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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:
 Specializations:
NormalToricVariety::AffineNormalToricVariety
: A normal toric variety that is affine, i.e., given by a cone.
Properties
Algebraic Geometry
Properties from algebraic geometry.

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

DEGENERATE
A toric variety is degenerate if the fan is not
FULL_DIM
. Type:

DIVISOR
 Type:
 Properties of DIVISOR:

MODULE_GENERATORS
Exponents of the fractional ideal of global sections. This property only makes sense for affine toric varieties.
 Type:
 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 P_{D} is nonempty.
 Type:

FANO
A toric variety is fano if the anticanonical divisor is
AMPLE
. Type:

MORI_CONE
The dual of the
NEF_CONE
. Type:

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 P_{D}; equivalently, a divisor is nef if the inequalities it defines are tight on P_{D}.
 Type:

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

RATIONAL_DIVISOR_CLASS_GROUP
The torsion free part of the class group.
 Type:

SMOOTH
A toric variety is smooth if the fan is smooth. Alias for property
SMOOTH_FAN
. Type:
Basic properties
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:

N_MAXIMAL_TORUS_ORBITS
The number of maximal torus orbits. Equals the number of rays of the fan.
 Type:

ORBIFOLD
A toric variety is an orbifold if the fan is
SIMPLICIAL
. Type:
Combinatorics
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: