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# 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