- Introduction and First Examples
- Ambients
- Affine and Projective Spaces
- 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
- SegreProduct(Xs) : SeqEnum[Sch] -> Sch, SeqEnum
- SegreEmbedding(X) : Sch -> Sch, MapIsoSch
- Example Scheme_sch:segre-embedding (H119E8)
- Functions and Homogeneity on Ambient Spaces
- Prelude to Points
- Constructing Schemes
- Scheme(X,f) : Sch,RngMPolElt -> Sch
- Cluster(X,f) : Sch,RngMPolElt -> Clstr
- Example Scheme_schemes-creation (H119E11)
- 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
- & join S : [Sch] -> Sch
- Difference(X, Y) : Sch, Sch -> Sch
- Complement(X, Y) : Sch, Sch -> Sch
- RemoveLinearRelations(X) : Sch -> Sch, MapIsoSch
- Blowup(X,Y) : Sch, Sch -> Sch, MapSch
- LocalBlowUp(X,Y) : Sch, Sch -> SeqEnum
- Example Scheme_remove (H119E12)
- Example Scheme_sch-blowup-ex (H119E13)
- Saturate(~X) : Sch ->
- AssignNames(~X,N) : Sch,SeqEnum ->
- X . i : Sch,RngIntElt -> RngMPolElt
- Different Types of Scheme
- Basic Attributes of Schemes
- 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 (H119E15)
- Restriction(f, Y) : FldFunFracSchElt, Sch -> FldFunFracSchElt
- GenericPoint(X) : Sch -> Pt
- Rational Points and Point Sets
- Zero-dimensional Schemes
- Local Geometry of Schemes
- Point Conditions
- Point Computations
- Analytically Hypersurface Singularities
- IsHypersurfaceSingularity(p,prec) : Pt, RngIntElt -> BoolElt, RngMPolElt, SeqEnum, Rec
- HypersurfaceSingularityExpandFurther(dat,prec,R): Rec, RngIntElt, RngMPol -> RngMPolElt
- HypersurfaceSingularityExpandFunction(dat,f,prec,R): Rec, FldFunRatMElt, RngIntElt, RngMPol -> RngMPolElt, RngMPolElt
- MilnorNumberAnalyticHypersurface(dat) : Rec -> RngIntElt
- Example Scheme_an-hyp-sing-ex (H119E18)
- Classification and Normal Forms of Singularities
- Global Geometry of Schemes
- 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 (H119E22)
- 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
- WeightedAffinePatch(P, i) : Prj, RngIntElt -> Sch, MapIsoSch
- Example Scheme_projective-closure (H119E23)
- Example Scheme_projective-closure-incorrect (H119E24)
- Example Scheme_weighted-patches (H119E25)
- 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
- 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 (H119E32)
- Example Scheme_map-fnfld (H119E33)
- Example Scheme_map-frobenius (H119E34)
- 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 (H119E35)
- Elimination(X,V) : Sch,SeqEnum -> Sch
- Inverse(f) : MapSch -> MapSch
- IsInvertible(f) : MapSch -> Bool, MapSch
- HasKnownInverse(f) : MapSch -> Bool
- Example Scheme_map_creation_inv (H119E36)
- g * f : MapSch,MapSch -> MapSch
- Components(f) : Map -> [Map]
- Example Scheme_hom-spaces (H119E37)
- 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 (H119E38)
- ImproveParametrization(p) : MapSch -> MapSch
- Example Scheme_improve_prm_ex (H119E39)
- Basic Attributes
- Maps and Points
- Maps and Schemes
- 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 (H119E45)
- Automorphisms
- Automorphism(X,F) : Sch,SeqEnum -> MapAutSch
- IdentityAutomorphism(X) : Sch -> MapAutSch
- IsEndomorphism(f) : MapSch -> BoolElt
- IsAutomorphism(f) : MapSch -> BoolElt,AutSch
- Example Scheme_automorphism-construction (H119E46)
- Example Scheme_aut-aff-jac (H119E47)
- 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 (H119E48)
- Automorphism(A,p) : Sch, RngMPolElt -> IsoSch
- AffineDecomposition(f) : MapSch -> MapSch,MapSch
- Example Scheme_decompose-automorphism (H119E49)
- NagataAutomorphism(A) : Aff -> MapSch
- Projectivity(A,M) : Aff,Mtrx -> MapAutSch
- Example Scheme_projectivity (H119E50)
- 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 (H119E51)
- 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 (H119E52)
- QuadraticTransformation(P) : Prj -> MapSch
- QuadraticTransformation(X) : Sch -> Sch, MapIsoSch
- Example Scheme_cremona-factorisation (H119E53)
- Scheme Graph Maps
- Tangent and Secant Varieties and Isomorphic Projections
- 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 (H119E58)
- ImageSystem(f,S,d) : MapSch,Sch,RngIntElt -> LinearSys
- Example Scheme_image-finder (H119E59)
- Geometric Restrictions: Points
- Geometric Restrictions: Schemes
- Geometric Restrictions: Affine Plane Curves with Non-ordinary Singularities
- LinearSystem(L, p, m, t) : LinearSys, Point, SeqEnum, SeqEnum[SeqEnum]) -> LinearSys
- LinearSystem(L, P, M, T) : LinearSys, Points, SeqEnum[SeqEnum], SeqEnum[SeqEnum[SeqEnum]]) -> LinearSys
- Example Scheme_tacnode (H119E67)
- Example Scheme_quadrifolium (H119E68)
- Example Scheme_cusp-sing (H119E69)
- Example Scheme_two-cubics (H119E70)
- Example Scheme_pencil-curves (H119E71)
- Geometric Restrictions: Trace on a Scheme
- Explicit Restrictions
- Basic Algebra of Linear Systems
- Linear Systems and Maps
- Divisors
- Divisor Groups
- Creation Of Divisors
- Ideals and Factorisations
- Basic Divisor Predicates
- Arithmetic of Divisors
- Further Divisor Properties
- IsCanonical(D) : DivSchElt -> BoolElt
- IsAnticanonical(D) : DivSchElt -> BoolElt
- IsCanonicalWithTwist(D) : DivSchElt -> BoolElt, RngIntElt
- IsPrincipal(D) : DivSchElt -> BoolElt, FldFunFracSchElt
- IsCartier(D) : DivSchElt -> BoolElt
- Example Scheme_divs-cartier-ex (H119E76)
- IsLinearlyEquivalent(D,E) : DivSchElt, DivSchElt -> BoolElt, FldFunFracSchElt
- BaseLocus(D) : DivSchElt -> Sch
- IntersectionNumber(D1,D2) : DivSchElt, DivSchElt-> FldRatElt
- SelfIntersection(D) : DivSchElt -> FldRatElt
- Degree(D) : DivSchElt -> FldRatElt
- IsNef(D) : DivSchElt -> BoolElt
- IsNefAndBig(D) : DivSchElt -> BoolElt
- NegativePrimeDivisors(D) : DivSchElt -> SeqEnum
- ZariskiDecomposition(D) : DivSchElt -> DivSchElt, DivSchElt
- Reduction(D,p) : DivSchElt, Any -> DivSchElt
- Riemann-Roch Spaces
- Isolated Points on Schemes
- Advanced Examples
- Bibliography
V2.28, 13 July 2023