Varieties : Table of Contents

• Varieties -- affine and projective algebraic geometry
  • Variety -- the class of all algebraic varieties
  • AffineVariety -- the class of all affine varieties
  • ProjectiveVariety -- the class of all projective varieties
  • CoherentSheaf -- the class of all coherent sheaves
  • SheafOfRings -- the class of sheaves of rings
  • SumOfTwists -- the class of all sums of twists
  • SheafMap -- the class of morphisms of coherent sheaves
    • map(CoherentSheaf,CoherentSheaf,Matrix) -- the constructor of morphisms of coherent sheaves
    • random(CoherentSheaf,CoherentSheaf) -- generate a random map of coherent sheaves
    • inducedMap(CoherentSheaf,CoherentSheaf) -- induced maps on coherent sheaves
• > ZZ
• >= ZZ
• AffineVariety ** AffineVariety -- Cartesian product of two affine varieties
• ambient(AffineVariety) -- the ambient affine or projective space of an embedded variety
• annihilator(CoherentSheaf) -- the annihilator ideal
• canonicalBundle -- the canonical bundle of a projective variety
• char(AffineVariety) -- the characteristic of the coordinate ring of a variety
• codim(CoherentSheaf) -- codimension of the support of a coherent sheaf on a projective variety
• codim(Variety) -- codimension of a variety
• coherent sheaves
• CoherentSheaf (*) -- sum of twists
• CoherentSheaf ** CoherentSheaf -- tensor produce of coherent sheaves
• CoherentSheaf ++ CoherentSheaf -- direct sum of coherent sheaves
• CoherentSheaf / CoherentSheaf -- quotient of coherent sheaves
• CoherentSheaf ^ ZZ -- direct sum
• CoherentSheaf ^** ZZ -- tensor power
• CoherentSheaf ZZ -- canonical twist of a coherent sheaf
• cotangentSheaf -- cotangent sheaf of a projective variety
• cotangentSheaf(ZZ,ProjectiveVariety) -- exterior powers of the cotangent sheaf of a projective variety
• cotangentSurjection
• degree(ProjectiveVariety)
• degrees(CoherentSheaf) -- degrees of a coherent sheaf
• dim(AffineVariety) -- dimension of the affine variety
• dim(ProjectiveVariety) -- dimension of the projective variety
• dual(CoherentSheaf) -- dual coherent sheaf
• embeddedToAbstract
• euler -- Euler characteristic
• euler(CoherentSheaf) -- Euler characteristic of coherent sheaf
• euler(Ideal)
• euler(Module)
• euler(ProjectiveVariety) -- topological Euler characteristic of a (smooth) projective variety
• euler(Ring)
• eulers -- list the sectional Euler characteristics
• eulers(CoherentSheaf)
• eulers(Ideal)
• eulers(Ring)
• eulerSequence (missing documentation)
• Ext^ZZ(CoherentSheaf,CoherentSheaf) -- global Ext
• Ext^ZZ(CoherentSheaf,SheafMap) -- maps on global Ext induced by a sheaf map
• Ext^ZZ(CoherentSheaf,SumOfTwists) -- global Ext
• exteriorPower(ZZ,CoherentSheaf)
• ExtLongExactSequence -- the long exact sequence of the Ext functor
• genera -- list of the successive linear sectional arithmetic genera
• genera(CoherentSheaf)
• genera(Ideal)
• genera(ProjectiveVariety)
• genera(Ring)
• genus -- arithmetic genus
• genus(CoherentSheaf)
• genus(ProjectiveVariety)
• genus(Ring)
• hh -- Hodge numbers of a smooth projective variety
• HH^ZZ SheafOfRings -- cohomology of a sheaf of rings on a projective variety
• HH^ZZ SumOfTwists -- coherent sheaf cohomology module
• HH^ZZ(AffineVariety,CoherentSheaf)
• HH^ZZ(ProjectiveVariety,CoherentSheaf) -- cohomology of a coherent sheaf on a projective variety
• HH^ZZ(ProjectiveVariety,SheafMap)
• hilbertPolynomial(CoherentSheaf) -- compute the Hilbert polynomial of the coherent sheaf
• hilbertPolynomial(ProjectiveVariety) -- compute the Hilbert polynomial of the projective variety
• Hom(CoherentSheaf,CoherentSheaf) -- global Hom
• ideal(Variety) -- returns the defining ideal
• idealSheaf -- ideal sheaf of a projective variety
• idealSheafSequence (missing documentation)
• isLocallyFree -- whether a coherent sheaf is locally free
• isProjective -- whether a variety is projective
• isSmooth(Variety) -- whether a variety is smooth
• isWellDefined(CoherentSheaf) -- whether a coherent sheaf is well-defined
• isWellDefined(SheafMap) -- whether a morphism of coherent sheaves is well-defined
• isWellDefined(Variety) -- whether a variety is well-defined
• LowerBound -- the class of lower bound objects
• matrix(SheafMap) -- the morphism of modules representing a morphisms of coherent sheaves
• minimalPresentation(CoherentSheaf) -- minimal presentation of a coherent sheaf or sheaf map
• module(CoherentSheaf) -- get the module defining a coherent sheaf
• module(SheafOfRings)
• numgens(CoherentSheaf) -- the number of generators of the underlying module
• OO -- the structure sheaf
• pdim(CoherentSheaf) -- calculate the projective dimension
• Proj -- make a projective variety
• quotient(SheafMap,SheafMap) -- factoring a morphism of coherent sheaves through another
• rank(CoherentSheaf) -- rank of a coherent sheaf
• RHom (missing documentation)
• ring(CoherentSheaf) -- the coordinate ring of the underlying variety
• ring(Variety) -- coordinate ring of the variety
• sheaf -- make a coherent sheaf
• sheaf(Matrix) -- the sheafification functor for morphisms
• sheaf(Module) -- make a coherent sheaf
• sheaf(Ring) -- make a coherent sheaf of rings
• sheaf(Variety) -- make a coherent sheaf
• sheaf(Variety,Module) -- make a coherent sheaf
• sheaf(Variety,Ring) -- make a coherent sheaf of rings
• sheafExt -- sheaf extension of coherent sheaves
• sheafHom -- sheaf Hom
• sheafHom(SheafMap,SheafMap) -- functor of sheaf Hom
• SheafMap ++ SheafMap -- takes the direct sum of two sheaf maps
• SheafMap ^ ZZ -- raises a SheafMap to a power
• SheafMap ^** ZZ -- repeatedly tensors a map
• SheafOfRings ^ List -- make a graded free coherent sheaf
• singularLocus(AffineVariety) -- the locus of singular points in a variety
• Spec -- make an affine variety
• tangentSheaf -- tangent sheaf of a projective variety
• varieties
• variety(CoherentSheaf) -- the underlying variety over which a coherent sheaf or morphism is defined.
• variety(Ideal) -- the closed projective subvariety defined by an ideal
• variety(Ring) -- the variety previously associated to a given ring
• yonedaSheafExtension (missing documentation)