirationalMap(MultiprojectiveVariety) -- identity map
• multirationalMap(MultiprojectiveVariety,MultiprojectiveVariety) -- get the natural inclusion
• multirationalMap(MultirationalMap,MultiprojectiveVariety) -- change the target of a multi-rational map
• parametrize(MultiprojectiveVariety) -- try to get a parametrization of a multi-projective variety
• permute(MultiprojectiveVariety,List) -- permute the factors of the ambient multi-projective space
• point(MultiprojectiveVariety) -- pick a random rational point on a multi-projective variety
• projectionMaps -- projections of a multi-projective variety
• projectionMaps(MultirationalMap) -- get the compositions of the multi-rational map with the projections of the target
• projections -- projections of a multi-projective variety
• projectiveVariety(...,MinimalGenerators=>...) -- whether to trim the ideal (intended for internal use only)
• random(List,MultiprojectiveVariety) -- get a random hypersurface of given multi-degree containing a multi-projective variety
• random(MultiprojectiveVariety) -- apply a random automorphism of the ambient multi-projective space
• RAT -- the hom-sets of rational maps between two multi-projective varieties
• RAT List -- define a multi-rational map
• RAT MultiprojectiveVariety -- rational map defined by a linear system of hypersurfaces through a variety
• RAT Tally -- rational map defined by an effective divisor
• rationalMap(MultiprojectiveVariety,Tally) -- rational map defined by an effective divisor
• ring(MultiprojectiveVariety) -- the coordinate ring of the variety
• Saturate -- whether to compute the multi-saturation of the ideal (intended for internal use only)
• schubertCycle -- take a random Schubert cycle
• sectionalGenus -- the sectional genus of an embedded projective variety
• segre(MultiprojectiveVariety) -- the Segre embedding of the variety
• segre(MultirationalMap) -- the composition of a multi-rational map with the Segre embedding of the target
• segreEmbedding -- the Segre embedding of the variety
• shape(MultiprojectiveVariety) -- shape of the ambient multi-projective space of a multi-projective variety
• shortcuts -- Some convenient shortcuts for multi-rational maps consisting of a single rational map
• show(MultirationalMap) -- display a multi-rational map
• singularLocus(MultiprojectiveVariety) -- the singular locus of the variety
• source(MultirationalMap) -- the source for a multi-rational map
• super(MultirationalMap) -- get the multi-rational map whose target is a product of projective spaces
• support(MultiprojectiveVariety) -- support of a multi-projective variety
• tangentCone(EmbeddedProjectiveVariety,EmbeddedProjectiveVariety) -- tangent cone to a projective variety at a point
• tangentialChowForm(EmbeddedProjectiveVariety,ZZ) -- higher Chow forms of a projective variety
• tangentSpace -- tangent space to a projective variety at a point
• target(MultirationalMap) -- the target for a multi-rational map
• topComponents(MultiprojectiveVariety) -- union of the top dimensional components of a multi-projective variety
• toRationalMap -- convert a multi-rational map consisting of a single rational map to a standard rational map
• trim(MultirationalMap) -- trim the target of a multi-rational map
• variety(EmbeddedProjectiveVariety) -- convert an embedded projective variety into a built-in projective variety
• WeightedProjectiveVariety -- the class of all weighted projective varieties
• WeightedRationalMap -- the class of all weighted-rational maps