MultiprojectiveVariety -- the class of all multi-projective varieties

Description

A multi-projective variety is a closed subvariety of a product of projective spaces $\mathbb{P}^{k_1}\times\mathbb{P}^{k_2}\times\cdots\times\mathbb{P}^{k_n}$. This is actually the class of all closed subschemes of products of projective spaces.

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

Types of multi-projective variety:

EmbeddedProjectiveVariety -- the class of all embedded projective varieties
WeightedProjectiveVariety -- the class of all weighted projective varieties

Functions and methods returning a multi-projective variety:

ambientVariety -- the ambient variety of a projective subvariety
baseLocus -- the base locus of a multi-rational map
fiberProduct -- fiber product of multi-projective varieties
projectiveVariety -- the closed multi-projective subvariety defined by a multi-homogeneous ideal

Methods that use a multi-projective variety:

ambient(MultiprojectiveVariety) -- the ambient multi-projective space of the variety
ambientVariety(MultiprojectiveVariety) -- see ambientVariety -- the ambient variety of a projective subvariety
codim(MultiprojectiveVariety) -- the codimension of the variety
coefficientRing(MultiprojectiveVariety) -- the coefficient ring of the variety
decompose(MultiprojectiveVariety) -- irreducible components of a variety
degree(MultiprojectiveVariety) -- the degree of the variety
degrees(MultiprojectiveVariety) -- degrees for the minimal generators
? MultiprojectiveVariety -- see describe(MultiprojectiveVariety) -- describe a multi-projective variety
describe(MultiprojectiveVariety) -- describe a multi-projective variety
dim(MultiprojectiveVariety) -- the dimension of the variety
euler(MultiprojectiveVariety) -- topological Euler characteristic of a (smooth) multi-projective variety
forceImage(MultirationalMap,MultiprojectiveVariety) -- declare which is the image of a multi-rational map
Hom(MultiprojectiveVariety,MultiprojectiveVariety) -- get the hom-set of rational maps between two multi-projective varieties
Hom(MultiprojectiveVariety,Nothing) -- see Hom(MultiprojectiveVariety,MultiprojectiveVariety) -- get the hom-set of rational maps between two multi-projective varieties
Hom(MultiprojectiveVariety,Symbol) -- see Hom(MultiprojectiveVariety,MultiprojectiveVariety) -- get the hom-set of rational maps between two multi-projective varieties
Hom(Nothing,MultiprojectiveVariety) -- see Hom(MultiprojectiveVariety,MultiprojectiveVariety) -- get the hom-set of rational maps between two multi-projective varieties
ideal(MultiprojectiveVariety) -- the defining ideal of the variety
isSubset(MultiprojectiveVariety,MultiprojectiveVariety) -- whether one variety is a subvariety of another
multidegree(MultiprojectiveVariety) -- the multidegree of the variety
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(List,MultiprojectiveVariety) -- see multirationalMap -- the multi-rational map defined by a list of rational maps
MultiprojectiveVariety << MultiprojectiveVariety -- see MultirationalMap << MultiprojectiveVariety -- force the change of the target in a multi-rational map
MultirationalMap << MultiprojectiveVariety -- force the change of the target in a multi-rational map
MultirationalMap ^** MultiprojectiveVariety -- inverse image via a multi-rational map
MultirationalMap | MultiprojectiveVariety -- restriction of a multi-rational map
RationalMap | MultiprojectiveVariety -- see MultirationalMap | MultiprojectiveVariety -- restriction of a multi-rational map
MultirationalMap || MultiprojectiveVariety -- restriction of a multi-rational map
RationalMap || MultiprojectiveVariety -- see MultirationalMap || MultiprojectiveVariety -- restriction of a multi-rational map
MultirationalMap MultiprojectiveVariety -- direct image via a multi-rational map
RationalMap MultiprojectiveVariety -- see MultirationalMap MultiprojectiveVariety -- direct image via a multi-rational map
multirationalMap(MultiprojectiveVariety) -- identity map
ZZ _ MultiprojectiveVariety -- see multirationalMap(MultiprojectiveVariety) -- identity map
multirationalMap(MultiprojectiveVariety,MultiprojectiveVariety) -- get the natural inclusion
rationalMap(MultiprojectiveVariety,MultiprojectiveVariety) -- see multirationalMap(MultiprojectiveVariety,MultiprojectiveVariety) -- get the natural inclusion
multirationalMap(MultirationalMap,MultiprojectiveVariety) -- change the target of a multi-rational map
rationalMap(MultirationalMap,MultiprojectiveVariety) -- see 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
|- MultiprojectiveVariety -- see point(MultiprojectiveVariety) -- pick a random rational point on a multi-projective variety
point(MultiprojectiveVariety) -- pick a random rational point on a multi-projective variety
projectionMaps(MultiprojectiveVariety) -- see projectionMaps -- projections of a multi-projective variety
projections(MultiprojectiveVariety) -- see projections -- projections of a multi-projective variety
random(List,MultiprojectiveVariety) -- get a random hypersurface of given multi-degree containing a multi-projective variety
random(ZZ,MultiprojectiveVariety) -- see 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 MultiprojectiveVariety -- rational map defined by a linear system of hypersurfaces through a variety
rationalMap(List,MultiprojectiveVariety) -- the multi-rational map defined by a list of rational maps
rationalMap(MultiprojectiveVariety,Tally) -- rational map defined by an effective divisor
rationalMap(MultiprojectiveVariety,Tally,List) -- see rationalMap(MultiprojectiveVariety,Tally) -- rational map defined by an effective divisor
ring(MultiprojectiveVariety) -- the coordinate ring of the variety
segre(MultiprojectiveVariety) -- the Segre embedding of the variety
segreEmbedding(MultiprojectiveVariety) -- see segreEmbedding -- the Segre embedding of the variety
shape(MultiprojectiveVariety) -- shape of the ambient multi-projective space of a multi-projective variety
rationalMap(MultiprojectiveVariety) -- see shortcuts -- Some convenient shortcuts for multi-rational maps consisting of a single rational map
rationalMap(MultiprojectiveVariety,List) -- see shortcuts -- Some convenient shortcuts for multi-rational maps consisting of a single rational map
rationalMap(MultiprojectiveVariety,ZZ) -- see shortcuts -- Some convenient shortcuts for multi-rational maps consisting of a single rational map
rationalMap(MultiprojectiveVariety,ZZ,ZZ) -- see shortcuts -- Some convenient shortcuts for multi-rational maps consisting of a single rational map
singularLocus(MultiprojectiveVariety) -- the singular locus of the variety
support(MultiprojectiveVariety) -- support of a multi-projective variety

For the programmer

The object MultiprojectiveVariety is a type, with ancestor classes MutableHashTable < HashTable < Thing.