Introduction and First Examples
Affine and Projective Spaces
AffineSpace(k,n) : Rng,RngIntElt -> Aff
ProjectiveSpace(k,n) : Rng,RngIntElt -> Prj
AffineSpace(R) : RngMPol -> Aff
ProjectiveSpace(R) : RngMPol -> Prj
AssignNames(~A,N) : Sch,[MonStgElt] ->
A . i : Sch,RngIntElt -> RngMPolElt
Example Scheme_affine-space-names (H83E1)
A eq B : Sch,Sch -> BoolElt
Scrolls and Products
DirectProduct(A,B) : Sch,Sch -> Sch,SeqEnum
RuledSurface(k,a,b) : Rng,RngIntElt,RngIntElt -> PrjScrl
RuledSurface(k,n) : Rng,RngIntElt -> PrjScrl
AbsoluteRationalScroll(k,N) : Rng,SeqEnum -> PrjScrl
ProductProjectiveSpace(k,N) : Rng,SeqEnum -> PrjScrl
Functions and Homogeneity on Ambient Spaces
CoordinateRing(A) : Sch -> Rng
FunctionField(A) : Sch -> FldFunG
IsAmbientFunction(A,f) : Sch,RngElt -> BoolElt, RngElt
IsAmbientRationalFunction(A,f) : Sch,RngElt -> BoolElt
Gradings(X) : Sch -> SeqEnum
NumberOfGradings(X) : Sch -> RngIntElt
NumberOfCoordinates(X) : Sch -> RngIntElt
Lengths(X) : Sch -> [RngIntElt]
IsHomogeneous(X,f) : Sch,RngMPolElt -> BoolElt
Multidegree(X,f) : Sch,RngMPolElt -> SeqEnum
Prelude to Points
A ! [a,b,...] : Sch,[RngElt] -> Pt
Example Scheme_schemes-points-example1 (H83E2)
Origin(A) : Aff -> Pt
Simplex(A) : Prj -> SeqEnum
Coordinates(p) : Pt -> SeqEnum
p[i] : Pt, RngIntElt -> RngElt
Evaluate(f,p) : RngElt,Pt -> RngElt
Example Scheme_evaluate-funfld-example (H83E3)
Constructing Schemes
Scheme(X,f) : Sch,RngMPolElt -> Sch
Example Scheme_schemes-creation (H83E4)
Spec(R) : RngMPolRes -> Sch,Aff
Proj(R) : RngMPolRes -> Sch,Prj
EmptySubscheme(X) : Sch -> Sch, MapSch
X meet Y : Sch,Sch -> Sch
X join Y : Sch,Sch -> Sch
AssignNames(~X,N) : Sch,SeqEnum ->
X . i : Sch,RngIntElt -> RngMPolElt
Different Types of Scheme
IsAffineSpace(X) : Sch -> BoolElt
IsProjectiveSpace(X) : Sch -> BoolElt
IsOrdinaryProjectiveSpace(X) : Sch -> BoolElt
IsCluster(X) : Sch -> BoolElt,Clstr
IsCurve(X) : Sch -> BoolElt,Crv
IsConic(X) : Sch -> BoolElt,CrvCon
IsRationalCurve(X) : Sch -> BoolElt,CrvRat
Functions of the Ambient Space
AmbientSpace(X) : Sch -> Sch
BaseRing(X) : Sch -> Rng
CoefficientRing(X) : Sch -> Fld
IsAffine(X) : Sch -> BoolElt
IsProjective(X) : Sch -> BoolElt
IsOrdinaryProjective(X) : Sch -> BoolElt
Functions of the Equations
DefiningPolynomials(X) : Sch -> SeqEnum
DefiningPolynomial(X) : Sch -> RngMPolElt
DefiningIdeal(X) : Sch -> RngMPol
CoordinateRing(X) : Sch -> RngMPol
Curve(X) : Sch -> Crv
GroebnerBasis(X) : Sch -> SeqEnum
MinimalBasis(X) : Sch -> [ RngMPolElt ]
IsHypersurface(X) : Sch -> BoolElt, RngMPolElt
JacobianIdeal(X) : Sch -> RngMPol
JacobianMatrix(X) : Sch -> ModMatRngElt
HessianMatrix(X) : Sch -> ModMatRngElt
X eq Y : Sch,Sch -> BoolElt
X subset Y : Sch,Sch -> BoolElt
Example Scheme_scheme-equality (H83E5)
Rational Points and Point Sets
X(L) : Sch,Rng -> SetPt
P eq Q : SetPt,SetPt -> BoolElt
Scheme(P) : SetPt -> Sch
Curve(P) : SetPt -> Crv
Ring(P) : SetPt -> Rng
X ! Q : Sch,SeqEnum -> Pt
p eq q : Pt,Pt -> BoolElt
p in X : Pt,Sch -> BoolElt
Scheme(p) : Pt -> Sch
Curve(p) : Pt -> Crv
Q in X : SeqEnum,Sch -> BoolElt
S subset X : Setq,Sch -> BoolElt
IsCoercible(X,Q) : Sch,SeqEnum -> BoolElt,Pt
RationalPoints(X) : Sch -> SetIndx
RationalPoints(X) : Sch -> SetIndx
HasNonsingularPoint(X) : Sch -> BoolElt,Pt
Example Scheme_scheme-points (H83E6)
Zero-dimensional Schemes
Scheme(p) : Pt -> Sch
RationalPoints(Z) : Sch -> SetEnum
PointsOverSplittingField(Z) : Clstr -> SetEnum
HasPointsOverExtension(X) : Sch -> BoolElt
Degree(Z) : Clstr -> RngIntElt
Example Scheme_cluster-degree5 (H83E7)
Point conditions
IsSingular(p) : Sch,Pt -> BoolElt
IsNonsingular(p) : Sch,Pt -> BoolElt
IsOrdinarySingularity(p) : Sch,Pt -> BoolElt
Point computations
Multiplicity(p) : Sch,Pt -> RngIntElt
TangentSpace(p) : Sch,Pt -> Sch
TangentCone(p) : Sch,Pt -> Sch
Global Geometry of Schemes
Dimension(X) : Sch -> RngIntElt
Codimension(X) : Sch -> RngIntElt
Degree(X) : Sch -> RngIntElt
IsEmpty(X) : Sch -> BoolElt
IsNonsingular(X) : Sch -> BoolElt
IsSingular(X) : Sch -> BoolElt
SingularSubscheme(X) : Sch -> Sch
PrimeComponents(X) : Sch -> SeqEnum
PrimaryComponents(X) : Sch -> SeqEnum
ReducedSubscheme(X) : Sch -> Sch, MapSch
IsIrreducible(X) : Sch -> BoolElt
IsReduced(X) : Sch -> BoolElt
Example Scheme_schemes-prime-components (H83E8)
Base Change for Schemes
BaseChange(A,K) : Sch,Rng -> Sch
BaseChange(A,m) : Sch, Map -> Sch
BaseChange(F,K) : SeqEnum,Rng -> SeqEnum
BaseChange(X,A) : Sch,Sch -> Sch
BaseChange(X, n) : Sch, RngIntElt -> Sch
Example Scheme_base-change-schemes (H83E9)
Projective Closure and Affine Patches
ProjectiveClosure(X) : Sch -> Sch
AffinePatch(X,i) : Sch,RngIntElt -> Sch
AffinePatch(X,p) : Sch,Pt -> Sch,Pt
Example Scheme_projective-closure (H83E10)
HyperplaneAtInfinity(X) : Sch -> Sch
ProjectiveClosureMap(A) : Aff -> MapSch
AffineDecomposition(P) : Prj -> [MapSch],Pt
CentredAffinePatch(S, p) : Sch, Pt -> Sch, MapSch
Creation of Maps
map< X -> Y | F > : Sch,Sch,SeqEnum -> MapSch
iso< X -> Y | F, G > : Sch,Sch,SeqEnum,SeqEnum -> MapAutSch
Example Scheme_map-creation (H83E11)
Example Scheme_map-frobenius (H83E12)
IdentityMap(X) : Sch -> MapSch
ConstantMap(X,Y,p) : Sch,Sch,Pt -> MapSch
Projection(X,Y) : Prj,Prj -> MapSch
Projection(X, Q) : Sch, Prj -> Sch, MapSch
ProjectionFromNonsingularPoint(X,p) : Sch,Pt -> Sch,MapSch,Sch
Elimination(X,V) : Sch,SeqEnum -> Sch
Inverse(f) : MapSch -> MapSch
g * f : MapSch,MapSch -> MapSch
Example Scheme_map-error (H83E13)
Example Scheme_hom-spaces (H83E14)
Restriction(f,X,Y) : MapSch,Sch,Sch -> MapSch
Trivial Attributes
Domain(f) : MapSch -> Sch
Codomain(f) : MapSch -> Sch
DefiningPolynomials(f) : MapSch -> SeqEnum
FactoredDefiningPolynomials(f) : MapSch -> SeqEnum
InverseDefiningPolynomials(f) : MapSch -> SeqEnum
FactoredInverseDefiningPolynomials(f) : MapSch -> SeqEnum
AlternateDefiningPolynomials(f) : MapSch -> SeqEnum
AlgebraMap(f) : MapSch -> Map
FunctionDegree(f) : MapSch -> RngIntElt
Basic Tests
f eq g : MapSch, MapSch -> BoolElt
IsRegular(f) : MapSch -> BoolElt
IsIsomorphism(f) : MapSch -> BoolElt, IsoSch
IsDominant(f) : AmbMap -> BoolElt
IsLinear(f) : MapSch -> BoolElt
IsAffineLinear(f) : MapSch -> BoolElt
Maps and Points
f(p) : MapSch,Pt -> Pt
p @@ f : Pt,MapSch -> Pt
f(K) : MapSch,Rng -> Map
Example Scheme_maps-point-image (H83E15)
Maps and Schemes
X @@ f : Sch, MapSch -> Sch
Image(f) : MapSch -> Sch
Image(f,X,d) : AmbProjMap,SchProj,RngIntElt -> []
Example Scheme_map-image1 (H83E16)
Example Scheme_map-image2 (H83E17)
BaseScheme(f) : MapSch -> Sch
BasePoints(f) : MapSch -> SetEnum
Example Scheme_map-base-points (H83E18)
Example Scheme_scroll-map-base-points (H83E19)
Maps and Closure
ProjectiveClosure(f) : MapSch -> MapSch
MakeProjectiveClosureMap(A, P, S) : Aff,Prj,SeqEnum ->
RestrictionToPatch(f,j) : MapSch,RngIntElt -> MapSch
RestrictionToPatch(f,i,j) : MapSch,RngIntElt,RngIntElt -> MapSch
Example Scheme_map-patches (H83E20)
Automorphisms
Automorphism(X,F) : Sch,SeqEnum -> MapAutSch
IdentityAutomorphism(X) : Sch -> MapAutSch
IsEndomorphism(f) : MapSch -> BoolElt
IsAutomorphism(f) : MapSch -> BoolElt,AutSch
Example Scheme_automorphism-construction (H83E21)
Example Scheme_aut-aff-jac (H83E22)
Affine Automorphisms
Automorphism(A,F) : Sch,SeqEnum -> MapSch
Automorphism(A,M) : Aff,Mtrx -> IsoSch
Translation(A,p) : Sch, Pt -> MapSch
PermutationAutomorphism(A,g) : Sch,GrpPermElt -> IsoSch
Example Scheme_aut-aff-perm (H83E23)
Automorphism(A,p) : Sch, RngMPolElt -> IsoSch
AffineDecomposition(f) : MapSch -> MapSch,MapSch
Example Scheme_decompose-automorphism (H83E24)
NagataAutomorphism(A) : Aff -> MapSch
Projectivity(A,M) : Aff,Mtrx -> MapAutSch
Example Scheme_projectivity (H83E25)
Projective Automorphisms
Automorphism(P,F) : Prj, SeqEnum -> MapSch
Matrix(f) : MapSch -> Mtrx
Automorphism(P,M) : Sch,Mtrx -> MapSch
Aut(P) : Prj -> PowAutSch
AutomorphismGroup(P) : P -> GrpMat,Map
Example Scheme_projective-automorphism-group (H83E26)
TranslationOfSimplex(P,Q) : Prj, [Pt] -> MapSch
Translation(P,Q) : Prj, [Pt] -> MapSch
Translation(P,p,q) : Prj, Pt, Pt -> MapSch
Translation(X,p) : Sch, Pt -> MapSch
Example Scheme_translation (H83E27)
QuadraticTransformation(P) : Prj -> MapSch
QuadraticTransformation(X) : Sch -> SchMap
Example Scheme_cremona-factorisation (H83E28)
Explicit Creation
LinearSystem(P,d) : Prj,RngIntElt -> LinSys
LinearSystem(P,F) : Prj,SeqEnum -> LinSys
Example Scheme_linsys-construction (H83E29)
ImageSystem(f,S,d) : AmbProjMap,SchProj,RngIntElt -> LinSys
Example Scheme_image-finder (H83E30)
Geometrical Restrictions
LinearSystem(L,p) : LinSys,Pt -> LinSys
LinearSystem(L,p,m) : LinSys,Pt,RngIntElt -> LinSys
Example Scheme_subsystems (H83E31)
LinearSystem(L,X) : LinSys,Sch -> LinSys
LinearSystemTrace(L,X) : LinearSys,Sch -> LinearSys
Example Scheme_trace (H83E32)
Explicit Restrictions
LinearSystem(L,F) : LinSys,SeqEnum -> LinSys
LinearSystem(L,V) : LinSys,ModTupFld -> LinSys
Basic Algebra of Linear Systems
Tests for Linear Systems
Ambient(L) : LinSys -> Prj
L eq K : LinSys,LinSys -> BoolElt
IsComplete(L) : LinSys -> BoolElt
IsBasePointFree(L) : LinSys -> BoolElt
Geometrical Properties
Sections(L) : LinSys -> SeqEnum
Degree(L) : LinSys -> RngIntElt
Dimension(L) : LinSys -> RngIntElt
BaseScheme(L) : LinSys -> SchProj
BaseComponent(L) : LinSys -> SchProj
Reduction(L) : LinSys -> LinSys
Example Scheme_ls-reduction (H83E33)
BasePoints(L) : LinSys -> SeqEnum
Multiplicity(L,p) : LinSys,Pt -> RngIntElt
Linear Algebra
CoefficientSpace(L) : LinSys -> ModTupFld
CoefficientMap(L) : LinSys -> ModTupFldElt
PolynomialMap(L) : LinSys -> RngMPolElt
Complement(L,K) : LinSys,LinSys -> LinSys
Complement(L,X) : LinSys,Sch -> LinSys
Example Scheme_creation-by-subspace (H83E34)
L meet K : LinSys,LinSys -> LinSys
X in L : Sch,LinSys -> BoolElt
f in L : RngMPolElt,LinSys -> BoolElt
K subset L : LinSys,LinSys -> BoolElt
Linear Systems and Maps
Pullback(f,L) : AmbProjMap,LinSys -> LinSys
A Pair of Twisted Cubics
Example Scheme_twisted-cubics (H83E35)