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 (missing documentation) 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 0.2 of Varieties.

Source code

The source code from which this documentation is derived is in the file Varieties.m2. The auxiliary files accompanying it are in the directory Varieties/.

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
- SheafExpression -- the class of sheaf expressions
- SheafMap (missing documentation)
- 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 (missing documentation)
- cotangentSheaf -- cotangent sheaf of a projective variety
- isLiftable (missing documentation)
- isProjective (missing documentation)
- Proj -- make a projective variety
- sheaf -- make a coherent sheaf
- sheafHom -- sheaf Hom
- sheafMap (missing documentation)
- Spec -- make an affine variety
- tangentSheaf -- tangent sheaf of a projective variety
- variety -- get the variety
Methods
- - SheafMap (missing documentation)
- > InfiniteNumber -- see > ZZ
- > ZZ
- >= InfiniteNumber -- see >= ZZ
- >= ZZ
- AffineVariety ** AffineVariety (missing documentation)
- AffineVariety ** Ring
- ambient(AffineVariety) (missing documentation)
- ambient(CoherentSheaf) (missing documentation)
- ambient(ProjectiveVariety) (missing documentation)
- ambient(SheafMap) (missing documentation)
- annihilator(CoherentSheaf) -- the annihilator ideal
- betti(CoherentSheaf) (missing documentation)
- canonicalBundle(ProjectiveVariety) (missing documentation)
- 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 ** SheafMap (missing documentation)
- CoherentSheaf ++ CoherentSheaf -- direct sum of coherent sheaves
- CoherentSheaf / CoherentSheaf -- quotient of coherent sheaves
- CoherentSheaf / Ideal -- see CoherentSheaf / CoherentSheaf -- quotient of coherent sheaves
- CoherentSheaf == CoherentSheaf (missing documentation)
- CoherentSheaf == ZZ (missing documentation)
- CoherentSheaf ^ Array (missing documentation)
- CoherentSheaf ^ List (missing documentation)
- CoherentSheaf ^ ZZ -- direct sum
- SheafOfRings ^ ZZ -- see CoherentSheaf ^ ZZ -- direct sum
- CoherentSheaf ^** ZZ -- tensor power
- CoherentSheaf _ Array (missing documentation)
- 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) -- 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) (missing documentation)
- 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(ProjectiveHilbertPolynomial) -- constant term of the Hilbert polynomial
- euler(ProjectiveVariety) -- topological Euler characteristic of a (smooth) projective variety
- euler(Ring)
- eulers(CoherentSheaf)
- eulers(Module) -- see eulers(CoherentSheaf)
- eulers(Ideal)
- eulers(Ring)
- 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) (missing documentation)
- Ext^ZZ(CoherentSheaf,SumOfTwists) -- global Ext
- Ext^ZZ(SheafOfRings,SumOfTwists) -- see Ext^ZZ(CoherentSheaf,SumOfTwists) -- global Ext
- exteriorPower(ZZ,CoherentSheaf)
- 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 SheafMap (missing documentation)
- HH^ZZ SheafOfRings -- cohomology of a sheaf of rings on a projective variety
- HH^ZZ SumOfTwists -- coherent sheaf cohomology module
- HH^ZZ(AffineVariety,CoherentSheaf) (missing documentation)
- 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(ProjectiveVariety,SheafMap) (missing documentation)
- HH^ZZ(ProjectiveVariety,SumOfTwists) (missing documentation)
- 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
- image(SheafMap) (missing documentation)
- inducedMap(CoherentSheaf,CoherentSheaf) (missing documentation)
- inverse(SheafMap) (missing documentation)
- isIsomorphism(SheafMap) (missing documentation)
- isLiftable(Matrix,Matrix) (missing documentation)
- isLiftable(SheafMap,ZZ) (missing documentation)
- isProjective(ProjectiveVariety) (missing documentation)
- isProjective(Variety) (missing documentation)
- isWellDefined(CoherentSheaf) (missing documentation)
- isWellDefined(SheafMap) (missing documentation)
- isWellDefined(Variety) (missing documentation)
- 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) (missing documentation)
- map(CoherentSheaf,CoherentSheaf,Matrix,InfiniteNumber) (missing documentation)
- map(CoherentSheaf,CoherentSheaf,Matrix,ZZ) (missing documentation)
- matrix(SheafMap) (missing documentation)
- minimalPresentation(CoherentSheaf) (missing documentation)
- minimalPresentation(SheafMap) (missing documentation)
- minimalPresentation(SheafOfRings) (missing documentation)
- module(CoherentSheaf) -- get the module defining a coherent sheaf
- module(SheafOfRings)
- numgens(CoherentSheaf) -- the number of generators of the underlying module
- pdim(CoherentSheaf) -- calculate the projective dimension
- Proj(Ring) -- see Proj -- make a projective variety
- prune(CoherentSheaf) (missing documentation)
- prune(SheafMap) (missing documentation)
- prune(SheafOfRings) (missing documentation)
- pullback(SheafMap,SheafMap) (missing documentation)
- pushout(SheafMap,SheafMap) (missing documentation)
- rank(CoherentSheaf) (missing documentation)
- 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(SheafMap) (missing documentation)
- ring(Variety) -- coordinate ring of the variety
- Module ~ -- 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(Variety) -- 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 Ext of coherent sheaves
- sheafExt^ZZ(CoherentSheaf,SheafOfRings) -- see sheafExt -- sheaf Ext of coherent sheaves
- sheafExt^ZZ(SheafOfRings,CoherentSheaf) -- see sheafExt -- sheaf Ext of coherent sheaves
- sheafExt^ZZ(SheafOfRings,SheafOfRings) -- see sheafExt -- sheaf Ext 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) (missing documentation)
- sheafHom(SheafMap,CoherentSheaf) (missing documentation)
- sheafHom(SheafMap,SheafMap) (missing documentation)
- sheafHom(SheafMap,SheafOfRings) (missing documentation)
- sheafHom(SheafOfRings,SheafMap) (missing documentation)
- SheafMap * SheafMap (missing documentation)
- SheafMap ** CoherentSheaf (missing documentation)
- SheafMap ** SheafMap (missing documentation)
- SheafMap ** SheafOfRings (missing documentation)
- SheafMap ++ SheafMap (missing documentation)
- SheafMap == ZZ (missing documentation)
- SheafMap ^ ZZ (missing documentation)
- SheafMap ZZ (missing documentation)
- sheafMap(Matrix) (missing documentation)
- sheafMap(Matrix,ZZ) (missing documentation)
- SheafOfRings ** SheafMap (missing documentation)
- SheafOfRings ^ List -- make a graded free coherent sheaf
- SheafOfRings Sequence (missing documentation)
- singularLocus(AffineVariety) (missing documentation)
- singularLocus(ProjectiveVariety) (missing documentation)
- 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)
- tangentSheaf(ProjectiveVariety) -- tangent sheaf of a projective variety
- target(SheafMap) (missing documentation)
- tensor(CoherentSheaf,CoherentSheaf) (missing documentation)
- tensor(SheafMap,SheafMap) (missing documentation)
- variety(CoherentSheaf) -- get the variety
- variety(Ideal) -- the closed projective subvariety defined by an ideal
- variety(Ring) -- the variety previously associated to a given ring
- variety(SheafMap) (missing documentation)
- variety(SheafOfRings) -- get the variety
- variety(SumOfTwists) (missing documentation)
- ZZ == CoherentSheaf (missing documentation)
- ZZ == SheafMap (missing documentation)
Symbols
- GlobalSectionLimit (missing documentation)
- SaturationMap (missing documentation)
- TorsionFree (missing documentation)
Other things
- hh -- Hodge numbers of a smooth projective variety
- OO -- the structure sheaf
- sheafExt -- sheaf Ext of coherent sheaves

For the programmer

The object Varieties is a package.

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 (missing documentation)