Hilbert Series of Polarised Varieties
- Introduction
- Hilbert Series and Graded Rings
- Baskets of Singularities
- Point Singularities
- Curve Singularities
- Example: Curve Singularity with Transverse Type
- Creation of Curve Singularities
Curve(d,p,m): FldRatElt, GRPtS, FldRatElt → GRCrvSCurve(d,p,N): FldRatElt, GRPtS, RngIntElt → GRCrvSCurve(d,p,N,t): FldRatElt, GRPtS, RngIntElt, RngIntElt → GRCrvS
- Accessing the Key Data and Testing Equality
- Baskets of Singularities
- Curves and Dissident Points
- Generic Polarised Varieties
- Subcanonical Curves
- Creation of Subcanonical Curves
SubcanonicalCurve(g,d,Q): RngIntElt, RngIntElt, SeqEnum → GRCrvKIsSubcanonicalCurve(g,d,Q): RngIntElt, RngIntElt, SeqEnum → BoolElt, GRCrvKHilbertPolynomialOfCurve(g,m): RngIntElt, RngIntElt → RngUPolEltIsEffective(C): GRCrvK → BoolElt
- Catalogue of Subcanonical Curves
- K3 Surfaces
- Weil Polynomials
SetVerbose("WeilPolynomials", v): MonStgElt, RngIntEltHasAllRootsOnUnitCircle(f): RngUPolElt → BoolEltFrobeniusTracesToWeilPolynomials(tr, q, i, deg): SeqEnum, RngIntElt, RngIntElt, RngIntElt → SeqEnumWeilPolynomialToRankBound(f, q): RngUPolElt, RngIntElt → RngIntEltArtinTateFormula(f, q, h20): RngUPolElt, RngIntElt, RngIntElt → RngIntElt, RngIntEltWeilPolynomialOverFieldExtension(f, deg): RngUPolElt, RngIntElt → RngUPolEltCheckWeilPolynomial(f, q, h20): RngUPolElt, RngIntElt, RngIntElt → BoolElt- Example: Weil Polynomial of a Quartic Surface
- Example: Bounding Picard Rank with Fewer Point Counts
- Point Counting on Degree Two K3 Surfaces
- The K3 Database
- Searching the K3 Database
- Working with the K3 Database
K3Surface(D,i): DB, RngIntElt → GRK3K3Surface(D,Q,i): DB, SeqEnum, RngIntElt → GRK3K3Surface(D,g,i): DB, RngIntElt, RngIntElt → GRK3K3Surface(D,g1,g2,i): DB, RngIntElt, RngIntElt, RngIntElt → GRK3K3Surface(D,W): DB, SeqEnum → GRK3K3Surface(D,g,B): DB, RngIntElt, GRBskt → GRK3K3Surface(D,g,B): DB, RngIntElt, SeqEnum → GRK3
- Fano 3-folds
- Example: Fano 3-Folds with the Same Basket
- Creation: \(f=1,2\) or \(\ge 3\)
- A Preliminary Fano Database
FanoDatabase() → DBFano(D,i): DB, RngIntElt → GRFanoFano(D,f,i): DB, RngIntElt, RngIntElt → GRFanoFano(D,f,Q,i): DB, SeqEnum, RngIntElt → GRFano
- Calabi–Yau 3-folds
CalabiYau(p1,p2,B): RngIntElt, RngIntElt, GRBskt → GRCYFindN(X): GRCY → RngIntElt, RngIntEltFindN(p1,p2,B): RngIntElt, RngIntElt, GRBskt → RngIntElt, RngIntElt
- Building Databases
- Bibliography