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

Description

This is a package for handling multi-projective varieties, that is, closed subvarieties of products of projective spaces, and rational maps between them. This extends the package Cremona, which treats rational maps from multi-projective varieties to standard projective varieties, $X\subseteq \mathbb{P}^{k_1}\times\mathbb{P}^{k_2}\times\cdots\times\mathbb{P}^{k_n}\dashrightarrow Y\subseteq\mathbb{P}^N$.

References

ArXiv preprint: Computations with rational maps between multi-projective varieties.

Author

Giovanni Staglianò <giovannistagliano@gmail.com>

Certification

Version 2.3 of this package was accepted for publication in volume 11 of The Journal of Software for Algebra and Geometry on 31 August 2021, in the article Computations with rational maps between multi-projective varieties. That version can be obtained from the journal or from the Macaulay2 source code repository.

Version

This documentation describes version 2.6 of MultiprojectiveVarieties.

Source code

The source code from which this documentation is derived is in the file MultiprojectiveVarieties.m2.

Exports

Types
- EmbeddedProjectiveVariety -- the class of all embedded projective varieties
- GrassmannianVariety -- the class of all Grassmannians of linear subspaces of projective spaces
- MultiprojectiveVariety -- the class of all multi-projective varieties
- MultirationalMap -- the class of all multi-rational maps
Functions and commands
- ∏ -- product of multi-projective varieties
- ⋂ -- intersection of multi-projective varieties
- ⋃ -- union of multi-projective varieties
- ambientVariety -- the ambient variety of a projective subvariety
- baseLocus -- the base locus of a multi-rational map
- coneOfLines -- cone of lines on a subvariety passing through a point
- cycleClass -- determine the expression of the class of a cycle as a linear combination of Schubert classes
- degreeSequence -- the (multi)-degree sequence of a (multi)-rational map
- fiberProduct -- fiber product of multi-projective varieties
- GG -- the Grassmannian of k-dimensional linear subspaces of an n-dimensional projective space
- inverse2 -- inverse of a birational map using a faster algorithm for a special class of maps
- linearlyNormalEmbedding -- get the linearly normal embedding
- linearSpan -- the linear span of an embedded projective variety
- multirationalMap -- the multi-rational map defined by a list of rational maps
- projections -- projections of a multi-projective variety
- projectiveVariety -- the closed multi-projective subvariety defined by a multi-homogeneous ideal
- schubertCycle -- take a random Schubert cycle
- sectionalGenus -- the sectional genus of an embedded projective variety
- segreEmbedding -- the Segre embedding of the variety
- tangentSpace -- tangent space to a projective variety at a point
- toRationalMap -- convert a multi-rational map consisting of a single rational map to a standard rational map
Methods
- "∏(List)" -- see ∏ -- product of multi-projective varieties
- "⋂(List)" -- see ⋂ -- intersection of multi-projective varieties
- "⋃(List)" -- see ⋃ -- union of multi-projective varieties
- ambient(MultiprojectiveVariety) -- the ambient multi-projective space of the variety
- "ambientVariety(MultiprojectiveVariety)" -- see ambientVariety -- the ambient variety of a projective subvariety
- "baseLocus(MultirationalMap)" -- see baseLocus -- the base locus of a multi-rational map
- "baseLocus(RationalMap)" -- see 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
- "clean(RationalMap)" -- see 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(EmbeddedProjectiveVariety,EmbeddedProjectiveVariety)" -- see coneOfLines -- cone of lines on a subvariety passing through a point
- conormalVariety(EmbeddedProjectiveVariety) -- the conormal variety of a projective variety
- "cycleClass(EmbeddedProjectiveVariety)" -- see 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
- "degreeMap(MultirationalMap)" -- see 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(MultirationalMap)" -- see degreeSequence -- the (multi)-degree sequence of a (multi)-rational map
- "degreeSequence(RationalMap)" -- see degreeSequence -- the (multi)-degree sequence of a (multi)-rational map
- "? MultiprojectiveVariety" -- see describe(MultiprojectiveVariety) -- describe a multi-projective variety
- describe(MultiprojectiveVariety) -- describe a multi-projective variety
- "? MultirationalMap" -- see describe(MultirationalMap) -- describe a multi-rational map
- 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 ! -- print a more detailed description of an embedded projective variety
- EmbeddedProjectiveVariety ++ EmbeddedProjectiveVariety -- join of projective varieties
- "EmbeddedProjectiveVariety <=== EmbeddedProjectiveVariety" -- see EmbeddedProjectiveVariety ===> EmbeddedProjectiveVariety -- try to find an isomorphism between two projective varieties
- EmbeddedProjectiveVariety ===> EmbeddedProjectiveVariety -- try to find an isomorphism between two projective varieties
- euler(MultiprojectiveVariety) -- topological Euler characteristic of a (smooth) multi-projective variety
- factor(MultirationalMap) -- the list of rational maps defining a multi-rational map
- "Fano(EmbeddedProjectiveVariety)" -- see Fano(ZZ,EmbeddedProjectiveVariety) -- Fano scheme of a projective variety
- Fano(ZZ,EmbeddedProjectiveVariety) -- Fano scheme of a projective variety
- "fiberProduct(RationalMap,RationalMap)" -- see fiberProduct -- fiber product of multi-projective varieties
- forceImage(MultirationalMap,MultiprojectiveVariety) -- declare which is the image of a multi-rational map
- "forceImage(MultirationalMap,ZZ)" -- see forceImage(MultirationalMap,MultiprojectiveVariety) -- declare which is the image of a multi-rational map
- "GG(EmbeddedProjectiveVariety)" -- see GG -- the Grassmannian of k-dimensional linear subspaces of an n-dimensional projective space
- "GG(Ring)" -- see GG -- the Grassmannian of k-dimensional linear subspaces of an n-dimensional projective space
- "GG(Ring,ZZ,ZZ)" -- see GG -- the Grassmannian of k-dimensional linear subspaces of an n-dimensional projective space
- "GG(ZZ,EmbeddedProjectiveVariety)" -- see GG -- the Grassmannian of k-dimensional linear subspaces of an n-dimensional projective space
- "GG(ZZ,ZZ)" -- see GG -- the Grassmannian of k-dimensional linear subspaces of an n-dimensional projective space
- GG(ZZ,MultirationalMap) -- induced automorphism of the Grassmannian
- "GG(ZZ,RationalMap)" -- see GG(ZZ,MultirationalMap) -- induced automorphism of the Grassmannian
- graph(MultirationalMap) -- the graph of a multi-rational map
- hilbertPolynomial(EmbeddedProjectiveVariety) -- the Hilbert polynomial of the variety
- ideal(MultiprojectiveVariety) -- the defining ideal of the variety
- image(MultirationalMap) -- image of a multi-rational map
- inverse(MultirationalMap) -- inverse of a birational map
- "inverse2(MultirationalMap)" -- see 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
- 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(EmbeddedProjectiveVariety)" -- see linearlyNormalEmbedding -- get the linearly normal embedding
- "linearSpan(EmbeddedProjectiveVariety)" -- see linearSpan -- the linear span of an embedded projective variety
- "linearSpan(List)" -- see 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
- "projectiveDegrees(MultirationalMap)" -- see 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(List)" -- see multirationalMap -- the multi-rational map defined by a list of rational maps
- "multirationalMap(List,MultiprojectiveVariety)" -- see multirationalMap -- the multi-rational map defined by a list of rational maps
- "compose(MultirationalMap,MultirationalMap)" -- see MultirationalMap * MultirationalMap -- composition of multi-rational maps
- MultirationalMap * MultirationalMap -- composition of multi-rational maps
- "MultirationalMap * RationalMap" -- see MultirationalMap * MultirationalMap -- composition of multi-rational maps
- "MultirationalMap ^ ZZ" -- see MultirationalMap * MultirationalMap -- composition of multi-rational maps
- "RationalMap * MultirationalMap" -- see MultirationalMap * MultirationalMap -- composition of multi-rational maps
- MultirationalMap ** Ring -- change the coefficient ring of a multi-rational map
- "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 <==> MultirationalMap -- equality of multi-rational maps with checks on internal data
- MultirationalMap == MultirationalMap -- equality of multi-rational maps
- "MultirationalMap == RationalMap" -- see MultirationalMap == MultirationalMap -- equality of multi-rational maps
- "MultirationalMap == ZZ" -- see MultirationalMap == MultirationalMap -- equality of multi-rational maps
- "RationalMap == MultirationalMap" -- see MultirationalMap == MultirationalMap -- equality of multi-rational maps
- "ZZ == MultirationalMap" -- see MultirationalMap == MultirationalMap -- equality of multi-rational maps
- "MultirationalMap ^*" -- see MultirationalMap ^** MultiprojectiveVariety -- inverse image via a multi-rational map
- MultirationalMap ^** MultiprojectiveVariety -- inverse image via a multi-rational map
- "RationalMap ^** EmbeddedProjectiveVariety" -- see MultirationalMap ^** MultiprojectiveVariety -- inverse image via a multi-rational map
- "MultirationalMap | List" -- 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 | MultirationalMap -- product of multi-rational maps
- "MultirationalMap | RationalMap" -- see MultirationalMap | MultirationalMap -- product of multi-rational maps
- "RationalMap | MultirationalMap" -- see MultirationalMap | MultirationalMap -- product of multi-rational maps
- "RationalMap | RationalMap" -- see MultirationalMap | MultirationalMap -- product of multi-rational maps
- "MultirationalMap || List" -- 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 || MultirationalMap -- product of multi-rational maps
- "MultirationalMap || RationalMap" -- see MultirationalMap || MultirationalMap -- product of multi-rational maps
- "RationalMap || MultirationalMap" -- see MultirationalMap || MultirationalMap -- product of multi-rational maps
- "RationalMap || RationalMap" -- see MultirationalMap || MultirationalMap -- product of multi-rational maps
- 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
- "projections(MultiprojectiveVariety)" -- see projections -- projections of a multi-projective variety
- "projectiveVariety(Ideal)" -- see projectiveVariety -- the closed multi-projective subvariety defined by a multi-homogeneous ideal
- "projectiveVariety(Matrix)" -- see projectiveVariety -- the closed multi-projective subvariety defined by a multi-homogeneous ideal
- "projectiveVariety(Ring)" -- see projectiveVariety -- the closed multi-projective subvariety defined by a multi-homogeneous ideal
- "projectiveVariety(RingElement)" -- see projectiveVariety -- the closed multi-projective subvariety defined by a multi-homogeneous ideal
- projectiveVariety(List,List,Ring) -- the Segre-Veronese variety
- "projectiveVariety(ZZ,ZZ,Ring)" -- see projectiveVariety(List,List,Ring) -- the Segre-Veronese variety
- projectiveVariety(List,Ring) -- product of projective spaces
- "projectiveVariety(ZZ,Ring)" -- see projectiveVariety(List,Ring) -- product of projective spaces
- projectiveVariety(MultidimensionalMatrix) -- the multi-projective variety defined by a multi-dimensional matrix
- 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
- rationalMap(List,MultiprojectiveVariety) -- the multi-rational map defined by a list of rational maps
- "rationalMap(MultirationalMap)" -- see rationalMap(List,MultiprojectiveVariety) -- the multi-rational map defined by a list of rational maps
- "rationalMap(EmbeddedProjectiveVariety,Tally,ZZ)" -- see rationalMap(MultiprojectiveVariety,Tally) -- rational map defined by an effective divisor
- 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
- "schubertCycle(VisibleList,GrassmannianVariety)" -- see schubertCycle -- take a random Schubert cycle
- "sectionalGenus(EmbeddedProjectiveVariety)" -- see 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(MultiprojectiveVariety)" -- see segreEmbedding -- the Segre embedding of the variety
- shape(MultiprojectiveVariety) -- shape of the ambient multi-projective space of a multi-projective variety
- "multirationalMap(RationalMap)" -- see shortcuts -- Some convenient shortcuts for multi-rational maps consisting of a single rational map
- "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
- show(MultirationalMap) -- display a multi-rational map
- "show(RationalMap)" -- see 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(EmbeddedProjectiveVariety,EmbeddedProjectiveVariety)" -- see 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(MultirationalMap)" -- see 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
- "trim(RationalMap)" -- see trim(MultirationalMap) -- trim the target of a multi-rational map
- variety(EmbeddedProjectiveVariety) -- convert an embedded projective variety into a built-in projective variety
Symbols
- Saturate -- whether to compute the multi-saturation of the ideal (intended for internal use only)
Other things
- PP -- product of projective spaces

For the programmer

The object MultiprojectiveVarieties is a package.