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Acknowledgements
Introduction and First Examples
Ambient Spaces
Schemes
Rational Points
Projective Closure
Maps
Linear Systems
Aside: Types of Schemes
Ambients
Affine and Projective Spaces
Scrolls and Products
Functions and Homogeneity on Ambient Spaces
Prelude to Points
Constructing Schemes
Different Types of Scheme
Basic Attributes of Schemes
Functions of the Ambient Space
Functions of the Equations
Function Fields and their Elements
Rational Points and Point Sets
Zero-dimensional Schemes
Local Geometry of Schemes
Point conditions
Point computations
Global Geometry of Schemes
Base Change for Schemes
Affine Patches and Projective Closure
Arithmetic Properties of Schemes and Points
Height
Restriction of Scalars
Local Solubility
Searching for Points
Maps between Schemes
Creation of Maps
Basic Attributes
Trivial Attributes
Basic Tests
Maps and Points
Maps and Schemes
Maps and Closure
Automorphisms
Affine Automorphisms
Projective Automorphisms
Tangent and Secant Varieties and Isomorphic Projections
Tangent Varieties
Secant Varieties
Isomorphic Projection to Subspaces
Linear Systems
Creation of Linear Systems
Explicit Creation
Geometrical Restrictions
Explicit Restrictions
Basic Algebra of Linear Systems
Tests for Linear Systems
Geometrical Properties
Linear Algebra
Linear Systems and Maps
Parametrization of Del Pezzo Surfaces
Advanced Examples
A Pair of Twisted Cubics
Curves in Space
Bibliography
DETAILS
Introduction and First Examples
Ambient Spaces
Schemes
Rational Points
Projective Closure
Maps
Linear Systems
Aside: Types of Schemes
Ambients
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 (H98E1)
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 -> FldFunFracSch
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 (H98E2)
Origin(A) : Aff -> Pt
Simplex(A) : Prj -> SeqEnum
Coordinates(p) : Pt -> SeqEnum
p[i] : Pt, RngIntElt -> RngElt
p @ f : Pt, FldFunFracSchElt -> RngElt
Example Scheme_evaluate-funfld-example (H98E3)
Constructing Schemes
Scheme(X,f) : Sch,RngMPolElt -> Sch
Cluster(X,f) : Sch,RngMPolElt -> Clstr
Example Scheme_schemes-creation (H98E4)
Spec(R) : RngMPolRes -> Sch,Aff
Proj(R) : RngMPolRes -> Sch,Prj
EmptyScheme(X) : Sch -> Sch
X meet Y : Sch,Sch -> Sch
X join Y : Sch,Sch -> Sch
Difference(X, Y) : Sch, Sch -> Sch
Example Scheme_remove (H98E5)
Saturate(~X) : Sch ->
AssignNames(~X,N) : Sch,SeqEnum ->
X . i : Sch,RngIntElt -> RngMPolElt
Different Types of Scheme
IsAffine(X) : Sch -> BoolElt
IsProjective(X) : Sch -> BoolElt
IsOrdinaryProjectiveSpace(X) : Sch -> BoolElt
IsAmbient(X) : Sch -> BoolElt
IsCluster(X) : Sch -> BoolElt,Clstr
IsCurve(X) : Sch -> BoolElt,Crv
IsPlaneCurve(X) : Sch -> BoolElt, CrvPln
IsConic(X) : Sch -> BoolElt,CrvCon
IsRationalCurve(X) : Sch -> BoolElt,CrvRat
IsHyperellipticCurve(X) : Sch -> BoolElt,CrvHyp
IsModularCurve(X) : Sch -> BoolElt,CrvMod
Basic Attributes of Schemes
Functions of the Ambient Space
AmbientSpace(X) : Sch -> Sch
SuperScheme(X) : Sch -> Sch
BaseRing(X) : Sch -> Rng
BaseField(X) : Sch -> Fld
IsAffine(X) : Sch -> BoolElt
IsProjective(X) : Sch -> BoolElt
IsOrdinaryProjective(X) : Sch -> BoolElt
IsPlanar(X) : Sch -> BoolElt
IsSaturated(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
IsSubscheme(X, Y) : Sch,Sch -> BoolElt
IsLinear(X) : Sch -> BoolElt
Example Scheme_scheme-equality (H98E6)
Function Fields and their Elements
Scheme(F) : FldFunFracSch -> Sch
IntegerRing(F) : RngFrac -> Rng
AssignNames(~F, S) : RngFrac, [MonStgElt] ->
F ! g : FldFunFracSch, RngElt -> FldFunFracSchElt
F . i : FldFunFracSch, RngIntElt -> FldFunFracSchElt
ProjectiveFunction(f) : FldFunFracSchElt -> FldFracElt
ProjectiveRationalFunction(f) : FldFunFracSchElt -> FldFunRatMElt
RestrictionToPatch(f, Xi) : FldFunFracSchElt, Sch -> FldFracElt
Numerator(f) : RngFracElt -> RngElt
IntegralSplit(f, X) : FldFunFracSchElt, Sch -> RngMPolElt, RngMPolElt
Numerator(f, X) : FldFunFracSchElt, Sch -> MPolElt
Denominator(f, X) : FldFunFracSchElt, Sch -> MPolElt
Example Scheme_scheme_fld_fun_elt (H98E7)
Restriction(f, Y) : FldFunFracSchElt, Sch -> FldFunFracSchElt
GenericPoint(X) : Sch -> Pt
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
RingMap(P) : SetPt -> Map
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
RationalPointsByFibration(X) : Sch -> SetIndx
HasNonsingularPoint(X) : Sch -> BoolElt,Pt
Example Scheme_scheme-points (H98E8)
Zero-dimensional Schemes
Cluster(p) : Pt -> Clstr
RationalPoints(Z) : Sch -> SetEnum
PointsOverSplittingField(Z) : Clstr -> SetEnum
HasPointsOverExtension(X) : Sch -> BoolElt
Degree(Z) : Clstr -> RngIntElt
Example Scheme_cluster-degree5 (H98E9)
Local Geometry of Schemes
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
ArithmeticGenus(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 (H98E10)
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 (H98E11)
Affine Patches and Projective Closure
ProjectiveClosure(X) : Sch -> Sch
AffinePatch(X,i) : Sch,RngIntElt -> Sch
AffinePatch(X,p) : Sch,Pt -> Sch,Pt
IsStandardAffinePatch(A) : Aff -> BoolElt, RngIntElt
NumberOfAffinePatches(X) : Sch -> BoolElt
HasAffinePatch(X, i) : Sch, RngIntElt -> BoolElt
Example Scheme_projective-closure (H98E12)
Example Scheme_projective-closure-incorrect (H98E13)
HyperplaneAtInfinity(X) : Sch -> Sch
ProjectiveClosureMap(A) : Aff -> MapSch
AffineDecomposition(P) : Prj -> [MapSch],Pt
CentredAffinePatch(S, p) : Sch, Pt -> Sch, MapSch
Arithmetic Properties of Schemes and Points
Height
HeightOnAmbient(P) : Pt -> FldReElt
Restriction of Scalars
RestrictionOfScalars(S, F) : Sch, Fld -> Sch, MapSch, UserProgram, Map
Local Solubility
IsEmpty(Xm) : SetPt -> BoolElt, Pt
Example Scheme_anf1 (H98E14)
Example Scheme_anf2 (H98E15)
IsLocallySolvable(X, p) : Sch, RngOrdIdl -> BoolElt, Pt
Example Scheme_anf-local-solv (H98E16)
LiftPoint(P, n) : Pt, RngIntElt -> Pt
Example Scheme_anf_lift (H98E17)
Searching for Points
PointSearch(S,H : parameters) : Sch[FldRat], RngIntElt -> SeqEnum
Example Scheme_point-count (H98E18)
Maps between Schemes
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 (H98E19)
Example Scheme_map-fnfld (H98E20)
Example Scheme_map-frobenius (H98E21)
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
ProjectiveMap(L, Y) : [FldFunFracSchElt], Sch -> MapSch
ProjectiveMap(f, Y) : FldFunFracSchElt, Sch -> MapSch
Example Scheme_map-creation-prj (H98E22)
Elimination(X,V) : Sch,SeqEnum -> Sch
Inverse(f) : MapSch -> MapSch
IsInvertible(f) : MapSch -> Bool, MapSch
HasKnownInverse(f) : MapSch -> Bool
Example Scheme_map_creation_inv (H98E23)
g * f : MapSch,MapSch -> MapSch
Components(f) : Map -> [Map]
Example Scheme_map-error (H98E24)
Example Scheme_hom-spaces (H98E25)
Restriction(f,X,Y) : MapSch,Sch,Sch -> MapSch
Expand(phi) : MapSch -> MapSch
Extend(phi) : MapSch -> MapSch
Prune(phi) : MapSch -> MapSch
Normalization(phi) : MapSch -> MapSch
Example Scheme_map_creation-comp_alt (H98E26)
Basic Attributes
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
AllDefiningPolynomials(f) : MapSch -> SeqEnum
AllInverseDefiningPolynomials(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) : MapSch -> BoolElt
IsLinear(f) : MapSch -> BoolElt
IsAffineLinear(f) : MapSch -> BoolElt
Maps and Points
f(p) : MapSch,Pt -> Pt
Pullback(f, p) : MapSch, Pt -> Any
p @@ f : Pt,MapSch -> Any
f(K) : MapSch,Rng -> Map
Example Scheme_maps-point-image (H98E27)
Maps and Schemes
Pullback(f, X) : MapSch, Sch -> Sch
Image(f) : MapSch -> Sch
Image(f,X,d) : MapSch,Sch,RngIntElt -> []
Example Scheme_map-image1 (H98E28)
Example Scheme_map-image2 (H98E29)
BaseScheme(f) : MapSch -> Sch
BasePoints(f) : MapSch -> SetEnum
Example Scheme_map-base-points (H98E30)
Example Scheme_scroll-map-base-points (H98E31)
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 (H98E32)
Automorphisms
Automorphism(X,F) : Sch,SeqEnum -> MapAutSch
IdentityAutomorphism(X) : Sch -> MapAutSch
IsEndomorphism(f) : MapSch -> BoolElt
IsAutomorphism(f) : MapSch -> BoolElt,AutSch
Example Scheme_automorphism-construction (H98E33)
Example Scheme_aut-aff-jac (H98E34)
Affine Automorphisms
Automorphism(A,F) : Sch,SeqEnum -> MapSch
Automorphism(A,M) : Sch,Mtrx -> MapIsoSch
Translation(A,p) : Sch, Pt -> MapSch
PermutationAutomorphism(A, g) : Sch,GrpPermElt -> MapIsoSch
Example Scheme_aut-aff-perm (H98E35)
Automorphism(A,p) : Sch, RngMPolElt -> IsoSch
AffineDecomposition(f) : MapSch -> MapSch,MapSch
Example Scheme_decompose-automorphism (H98E36)
NagataAutomorphism(A) : Aff -> MapSch
Projectivity(A,M) : Aff,Mtrx -> MapAutSch
Example Scheme_projectivity (H98E37)
Projective Automorphisms
Automorphism(P,F) : Prj, SeqEnum -> MapSch
Matrix(f) : MapSch -> Mtrx
Automorphism(P,M) : Sch,Mtrx -> MapSch
Aut(P) : Prj -> PowAutSch
AutomorphismGroup(P) : Prj -> GrpMat,Map
Example Scheme_projective-automorphism-group (H98E38)
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 (H98E39)
QuadraticTransformation(P) : Prj -> MapSch
QuadraticTransformation(X) : Sch -> Sch, MapIsoSch
Example Scheme_cremona-factorisation (H98E40)
Tangent and Secant Varieties and Isomorphic Projections
Tangent Varieties
TangentVariety(X) : Sch -> Sch
IsInTangentVariety(X,P) : Sch,Pt -> BoolElt
Example Scheme_TangentVariety (H98E41)
Secant Varieties
SecantVariety(X) : Sch -> Sch
IsInSecantVariety(X,P) : Sch,Pt -> BoolElt
Example Scheme_SecantVariety (H98E42)
Isomorphic Projection to Subspaces
IsomorphicProjectionToSubspace(X) : Sch -> Sch, MapSch
EmbedPlaneCurveInP3(C) : Crv -> Sch, MapSch
Example Scheme_EmbeddingACurve (H98E43)
Linear Systems
Creation of Linear Systems
Explicit Creation
LinearSystem(P,d) : Sch,RngIntElt -> LinearSys
LinearSystem(P, d) : Sch, [RngIntElt] -> LinearSys
LinearSystem(P,F) : Sch,SeqEnum[RngMPolElt] -> LinearSys
MonomialsOfWeightedDegree(X, D) : Sch, [RngIntElt] -> SetIndx
Example Scheme_linsys-construction (H98E44)
ImageSystem(f,S,d) : MapSch,Sch,RngIntElt -> LinearSys
Example Scheme_image-finder (H98E45)
Geometrical Restrictions
LinearSystem(L,p) : LinearSys,Pt -> LinearSys
LinearSystem(L,p,m) : LinearSys,Pt,RngIntElt -> LinearSys
Example Scheme_subsystems (H98E46)
LinearSystem(L,X) : LinearSys,Sch -> LinearSys
LinearSystemTrace(L,X) : LinearSys,Sch -> LinearSys
Example Scheme_trace (H98E47)
Explicit Restrictions
LinearSystem(L,F) : LinearSys,SeqEnum -> LinearSys
LinearSystem(L,V) : LinearSys,ModTupFld -> LinearSys
Basic Algebra of Linear Systems
Tests for Linear Systems
Ambient(L) : LinearSys -> Prj
L eq K : LinearSys,LinearSys -> BoolElt
IsComplete(L) : LinearSys -> BoolElt
IsBasePointFree(L) : LinearSys -> BoolElt
Geometrical Properties
Sections(L) : LinearSys -> SeqEnum
Random(LS) : LinearSys -> RngMPolElt
Degree(L) : LinearSys -> RngIntElt
Dimension(L) : LinearSys -> RngIntElt
BaseScheme(L) : LinearSys -> SchProj
BaseComponent(L) : LinearSys -> SchProj
Reduction(L) : LinearSys -> LinearSys
Example Scheme_ls-reduction (H98E48)
BasePoints(L) : LinearSys -> SeqEnum
Multiplicity(L,p) : LinearSys,Pt -> RngIntElt
Linear Algebra
CoefficientSpace(L) : LinearSys -> ModTupFld
CoefficientMap(L) : LinearSys -> ModTupFldElt
PolynomialMap(L) : LinearSys -> RngMPolElt
Complement(L,K) : LinearSys,LinearSys -> LinearSys
Complement(L,X) : LinearSys,Sch -> LinearSys
Example Scheme_creation-by-subspace (H98E49)
L meet K : LinearSys,LinearSys -> LinearSys
X in L : Sch,LinearSys -> BoolElt
f in L : RngMPolElt,LinearSys -> BoolElt
K subset L : LinearSys,LinearSys -> BoolElt
Linear Systems and Maps
Pullback(f,L) : MapSch,LinearSys -> LinearSys
Parametrization of Del Pezzo Surfaces
SetVerbose("ParamDP", v) : MonStgElt, RngIntElt ->
ParametrizeDegree9DelPezzo(X) : Sch -> BoolElt, MapSch
ParametrizeDegree8DelPezzo(X) : Sch -> BoolElt, MapSch
Example Scheme_del-pezzo-ex-8 (H98E50)
ParametrizeDegree6DelPezzo(X) : Sch -> BoolElt, MapSch
Degree6DelPezzoType2_1(K,pt) : FldNum, Pt -> Sch
Example Scheme_del-pezzo-ex-6 (H98E51)
Advanced Examples
A Pair of Twisted Cubics
Example Scheme_twisted-cubics (H98E52)
Curves in Space
Bibliography
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