Schubert2 : Index
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- AbstractSheaf -- negation of an abstract sheaf
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A cubic fourfold containing a degree-5 del Pezzo surface -- an example
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abstractProjectiveSpace -- make a projective space
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abstractProjectiveSpace' -- make a projective space
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abstractProjectiveSpace'(...,VariableName=>...) -- make a projective space
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abstractProjectiveSpace'(ZZ) -- make a projective space
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abstractProjectiveSpace'(ZZ,AbstractVariety) -- make a projective space
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abstractProjectiveSpace(...,VariableName=>...) -- make a projective space
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abstractProjectiveSpace(ZZ) -- make a projective space
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abstractProjectiveSpace(ZZ,AbstractVariety) -- make a projective space
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AbstractSheaf -- the class of all abstract sheaves
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abstractSheaf -- make an abstract sheaf
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AbstractSheaf * AbstractSheaf -- (tensor) product of two abstract sheaves
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AbstractSheaf * QQ -- (tensor) product of two abstract sheaves
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AbstractSheaf * RingElement -- (tensor) product of two abstract sheaves
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AbstractSheaf * ZZ -- (tensor) product of two abstract sheaves
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AbstractSheaf ** AbstractSheaf -- (tensor) product of two abstract sheaves
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AbstractSheaf + AbstractSheaf -- (direct) sum of two abstract sheaves
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AbstractSheaf + QQ -- (direct) sum of two abstract sheaves
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AbstractSheaf + RingElement -- (direct) sum of two abstract sheaves
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AbstractSheaf + ZZ -- (direct) sum of two abstract sheaves
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AbstractSheaf ++ AbstractSheaf -- (direct) sum of two abstract sheaves
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AbstractSheaf ++ QQ -- (direct) sum of two abstract sheaves
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AbstractSheaf ++ RingElement -- (direct) sum of two abstract sheaves
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AbstractSheaf ++ ZZ -- (direct) sum of two abstract sheaves
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AbstractSheaf - AbstractSheaf -- difference of two abstract sheaves
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AbstractSheaf - QQ -- difference of two abstract sheaves
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AbstractSheaf - RingElement -- difference of two abstract sheaves
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AbstractSheaf - ZZ -- difference of two abstract sheaves
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AbstractSheaf ^ ZZ -- power of an abstract sheaf
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AbstractSheaf ^** QQ -- tensor power of an abstract sheaf
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AbstractSheaf ^** RingElement -- tensor power of an abstract sheaf
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AbstractSheaf ^** ZZ -- tensor power of an abstract sheaf
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AbstractSheaf QQ -- twist by a divisor class
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AbstractSheaf RingElement -- twist by a divisor class
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AbstractSheaf ZZ -- twist by a divisor class
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abstractSheaf(...,ChernCharacter=>...) -- make an abstract sheaf
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abstractSheaf(...,ChernClass=>...) -- make an abstract sheaf
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abstractSheaf(...,Name=>...) -- make an abstract sheaf
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abstractSheaf(...,Rank=>...) -- make an abstract sheaf
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abstractSheaf(AbstractVariety) -- make an abstract sheaf
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abstractSheaf(AbstractVariety,QQ) -- make an abstract sheaf
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abstractSheaf(AbstractVariety,RingElement) -- make an abstract sheaf
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abstractSheaf(AbstractVariety,ZZ) -- make an abstract sheaf
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AbstractVariety -- the class of all abstract varieties
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abstractVariety -- make an abstract variety
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AbstractVariety / AbstractVariety -- get a structure map
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abstractVariety(...,DefaultIntegral=>...) -- make an abstract variety
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abstractVariety(...,ReturnType=>...) -- make an abstract variety
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abstractVariety(ZZ,Ring) -- make an abstract variety
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AbstractVarietyMap -- the class of all abstract variety maps
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abstractVarietyMap -- make an abstract variety morphism
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AbstractVarietyMap * AbstractVarietyMap -- composition of maps of abstract varieties
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AbstractVarietyMap ^* -- pull back operator associated to a map of varieties
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AbstractVarietyMap _* -- push forward operator associated to a map of varieties
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abstractVarietyMap(...,DefaultPullBack=>...) -- make an abstract variety morphism
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abstractVarietyMap(...,DefaultPushForward=>...) -- make an abstract variety morphism
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abstractVarietyMap(...,SectionClass=>...) -- make an abstract variety morphism
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abstractVarietyMap(...,TangentBundle=>...) -- make an abstract variety morphism
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abstractVarietyMap(AbstractVariety,AbstractVariety,MethodFunction,MethodFunction) -- make an abstract variety morphism
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abstractVarietyMap(AbstractVariety,AbstractVariety,RingMap,MethodFunction) -- make an abstract variety morphism
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adams -- compute Adams operations
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adams(ZZ,AbstractSheaf) -- compute Adams operations
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adams(ZZ,RingElement) -- compute Adams operations
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Base -- get the base variety of a flag bundle
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base -- make an abstract variety from nothing, equipped with some parameters and some bundles
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base(Sequence) -- make an abstract variety from nothing, equipped with some parameters and some bundles
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base(Thing) -- make an abstract variety from nothing, equipped with some parameters and some bundles
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blowup -- blow up an abstract variety along a subvariety
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blowup(AbstractVarietyMap) -- blow up an abstract variety along a subvariety
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Bundle -- make an abstract variety from nothing, equipped with some parameters and some bundles
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BundleRanks -- get the ranks of the tautological sheaves on a flag bundle
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Bundles -- a symbol used internally as a key
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bundles -- get the list of tautological line bundles on a flag bundle
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bundles(FlagBundle) -- get the list of tautological line bundles on a flag bundle
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ch -- Chern character of an abstract sheaf
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ch(AbstractSheaf) -- total Chern character of an abstract sheaf
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ch(ZZ,AbstractSheaf) -- the i-th Chern character of an abstract sheaf
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chern -- compute Chern classes of a sheaf
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chern(AbstractSheaf) -- compute the total Chern class of a sheaf
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chern(ZZ,AbstractSheaf) -- compute a Chern class of a sheaf
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chern(ZZ,ChernClassVariableTable) -- get value of a Chern class variable
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chern(ZZ,Symbol) -- make a Chern class symbol
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chern(ZZ,ZZ,AbstractSheaf) -- compute several Chern classes of an abstract sheaf
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ChernCharacter -- make an abstract sheaf
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ChernClass -- make an abstract sheaf
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ChernClassVariable -- the class of all Chern class variables
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ChernClassVariable .. ChernClassVariable -- generate a sequence of Chern class variables
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ChernClassVariable <- Thing -- assignment to a Chern class variable
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ChernClassVariableTable -- the class of all Chern class variable tables
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chi(AbstractSheaf) -- compute the Euler characteristic of an abstract sheaf
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Conics on a quintic threefold -- an example
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Correspondence -- the class of all correspondences
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Correspondence * Correspondence -- composition of correspondences
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Correspondence ^* -- pullback along a correspondence
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Correspondence _* -- pushforward along a correspondence
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cotangentBundle -- get the cotangent bundle
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cotangentBundle(AbstractVariety) -- get the cotangent bundle
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cotangentBundle(AbstractVarietyMap) -- get the cotangent bundle
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ctop -- the top Chern class of an abstract sheaf
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ctop(AbstractSheaf) -- the top Chern class of an abstract sheaf
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DefaultIntegral -- make an abstract variety
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DefaultPullBack -- make an abstract variety morphism
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DefaultPushForward -- make an abstract variety morphism
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degeneracyLocus
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degeneracyLocus(ZZ,AbstractSheaf,AbstractSheaf)
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degeneracyLocus2
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degeneracyLocus2(ZZ,AbstractSheaf,AbstractSheaf)
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determinant(AbstractSheaf) -- determinant of an abstract sheaf
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dim(AbstractVariety) -- the dimension of an abstract variety
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dim(AbstractVarietyMap) -- the dimension of an abstract variety
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dual(AbstractSheaf) -- the dual of an abstract sheaf
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euler(AbstractVariety)
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Example from Schubert: Conics on a quintic threefold
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Example from Schubert: Count the number of space conics intersecting 8 given lines
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Example from Schubert: Euler characteristic of Horrocks-Mumford bundle
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Example from Schubert: Generation of formulas
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Example from Schubert: Grassmannian of lines in P3
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Example from Schubert: Hilbert polynomial and Todd class of projective 3-space
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Example from Schubert: Lines on a quintic threefold
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Example from Schubert: Riemann-Roch formulas
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Example from Schubert: The number of elliptic cubics on a sextic 4-fold
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Examples from Schubert, translated
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exceptionalDivisor -- get the exceptional divisor of a blowup
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exceptionalDivisor(AbstractVariety) -- get the exceptional divisor of a blowup
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extensionAlgebra -- extend a graded algebra by a graded module
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extensionAlgebra(...,Codimension=>...) -- extend a graded algebra by a graded module
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extensionAlgebra(...,CoefficientRing=>...) -- extend a graded algebra by a graded module
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extensionAlgebra(RingMap,RingElement) -- extend a graded algebra by a graded module
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exteriorPower(AbstractSheaf) -- alternating sum of exterior powers
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exteriorPower(ZZ,AbstractSheaf)
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FlagBundle -- the class of all flag bundles
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flagBundle -- make a flag bundle
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FlagBundle _ List -- compute Schubert Cycles on a Grassmannian, Grothendieck-style
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FlagBundle _ Sequence -- compute Schubert Cycles on a Grassmannian, Grothendieck-style
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flagBundle(...,Isotropic=>...) -- make a flag bundle
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flagBundle(...,QuotientBundles=>...) -- make a flag bundle
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flagBundle(...,VariableNames=>...) -- make a flag bundle
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flagBundle(List) -- make a flag bundle
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flagBundle(List,AbstractSheaf) -- make a flag bundle
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flagBundle(List,AbstractVariety) -- make a flag bundle
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flagBundle(List,ZZ) -- make a flag bundle
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Hom(AbstractSheaf,AbstractSheaf) -- Hom between abstract sheaves
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id _ AbstractVariety -- the identity map of an abstract variety
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Ideas for future development
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IncidenceCorrespondence -- the class of all incidence correspondences
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incidenceCorrespondence -- build containment correspondence between two Grassmannians
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incidenceCorrespondence(FlagBundle,FlagBundle) -- build containment correspondence between two Grassmannians
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incidenceCorrespondence(List) -- build containment correspondence between two Grassmannians
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incidenceCorrespondence(List,AbstractSheaf) -- build containment correspondence between two Grassmannians
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inclusion -- build the freest possible inclusion map
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inclusion(...,Base=>...) -- build the freest possible inclusion map
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inclusion(...,Codimension=>...) -- build the freest possible inclusion map
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inclusion(...,NormalClass=>...) -- build the freest possible inclusion map
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inclusion(...,SubDimension=>...) -- build the freest possible inclusion map
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inclusion(...,SubTangent=>...) -- build the freest possible inclusion map
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inclusion(...,SuperDimension=>...) -- build the freest possible inclusion map
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inclusion(...,SuperTangent=>...) -- build the freest possible inclusion map
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inclusion(RingMap) -- build the freest possible inclusion map
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integral -- compute an integral (push forward a cycle to the base)
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intermediates
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intermediates(IncidenceCorrespondence)
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IntersectionRing -- a key used for storing the intersection ring in an abstract variety
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intersectionRing -- get the intersection ring of an abstract variety
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intersectionRing(AbstractVariety) -- get the intersection ring of an abstract variety
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Isotropic -- make an isotropic flag bundle
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kernelBundle
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kernelBundle(ZZ,AbstractSheaf,AbstractSheaf)
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Lines on hypersurfaces -- an example
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map(FlagBundle,AbstractVariety,AbstractSheaf) -- maps to projective bundles
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map(FlagBundle,AbstractVariety,List) -- make a map from an abstract variety to a flag bundle
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map(FlagBundle,AbstractVarietyMap,List) -- make a map from an abstract variety to a flag bundle
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map(FlagBundle,FlagBundle) -- forgetful maps of flag varieties
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NormalClass -- build the freest possible inclusion map
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OO _ AbstractVariety -- the structure sheaf of an abstract variety
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OO _ RingElement -- the abstract structure sheaf of a divisor class
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OO RingElement -- twist by a divisor class
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point -- the default base variety of dimension 0
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projectiveBundle -- make a projective bundle from an abstract sheaf
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projectiveBundle' -- make a projective bundle from an abstract sheaf
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projectiveBundle'(...,VariableNames=>...) -- make a projective bundle from an abstract sheaf
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projectiveBundle'(AbstractSheaf) -- make a projective bundle from an abstract sheaf
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projectiveBundle'(ZZ) -- make a projective bundle from an abstract sheaf
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projectiveBundle'(ZZ,AbstractVariety) -- make a projective bundle from an abstract sheaf
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projectiveBundle(...,VariableNames=>...) -- make a projective bundle from an abstract sheaf
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projectiveBundle(AbstractSheaf) -- make a projective bundle from an abstract sheaf
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projectiveBundle(ZZ) -- make a projective bundle from an abstract sheaf
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projectiveBundle(ZZ,AbstractVariety) -- make a projective bundle from an abstract sheaf
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PullBack -- a symbol used internally as a key for storing the pull back map in an abstract variety map
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QQ * AbstractSheaf -- (tensor) product of two abstract sheaves
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QQ + AbstractSheaf -- (direct) sum of two abstract sheaves
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QQ ++ AbstractSheaf -- (direct) sum of two abstract sheaves
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QQ - AbstractSheaf -- difference of two abstract sheaves
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QuotientBundles -- get the quotient bundles of the tautological filtration on a flag bundle
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Rank -- make an abstract sheaf
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rank(AbstractSheaf)
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ReturnType -- make an abstract variety
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Riemann-Roch on a curve
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Riemann-Roch on a surface
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Riemann-Roch without denominators
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Ring _ ChernClassVariable -- get a ring generator corresponding to a Chern class variable
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RingElement * AbstractSheaf -- (tensor) product of two abstract sheaves
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RingElement + AbstractSheaf -- (direct) sum of two abstract sheaves
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RingElement ++ AbstractSheaf -- (direct) sum of two abstract sheaves
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RingElement - AbstractSheaf -- difference of two abstract sheaves
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Schubert2 -- computation in intersection theory
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schubertCycle -- compute Schubert Cycles on a Grassmannian, Fulton-style
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schubertCycle' -- compute Schubert Cycles on a Grassmannian, Grothendieck-style
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schubertCycle'(List,FlagBundle) -- compute Schubert Cycles on a Grassmannian, Grothendieck-style
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schubertCycle'(Sequence,FlagBundle) -- compute Schubert Cycles on a Grassmannian, Grothendieck-style
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schubertCycle(List,FlagBundle) -- compute Schubert Cycles on a Grassmannian, Fulton-style
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schubertCycle(Sequence,FlagBundle) -- compute Schubert Cycles on a Grassmannian, Fulton-style
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SchubertRing -- a symbol used internally as a key
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schubertRing -- get the Schubert ring of a Grassmannian
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schubertRing(FlagBundle) -- get the Schubert ring of a Grassmannian
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schur -- apply a Schur functor to an abstract sheaf
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schur(List,AbstractSheaf) -- apply a Schur functor to an abstract sheaf
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SectionClass -- a symbol used internally as a key
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sectionClass -- get the class of the image of a section of a map of abstract varieties
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sectionClass(AbstractVarietyMap) -- get the class of the image of a section of a map of abstract varieties
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sectionZeroLocus
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sectionZeroLocus(AbstractSheaf)
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segre -- compute Segre classes of a sheaf
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segre(AbstractSheaf) -- compute the total Segre class of a sheaf
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segre(ZZ,AbstractSheaf) -- compute a Segre class of a sheaf
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source(AbstractVarietyMap) -- the source of a map of abstract varieties
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source(Correspondence) -- the source of a correspondence
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StructureMap -- get the structure map of an abstract variety
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SubBundles -- get the subbundles of the tautological filtration on a flag bundle
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SubDimension -- build the freest possible inclusion map
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SubTangent -- build the freest possible inclusion map
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SuperDimension -- build the freest possible inclusion map
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SuperTangent -- build the freest possible inclusion map
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symmetricPower(RingElement,AbstractSheaf) -- symmetric power of an abstract sheaf
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TangentBundle -- a symbol used internally as a key
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tangentBundle -- get the tangent bundle
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tangentBundle(AbstractVariety) -- get the tangent bundle
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tangentBundle(AbstractVarietyMap) -- get the tangent bundle
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target(AbstractVarietyMap) -- the target of a map of abstract varieties
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target(Correspondence) -- the target of a correspondence
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TautologicalLineBundle -- a symbol used internally as a key
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tautologicalLineBundle -- get the tautological line bundle of a flag bundle or abstract variety
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tautologicalLineBundle(AbstractVariety) -- get the tautological line bundle of a flag bundle or abstract variety
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tautologicalLineBundle(FlagBundle) -- get the tautological line bundle of a flag bundle or abstract variety
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The Horrocks-Mumford bundle -- an example
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todd -- compute the Todd class of an abstract sheaf, variety, map
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todd(AbstractSheaf) -- compute the Todd class of an abstract sheaf, variety, map
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todd(AbstractVariety) -- compute the Todd class of an abstract sheaf, variety, map
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todd(AbstractVarietyMap) -- compute the Todd class of an abstract sheaf, variety, map
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toSchubertBasis -- express cycles on a Grassmannian in terms of the Schubert basis
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toSchubertBasis(RingElement) -- express cycles on a Grassmannian in terms of the Schubert basis
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transpose(Correspondence) -- reverse the direction of a correspondence
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transpose(IncidenceCorrespondence) -- reverse the direction of a correspondence
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use(AbstractVariety) -- assign values to variables associated with a variety
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value(ChernClassVariable) -- the value of a Chern class symbol
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VariableName -- make a projective space
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VariableNames -- make a flag bundle
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variety(AbstractSheaf) -- the variety of an abstract sheaf
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VarietyDimension -- get the dimension of a variety from its intersection ring
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ZZ * AbstractSheaf -- (tensor) product of two abstract sheaves
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ZZ + AbstractSheaf -- (direct) sum of two abstract sheaves
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ZZ ++ AbstractSheaf -- (direct) sum of two abstract sheaves
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ZZ - AbstractSheaf -- difference of two abstract sheaves