documentation:latest:fulton:normaltoricvariety

Available versions of this document: latest release, release 4.0, release 3.6, release 3.5, nightly master

Reference documentation for older polymake versions: release 3.4, release 3.3, release 3.2

BigObject NormalToricVariety

from application fulton

A normal toric variety given by a fan.

derived from:
PolyhedralFan
Specializations:

NormalToricVariety::AffineNormalToricVariety: A normal toric variety that is affine, i.e., given by a cone.

Properties from algebraic geometry.

AFFINE

A toric variety is affine if the fan consists of a single cone.

Type:
Bool

DEGENERATE

A toric variety is degenerate if the fan is not FULL_DIM.

Type:
Bool

DIVISOR
Type:
TDivisor
Properties of DIVISOR:
MODULE_GENERATORS

Exponents of the fractional ideal of global sections. This property only makes sense for affine toric varieties.

Type:
Matrix<Integer,NonSymmetric>
depends on extension:

EFFECTIVE_CONE

The cone of effective divisors in the rational class group. A class of divisors D is effective if the corresponding polytope PD is non-empty.

Type:
Cone<Rational>

FANO

A toric variety is fano if the anticanonical divisor is AMPLE.

Type:
Bool

MORI_CONE

The dual of the NEF_CONE.

Type:
Cone<Rational>

NEF_CONE

The cone of nef divisors in the rational class group. This is the closure of the cone of ample divisors, where a divisor of the variety is ample if the fan coincides with the normal fan of PD; equivalently, a divisor is nef if the inequalities it defines are tight on PD.

Type:
Cone<Rational>

PROJECTIVE

A toric variety is projective if the corresponding fan is the normal fan of some polytope. Alias for property REGULAR.

Type:
Bool

RATIONAL_DIVISOR_CLASS_GROUP

The torsion free part of the class group.

Type:
RationalDivisorClassGroup

SMOOTH

A toric variety is smooth if the fan is smooth. Alias for property SMOOTH_FAN.

Type:
Bool

Properties defining a normal toric variety.

GENERATING_POLYTOPE

Polytope such that the fan of the toric variety is the normal fan of this polytope. This does not necessarily exist. For determining existence and computation we use Shepards theorem.

Type:
Polytope<Rational>

N_MAXIMAL_TORUS_ORBITS

The number of maximal torus orbits. Equals the number of rays of the fan.

Type:
Int

ORBIFOLD

A toric variety is an orbifold if the fan is SIMPLICIAL.

Type:
Bool

These properties capture combinatorial information of the object. Combinatorial properties only depend on combinatorial data of the object like, e.g., the face lattice.

WEIGHT_CONE

The weight cone of the algebra yielding the affine normal toric variety. I.e., intersect this cone with the lattice, take the semigroup algebra over it and take the spectrum.

Type:
Cone<Rational>

• documentation/latest/fulton/normaltoricvariety.txt