Schubert2 : Table of Contents

• Schubert2 -- computation in intersection theory

  • Lines on hypersurfaces -- an example
  • Conics on a quintic threefold -- an example
  • The Horrocks-Mumford bundle -- an example
  • Riemann-Roch on a curve
  • Riemann-Roch on a surface
  • Riemann-Roch without denominators
  • A cubic fourfold containing a degree-5 del Pezzo surface -- an example
  • Examples from Schubert, translated

    • Example from Schubert: Hilbert polynomial and Todd class of projective 3-space
    • Example from Schubert: Generation of formulas
    • Example from Schubert: Grassmannian of lines in P3
    • Example from Schubert: Lines on a quintic threefold
    • 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: Riemann-Roch formulas
    • Example from Schubert: The number of elliptic cubics on a sextic 4-fold
  • Ideas for future development
  • OO _ RingElement -- the abstract structure sheaf of a divisor class
  • AbstractSheaf RingElement -- twist by a divisor class
• - AbstractSheaf -- negation of an abstract sheaf
• abstractProjectiveSpace -- make a projective space
• abstractProjectiveSpace' -- make a projective space
• AbstractSheaf -- the class of all abstract sheaves
• AbstractSheaf * AbstractSheaf -- (tensor) product of two abstract sheaves
• AbstractSheaf + AbstractSheaf -- (direct) sum of two abstract sheaves
• AbstractSheaf - AbstractSheaf -- difference of two abstract sheaves
• AbstractSheaf ^ ZZ -- power of an abstract sheaf
• AbstractSheaf ^** RingElement -- tensor power of an abstract sheaf
• abstractSheaf(AbstractVariety,RingElement) -- make an abstract sheaf
• AbstractVariety -- the class of all abstract varieties
• AbstractVariety / AbstractVariety -- get a structure map
• 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
• 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
• blowup -- blow up an abstract variety along a subvariety
• 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
• 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
• 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
• 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
• ctop(AbstractSheaf) -- the top Chern class of an abstract sheaf
• degeneracyLocus(ZZ,AbstractSheaf,AbstractSheaf)
• 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)
• exceptionalDivisor -- get the exceptional divisor of a blowup
• extensionAlgebra -- extend a graded algebra by a graded module
• exteriorPower(AbstractSheaf) -- alternating sum of exterior powers
• exteriorPower(ZZ,AbstractSheaf)
• FlagBundle -- the class of all flag bundles
• flagBundle -- make a flag bundle
• Hom(AbstractSheaf,AbstractSheaf) -- Hom between abstract sheaves
• id _ AbstractVariety -- the identity map of an abstract variety
• IncidenceCorrespondence -- the class of all incidence correspondences
• incidenceCorrespondence -- 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
• integral -- compute an integral (push forward a cycle to the base)
• intermediates
• IntersectionRing -- a key used for storing the intersection ring in an abstract variety
• intersectionRing(AbstractVariety) -- get the intersection ring of an abstract variety
• Isotropic -- make an isotropic flag bundle
• kernelBundle(ZZ,AbstractSheaf,AbstractSheaf)
• map(FlagBundle,AbstractVariety,AbstractSheaf) -- maps to projective bundles
• map(FlagBundle,AbstractVarietyMap,List) -- make a map from an abstract variety to a flag bundle
• map(FlagBundle,FlagBundle) -- forgetful maps of flag varieties
• OO _ AbstractVariety -- the structure sheaf of an abstract variety
• point -- the default base variety of dimension 0
• projectiveBundle -- make a projective bundle from an abstract sheaf
• projectiveBundle' -- make a projective bundle from an abstract sheaf
• PullBack -- a symbol used internally as a key for storing the pull back map in an abstract variety map
• QuotientBundles -- get the quotient bundles of the tautological filtration on a flag bundle
• rank(AbstractSheaf)
• Ring _ ChernClassVariable -- get a ring generator corresponding to a Chern class variable
• schubertCycle -- compute Schubert Cycles on a Grassmannian, Fulton-style
• schubertCycle' -- compute Schubert Cycles on a Grassmannian, Grothendieck-style
• SchubertRing -- a symbol used internally as a key
• schubertRing -- get the Schubert ring of a Grassmannian
• schur(List,AbstractSheaf) -- apply a Schur functor to an abstract sheaf
• SectionClass -- a symbol used internally as a key
• sectionClass(AbstractVarietyMap) -- get the class of the image of a section of a map of abstract varieties
• 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
• symmetricPower(RingElement,AbstractSheaf) -- symmetric power of an abstract sheaf
• TangentBundle -- a symbol used internally as a key
• tangentBundle -- 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
• todd -- compute the Todd class of an abstract sheaf, variety, map
• toSchubertBasis -- express cycles on a Grassmannian in terms of the Schubert basis
• transpose(Correspondence) -- 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
• variety(AbstractSheaf) -- the variety of an abstract sheaf
• VarietyDimension -- get the dimension of a variety from its intersection ring