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documentation:master:fulton [2019/12/11 19:34] (current)
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 +====== 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:​BinomialIdeal |BinomialIdeal]]'':​\\ UNDOCUMENTED
 +  ** ''​[[.:​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.
 +----
 +{{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
 +
 +
 +----
 +
 +==== Commutative Algebra ====
 + These methods help with translating combinatorics to commutative algebra.
 +----
 +{{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.
 +
 +
 +----
 +
 +==== Continued fractions ====
 + Two simple methods for switching between rational numbers and continued fractions.
 +----
 +{{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]]''​
 +
 +
 +----
 +{{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]]>''​
 +
 +
 +----
 +
 +==== Producing a normal toric variety ====
 + With these clients you can create a normal toric variety from various input data.
 +----
 +{{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]]''​
 +
 +
 +----
 +{{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]]''​
 +
 +
 +----
 +{{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]]''​
 +
 +
 +----
 +{{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]]''​
 +
 +
 +----
  
  • documentation/master/fulton.txt
  • Last modified: 2019/12/11 19:34
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