Introduction and First Examples
Ambient Spaces
Example Scheme_EXAMPLE (H112E1)
Schemes
Example Scheme_ex2 (H112E2)
Rational Points
Example Scheme_ex3 (H112E3)
Projective Closure
Example Scheme_ex4 (H112E4)
Maps
Example Scheme_ex5 (H112E5)
Linear Systems
Example Scheme_ex6 (H112E6)
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 (H112E7)
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
SegreProduct(Xs) : SeqEnum[Sch] -> Sch, SeqEnum
SegreEmbedding(X) : Sch -> Sch, MapIsoSch
Example Scheme_sch:segre-embedding (H112E8)
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 (H112E9)
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 (H112E10)
Constructing Schemes
Scheme(X,f) : Sch,RngMPolElt -> Sch
Cluster(X,f) : Sch,RngMPolElt -> Clstr
Example Scheme_schemes-creation (H112E11)
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
RemoveLinearRelations(X) : Sch -> Sch, MapIsoSch
BlowUp(X,Y) : Sch, Sch -> Sch, MapSch
Example Scheme_remove (H112E12)
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
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 (H112E13)
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 (H112E14)
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
RationalPointsByFibration(X) : Sch -> SetIndx
Random(S) : SetPt -> Pt
HasNonsingularPoint(X) : Sch -> BoolElt,Pt
Example Scheme_scheme-points (H112E15)
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 (H112E16)
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
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
Example Scheme_an-hyp-sing-ex (H112E17)
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
IsCohenMacaulay(X) : Sch -> BoolElt
Example Scheme_schemes-prime-components (H112E18)
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 (H112E19)
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 (H112E20)
Example Scheme_projective-closure-incorrect (H112E21)
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 (H112E22)
Example Scheme_anf2 (H112E23)
IsLocallySolvable(X, p) : Sch, RngOrdIdl -> BoolElt, Pt
Example Scheme_anf-local-solv (H112E24)
LiftPoint(P, n) : Pt, RngIntElt -> Pt
Example Scheme_anf_lift (H112E25)
Searching for Points
PointSearch(S,H : parameters) : Sch[FldRat], RngIntElt -> SeqEnum
Example Scheme_point-count (H112E26)
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 (H112E27)
Example Scheme_map-fnfld (H112E28)
Example Scheme_map-frobenius (H112E29)
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 (H112E30)
Elimination(X,V) : Sch,SeqEnum -> Sch
Inverse(f) : MapSch -> MapSch
IsInvertible(f) : MapSch -> Bool, MapSch
HasKnownInverse(f) : MapSch -> Bool
Example Scheme_map_creation_inv (H112E31)
g * f : MapSch,MapSch -> MapSch
Components(f) : Map -> [Map]
Example Scheme_hom-spaces (H112E32)
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 (H112E33)
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 (H112E34)
Maps and Schemes
Pullback(f, X) : MapSch, Sch -> Sch
Image(f) : MapSch -> Sch
Image(f,X,d) : MapSch,Sch,RngIntElt -> []
Example Scheme_map-image1 (H112E35)
Example Scheme_map-image2 (H112E36)
BaseScheme(f) : MapSch -> Sch
BasePoints(f) : MapSch -> SetEnum
Example Scheme_map-base-points (H112E37)
Example Scheme_scroll-map-base-points (H112E38)
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 (H112E39)
Automorphisms
Automorphism(X,F) : Sch,SeqEnum -> MapAutSch
IdentityAutomorphism(X) : Sch -> MapAutSch
IsEndomorphism(f) : MapSch -> BoolElt
IsAutomorphism(f) : MapSch -> BoolElt,AutSch
Example Scheme_automorphism-construction (H112E40)
Example Scheme_aut-aff-jac (H112E41)
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 (H112E42)
Automorphism(A,p) : Sch, RngMPolElt -> IsoSch
AffineDecomposition(f) : MapSch -> MapSch,MapSch
Example Scheme_decompose-automorphism (H112E43)
NagataAutomorphism(A) : Aff -> MapSch
Projectivity(A,M) : Aff,Mtrx -> MapAutSch
Example Scheme_projectivity (H112E44)
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 (H112E45)
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 (H112E46)
QuadraticTransformation(P) : Prj -> MapSch
QuadraticTransformation(X) : Sch -> Sch, MapIsoSch
Example Scheme_cremona-factorisation (H112E47)
Scheme Graph Maps
SchemeGraphMap(X, Y, I) : Sch, Sch, RngMPol -> MapSchGrph
SchemeGraphMapToSchemeMap(f) : MapSchGrph -> MapSch
IsInvertible(f) : MapSchGrph -> BoolElt, MapSchGrph
Example Scheme_graph_maps (H112E48)
Tangent and Secant Varieties and Isomorphic Projections
Tangent Varieties
TangentVariety(X) : Sch -> Sch
IsInTangentVariety(X,P) : Sch,Pt -> BoolElt
Example Scheme_TangentVariety (H112E49)
Secant Varieties
SecantVariety(X) : Sch -> Sch
IsInSecantVariety(X,P) : Sch,Pt -> BoolElt
Example Scheme_SecantVariety (H112E50)
Isomorphic Projection to Subspaces
IsomorphicProjectionToSubspace(X) : Sch -> Sch, MapSch
EmbedPlaneCurveInP3(C) : Crv -> Sch, MapSch
Example Scheme_EmbeddingACurve (H112E51)
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 (H112E52)
ImageSystem(f,S,d) : MapSch,Sch,RngIntElt -> LinearSys
Example Scheme_image-finder (H112E53)
Geometrical Restrictions
LinearSystem(L,p) : LinearSys,Pt -> LinearSys
LinearSystem(L,p,m) : LinearSys,Pt,RngIntElt -> LinearSys
Example Scheme_subsystems (H112E54)
LinearSystem(L,X) : LinearSys,Sch -> LinearSys
LinearSystemTrace(L,X) : LinearSys,Sch -> LinearSys
Example Scheme_trace (H112E55)
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 (H112E56)
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 (H112E57)
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
Divisor Groups
DivisorGroup(X) : Sch -> DivSch
Variety(G) : DivSch -> Sch
G1 eq G2: DivSch, DivSch -> BoolElt
Creation Of Divisors
Divisor(X,f) : Sch, FldFunFracSchElt -> DivSchElt
Divisor(X,Q) : Sch, SeqEnum -> DivSchElt
HyperplaneSectionDivisor(X) : Sch -> DivSchElt
ZeroDivisor(X) : Sch -> DivSchElt
CanonicalDivisor(X) : Sch -> DivSchElt
SheafToDivisor(S) : ShfCoh -> DivSchElt
RoundDownDivisor(D) : DivSchElt -> DivSchElt
RoundUpDivisor(D) : DivSchElt -> DivSchElt
FractionalPart(D) : DivSchElt -> DivSchElt
IntegralMultiple(D) : DivSchElt -> DivSchElt,RngIntElt
Ideals and Factorisations
Ideal(D) : DivSchElt -> RngMPol
Support(D) : DivSchElt -> Sch
IdealOfSupport(D) : DivSchElt -> RngMPol
SignDecomposition(D) : DivSchElt -> DivSchElt, DivSchElt
IdealFactorisation(D) : DivSchElt -> SeqEnum
CombineIdealFactorisation(~D) : DivSchElt ->
ComputeReducedFactorisation(~D) : DivSchElt ->
ComputePrimeFactorisation(~D) : DivSchElt ->
Multiplicity(D,E) : DivSchElt, DivSchElt -> FldRatElt
Basic Divisor Predicates
IsZeroDivisor(D) : DivSchElt -> BoolElt
IsIntegral(D) : DivSchElt -> BoolElt
IsEffective(D) : DivSchElt -> BoolElt
IsPrime(D) : DivSchElt -> BoolElt
IsFactorisationPrime(D) : DivSchElt -> BoolElt
IsDivisible(D) : DivSchElt -> BoolElt, RngIntElt
Arithmetic of Divisors
D1 + D2 : DivSchElt, DivSchElt -> DivSchElt
n * D : RngIntElt, DivSchElt -> DivSchElt
D1 eq D2 : DivSchElt, DivSchElt -> BoolElt
Further Divisor Properties
IsCanonical(D) : DivSchElt -> BoolElt
IsAnticanonical(D) : DivSchElt -> BoolElt
IsCanonicalWithTwist(D) : DivSchElt -> BoolElt, RngIntElt
IsPrincipal(D) : DivSchElt -> BoolElt, FldFunFracSchElt
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
Riemann-Roch Spaces
Sheaf(D) : DivSchElt -> ShfCoh
RiemannRochBasis(D) : DivSchElt -> SeqEnum
RiemannRochCoordinates(f,D) : Any, DivSchElt -> BoolElt, SeqEnum
IsLinearSystemNonEmpty(D) : DivSchElt -> BoolElt, DivSchElt
Isolated Points on Schemes
LinearElimination(S) : Sch -> Map
IsolatedPointsFinder(S,P) : Sch, SeqEnum -> List
IsolatedPointsLifter(S,P) : Sch, SeqEnum -> BoolElt, Pt
IsolatedPointsLiftToMinimalPolynomials(S,P) : Sch, SeqEnum -> BoolElt, SeqEnum
Example Scheme_ec-large-int-pts (H112E58)
Example Scheme_halls-conjecture (H112E59)
Example Scheme_random-linear-scheme (H112E60)
Example Scheme_mathieu-monodromy (H112E61)
A Pair of Twisted Cubics
Example Scheme_twisted-cubics (H112E62)
Curves in Space
Example Scheme_curves-in-space (H112E63)
Bibliography
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Version: V2.19 of
Wed Apr 24 15:09:57 EST 2013