Regular coordinates on the dual semigroup of a toric cone #
Let σ be a toric cone and let b be an integral basis of N, indexed by ToricRay σ ⊕ ι,
whose vectors at the ray indices are the primitive ray generators of σ. Evaluating integral
characters on b identifies N →+ ℤ with the integer-valued functions on the index set, and an
integral character lies in the dual semigroup of σ exactly when it is nonnegative on every
primitive ray generator. The dual semigroup is therefore the monoid of functions that are
nonnegative at the ray indices and arbitrary at the complementary indices, that is, to
(ToricRay σ →₀ ℕ) × (ι →₀ ℤ).
This is the coordinate model (ι →₀ ℕ) × (κ →₀ ℤ) consumed by
TauCeti.Toric.regularAffinePointEquiv, so for a regular cone it exhibits the complex points of
the affine toric chart as ℂ ^ k × (ℂ ^ *) ^ (n - k), with k the number of rays and n the
rank of the lattice. Finite generation for all lattice-rational cones is proved in
TauCeti.Geometry.Toric.Algebraic.DualSemigroup.Finiteness.
Main declarations #
TauCeti.Toric.regularDualSemigroupEquiv: the regular-coordinate equivalence of the dual semigroup attached to a basis extending the primitive ray generators, withTauCeti.Toric.coe_regularDualSemigroupEquiv_fst_apply,TauCeti.Toric.realCharacter_apply_primitiveGenerator,TauCeti.Toric.regularDualSemigroupEquiv_snd_apply,TauCeti.Toric.regularDualSemigroupEquiv_symm_apply_inlandTauCeti.Toric.regularDualSemigroupEquiv_symm_apply_inrcomputing both directions.TauCeti.Toric.dualSemigroupCoord: the coordinate functional of an extending basis, as an element of the dual semigroup, withTauCeti.Toric.regularDualSemigroupEquiv_fst_dualSemigroupCoord_inl,TauCeti.Toric.regularDualSemigroupEquiv_fst_dualSemigroupCoord_inrandTauCeti.Toric.regularDualSemigroupEquiv_snd_dualSemigroupCoordcomputing the regular coordinates, with respect to one extending basis, of the dual basis characters of another.TauCeti.Toric.IsRegularCone.nonempty_dualSemigroup_addEquiv: the dual semigroup of a regular cone withkrays in a lattice of ranknis isomorphic toℕ ^ k × ℤ ^ (n - k).TauCeti.Toric.IsRegularCone.mem_iff_forall_realCharacter_nonneg: a regular cone is the set of points at which every character of its dual semigroup is nonnegative, andTauCeti.Toric.IsRegularCone.mem_iff_forall_realCharacter_nonneg_of_closure_range_eq_top: it suffices to test a generating family.
References #
- W. Fulton, Introduction to Toric Varieties, §§1.2 and 2.1.
- D. Cox, J. Little and H. Schenck, Toric Varieties, §1.2 and Example 1.2.21.
The regular coordinates of the dual semigroup of a toric cone, attached to an integral basis
indexed by ToricRay σ ⊕ ι whose ray vectors are the primitive ray generators. A character
corresponds to its values on the basis: natural numbers at the ray indices, since it is
nonnegative on the primitive ray generators, and arbitrary integers at the complementary
indices.
Instances For
The ray coordinates of a character in the dual semigroup are its values on the primitive ray generators, which are natural numbers.
Evaluating the real character of a dual-semigroup element at a primitive ray generator gives
the corresponding regular ray coordinate, viewed in ℝ.
The complementary coordinates of a character in the dual semigroup are its values on the complementary basis vectors.
The character with prescribed regular coordinates takes the prescribed natural value on each primitive ray generator.
The character with prescribed regular coordinates takes the prescribed integral value on each complementary basis vector.
The dual basis characters #
The coordinate functional of a basis extending the primitive ray generators, as an element of
the dual semigroup of the cone: its values on the basis vectors are 0 and 1, so it is
nonnegative on every primitive ray generator. These characters are the monomials whose values on a
complex point are the regular coordinates of the affine chart of the cone, the ray indices giving
the coordinates that may vanish.
Instances For
A dual basis character takes the value 1 on its own basis vector.
A dual basis character vanishes on every other basis vector.
The ray coordinates of the dual basis character of a ray index of a second extending basis form the standard generator at that ray: both bases carry the primitive generator of a ray at the index of that ray, so the ray block of the transition matrix is the identity.
The dual basis character of a complementary index of a second extending basis has no ray coordinates: it vanishes on every primitive ray generator, since both bases carry the primitive generator of a ray at the index of that ray.
The complementary coordinates of the dual basis character of a second extending basis are the entries of the transition matrix between the two bases.
The character with a single standard ray coordinate is the dual basis character of that ray index.
The character with a single standard complementary coordinate is the dual basis character of that complementary index.
The dual semigroup of a regular cone with k rays in a lattice of rank n is isomorphic to
ℕ ^ k × ℤ ^ (n - k), the ray coordinates being indexed by the rays themselves.
A regular cone is cut out by its dual semigroup: a point of the ambient real space lies in the cone exactly when the real extension of every character of the dual semigroup is nonnegative at it.
A regular cone is cut out by any generating family of its dual semigroup: a point lies in the cone exactly when the real extension of every character of the family is nonnegative at it.