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