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 \(P^\vee\) 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 \({\mathbb{Q}}\)-Gorenstein.)

GorensteinIndex(P): TorPol -> RngIntElt

The Gorenstein index of the lattice polytope \(P\); i.e. the smallest positive integer \(k\) such that \(kP^\vee\) 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).

Example: Toric Terminal Polytope Example (ex-ee8303)

We make the cone corresponding to the (affine) terminal quotient singularity \({\mathbb{C}}^3/({\mathbb{Z}}/5)\) where \({\mathbb{Z}}/5\) acts as the \(5\)th roots of unity in the diagonal representation \({\rm diag}(1,2,3)\).

> 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)

Run in calculatorCopy input

We can check that this really is terminal and compute its Gorenstein index, the least positive multiple of the canonical class that is Cartier.

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

Run in calculatorCopy input

We can compute a resolution of singularities of this cone, the analogue of a simplicial subdivision for cones, although we must treat it as a fan to do so.

> 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

Run in calculatorCopy input

Note that this is not a minimal resolution: such a resolution would only need to subdivide at the four additional rays at the (original) lattice points \(1/5(1,2,3)\), \(1/5(2,4,1)\), \(1/5(3,1,4)\) and \(1/5(4,3,2)\).