Available versions of this document: latest release, release 3.5, nightly master
Reference documentation for older polymake versions: release 3.4, release 3.3, release 3.2
application fulton
This application deals with normal toric varieties as discussed in the famous book William Fulton: Introduction to toric varieties.
imports from:
uses:
Objects
BinomialIdeal
:
UNDOCUMENTEDCyclicQuotient
:
An affine normal toric variety given by a twodimensional cone in twodimensional space.NormalToricVariety
:
A normal toric variety given by a fan.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_RAYSDIM. 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 nonsingular.TDivisor
:
A Tinvariant divisor on a normal toric variety.VersalComponent
:
A component of the versal deformation of aCyclicQuotient
singularity.
Functions
Combinatorics
Combinatorial functions.

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.

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.

cf2rational(Vector<Integer> v)
Compute the rational number corresponding to a continued fraction.
 Parameters:
 Returns:

rational2cf(Rational r)
Compute the continued fraction corresponding to a rational number r.
 Parameters:
Rational
r
 Returns:
Producing a normal toric variety
With these clients you can create a normal toric variety from various input data.

hirzebruch_surface(Integer r)
Takes one parameter r and returns the polyhedral fan corresponding the the Hirzebruch surface H_{r}.
 Parameters:
Integer
r
: Parameter Returns:

polarized_toric_variety(Polytope<Rational> P, String name)
Creates a toric variety from the normal fan of a polytope and adds the defining divisor of the polytope
 Parameters:
String
name
: : a name for the divisor Returns:

projective_space(Int d)
Takes one parameter d and returns the fan corresponding to the ddimensional projective space.
 Parameters:
Int
d
: Dimension Returns: