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MultiprojectiveVarieties : Table of Contents
MultiprojectiveVarieties
 Multiprojective varieties and multirational maps
MultiprojectiveVariety
 the class of all multiprojective varieties
projectiveVariety
 the closed multiprojective subvariety defined by a multihomogeneous ideal
projectiveVariety(List,Ring)
 product of projective spaces
projectiveVariety(List,List,Ring)
 the SegreVeronese variety
projectiveVariety(MultidimensionalMatrix)
 the multiprojective variety defined by a multidimensional matrix
MultirationalMap
 the class of all multirational maps
multirationalMap
 the multirational map defined by a list of rational maps
rationalMap(List,MultiprojectiveVariety)
 the multirational map defined by a list of rational maps
∏
 product of multiprojective varieties
⋂
 intersection of multiprojective varieties
⋃
 union of multiprojective varieties
ambient(MultiprojectiveVariety)
 the ambient multiprojective space of the variety
ambientVariety
 the ambient variety of a projective subvariety
baseLocus
 the base locus of a multirational map
check(MultirationalMap)
 check that a multirational map is welldefined
chowForm(EmbeddedProjectiveVariety)
 chow forms of a projective variety
clean(MultirationalMap)
 clean the internal information of a multirational map
codim(MultiprojectiveVariety)
 the codimension of the variety
coefficientRing(MultiprojectiveVariety)
 the coefficient ring of the variety
coefficientRing(MultirationalMap)
 the coefficient ring of a multirational 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 multirational map
degree(MultirationalMap,Option)
 degree of a multirational 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 multiprojective variety
describe(MultirationalMap)
 describe a multirational 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) multiprojective variety
factor(MultirationalMap)
 the list of rational maps defining a multirational map
Fano(ZZ,EmbeddedProjectiveVariety)
 Fano scheme of a projective variety
fiberProduct
 fiber product of multiprojective varieties
forceImage(MultirationalMap,MultiprojectiveVariety)
 declare which is the image of a multirational map
GG
 the Grassmannian of kdimensional linear subspaces of an ndimensional projective space
GG(ZZ,MultirationalMap)
 induced automorphism of the Grassmannian
graph(MultirationalMap)
 the graph of a multirational 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 homset of rational maps between two multiprojective varieties
ideal(MultiprojectiveVariety)
 the defining ideal of the variety
image(MultirationalMap)
 image of a multirational 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 homset of rational maps
isMorphism(MultirationalMap)
 whether a multirational map is a morphism
isSubset(MultiprojectiveVariety,MultiprojectiveVariety)
 whether one variety is a subvariety of another
isWellDefined(MultirationalMap)
 whether a multirational map is welldefined
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 multirational map
multidegree(ZZ,MultirationalMap)
 ith projective degree of a multirational map using a probabilistic approach
MultiprojectiveVariety % MultiprojectiveVariety
 subvariety of a projective variety
MultiprojectiveVariety * MultiprojectiveVariety
 intersection of two multiprojective varieties
MultiprojectiveVariety ** MultiprojectiveVariety
 product of two multiprojective varieties
MultiprojectiveVariety ** Ring
 change the coefficient ring of a multiprojective variety
MultiprojectiveVariety + MultiprojectiveVariety
 union of two multiprojective varieties
MultiprojectiveVariety == MultiprojectiveVariety
 equality of multiprojective varieties
MultiprojectiveVariety \ MultiprojectiveVariety
 difference of multiprojective varieties
MultiprojectiveVariety \\ MultiprojectiveVariety
 difference of multiprojective varieties
MultiprojectiveVariety ^ ZZ
 power of a multiprojective variety
MultirationalMap * MultirationalMap
 composition of multirational maps
MultirationalMap ** Ring
 change the coefficient ring of a multirational map
MultirationalMap << MultiprojectiveVariety
 force the change of the target in a multirational map
MultirationalMap <==> MultirationalMap
 equality of multirational maps with checks on internal data
MultirationalMap == MultirationalMap
 equality of multirational maps
MultirationalMap ^** MultiprojectiveVariety
 inverse image via a multirational map
MultirationalMap  MultiprojectiveVariety
 restriction of a multirational map
MultirationalMap  MultirationalMap
 product of multirational maps
MultirationalMap  MultiprojectiveVariety
 restriction of a multirational map
MultirationalMap  MultirationalMap
 product of multirational maps
MultirationalMap MultiprojectiveVariety
 direct image via a multirational map
multirationalMap(MultiprojectiveVariety)
 identity map
multirationalMap(MultiprojectiveVariety,MultiprojectiveVariety)
 get the natural inclusion
multirationalMap(MultirationalMap,MultiprojectiveVariety)
 change the target of a multirational map
parametrize(MultiprojectiveVariety)
 try to get a parametrization of a multiprojective variety
permute(MultiprojectiveVariety,List)
 permute the factors of the ambient multiprojective space
point(MultiprojectiveVariety)
 pick a random rational point on a multiprojective variety
projectionMaps
 projections of a multiprojective variety
projectionMaps(MultirationalMap)
 get the compositions of the multirational map with the projections of the target
projections
 projections of a multiprojective variety
projectiveVariety(...,MinimalGenerators=>...)
 whether to trim the ideal (intended for internal use only)
random(List,MultiprojectiveVariety)
 get a random hypersurface of given multidegree containing a multiprojective variety
random(MultiprojectiveVariety)
 apply a random automorphism of the ambient multiprojective space
RAT
 the homsets of rational maps between two multiprojective varieties
RAT List
 define a multirational 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 multisaturation 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 multirational map with the Segre embedding of the target
segreEmbedding
 the Segre embedding of the variety
shape(MultiprojectiveVariety)
 shape of the ambient multiprojective space of a multiprojective variety
shortcuts
 Some convenient shortcuts for multirational maps consisting of a single rational map
show(MultirationalMap)
 display a multirational map
singularLocus(MultiprojectiveVariety)
 the singular locus of the variety
source(MultirationalMap)
 the source for a multirational map
super(MultirationalMap)
 get the multirational map whose target is a product of projective spaces
support(MultiprojectiveVariety)
 support of a multiprojective 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 multirational map
topComponents(MultiprojectiveVariety)
 union of the top dimensional components of a multiprojective variety
toRationalMap
 convert a multirational map consisting of a single rational map to a standard rational map
trim(MultirationalMap)
 trim the target of a multirational map
variety(EmbeddedProjectiveVariety)
 convert an embedded projective variety into a builtin projective variety
WeightedProjectiveVariety
 the class of all weighted projective varieties
WeightedRationalMap
 the class of all weightedrational maps