Schemes
- Introduction and First Examples
- Ambients
- Affine and Projective Spaces
AffineSpace(k,n): Rng, RngIntElt → AffProjectiveSpace(k,n): Rng, RngIntElt → PrjProjectiveSpace(k,W): Rng, [RngIntElt] → PrjProjectiveLine(k): RngIntRes → SetIndx, UserProgramProjectiveLine(k): RngOrdRes → SetIndx, UserProgramProjectiveLineProcess(V): ModTupFld[FldFin] → ProcPLProjectiveLineProcess(k, n): FldFin, RngIntElt → ProcPL# P: ProcPL → RngIntEltNext(P): ProcPL → ModTupFldEltAffineSpace(R): RngMPol → AffSpec(R): RngMPol → AffProjectiveSpace(R): RngMPol → PrjProj(R): RngMPol → PrjAssignNames(~A,N): Sch, [MonStgElt]A . i: Sch, RngIntElt → RngMPolEltName(A,i): Sch, RngIntElt → RngMPolEltExample: Affine Space NamesA eq B: Sch, Sch → BoolElt
- Scrolls and Products
DirectProduct(A,B): Sch, Sch → Sch, SeqEnumRuledSurface(k,a,b): Rng, RngIntElt, RngIntElt → PrjScrlRuledSurface(k,n): Rng, RngIntElt → PrjScrlAbsoluteRationalScroll(k,N): Rng, SeqEnum → PrjScrlProductProjectiveSpace(k,N): Rng, SeqEnum → PrjScrlSegreProduct(Xs): SeqEnum[Sch] → Sch, SeqEnumSegreEmbedding(X): Sch → Sch, MapIsoSchExample: Sch:segre Embedding
- Functions and Homogeneity on Ambient Spaces
- Prelude to Points
A ! [a,b,...]: Sch, [RngElt] → PtA(L) ! [a,b,...]: SetPt, [RngElt] → PtExample: Schemes Points Example1Origin(A): Aff → PtSimplex(A): Prj → SeqEnumCoordinates(p): Pt → SeqEnump[i]: Pt, RngIntElt → RngEltCoordinate(p,i): Pt, RngIntElt → SeqEnump @ f: Pt, FldFunFracSchElt → RngEltf(p): Pt, FldFunFracSchElt → RngEltEvaluate(f, p): RngElt, Pt → RngEltExample: Evaluate Funfld Example
- Constructing Schemes
Scheme(X,f): Sch, RngMPolElt → SchScheme(X,F): Sch, SeqEnum → SchScheme(X,I): Sch, RngMPol → SchScheme(X,Q): Sch, RngMPolRes → SchCluster(X,f): Sch, RngMPolElt → ClstrCluster(X,F): Sch, SeqEnum → ClstrCluster(X,I): Sch, RngMPol → ClstrCluster(X,Q): Sch, RngMPolRes → ClstrExample: Schemes CreationSpec(R): RngMPolRes → Sch, AffProj(R): RngMPolRes → Sch, PrjEmptyScheme(X): Sch → SchEmptySubscheme(X): Sch → Sch, MapSchX meet Y: Sch, Sch → SchIntersection(X,Y): Sch, Sch → SchX join Y: Sch, Sch → SchUnion(X,Y): Sch, Sch → Sch& join S: [Sch] → SchDifference(X, Y): Sch, Sch → SchComplement(X, Y): Sch, Sch → SchRemoveLinearRelations(X): Sch → Sch, MapIsoSchBlowup(X,Y): Sch, Sch → Sch, MapSchBlowup(X,p): Sch, Pt → Sch, MapSchLocalBlowUp(X,Y): Sch, Sch → SeqEnumExample: removeExample: Sch Blowup ExSaturate(~X): SchAssignNames(~X,N): Sch, SeqEnumX . i: Sch, RngIntElt → RngMPolEltName(X,i): Sch, RngIntElt → RngMPolElt
- Different Types of Scheme
- Basic Attributes of Schemes
- Function Fields and their Elements
Scheme(F): FldFunFracSch → SchIntegerRing(F): RngFrac → RngIntegers(F): RngFrac → RngAssignNames(~F, S): RngFrac, [MonStgElt]F ! g: FldFunFracSch, RngElt → FldFunFracSchEltF . i: FldFunFracSch, RngIntElt → FldFunFracSchEltProjectiveFunction(f): FldFunFracSchElt → FldFracEltProjectiveRationalFunction(f): FldFunFracSchElt → FldFunRatMEltRestrictionToPatch(f, Xi): FldFunFracSchElt, Sch → FldFracEltNumerator(f): RngFracElt → RngEltDenominator(f): RngFracElt → RngEltf * g: RngFracElt, RngFracElt → RngFracEltf + g: RngFracElt, RngFracElt → RngFracEltf - g: RngFracElt, RngFracElt → RngFracElt- f: RngFracElt → RngFracEltf / g: RngFracElt, RngFracElt → RngFracEltf ^ n: RngFracElt, RngIntElt → RngFracEltf eq g: RngFracElt, RngFracElt → BoolEltIsZero(f): RngFracElt → BoolEltIsOne(f): RngFracElt → BoolEltIsMinusOne(f): RngFracElt → BoolEltIsUnit(f): RngFracElt → BoolEltIntegralSplit(f, X): FldFunFracSchElt, Sch → RngMPolElt, RngMPolEltNumerator(f, X): FldFunFracSchElt, Sch → MPolEltDenominator(f, X): FldFunFracSchElt, Sch → MPolEltExample: Scheme Fld Fun EltRestriction(f, Y): FldFunFracSchElt, Sch → FldFunFracSchEltGenericPoint(X): Sch → Pt
- Rational Points and Point Sets
X(L): Sch, Rng → SetPtPointSet(X,L): Sch, Rng → SetPtX(m): Sch, Map → SetPtPointSet(X,m): Sch, Map → SetPtP eq Q: SetPt, SetPt → BoolEltScheme(P): SetPt → SchCurve(P): SetPt → CrvRing(P): SetPt → RngRingMap(P): SetPt → MapX ! Q: Sch, SeqEnum → PtX(L) ! Q: SetPt, SeqEnum → Ptp eq q: Pt, Pt → BoolEltp in X: Pt, Sch → BoolEltScheme(p): Pt → SchCurve(p): Pt → CrvQ in X: SeqEnum, Sch → BoolEltS subset X: Setq, Sch → BoolEltRationalPoints(X): Sch → SetIndxRationalPoints(X,L): Sch, Rng → SetIndxPoints(X): Sch → SetIndxPoints(X,L): Sch, Rng → SetIndxRationalPointsByFibration(X): Sch → SetIndxRandom(S): SetPt → PtHasNonsingularPoint(X): Sch → BoolElt, PtHasNonsingularPoint(X,L): Sch, FldFin → BoolElt, PtPointsModPN(S, P, N): Sch, ., RngIntElt → SetIndxPointsModPN(S, M): Sch, RngIntElt → SetIndxPointsModPN(S, M): Sch, RngOrdIdl → SetIndxExample: Scheme Points
- Zero-dimensional Schemes
Cluster(p): Pt → ClstrCluster(X, p): Sch, Pt → Clstr;Cluster(S): SetEnum → ClstrCluster(S): SetIndx → ClstrCluster(X, S): Sch, SetEnum → ClstrCluster(X, S): Sch, SetIndx → ClstrRationalPoints(Z): Sch → SetEnumRationalPoints(Z,L): Sch, Rng → SetEnumPointsOverSplittingField(Z): Clstr → SetEnumHasPointsOverExtension(X): Sch → BoolEltHasPointsOverExtension(X,L): Sch, Rng → BoolEltDegree(Z): Clstr → RngIntEltExample: Cluster Degree5
- Local Geometry of Schemes
- Point Conditions
- Point Computations
- Analytically Hypersurface Singularities
IsHypersurfaceSingularity(p,prec): Pt, RngIntElt → BoolElt, RngMPolElt, SeqEnum, RecHypersurfaceSingularityExpandFurther(dat,prec,R): Rec, RngIntElt, RngMPol → RngMPolEltHypersurfaceSingularityExpandFunction(dat,f,prec,R): Rec, FldFunRatMElt, RngIntElt, RngMPol → RngMPolElt, RngMPolEltMilnorNumberAnalyticHypersurface(dat): Rec → RngIntEltTjurinaNumberAnalyticHypersurface(dat): Rec → RngIntEltExample: An Hyp Sing Ex
- Classification and Normal Forms of Singularities
- Global Geometry of Schemes
- Base Change for Schemes
BaseChange(A,K): Sch, Rng → SchBaseExtend(A,K): Sch, Rng → SchBaseChange(A,m): Sch, Map → SchBaseExtend(A,m): Sch, Map → SchBaseChange(F,K): SeqEnum, Rng → SeqEnumBaseExtend(F,K): SeqEnum, Rng → SeqEnumBaseChange(F,m): SeqEnum, Rng → SeqEnumBaseExtend(F,m): SeqEnum, Rng → SeqEnumBaseChange(X,A): Sch, Sch → SchBaseExtend(X,A): Sch, Sch → SchBaseChange(X,A,m): Sch, Sch, Map → SchBaseExtend(X,A,m): Sch, Sch, Map → SchBaseChange(X, n): Sch, RngIntElt → SchBaseExtend(X, n): Sch, RngIntElt → SchExample: Base Change Schemes
- Affine Patches and Projective Closure
ProjectiveClosure(X): Sch → SchAffinePatch(X,i): Sch, RngIntElt → SchAffinePatch(X,p): Sch, Pt → Sch, PtIsStandardAffinePatch(A): Aff → BoolElt, RngIntEltNumberOfAffinePatches(X): Sch → BoolEltHasAffinePatch(X, i): Sch, RngIntElt → BoolEltWeightedAffinePatch(P, i): Prj, RngIntElt → Sch, MapIsoSchExample: Projective ClosureExample: Projective Closure IncorrectExample: Weighted PatchesHyperplaneAtInfinity(X): Sch → SchProjectiveClosureMap(A): Aff → MapSchPCMap(A): Aff → MapSchAffineDecomposition(P): Prj → [MapSch], PtCentredAffinePatch(S, p): Sch, Pt → Sch, MapSch
- Arithmetic Properties of Schemes and Points
- Height
- Restriction of Scalars
RestrictionOfScalars(S, F): Sch, Fld → Sch, MapSch, UserProgram, MapWeilRestriction(S, F): Sch, Fld → Sch, MapSch, UserProgram, Map
- Local Solubility
IsEmpty(Xm): SetPt → BoolElt, PtExample: anf1Example: anf2IsLocallySolvable(X, p): Sch, RngOrdIdl → BoolElt, PtIsLocallySolvable(X, p): Sch, RngIntElt → BoolElt, PtIsLocallySolvable(X,pl): Sch, PlcFunElt → BoolElt, PtExample: Anf Local SolvLiftPoint(P, n): Pt, RngIntElt → PtLiftPoint(F, d, P, n): [RngMPolElt], RngIntElt, [FldLocElt], RngIntElt → SeqEnumExample: Anf Lift
- Searching for Points
- Reduction Mod \(p\)
- Maps between Schemes
- Creation of Maps
map< X -> Y | F >: Sch, Sch, SeqEnum → MapSchmap< X -> Y | F, G >: Sch, Sch, SeqEnum → MapSchmap< X -> Y | u, F >: Sch, Sch, Map, SeqEnum → MapSchiso< X -> Y | F, G >: Sch, Sch, SeqEnum, SeqEnum → MapAutSchExample: Map CreationExample: Map FnfldExample: Map FrobeniusIdentityMap(X): Sch → MapSchConstantMap(X,Y,p): Sch, Sch, Pt → MapSchmap< X -> Y | Q >: Sch, Sch, SeqEnum → MapSchProjection(X,Y): Prj, Prj → MapSchProjection(X, Q): Sch, Prj → Sch, MapSchProjection(X): Sch → Sch, MapSchProjection(X, p): Sch, Pt → Sch, MapSchProjectionFromNonsingularPoint(X,p): Sch, Pt → Sch, MapSch, SchProjectiveMap(L, Y): [FldFunFracSchElt], Sch → MapSchProjectiveMap(L): [FldFunFracSchElt] → MapSchProjectiveMap(f, Y): FldFunFracSchElt, Sch → MapSchProjectiveMap(f): FldFunFracSchElt → MapSchExample: Map Creation PrjElimination(X,V): Sch, SeqEnum → SchInverse(f): MapSch → MapSchIsInvertible(f): MapSch → Bool, MapSchHasKnownInverse(f): MapSch → BoolExample: Map Creation Invg * f: MapSch, MapSch → MapSchComponents(f): Map → [Map]Example: Hom SpacesRestriction(f,X,Y): MapSch, Sch, Sch → MapSchExpand(phi): MapSch → MapSchExtend(phi): MapSch → MapSchPrune(phi): MapSch → MapSchNormalization(phi): MapSch → MapSchNormalisation(phi): MapSch → MapSchExample: Map Creation Comp AltImproveParametrization(p): MapSch → MapSchExample: Improve Prm Ex
- Basic Attributes
- Maps and Points
- Maps and Schemes
Pullback(f, X): MapSch, Sch → SchX @@ f: Sch, MapSch → SchImage(f): MapSch → Schf(X): Sch, MapSch → SchImage(f,X,d): MapSch, Sch, RngIntElt → []Example: Map Image1Example: Map Image2BaseScheme(f): MapSch → SchBasePoints(f): MapSch → SetEnumBasePoints(f,L): MapSch, Fld → SetEnumExample: Map Base PointsExample: Scroll Map Base Points
- Maps and Closure
ProjectiveClosure(f): MapSch → MapSchMakeProjectiveClosureMap(A, P, S): Aff, Prj, SeqEnumMakePCMap(A, P, S): Aff, Prj, SeqEnumMakeProjectiveClosureMap(m): MapMakePCMap(m): MapRestrictionToPatch(f,j): MapSch, RngIntElt → MapSchRestrictionToPatch(f,i,j): MapSch, RngIntElt, RngIntElt → MapSchExample: Map Patches
- Automorphisms
Automorphism(X,F): Sch, SeqEnum → MapAutSchIdentityAutomorphism(X): Sch → MapAutSchIdentityMap(X): Sch → MapAutSchIsEndomorphism(f): MapSch → BoolEltIsAutomorphism(f): MapSch → BoolElt, AutSchExample: Automorphism ConstructionExample: Aut Aff Jac- Affine Automorphisms
Automorphism(A,F): Sch, SeqEnum → MapSchAutomorphism(A,M): Sch, Mtrx → MapIsoSchTranslation(A,p): Sch, Pt → MapSchPermutationAutomorphism(A, g): Sch, GrpPermElt → MapIsoSchAutomorphism(A, g): Sch, GrpPermElt → MapIsoSchExample: Aut Aff PermAutomorphism(A,p): Sch, RngMPolElt → IsoSchAffineDecomposition(f): MapSch → MapSch, MapSchExample: Decompose AutomorphismNagataAutomorphism(A): Aff → MapSchProjectivity(A,M): Aff, Mtrx → MapAutSchExample: projectivity
- Projective Automorphisms
Automorphism(P,F): Prj, SeqEnum → MapSchMatrix(f): MapSch → MtrxAutomorphism(P,M): Sch, Mtrx → MapSchAut(P): Prj → PowAutSchAutomorphismGroup(P): Prj → GrpMat, MapExample: Projective Automorphism GroupTranslationOfSimplex(P,Q): Prj, [Pt] → MapSchTranslation(P,Q): Prj, [Pt] → MapSchTranslation(P,p,q): Prj, Pt, Pt → MapSchTranslation(X,p): Sch, Pt → MapSchExample: translationQuadraticTransformation(P): Prj → MapSchQuadraticTransformation(P,Q): Prj, SeqEnum → MapSchQuadraticTransformation(X): Sch → Sch, MapIsoSchQuadraticTransformation(X,Q): Sch, [Pt] → Sch, MapIsoSchExample: Cremona Factorisation
- Scheme Graph Maps
- Tangent and Secant Varieties and Isomorphic Projections
- Linear Systems
- Creation of Linear Systems
- Explicit Creation
LinearSystem(P, d): Sch, RngIntElt → LinearSysLinearSystem(P, d): Sch, [RngIntElt] → LinearSysLinearSystem(P, F): Sch, SeqEnum[RngMPolElt] → LinearSysMonomialsOfWeightedDegree(X, D): Sch, [RngIntElt] → SetIndxExample: Linsys ConstructionImageSystem(f,S,d): MapSch, Sch, RngIntElt → LinearSysExample: Image Finder
- Geometric Restrictions: Points
LinearSystem(L, p): LinearSys, Point → LinearSysLinearSystem(L, P): LinearSys, SeqEnum[Point) → LinearSysExample: subsystemsLinearSystem(L, p, m): LinearSys, Point, RngIntElt → LinearSysLinearSystem(L, P, M): LinearSys, SeqEnum[Point), SeqEnum(RngIntElt) → LinearSysExample: Subsystems MultExample: Subsystems Speed
- Geometric Restrictions: Schemes
- Geometric Restrictions: Affine Plane Curves with Non-ordinary Singularities
LinearSystem(L, p, m, t): LinearSys, Point, SeqEnum, SeqEnum[SeqEnum]) → LinearSysLinearSystem(L, P, M, T): LinearSys, Points, SeqEnum[SeqEnum], SeqEnum[SeqEnum[SeqEnum]]) → LinearSysExample: tacnodeExample: quadrifoliumExample: Cusp SingExample: Two CubicsExample: Pencil Curves
- Geometric Restrictions: Trace on a Scheme
- Explicit Restrictions
- Basic Algebra of Linear Systems
- Tests for Linear Systems
- Geometrical Properties
- Linear Algebra
CoefficientSpace(L): LinearSys → ModTupFldCoefficientMap(L): LinearSys → ModTupFldEltPolynomialMap(L): LinearSys → RngMPolEltComplement(L,K): LinearSys, LinearSys → LinearSysComplement(L,X): LinearSys, Sch → LinearSysExample: Creation By SubspaceL meet K: LinearSys, LinearSys → LinearSysIntersection(L,K): LinearSys, LinearSys → LinearSysX in L: Sch, LinearSys → BoolEltf in L: RngMPolElt, LinearSys → BoolEltK subset L: LinearSys, LinearSys → BoolEltIsSubsystem(L,K): LinearSys, LinearSys → BoolElt
- Linear Systems and Maps
- Divisors
- Divisor Groups
- Creation Of Divisors
Divisor(X,f): Sch, FldFunFracSchElt → DivSchEltDivisor(X,f): Sch, FldFunRatMElt → DivSchEltDivisor(X,f): Sch, RngMPolElt → DivSchEltDivisor(X,Q): Sch, SeqEnum → DivSchEltDivisor(X,Y): Sch, Sch → DivSchEltDivisor(X,I): Sch, RngMPol → DivSchEltHyperplaneSectionDivisor(X): Sch → DivSchEltZeroDivisor(X): Sch → DivSchEltCanonicalDivisor(X): Sch → DivSchEltSheafToDivisor(S): ShfCoh → DivSchEltRoundDownDivisor(D): DivSchElt → DivSchEltRoundUpDivisor(D): DivSchElt → DivSchEltFractionalPart(D): DivSchElt → DivSchEltIntegralMultiple(D): DivSchElt → DivSchElt, RngIntEltEffectiveHypersurfaceTwist(D): DivSchElt → DivSchElt, RngMPolElt
- Ideals and Factorisations
- Basic Divisor Predicates
- Arithmetic of Divisors
- Further Divisor Properties
IsCanonical(D): DivSchElt → BoolEltIsAnticanonical(D): DivSchElt → BoolEltIsCanonicalWithTwist(D): DivSchElt → BoolElt, RngIntEltIsPrincipal(D): DivSchElt → BoolElt, FldFunFracSchEltIsCartier(D): DivSchElt → BoolEltExample: Divs Cartier ExIsLinearlyEquivalent(D,E): DivSchElt, DivSchElt → BoolElt, FldFunFracSchEltBaseLocus(D): DivSchElt → SchIsBasePointFree(D): DivSchElt → BoolEltIsMobile(D): DivSchElt → BoolEltIntersectionNumber(D1,D2): DivSchElt, DivSchElt → FldRatEltSelfIntersection(D): DivSchElt → FldRatEltDegree(D): DivSchElt → FldRatEltDegree(D,H): DivSchElt, DivSchElt → FldRatEltIsNef(D): DivSchElt → BoolEltIsNefAndBig(D): DivSchElt → BoolEltNegativePrimeDivisors(D): DivSchElt → SeqEnumZariskiDecomposition(D): DivSchElt → DivSchElt, DivSchEltReduction(D,p): DivSchElt, Any → DivSchElt
- Riemann-Roch Spaces
- Isolated Points on Schemes
- Advanced Examples
