Varieties -- affine and projective algebraic geometry

Description

This package provides routines for working with affine and projective varieties and coherent sheaves on them.

Acknowledgement

Large portions of this package were originally in m2/varieties.m2. Work on the type SheafMap and related algorithms began at a Macaulay2 workshop at the American Institute of Mathematics in September 2023.

Contributors

Claudiu Raicu contributed to the development of this package.

Authors

Version

This documentation describes version 0.3 of Varieties.

Citation

If you have used this package in your research, please cite it as follows:

@misc{VarietiesSource,
  title = {{Varieties: A \emph{Macaulay2} package. Version~0.3}},
  author = {Devlin Mallory and Ritvik Ramkumar and Mahrud Sayrafi and Gregory G. Smith and Keller VandeBogert and John Cobb},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages}}
}

Exports

Types
- AffineVariety -- the class of all affine varieties
- CoherentSheaf -- the class of all coherent sheaves
- LowerBound -- the class of lower bound objects
- ProjectiveVariety -- the class of all projective varieties
- SheafMap -- the class of morphisms of coherent sheaves
- SheafOfRings -- the class of sheaves of rings
- SumOfTwists -- the class of all sums of twists
- Variety -- the class of all algebraic varieties
Functions and commands
- canonicalBundle -- the canonical bundle of a projective variety
- cotangentSheaf -- cotangent sheaf of a projective variety
- idealSheaf -- ideal sheaf of a projective variety
- isLocallyFree -- whether a coherent sheaf is locally free
- isProjective -- whether a variety is projective
- Proj -- make a projective variety
- sheaf -- make a coherent sheaf
- sheafHom -- sheaf Hom
- Spec -- make an affine variety
- tangentSheaf -- tangent sheaf of a projective variety
- variety -- see variety(CoherentSheaf) -- the underlying variety over which a coherent sheaf or morphism is defined.
Methods
- > InfiniteNumber -- see > ZZ
- > ZZ
- >= InfiniteNumber -- see >= ZZ
- >= List -- see >= ZZ
- >= ZZ
- AffineVariety ** AffineVariety -- Cartesian product of two affine varieties
- AffineVariety ** Ring -- see AffineVariety ** AffineVariety -- Cartesian product of two affine varieties
- ambient(AffineVariety) -- the ambient affine or projective space of an embedded variety
- ambient(ProjectiveVariety) -- see ambient(AffineVariety) -- the ambient affine or projective space of an embedded variety
- ambient(CoherentSheaf) (missing documentation)
- ambient(SheafMap) (missing documentation)
- annihilator(CoherentSheaf) -- the annihilator ideal
- betti(CoherentSheaf) (missing documentation)
- canonicalBundle(ProjectiveVariety) -- see canonicalBundle -- the canonical bundle of a projective variety
- char(AffineVariety) -- the characteristic of the coordinate ring of a variety
- char(ProjectiveVariety) -- see 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
- CoherentSheaf (*) -- sum of twists
- SheafOfRings (*) -- see 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 / Ideal -- see CoherentSheaf / CoherentSheaf -- quotient of coherent sheaves
- CoherentSheaf == Module (missing documentation)
- CoherentSheaf ^ List -- see CoherentSheaf ^ ZZ -- direct sum
- CoherentSheaf ^ ZZ -- direct sum
- CoherentSheaf ^** ZZ -- tensor power
- CoherentSheaf Sequence (missing documentation)
- CoherentSheaf ZZ -- canonical twist of a coherent sheaf
- SheafOfRings ZZ -- see CoherentSheaf ZZ -- canonical twist of a coherent sheaf
- coimage(SheafMap) (missing documentation)
- cokernel(SheafMap) (missing documentation)
- components(CoherentSheaf) (missing documentation)
- components(SheafMap) (missing documentation)
- cotangentSheaf(ProjectiveVariety) -- see cotangentSheaf -- cotangent sheaf of a projective variety
- cotangentSheaf(ZZ,ProjectiveVariety) -- exterior powers of the cotangent sheaf of a projective variety
- cover(CoherentSheaf) (missing documentation)
- cover(SheafMap) (missing documentation)
- degree(ProjectiveVariety)
- degree(SheafMap) (missing documentation)
- degrees(CoherentSheaf) -- degrees of a coherent sheaf
- determinant(CoherentSheaf) (missing documentation)
- dim(AffineVariety) -- dimension of the affine variety
- dim(ProjectiveVariety) -- dimension of the projective variety
- directSum(CoherentSheaf) (missing documentation)
- directSum(SheafMap) (missing documentation)
- dual(CoherentSheaf) -- dual coherent sheaf
- dual(SheafMap) (missing documentation)
- euler(CoherentSheaf) -- Euler characteristic of coherent sheaf
- euler(Ideal)
- euler(Module)
- euler(ProjectiveVariety) -- topological Euler characteristic of a (smooth) projective variety
- euler(Ring)
- eulers(CoherentSheaf)
- eulers(Module) -- see eulers(CoherentSheaf)
- eulers(Ideal)
- eulers(Ring)
- expression(CoherentSheaf) (missing documentation)
- expression(SheafMap) (missing documentation)
- expression(SheafOfRings) (missing documentation)
- expression(SumOfTwists) (missing documentation)
- expression(Variety) (missing documentation)
- Ext^ZZ(CoherentSheaf,CoherentSheaf) -- global Ext
- Ext^ZZ(CoherentSheaf,SheafOfRings) -- see Ext^ZZ(CoherentSheaf,CoherentSheaf) -- global Ext
- Ext^ZZ(SheafOfRings,CoherentSheaf) -- see Ext^ZZ(CoherentSheaf,CoherentSheaf) -- global Ext
- Ext^ZZ(SheafOfRings,SheafOfRings) -- see 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
- Ext^ZZ(SheafOfRings,SumOfTwists) -- see Ext^ZZ(CoherentSheaf,SumOfTwists) -- global Ext
- exteriorPower(ZZ,CoherentSheaf)
- exteriorPower(ZZ,SheafMap) (missing documentation)
- genera(CoherentSheaf)
- genera(Module) -- see genera(CoherentSheaf)
- genera(Ideal)
- genera(ProjectiveVariety)
- genera(Ring)
- genus(CoherentSheaf)
- genus(Ideal) -- see genus(CoherentSheaf)
- genus(Module) -- see genus(CoherentSheaf)
- genus(ProjectiveVariety)
- genus(Ring)
- hh(Sequence,ProjectiveVariety) -- see 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(ProjectiveVariety,SumOfTwists) -- see HH^ZZ SumOfTwists -- coherent sheaf cohomology module
- HH^ZZ(AffineVariety,CoherentSheaf)
- HH^ZZ CoherentSheaf -- see HH^ZZ(ProjectiveVariety,CoherentSheaf) -- cohomology of a coherent sheaf on a projective variety
- HH^ZZ(ProjectiveVariety,CoherentSheaf) -- cohomology of a coherent sheaf on a projective variety
- HH^ZZ SheafMap -- see HH^ZZ(ProjectiveVariety,SheafMap)
- HH^ZZ(ProjectiveVariety,SheafMap)
- HH^ZZ(Variety,SheafOfRings) (missing documentation)
- 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
- Hom(CoherentSheaf,SheafOfRings) -- see Hom(CoherentSheaf,CoherentSheaf) -- global Hom
- Hom(SheafOfRings,CoherentSheaf) -- see Hom(CoherentSheaf,CoherentSheaf) -- global Hom
- Hom(SheafOfRings,SheafOfRings) -- see Hom(CoherentSheaf,CoherentSheaf) -- global Hom
- homology(SheafMap,SheafMap) (missing documentation)
- homomorphism'(SheafMap) (missing documentation)
- Ideal * CoherentSheaf (missing documentation)
- ideal(Variety) -- returns the defining ideal
- idealSheaf(ProjectiveVariety) -- see idealSheaf -- ideal sheaf of a projective variety
- image(SheafMap) (missing documentation)
- inducedMap(CoherentSheaf,CoherentSheaf) -- induced maps on coherent sheaves
- inducedMap(CoherentSheaf,CoherentSheaf,SheafMap) -- see inducedMap(CoherentSheaf,CoherentSheaf) -- induced maps on coherent sheaves
- inverse(SheafMap) (missing documentation)
- isCommutative(SheafOfRings) (missing documentation)
- isIsomorphic(CoherentSheaf,CoherentSheaf) (missing documentation)
- isIsomorphic(SheafMap,SheafMap) (missing documentation)
- isIsomorphism(SheafMap) (missing documentation)
- isLocallyFree(CoherentSheaf) -- see isLocallyFree -- whether a coherent sheaf is locally free
- isLocallyFree(SheafOfRings) -- see isLocallyFree -- whether a coherent sheaf is locally free
- isLocallyFree(SumOfTwists) -- see isLocallyFree -- whether a coherent sheaf is locally free
- isomorphism(CoherentSheaf,CoherentSheaf) (missing documentation)
- isProjective(ProjectiveVariety) -- see isProjective -- whether a variety is projective
- isProjective(Variety) -- see isProjective -- whether a variety is projective
- isSmooth(Variety) -- whether a variety is smooth
- isSurjective(SheafMap) (missing documentation)
- isWellDefined(CoherentSheaf) -- whether a coherent sheaf is well-defined
- isWellDefined(SheafMap) -- whether a morphism of coherent sheaves is well-defined
- isWellDefined(SheafOfRings) (missing documentation)
- isWellDefined(Variety) -- whether a variety is well-defined
- kernel(SheafMap) (missing documentation)
- lift(SheafMap) (missing documentation)
- CoherentSheaf LowerBound -- see LowerBound -- the class of lower bound objects
- SheafOfRings LowerBound -- see LowerBound -- the class of lower bound objects
- map(CoherentSheaf,CoherentSheaf,Matrix) -- the constructor of morphisms of coherent sheaves
- map(CoherentSheaf,CoherentSheaf,Matrix,InfiniteNumber) -- see map(CoherentSheaf,CoherentSheaf,Matrix) -- the constructor of morphisms of coherent sheaves
- map(CoherentSheaf,CoherentSheaf,Matrix,ZZ) -- see map(CoherentSheaf,CoherentSheaf,Matrix) -- the constructor of morphisms of coherent sheaves
- map(CoherentSheaf,Nothing,Matrix) -- see map(CoherentSheaf,CoherentSheaf,Matrix) -- the constructor of morphisms of coherent sheaves
- map(Nothing,CoherentSheaf,Matrix) -- see map(CoherentSheaf,CoherentSheaf,Matrix) -- the constructor of morphisms of coherent sheaves
- map(CoherentSheaf,CoherentSheaf,SheafMap) (missing documentation)
- map(CoherentSheaf,CoherentSheaf,ZZ) (missing documentation)
- map(CoherentSheaf,Module,ZZ) (missing documentation)
- map(Module,CoherentSheaf,ZZ) (missing documentation)
- mathML(CoherentSheaf) (missing documentation)
- mathML(SheafOfRings) (missing documentation)
- mathML(Variety) (missing documentation)
- matrix(SheafMap) -- the morphism of modules representing a morphisms of coherent sheaves
- minimalPresentation(CoherentSheaf) -- minimal presentation of a coherent sheaf or sheaf map
- minimalPresentation(SheafMap) -- see minimalPresentation(CoherentSheaf) -- minimal presentation of a coherent sheaf or sheaf map
- prune(CoherentSheaf) -- see minimalPresentation(CoherentSheaf) -- minimal presentation of a coherent sheaf or sheaf map
- prune(SheafMap) -- see minimalPresentation(CoherentSheaf) -- minimal presentation of a coherent sheaf or sheaf map
- minimalPresentation(SheafOfRings) (missing documentation)
- Module == CoherentSheaf (missing documentation)
- module(CoherentSheaf) -- get the module defining a coherent sheaf
- module(SheafOfRings)
- net(CoherentSheaf) (missing documentation)
- net(SheafExpression) (missing documentation)
- net(SheafMap) (missing documentation)
- net(SheafOfRings) (missing documentation)
- net(SumOfTwists) (missing documentation)
- net(Variety) (missing documentation)
- numgens(CoherentSheaf) -- the number of generators of the underlying module
- pdim(CoherentSheaf) -- calculate the projective dimension
- Proj(Ring) -- see Proj -- make a projective variety
- promote(SheafMap,type of Nothing) (missing documentation)
- promote(SheafMap,type of RingElement) (missing documentation)
- prune(SheafOfRings) (missing documentation)
- pullback(SheafMap,SheafMap) (missing documentation)
- pushout(SheafMap,SheafMap) (missing documentation)
- quotient'(SheafMap,SheafMap) -- see quotient(SheafMap,SheafMap) -- factoring a morphism of coherent sheaves through another
- quotient(SheafMap,SheafMap) -- factoring a morphism of coherent sheaves through another
- SheafMap // SheafMap -- see quotient(SheafMap,SheafMap) -- factoring a morphism of coherent sheaves through another
- SheafMap \\ SheafMap -- see quotient(SheafMap,SheafMap) -- factoring a morphism of coherent sheaves through another
- random(CoherentSheaf,CoherentSheaf) -- generate a random map of coherent sheaves
- rank(CoherentSheaf) -- rank of a coherent sheaf
- ring(CoherentSheaf) -- the coordinate ring of the underlying variety
- ring(SheafMap) -- see ring(CoherentSheaf) -- the coordinate ring of the underlying variety
- ring(SheafOfRings) -- see ring(CoherentSheaf) -- the coordinate ring of the underlying variety
- ring(SumOfTwists) -- see ring(CoherentSheaf) -- the coordinate ring of the underlying variety
- ring(Variety) -- coordinate ring of the variety
- sheaf(Matrix) -- the sheafification functor for morphisms
- sheaf(Matrix,ZZ) -- see sheaf(Matrix) -- the sheafification functor for morphisms
- sheaf(Variety,Matrix) -- see sheaf(Matrix) -- the sheafification functor for morphisms
- sheaf(Variety,Matrix,ZZ) -- see sheaf(Matrix) -- the sheafification functor for morphisms
- Ideal ^~ -- see sheaf(Module) -- make a coherent sheaf
- Module ^~ -- see sheaf(Module) -- make a coherent sheaf
- sheaf(Ideal) -- see sheaf(Module) -- make a coherent sheaf
- sheaf(Module) -- make a coherent sheaf
- Ring ^~ -- see sheaf(Ring) -- make a coherent sheaf of rings
- sheaf(Ring) -- make a coherent sheaf of rings
- sheaf(SheafMap) (missing documentation)
- sheaf(Variety) -- make a coherent sheaf
- sheaf(Variety,Ideal) -- see sheaf(Variety,Module) -- make a coherent sheaf
- sheaf(Variety,Module) -- make a coherent sheaf
- sheaf(Variety,Ring) -- make a coherent sheaf of rings
- sheafExt^ZZ(CoherentSheaf,CoherentSheaf) -- see sheafExt -- sheaf extension of coherent sheaves
- sheafExt^ZZ(CoherentSheaf,SheafOfRings) -- see sheafExt -- sheaf extension of coherent sheaves
- sheafExt^ZZ(SheafOfRings,CoherentSheaf) -- see sheafExt -- sheaf extension of coherent sheaves
- sheafExt^ZZ(SheafOfRings,SheafOfRings) -- see sheafExt -- sheaf extension of coherent sheaves
- sheafHom(CoherentSheaf,CoherentSheaf) -- see sheafHom -- sheaf Hom
- sheafHom(CoherentSheaf,SheafOfRings) -- see sheafHom -- sheaf Hom
- sheafHom(SheafOfRings,CoherentSheaf) -- see sheafHom -- sheaf Hom
- sheafHom(SheafOfRings,SheafOfRings) -- see sheafHom -- sheaf Hom
- sheafHom(CoherentSheaf,SheafMap) -- see sheafHom(SheafMap,SheafMap) -- functor of sheaf Hom
- sheafHom(SheafMap,CoherentSheaf) -- see sheafHom(SheafMap,SheafMap) -- functor of sheaf Hom
- sheafHom(SheafMap,SheafMap) -- functor of sheaf Hom
- sheafHom(SheafMap,SheafOfRings) -- see sheafHom(SheafMap,SheafMap) -- functor of sheaf Hom
- sheafHom(SheafOfRings,SheafMap) -- see sheafHom(SheafMap,SheafMap) -- functor of sheaf Hom
- SheafMap * RingElement (missing documentation)
- SheafMap * ZZ (missing documentation)
- 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
- SheafOfRings ^ ZZ -- see SheafOfRings ^ List -- make a graded free coherent sheaf
- SheafOfRings Sequence (missing documentation)
- singularLocus(AffineVariety) -- the locus of singular points in a variety
- singularLocus(ProjectiveVariety) -- see singularLocus(AffineVariety) -- the locus of singular points in a variety
- source(SheafMap) (missing documentation)
- Spec(Ring) -- see Spec -- make an affine variety
- super(CoherentSheaf) (missing documentation)
- super(SheafMap) (missing documentation)
- symmetricPower(ZZ,CoherentSheaf) (missing documentation)
- symmetricPower(ZZ,SheafMap) (missing documentation)
- tangentSheaf(ProjectiveVariety) -- see tangentSheaf -- tangent sheaf of a projective variety
- target(SheafMap) (missing documentation)
- tensor(CoherentSheaf,CoherentSheaf) (missing documentation)
- tensor(SheafMap,SheafMap) (missing documentation)
- texMath(CoherentSheaf) (missing documentation)
- texMath(SheafExpression) (missing documentation)
- texMath(SheafMap) (missing documentation)
- texMath(SheafOfRings) (missing documentation)
- texMath(SumOfTwists) (missing documentation)
- texMath(Variety) (missing documentation)
- toExternalString(SheafMap) (missing documentation)
- toExternalString(Variety) (missing documentation)
- toString(CoherentSheaf) (missing documentation)
- toString(SheafMap) (missing documentation)
- toString(SheafOfRings) (missing documentation)
- toString(SumOfTwists) (missing documentation)
- toString(Variety) (missing documentation)
- variety(CoherentSheaf) -- the underlying variety over which a coherent sheaf or morphism is defined.
- variety(SheafMap) -- see variety(CoherentSheaf) -- the underlying variety over which a coherent sheaf or morphism is defined.
- variety(SheafOfRings) -- see variety(CoherentSheaf) -- the underlying variety over which a coherent sheaf or morphism is defined.
- variety(SumOfTwists) -- see 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
Symbols
- cotangentSurjection
- embeddedToAbstract
- eulerSequence (missing documentation)
- ExtLongExactSequence -- the long exact sequence of the Ext functor
- GlobalSectionLimit -- see HH^ZZ SumOfTwists -- coherent sheaf cohomology module
- SaturationMap -- see HH^ZZ SumOfTwists -- coherent sheaf cohomology module
- TorsionFree -- see HH^ZZ SumOfTwists -- coherent sheaf cohomology module
- idealSheafSequence (missing documentation)
- RHom (missing documentation)
- yonedaSheafExtension (missing documentation)
Other things
- hh -- Hodge numbers of a smooth projective variety
- OO -- the structure sheaf
- sheafExt -- sheaf extension of coherent sheaves

For the programmer

The object Varieties is a package, defined in Varieties.m2, with auxiliary files in Varieties/.

The source of this document is in Varieties.m2:878:0.