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 Pv 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 kPv 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_toric-terminal-polytope-example (H126E10)

We make the cone corresponding to the (affine) terminal quotient singularity C3/(Z/5) where Z/5 acts as the 5th roots of unity in the diagonal representation (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)
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)
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
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).
V2.28, 13 July 2023