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.

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AFFINE

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

Type:

Bool

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DEGENERATE

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

Type:

Bool

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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:

4ti2 or libnormaliz

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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 non-empty.

Type:

Cone<Rational>

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FANO

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

Type:

Bool

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MORI_CONE

The dual of the NEF_CONE.

Type:

Cone<Rational>

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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:

Cone<Rational>

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PROJECTIVE

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

Type:

Bool

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RATIONAL_DIVISOR_CLASS_GROUP

The torsion free part of the class group.

Type:

RationalDivisorClassGroup

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SMOOTH

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

Type:

Bool

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Properties defining a normal toric variety.

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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>

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N_MAXIMAL_TORUS_ORBITS

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

Type:

Int

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ORBIFOLD

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

Type:

Bool

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

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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>

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TORIC_IDEAL

The toric ideal defining the variety.

Type:

Ideal

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