# application: fulton

This application deals with normal toric varieties as discussed in the famous book

William Fulton: Introduction to toric varieties.

imports from: common, fan, graph, ideal, polytope
uses: group, topaz

## Objects

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### CyclicQuotient

An affine normal toric variety given by a two-dimensional cone in two-dimensional space.

derived from: NormalToricVariety

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### Algebraic Geometry

Properties relevant for the algebro-geometric side of CQS.

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### Input properties

Properties defining a cyclic quotient singularity. Please be careful in checking the consistency if you give multiple input properties.

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### NormalToricVariety

A normal toric variety given by a fan.

derived from: fan::PolyhedralFan<Rational>

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### 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 N_RAYS-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.

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### TDivisor

A T-invariant divisor on a normal toric variety.

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### VersalComponent

A component of the versal deformation of a CyclicQuotient singularity.

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### Algebraic Geometry

Properties relevant for the algebro-geometric side of the components of the versal deformation of CQS.

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### 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.

## User Functions

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### Commutative Algebra

These methods help with translating combinatorics to commutative algebra.

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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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cf2rational (v) → Rational

Compute the rational number corresponding to a continued fraction.

##### Parameters
 Vector v
##### Returns
 Rational
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rational2cf (r) → Vector<Integer>

Compute the continued fraction corresponding to a rational number r.

##### Parameters
 Rational r
##### Returns
 Vector
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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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hirzebruch_surface (r) → NormalToricVariety

Takes one parameter r and returns the polyhedral fan corresponding the the Hirzebruch surface //Hr//.

##### Parameters
 Integer r Parameter
##### Returns
 NormalToricVariety
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projective_space (d) → NormalToricVariety

Takes one parameter d and returns the fan corresponding to the d-dimensional projective space.

##### Parameters
 Int d Dimension
##### Returns
 NormalToricVariety