Algebraic Surfaces
- Introduction
- Generalities
- General Surfaces
- Introduction
- Creation Functions
Surface(A,I): Sch, ModMPol → SrfcSurface(A,f): Sch, RngMPolElt → SrfcSurface(A,S): Sch, SeqEnum → SrfcRationalRuledSurface(P,n): Prj, RngIntElt → Srfc, MapSchRandomCompleteIntersection(P,ds): Prj, SeqEnum[RngIntElt] → SchKummerSurfaceScheme(C): CrvHyp → SrfcExample: Srfcs Creation
- Invariants
GeometricGenus(S): Srfc → RngIntEltPlurigenus(S,n): Srfc, RngIntElt → RngIntEltArithmeticGenus(S): Srfc → RngIntEltIrregularity(S): Srfc → RngIntEltChernNumber(S,n): Srfc, RngIntElt → RngIntEltMinimalChernNumber(S,n): Srfc, RngIntElt → RngIntEltHodgeNumber(S,i,j): Srfc, RngIntElt, RngIntElt → RngIntEltExample: Srfcs Invs
- Singularity Properties
- Kodaira-Enriques Classification
- Minimal Models
- Special Surfaces in Projective 4-space
- Desingularisation by Blow Up
- Introduction
- Accessor Functions
- Multiplicities, Intersections and Restricted Linear Systems
IntersectionMatrix(dsd): DesingData → MtrxMultiplicities(S,D): Srfc, DivSchElt → SeqEnumMultiplicities(S,D): Srfc, Sch → SeqEnumMultiplicities(S,dsd,D): Srfc, DesingData, Sch → SeqEnumMultiplicitiesAndIntersections(S,D): Srfc, DivSchElt → SeqEnum, SeqEnumMultiplicitiesAndIntersections(S,D): Srfc, Sch → SeqEnum, SeqEnumMultiplicitiesAndIntersections(S,dsd,D): Srfc, DesingData, Sch → SeqEnum, SeqEnumIntersectionNumberOfStrictTransforms(S,D1,D2): Srfc, Sch, Sch → RngIntEltSelfIntersectionOfStrictTransform(S,D): Srfc, Sch → RngIntEltLinearSystemDivisorRestriction(S,B,ms): Srfc, SeqEnum[RngMPolElt], SeqEnum[SeqEnum] → SeqEnum
- Canonical Divisor Functionality
DifferentialMultiplicities(dsd): DesingData → SeqEnumBasisOfHolomorphicTwoForms(S): Srfc → SeqEnum[RngMPolElt], RngMPolElt, SeqEnum, ListPluriCanonicalBasis(S,m): Srfc, RngIntElt → SeqEnumExample: Srfc Sing Dp4 2frms N PluriFirstChernClassOfDesingularization(S): Srfc → RngIntElt[RngMPolElt]CanonicalIntersection(S,D): Srfc, DivSchElt → RngIntEltCanonicalIntersection(S,D): Srfc, Sch → RngIntEltCanonicalIntersection(dsd,i): DesingData, RngIntElt → RngIntEltExceptionalDivisors(S): Srfc → List, ListExample: Srfc Ex Divs Intr
- Dual Resolution Graphs
DualResolutionGraph(dsd): DesingData → GrphMultUnd, SeqEnum, SeqEnumMinimalDualResolutionGraph(dsd): DesingData → GrphMultUnd, SeqEnum, SeqEnum, MtrxExample: Srfc Dual Desing Grph 1Example: Srfc Dual Desing Grph 2SelfIntersection(dsd,i): DesingData, RngIntElt → RngIntEltSelfIntersection(G,i): GrphMultUnd, RngIntElt → RngIntEltCanonicalIntersection(dsd,i): DesingData, RngIntElt → RngIntEltCanonicalIntersection(G,i): GrphMultUnd, RngIntElt → RngIntEltArithmeticGenus(dsd,i): DesingData, RngIntElt → RngIntEltArithmeticGenus(G,i): GrphMultUnd, RngIntElt → RngIntEltGenus(dsd,i): DesingData, RngIntElt → RngIntEltGenus(G,i): GrphMultUnd, RngIntElt → RngIntEltLocalIntersectionNumber(e): GrphEdge → RngIntElt- Point counting on resolution fibres
- Extended Examples
- Surfaces in \({\mathbb{P}}^3\)
- Del Pezzo Surfaces
- Introduction
- Creation of General Del Pezzos
- Some Auxiliary Intrinsics
- Parametrization of Del Pezzo Surfaces
SetVerbose("ParamDP", v): MonStgElt, RngIntEltParametrizeDegree9DelPezzo(X): Sch → BoolElt, MapIsoSchParametrizeDegree8DelPezzo(X): Sch → BoolElt, MapSchExample: Del Pezzo Ex 8ParametrizeDegree7DelPezzo(X): Sch → MapIsoSchParametrizeDegree6DelPezzo(X): Sch → BoolElt, MapIsoSchDegree6DelPezzoType2_1(K,pt): FldNum, Pt → SchDegree6DelPezzoType2_2(K,pt): FldNum, Pt → SchDegree6DelPezzoType2_3(K,pt): FldNum, Pt → SchDegree6DelPezzoType3(K,pt): FldNum, Pt → SchDegree6DelPezzoType4(K,K1,pt): FldNum, Fld, Pt → SchDegree6DelPezzoType6(K,pt): FldNum, Pt → SchParametrizeDelPezzoDeg6(X): Sch → BoolElt, MapIsoSchExample: Del Pezzo Ex 6ParametrizeDegree5DelPezzo(X): Sch → MapIsoSchParametrizeSingularDegree3DelPezzo(X,P2): Sch, Prj → BoolElt, MapIsoSchParametrizeSingularDegree4DelPezzo(X,P2): Sch, Prj → BoolElt, MapIsoSchExample: Del Pezzo Ex 3 Sing
- Minimization and Reduction of Surfaces
MinimizeCubicSurface(f, p): RngMPolElt, RngIntElt → RngMPolElt, MtrxReduceCubicSurface(f): RngMPolElt → RngMPolElt, MtrxMinimizeReduceCubicSurface(f): MPolElt → RngMPolElt, MtrxMinimizeDeg4delPezzo(f, p): SeqEnum, RngIntElt → SeqEnum, MtrxMinimizeReduceDeg4delPezzo(f): SeqEnum → SeqEnum, MtrxMinimizeReduce(S): SrfDelPezzo → SrfDelPezzo, MtrxExample: dp34
- Cubic Surfaces over Finite Fields
- Construction of Cubic Surfaces
- Invariant Theory of Cubic Surfaces
- The Pentahedron of a Cubic Surface
- Degree \(2\) K3 Surfaces
- Introduction
- Creation Functions
- Split Divisors and Intersection Matrices
- Elliptic Fibrations
KodairaConfigurations(imat): Mtrx → ListExample: Deg2K3 Int PairingsEllipticFibrationRRSpaceDeg2K3(S,divlst,exdivlst): Srfc, SeqEnum, SeqEnum → SeqEnum[RngMPolElt], RngMPolEltEllipticGeneralFibreDeg2K3(S,B,secs): Srfc, SeqEnum[RngMPolElt], SeqEnum[Sch] → Crv, SeqEnum, SeqEnumExample: Deg2K3 Ell Fib Ex
