====== application fulton ======
 This application deals with normal toric varieties as discussed in the famous book William Fulton: Introduction to toric varieties.

imports from:
    * application [[.:common|common]]
    * application [[.:fan|fan]]
    * application [[.:graph|graph]]
    * application [[.:ideal|ideal]]
    * application [[.:polytope|polytope]]
uses:
    * application [[.:group|group]]
    * application [[.:topaz|topaz]]

===== Objects =====
  ** ''[[.:fulton:CyclicQuotient |CyclicQuotient]]'':\\  An affine normal toric variety given by a two-dimensional cone in two-dimensional space.
  ** ''[[.:fulton:NormalToricVariety |NormalToricVariety]]'':\\  A normal toric variety given by a fan.
  ** ''[[.:fulton:RationalDivisorClassGroup |RationalDivisorClassGroup]]'':\\  The class group Cl(X) of Weil divisors on the toric variety defined by the fan is a finitely generated abelian group of rank [[.:fan:PolyhedralFan#N_RAYS |N_RAYS]]-[[.:fan:PolyhedralFan#DIM |DIM]]. It usually contains torsion. The rational divisor class group is the tensor product of Cl(X) with Q over Z. This group is torsion free and corresponds to the Picard group if the variety is non-singular.
  ** ''[[.:fulton:TDivisor |TDivisor]]'':\\  A //T//-invariant divisor on a normal toric variety.
  ** ''[[.:fulton:VersalComponent |VersalComponent]]'':\\  A component of the versal deformation of a ''[[.:fulton:CyclicQuotient |CyclicQuotient]]'' singularity.

===== Functions =====

==== Combinatorics ====
 Combinatorial functions.
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{{anchor:polytope_of_divisor_class:}}
  ?  **''polytope_of_divisor_class''**
  :: return the polytope defined by an element of the nef or effective cone first argument is the fan, second the Vector defining the divisor class


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==== Commutative Algebra ====
 These methods help with translating combinatorics to commutative algebra.
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{{anchor:lower_lattice_points:}}
  ?  **''lower_lattice_points''**
  :: Find all lattice points of a polytope P that are not reachable from some other lattice point via the tail cone.


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==== Continued fractions ====
 Two simple methods for switching between rational numbers and continued fractions.
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{{anchor:cf2rational:}}
  ?  **''cf2rational([[.:common#Vector |Vector]]<[[.:common#Integer |Integer]]> v)''**
  :: Compute the rational number corresponding to a continued fraction.
    ? Parameters:
    :: ''[[.:common#Vector |Vector]]<[[.:common#Integer |Integer]]>'' ''v''
    ? Returns:
    :''[[.:common#Rational |Rational]]''


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{{anchor:rational2cf:}}
  ?  **''rational2cf([[.:common#Rational |Rational]] r)''**
  :: Compute the continued fraction corresponding to a rational number //r//.
    ? Parameters:
    :: ''[[.:common#Rational |Rational]]'' ''r''
    ? Returns:
    :''[[.:common#Vector |Vector]]<[[.:common#Integer |Integer]]>''


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==== Producing a normal toric variety ====
 With these clients you can create a normal toric variety from various input data.
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{{anchor:hirzebruch_surface:}}
  ?  **''hirzebruch_surface([[.:common#Integer |Integer]] r)''**
  :: Takes one parameter //r// and returns the polyhedral fan corresponding the the Hirzebruch surface //H<sub>r</sub>//.
    ? Parameters:
    :: ''[[.:common#Integer |Integer]]'' ''r'': Parameter
    ? Returns:
    :''[[.:fulton:NormalToricVariety |NormalToricVariety]]''


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{{anchor:polarized_toric_variety:}}
  ?  **''polarized_toric_variety([[.:polytope:Polytope |Polytope]]<[[.:common#Rational |Rational]]> P, [[.:common#String |String]] name)''**
  :: Creates a toric variety from the normal fan of a polytope and adds the defining divisor of the polytope
    ? Parameters:
    :: ''[[.:polytope:Polytope |Polytope]]<[[.:common#Rational |Rational]]>'' ''P'': : the input polytope
    :: ''[[.:common#String |String]]'' ''name'': : a name for the divisor
    ? Returns:
    :''[[.:fulton:NormalToricVariety |NormalToricVariety]]''


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{{anchor:projective_space:}}
  ?  **''projective_space([[.:common#Int |Int]] d)''**
  :: Takes one parameter //d// and returns the fan corresponding to the //d//-dimensional projective space.
    ? Parameters:
    :: ''[[.:common#Int |Int]]'' ''d'': Dimension
    ? Returns:
    :''[[.:fulton:NormalToricVariety |NormalToricVariety]]''


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{{anchor:weighted_projective_space:}}
  ?  **''weighted_projective_space([[.:common#Vector |Vector]]<[[.:common#Int |Int]]> a)''**
  :: Takes a vector //a// and returns the fan corresponding to the weighted projective space associated to //a//.
    ? Parameters:
    :: ''[[.:common#Vector |Vector]]<[[.:common#Int |Int]]>'' ''a'': the weights
    ? Returns:
    :''[[.:fulton:NormalToricVariety |NormalToricVariety]]''


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==== no category ====
{{anchor:markov_basis:}}
  ?  **''markov_basis([[.:common#Set |Set]]<[[.:common#Vector |Vector]]<[[.:common#Integer |Integer]]%%>>%% S)''**
  :: Implementation of Project and Lift algorithm by Hemmecke and Malkin. Given a spanning set of a lattice returns a markov basis.
    ? Parameters:
    :: ''[[.:common#Set |Set]]<[[.:common#Vector |Vector]]<[[.:common#Integer |Integer]]%%>>%%'' ''S''
    ? Returns:
    :''[[.:common#Set |Set]]<[[.:common#Vector |Vector]]<[[.:common#Integer |Integer]]%%>>%%''
    ? Example:
    :: <code perl> > $s = new Set<Vector<Integer>>([1, -2, 1], [1, 1, -1]);
 > print markov_basis($s);
 {<-2 1 0> <-1 -1 1> <0 -3 2>}
</code>
  ?  **''markov_basis([[.:common#Matrix |Matrix]]<[[.:common#Integer |Integer]]> M)''**
  :: Implementation of Project and Lift algorithm by Hemmecke and Malkin. Given a Matrix whose rows form a spanning set of a lattice return markov basis as rows of an Integer Matrix,  or if use_kernel = true, returns a markov basis of integer kernel of given Matrix as rows of an Integer Matrix.
    ? Parameters:
    :: ''[[.:common#Matrix |Matrix]]<[[.:common#Integer |Integer]]>'' ''M''
    ? Options:
    : 
    :: ''[[.:common#Bool |Bool]]'' ''use_kernel'': = false
    ? Returns:
    :''[[.:common#Matrix |Matrix]]<[[.:common#Integer |Integer]]>''
    ? Example:
    :: <code perl> > $M = new Matrix<Integer>([[1, 1, 2, -2], [-1, 3, -2, 1]]);
 > print markov_basis($M, {"use_kernel" => true});
 0 2 7 8
 1 1 3 4
</code>
    :: <code perl> > $M = new Matrix<Integer>([[1, 1, 2, -2], [-1, 3, -2, 1]]);
 > print markov_basis($M);
 0 -4 0 1
 1 -7 2 0
 1 1 2 -2
</code>
  ?  **''markov_basis([[.:polytope:Polytope |Polytope]]<[[.:common#Rational |Rational]]> P)''**
  :: Implementation of Project and Lift algorithm by Hemmecke and Malkin. Given a polytope return the markov basis of the lattice spanned by it's lattice points as rows of an Integer Matrix,
    ? Parameters:
    :: ''[[.:polytope:Polytope |Polytope]]<[[.:common#Rational |Rational]]>'' ''P''
    ? Returns:
    :''[[.:common#Matrix |Matrix]]<[[.:common#Integer |Integer]]>''
    ? Example:
    :: <code perl> > $P = new Polytope(VERTICES=>[[1, 1, 0], [1, 0, 1], [1, 1, 1], [1, 0, 0]]);
 > print markov_basis($P);
 1 -1 -1 1
</code>


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