RationalMap -- the class of all rational maps between absolutely irreducible projective varieties over a field

Description

An object of the class RationalMap can be basically replaced by a homogeneous ring map of quotients of polynomial rings by homogeneous ideals. One main advantage to using this class is that things computed using non-probabilistic algorithms are stored internally (or partially stored).

The constructor for the class is rationalMap, which works quite similar to toMap. See in particular the methods: rationalMap(RingMap), rationalMap(Ideal,ZZ,ZZ), rationalMap(Tally), and rationalMap(PolynomialRing,List).

In the package MultiprojectiveVarieties (missing documentation) , this class has been extended to provide support to rational maps between multi-projective varieties, see MultirationalMap.

Functions and methods returning a rational map:

quadroQuadricCremonaTransformation -- quadro-quadric Cremona transformations
segre -- Segre embedding
specialCremonaTransformation -- special Cremona transformations whose base locus has dimension at most three
specialCubicTransformation -- special cubic transformations whose base locus has dimension at most three
specialQuadraticTransformation -- special quadratic transformations whose base locus has dimension three

Methods that use a rational map:

abstractRationalMap(RationalMap) -- see abstractRationalMap -- make an abstract rational map
approximateInverseMap(RationalMap) -- see approximateInverseMap -- random map related to the inverse of a birational map
approximateInverseMap(RationalMap,ZZ) -- see approximateInverseMap -- random map related to the inverse of a birational map
coefficientRing(RationalMap) -- coefficient ring of a rational map
coefficients(RationalMap) -- coefficient matrix of a rational map
degree(RationalMap) -- degree of a rational map
degreeMap(RationalMap) -- degree of a rational map
degrees(RationalMap) -- projective degrees of a rational map
multidegree(RationalMap) -- see degrees(RationalMap) -- projective degrees of a rational map
describe(RationalMap) -- describe a rational map
entries(RationalMap) -- the entries of the matrix associated to a rational map
exceptionalLocus(RationalMap) -- see exceptionalLocus -- exceptional locus of a birational map
flatten(RationalMap) -- write source and target as nondegenerate varieties
forceImage(RationalMap,Ideal) -- see forceImage -- declare which is the image of a rational map
forceInverseMap(RationalMap,RationalMap) -- see forceInverseMap -- declare that two rational maps are one the inverse of the other
graph(RationalMap) -- see graph -- closure of the graph of a rational map
ideal(RationalMap) -- base locus of a rational map
image(RationalMap,String) -- closure of the image of a rational map using the F4 algorithm (experimental)
image(RationalMap) -- see image(RationalMap,ZZ) -- closure of the image of a rational map
image(RationalMap,ZZ) -- closure of the image of a rational map
inverse(RationalMap) -- inverse of a birational map
inverse(RationalMap,Option) -- see inverse(RationalMap) -- inverse of a birational map
inverseMap(RationalMap) -- see inverseMap -- inverse of a birational map
isBirational(RationalMap) -- see isBirational -- whether a rational map is birational
isDominant(RationalMap) -- see isDominant -- whether a rational map is dominant
isInverseMap(RationalMap,RationalMap) -- checks whether two rational maps are one the inverse of the other
isIsomorphism(RationalMap) -- whether a birational map is an isomorphism
isMorphism(RationalMap) -- see isMorphism -- whether a rational map is a morphism
map(RationalMap) -- get the ring map defining a rational map
map(ZZ,RationalMap) -- see map(RationalMap) -- get the ring map defining a rational map
matrix(RationalMap) -- the matrix associated to a rational map
matrix(ZZ,RationalMap) -- see matrix(RationalMap) -- the matrix associated to a rational map
projectiveDegrees(RationalMap) -- projective degrees of a rational map
RationalMap ! -- calculates every possible thing
compose(RationalMap,RationalMap) -- see RationalMap * RationalMap -- composition of rational maps
RationalMap * RationalMap -- composition of rational maps
RationalMap ** Ring -- change the coefficient ring of a rational map
RationalMap == RationalMap -- equality of rational maps
RationalMap == ZZ -- see RationalMap == RationalMap -- equality of rational maps
ZZ == RationalMap -- see RationalMap == RationalMap -- equality of rational maps
RationalMap ^ ZZ -- power
RationalMap ^* -- see RationalMap ^** Ideal -- inverse image via a rational map
RationalMap ^** Ideal -- inverse image via a rational map
RationalMap _* -- direct image via a rational map
RationalMap Ideal -- see RationalMap _* -- direct image via a rational map
RationalMap | Ideal -- restriction of a rational map
RationalMap | Ring -- see RationalMap | Ideal -- restriction of a rational map
RationalMap | RingElement -- see RationalMap | Ideal -- restriction of a rational map
RationalMap || Ideal -- restriction of a rational map
RationalMap || Ring -- see RationalMap || Ideal -- restriction of a rational map
RationalMap || RingElement -- see RationalMap || Ideal -- restriction of a rational map
segre(RationalMap) -- see segre -- Segre embedding
SegreClass(RationalMap) -- see SegreClass -- Segre class of a closed subscheme of a projective variety
source(RationalMap) -- coordinate ring of the source for a rational map
substitute(RationalMap,PolynomialRing,PolynomialRing) -- substitute the ambient projective spaces of source and target
rationalMap(RationalMap) -- see super(RationalMap) -- get the rational map whose target is a projective space
super(RationalMap) -- get the rational map whose target is a projective space
target(RationalMap) -- coordinate ring of the target for a rational map
toExternalString(RationalMap) -- convert to a readable string

For the programmer

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