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— | documentation:master:fulton [2023/01/27 16:59] (current) – created - external edit 127.0.0.1 | ||
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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 [[.: | ||
+ | * application [[.: | ||
+ | * application [[.: | ||
+ | * application [[.: | ||
+ | * application [[.: | ||
+ | uses: | ||
+ | * application [[.: | ||
+ | * application [[.: | ||
+ | |||
+ | ===== Objects ===== | ||
+ | ** '' | ||
+ | ** '' | ||
+ | ** '' | ||
+ | ** '' | ||
+ | ** '' | ||
+ | |||
+ | ===== Functions ===== | ||
+ | |||
+ | ==== Combinatorics ==== | ||
+ | | ||
+ | ---- | ||
+ | {{anchor: | ||
+ | ? **'' | ||
+ | :: 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: | ||
+ | ? **'' | ||
+ | :: 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: | ||
+ | ? **'' | ||
+ | :: Compute the rational number corresponding to a continued fraction. | ||
+ | ? Parameters: | ||
+ | :: '' | ||
+ | ? Returns: | ||
+ | :'' | ||
+ | |||
+ | |||
+ | ---- | ||
+ | {{anchor: | ||
+ | ? **'' | ||
+ | :: Compute the continued fraction corresponding to a rational number //r//. | ||
+ | ? Parameters: | ||
+ | :: '' | ||
+ | ? Returns: | ||
+ | :'' | ||
+ | |||
+ | |||
+ | ---- | ||
+ | |||
+ | ==== Producing a normal toric variety ==== | ||
+ | With these clients you can create a normal toric variety from various input data. | ||
+ | ---- | ||
+ | {{anchor: | ||
+ | ? **'' | ||
+ | :: Takes one parameter //r// and returns the polyhedral fan corresponding the the Hirzebruch surface // | ||
+ | ? Parameters: | ||
+ | :: '' | ||
+ | ? Returns: | ||
+ | :'' | ||
+ | |||
+ | |||
+ | ---- | ||
+ | {{anchor: | ||
+ | ? **'' | ||
+ | :: Creates a toric variety from the normal fan of a polytope and adds the defining divisor of the polytope | ||
+ | ? Parameters: | ||
+ | :: '' | ||
+ | :: '' | ||
+ | ? Returns: | ||
+ | :'' | ||
+ | |||
+ | |||
+ | ---- | ||
+ | {{anchor: | ||
+ | ? **'' | ||
+ | :: Takes one parameter //d// and returns the fan corresponding to the // | ||
+ | ? Parameters: | ||
+ | :: '' | ||
+ | ? Returns: | ||
+ | :'' | ||
+ | |||
+ | |||
+ | ---- | ||
+ | {{anchor: | ||
+ | ? **'' | ||
+ | :: Takes a vector //a// and returns the fan corresponding to the weighted projective space associated to //a//. | ||
+ | ? Parameters: | ||
+ | :: '' | ||
+ | ? Returns: | ||
+ | :'' | ||
+ | |||
+ | |||
+ | ---- | ||
+ | |||
+ | ==== no category ==== | ||
+ | {{anchor: | ||
+ | ? **'' | ||
+ | :: Implementation of Project and Lift algorithm by Hemmecke and Malkin. Given a spanning set of a lattice returns a markov basis. | ||
+ | ? Parameters: | ||
+ | :: '' | ||
+ | ? Returns: | ||
+ | :'' | ||
+ | ? Example: | ||
+ | :: <code perl> > $s = new Set< | ||
+ | > print markov_basis($s); | ||
+ | | ||
+ | </ | ||
+ | ? **'' | ||
+ | :: 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, | ||
+ | ? Parameters: | ||
+ | :: '' | ||
+ | ? Options: | ||
+ | : | ||
+ | :: '' | ||
+ | ? Returns: | ||
+ | :'' | ||
+ | ? Example: | ||
+ | :: <code perl> > $M = new Matrix< | ||
+ | > print markov_basis($M, | ||
+ | 0 2 7 8 | ||
+ | 1 1 3 4 | ||
+ | </ | ||
+ | :: <code perl> > $M = new Matrix< | ||
+ | > print markov_basis($M); | ||
+ | 0 -4 0 1 | ||
+ | 1 -7 2 0 | ||
+ | 1 1 2 -2 | ||
+ | </ | ||
+ | ? **'' | ||
+ | :: 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: | ||
+ | :: '' | ||
+ | ? Returns: | ||
+ | :'' | ||
+ | ? Example: | ||
+ | :: <code perl> > $P = new Polytope(VERTICES=> | ||
+ | > print markov_basis($P); | ||
+ | 1 -1 -1 1 | ||
+ | </ | ||
+ | |||
+ | |||
+ | ---- | ||