BigObject NormalToricVariety [polymake wiki]

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====== BigObject NormalToricVariety ======
//from application [[..:fulton|fulton]]//\\
\\
 A normal toric variety given by a fan.

  ? derived from:
  : ''[[..:fan:PolyhedralFan |PolyhedralFan]]''
  ? Specializations:
  :: ''NormalToricVariety::AffineNormalToricVariety'':  A normal toric variety that is affine, i.e., given by a cone.
===== Properties =====

==== Algebraic Geometry ====
 Properties from algebraic geometry.
----
{{anchor:affine:}}
  ?  **''AFFINE''**
  :: A toric variety is __affine__ if the fan consists of a single cone.
    ? Type:
    :''[[..:common#Bool |Bool]]''


----
{{anchor:degenerate:}}
  ?  **''DEGENERATE''**
  :: A toric variety is __degenerate__ if the fan is not ''[[..:fan:PolyhedralFan#FULL_DIM |FULL_DIM]]''.
    ? Type:
    :''[[..:common#Bool |Bool]]''


----
{{anchor:divisor:}}
  ?  **''DIVISOR''**
  ::
    ? Type:
    :''[[..:fulton:TDivisor |TDivisor]]''
    ? Properties of DIVISOR:
    :
      ?  **''MODULE_GENERATORS''**
      :: Exponents of the fractional ideal of global sections. This property only makes sense for affine toric varieties.
        ? Type:
        :''[[..:common#Matrix |Matrix]]<[[..:common#Integer |Integer]],[[..:common#NonSymmetric |NonSymmetric]]>''
        ? depends on extension:
        : [[:external_software|4ti2 or libnormaliz]]


----
{{anchor:effective_cone:}}
  ?  **''EFFECTIVE_CONE''**
  :: The cone of effective divisors in the rational class group. A class of divisors D is effective if the corresponding polytope P<sub>D</sub> is non-empty.
    ? Type:
    :''[[..:polytope:Cone |Cone]]<[[..:common#Rational |Rational]]>''


----
{{anchor:fano:}}
  ?  **''FANO''**
  :: A toric variety is __fano__ if the anticanonical divisor is ''[[..:fulton:TDivisor#AMPLE |AMPLE]]''.
    ? Type:
    :''[[..:common#Bool |Bool]]''


----
{{anchor:mori_cone:}}
  ?  **''MORI_CONE''**
  :: The dual of the ''[[..:fulton:NormalToricVariety#NEF_CONE |NEF_CONE]]''.
    ? Type:
    :''[[..:polytope:Cone |Cone]]<[[..:common#Rational |Rational]]>''


----
{{anchor:nef_cone:}}
  ?  **''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<sub>D</sub>; equivalently, a divisor is nef if the inequalities it defines are tight on P<sub>D</sub>.
    ? Type:
    :''[[..:polytope:Cone |Cone]]<[[..:common#Rational |Rational]]>''


----
{{anchor:projective:}}
  ?  **''PROJECTIVE''**
  :: A toric variety is __projective__ if the corresponding fan is the [[..:fan#normal_fan |normal fan]] of some polytope. Alias for property ''[[..:fan:PolyhedralFan#REGULAR |REGULAR]]''.
    ? Type:
    :''[[..:common#Bool |Bool]]''


----
{{anchor:rational_divisor_class_group:}}
  ?  **''RATIONAL_DIVISOR_CLASS_GROUP''**
  :: The torsion free part of the class group.
    ? Type:
    :''[[..:fulton:RationalDivisorClassGroup |RationalDivisorClassGroup]]''


----
{{anchor:smooth:}}
  ?  **''SMOOTH''**
  :: A toric variety is __smooth__ if the fan is [[..:fan:PolyhedralFan#SMOOTH_FAN |smooth]]. Alias for property ''[[..:fan:PolyhedralFan#SMOOTH_FAN |SMOOTH_FAN]]''.
    ? Type:
    :''[[..:common#Bool |Bool]]''


----

==== Basic properties ====
 Properties defining a normal toric variety.
----
{{anchor:generating_polytope:}}
  ?  **''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:Polytope |Polytope]]<[[..:common#Rational |Rational]]>''


----
{{anchor:n_maximal_torus_orbits:}}
  ?  **''N_MAXIMAL_TORUS_ORBITS''**
  :: The number of maximal torus orbits. Equals the number of rays of the fan.
    ? Type:
    :''[[..:common#Int |Int]]''


----
{{anchor:orbifold:}}
  ?  **''ORBIFOLD''**
  :: A toric variety is an __orbifold__ if the fan is ''[[..:fan:PolyhedralFan#SIMPLICIAL |SIMPLICIAL]]''.
    ? Type:
    :''[[..:common#Bool |Bool]]''


----

==== 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.
----
{{anchor:weight_cone:}}
  ?  **''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:
    :''[[..:polytope:Cone |Cone]]<[[..:common#Rational |Rational]]>''


----

==== no category ====
{{anchor:toric_ideal:}}
  ?  **''TORIC_IDEAL''**
  :: The toric ideal defining the variety.
    ? Type:
    :''[[..:ideal:Ideal |Ideal]]''


----