documentation:master:fulton

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 — documentation:master:fulton [2019/12/11 19:34] (current) Line 1: Line 1: + ====== 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//​. + ? 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