# 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**

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

**derived from:**NormalToricVariety#### Properties of CyclicQuotient

Properties relevant for the algebro-geometric side of CQS.

**VERSAL_COMPONENTS**: common::Matrix<Rational, NonSymmetric>The continued fractions equivalent to zero that index the components of the versal deformation.

See

Jan Arthur Christophersen: On the components and discriminant of the versal base space of cyclic quotient singularities.

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

**CONTINUED_FRACTION**: common::Vector<Integer>Representation of the number n/q as a Hirzebruch-Jung continued fraction. Take care that this property agrees with the dual property.

**DUAL_CONTINUED_FRACTION**: common::Vector<Integer>Representation of the number n/(n-q) as a Hirzebruch-Jung continued fraction. Take care that this property agrees with the dual property.

**N**: common::IntegerEvery cyclic quotient variety corresponds to a cone given by the rays (1,0) and (-q,n).

**Q**: common::IntegerEvery cyclic quotient variety corresponds to a cone given by the rays (1,0) and (-q,n).

A normal toric variety given by a fan.

**derived from:**fan::PolyhedralFan<Rational>#### Specializations of NormalToricVariety

**NormalToricVariety::AffineNormalToricVariety**A normal toric variety that is affine, i.e., given by a cone.

#### Properties of NormalToricVariety

Properties from algebraic geometry.

**DIVISOR**: TDivisor as NormalToricVariety::DIVISORA toric invariant divisor on the variety given by the fan. It is represented by an integer vector with entries corresponding to the rays of the fan. The actual divisor is stored in the property COEFFICIENTS.

#### Properties of DIVISOR

These properties capture combinatorial information of the object. Combinatorial properties only depend on combinatorial data of the object like, e.g., the face lattice.

**MODULE_GENERATORS**: common::Matrix<Integer, NonSymmetric>**Only defined for NormalToricVariety::AffineNormalToricVariety**Exponents of the fractional ideal of global sections. This property only makes sense for affine toric varieties.

**Depends on:**4ti2 or libnormaliz

**EFFECTIVE_CONE**: polytope::Cone<Rational>The cone of effective divisors in the rational class group. A class of divisors D is effective if the corresponding polytope P

_{D}is non-empty.**NEF_CONE**: polytope::Cone<Rational>The cone of nef divisors in the rational class group. This is the closure of the cone of ample divisors, where a divisor of the variety is ample if the fan coincides with the normal fan of P

_{D}; equivalently, a divisor is nef if the inequalities it defines are tight on P_{D}.**PROJECTIVE**: common::BoolA toric variety is

*projective*if the corresponding fan is the normal fan of some polytope. Alias for property fan::PolyhedralFan::REGULAR.**SMOOTH**: common::BoolA toric variety is

*smooth*if the fan is smooth. Alias for property fan::PolyhedralFan::SMOOTH_FAN.

Properties defining a normal toric variety.

**GENERATING_POLYTOPE**: polytope::Polytope<Rational>Polytope such that the fan of the toric variety is the normal fan of this polytope. This does not necessarily exist. For determining existence and computation we use Shepards theorem.

**N_MAXIMAL_TORUS_ORBITS**: common::IntThe number of maximal torus orbits. Equals the number of rays of the fan.

These properties capture combinatorial information of the object. Combinatorial properties only depend on combinatorial data of the object like, e.g., the face lattice.

**WEIGHT_CONE**: polytope::Cone<Rational>**Only defined for NormalToricVariety::AffineNormalToricVariety**The weight cone of the algebra yielding the affine normal toric variety. I.e., intersect this cone with the lattice, take the semigroup algebra over it and take the spectrum.

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.

#### Properties of RationalDivisorClassGroup

Properties from algebraic geometry.

**LIFTING**: common::Matrix<Integer, NonSymmetric>Lifts a divisor in the class group onto a divisor on the fan.

**PROJECTION**: common::Matrix<Integer, NonSymmetric>Maps a divisor on the variety onto its representation in the class group.

A

*T*-invariant divisor on a normal toric variety.#### Properties of TDivisor

Properties from algebraic geometry.

These properties capture combinatorial information of the object. Combinatorial properties only depend on combinatorial data of the object like, e.g., the face lattice.

**CARTIER_DATA**: common::Map<Set<Int>, Vector<Rational>>Contains the

*Cartier data*of the divisor if it is CARTIER, i.e., contains a list of vertices of the lattice polytope defined by the divisor and the variety. The vertices appear in the same order as the maximal cones of the fan.**COEFFICIENTS**: common::Vector<Rational>The divisor on a toric variety, given as a list of coefficients for the torus invariant divisors corresponding to the RAYS of the fan. Take care of labeling of the Rays.

**SECTION_POLYTOPE**: polytope::Polytope<Rational>The polytope whose lattice points correspond to the global sections of the divisor.

A component of the versal deformation of a CyclicQuotient singularity.

#### Properties of VersalComponent

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

**ALPHA_LATTICE**: common::Map<Vector<Integer>, Vector<Integer>>Lattice of alphas as in below article.

Jan Arthur Christophersen: On the components and discriminant of the versal base space of cyclic quotient singularities.**CONTINUED_FRACTION**: common::Vector<Integer>The continued fraction equivalent to zero describing the versal component.

See

Jan Arthur Christophersen: On the components and discriminant of the versal base space of cyclic quotient singularities.and

Jan Stevens: On the versal deformation of cyclic quotient singularities.**CQS_MATRIX**: common::Matrix<Rational, NonSymmetric>The CQS matrix as described in Def 6.10 of the below article.

Theo de Jong, Duco van Straten: Deformation theory of sandwiched singularities.**N_GON_TRIANGULATION**: graph::Graph<Undirected>Triangulation of the N-gon corresponding to this component.

See

Jan Stevens: On the versal deformation of cyclic quotient singularities.**P_RESOLUTION**: fan::PolyhedralFan<Rational>The P-resolution fan corresponding to this component.

See

Klaus Altmann: P-Resolutions of Cyclic Quotients from the Toric Viewpoint.**TRIANGLES**: common::Matrix<Integer, NonSymmetric>The triangles in the N-gon triangulation corresponding to this component.

See

Jan Stevens: On the versal deformation of cyclic quotient singularities.

## User Functions

These methods help with translating combinatorics to commutative algebra.

**lower_lattice_points**()Find all lattice points of a polytope P that are not reachable from some other lattice point via the tail cone.

Two simple methods for switching between rational numbers and continued fractions.

**cf2rational**(v) → RationalCompute the rational number corresponding to a continued fraction.

**rational2cf**(r) → Vector<Integer>Compute the continued fraction corresponding to a rational number

*r*.

With these clients you can create a normal toric variety from various input data.

**hirzebruch_surface**(r) → NormalToricVarietyTakes one parameter

*r*and returns the polyhedral fan corresponding the the Hirzebruch surface //H_{r}//.**projective_space**(d) → NormalToricVarietyTakes one parameter

*d*and returns the fan corresponding to the*d*-dimensional projective space.