Geometrical Properties of Cones and Polyhedra

IsSingular(C) : TorCon -> BoolElt

Return true if and only if the affine variety associated with the cone C is singular.

IsNonsingular(C) : TorCon -> BoolElt

Return true if and only if the affine variety associated with the cone C is nonsingular.

IsSmooth(P) : TorPol -> BoolElt

Return true if and only if the polyhedron P is a smooth polytope.

IsGorenstein(C) : TorCon -> BoolElt

Return true if and only if the cone C has (the primitive points on its) rays contained in an affine hyperplane that is defined by an integral equation.

IsReflexive(P) : TorPol -> BoolElt

Return true if and only if the polyhedron P is reflexive; i.e. P and its dual P^v are both integral polytopes.

IsQGorenstein(C) : TorCon -> BoolElt

Return true if and only if the cone C has (the primitive points on its) rays contained in an affine hyperplane.

GorensteinIndex(C) : TorCon -> RngIntElt,TorLatElt

The Gorenstein index of the affine variety corresponding to the cone C together with the dual vector determining the equation of the hyperplane. (It is an error if C is not Q-Gorenstein.)

GorensteinIndex(P) : TorPol -> RngIntElt

The Gorenstein index of the lattice polytope P; i.e. the smallest positive integer k such that kP^v is an integral polytope.

IsIsolated(C) : TorCon -> BoolElt

Return true if and only if the singularity of the affine variety associated to the cone C is isolated.

IsQFactorial(C) : TorCon -> BoolElt

IsSimplicial(P) : TorPol -> BoolElt

Return true if and only if the cone C or polytope P is simplicial.

IsTerminal(C) : TorCon -> BoolElt

Return true if and only if the singularity of the affine variety associated to the cone C is (at worst) terminal.

IsCanonical(C) : TorCon -> BoolElt

Return true if and only if the singularity of the affine variety associated to the cone C is (at worst) canonical.

IsFano(P) : TorPol -> BoolElt

Return true if and only if the polyhedron P is a Fano polytope (i.e. of maximum dimension in the ambient lattice, containing the origin strictly in its interior, with primitive lattice vertices).

> L := ToricLattice(3);
> v := L ! [1/5,2/5,3/5];
> LL,emb := AddVectorToLattice(v);
> C := PositiveQuadrant(L);
> CC := Image(emb,C);
> CC;
Cone CC with 3 generators:
    (1, 0, 0),
    (0, 1, 0),
    (3, 1, 5)

> IsTerminal(CC);
true
> GorensteinIndex(CC);
5 (1, 1, -3/5)

> F := Fan(CC);
> F;
Fan F with 3 rays:
    (0, 1, 0),
    (1, 0, 0),
    (3, 1, 5)
and one cone with indices:
    [ 1, 2, 3 ]
> Resolution(F);
Fan with 8 rays:
    (0, 1, 0),
    (1, 0, 0),
    (3, 1, 5),
    (2, 1, 2),
    (1, 1, 1),
    (3, 1, 3),
    (3, 1, 4),
    (2, 1, 3)
and 11 cones