Toric Geometry

Authors: Martin Bies and Lars Kastner

2 Preliminaries

2.1 Polytopes, Cones, Fans and Toric Varieties

julia> sigma = positive_hull([1 0; -1 2])
Polyhedral cone in ambient dimension 2

julia> Usigma = affine_normal_toric_variety(sigma)
Normal toric variety

julia> toric_ideal(Usigma)
Ideal generated by
  -x1*x2 + x3^2

julia> hilbert_basis(weight_cone(Usigma))
3-element SubObjectIterator{PointVector{ZZRingElem}}:
 [2, 1]
 [0, 1]
 [1, 1]
julia> R = [1 0; 0 -1; -1 0; 0 1];

julia> IM = IncidenceMatrix([[1,2],[2,3],[3,4],[1,4]]);

julia> ntv = normal_toric_variety(IM, R)
Normal toric variety

julia> is_complete(ntv)
true

julia> is_smooth(ntv)
true
julia> c = cube(3)
Polytope in ambient dimension 3

julia> tv = normal_toric_variety(c)
Normal toric variety

2.3 Homogeneous Coordinate Ring and Chow Ring

julia> PP2 = projective_space(NormalToricVariety, 2)
Normal toric variety

julia> irrelevant_ideal(PP2)
Ideal generated by
  x3
  x1
  x2

julia> stanley_reisner_ideal(PP2)
Ideal generated by
  x1*x2*x3
julia> v = normal_toric_variety(IncidenceMatrix([[1], [2], [3]]), [[1, 0], [0, 1], [-1, -1]])
Normal toric variety

julia> is_complete(v)
false

julia> chow_ring(v)
Quotient
  of multivariate polynomial ring in 3 variables x1, x2, x3
    over rational field
  by ideal (x1 - x3, x2 - x3, x1*x2, x1*x3, x2*x3)

julia> M = cycle_matroid(complete_graph(3))
Matroid of rank 2 on 3 elements

julia> chow_ring(M)
Quotient
  of multivariate polynomial ring in 3 variables x_{Edge(2, 1)}, x_{Edge(3, 1)}, x_{Edge(3, 2)}
    over rational field
  by ideal with 5 generators

3 Data Types and Tools from Combinatorics

julia> pp1 = projective_space(NormalToricVariety, 1)
Normal toric variety

julia> tv = pp1 * pp1
Normal toric variety

julia> rays(tv)
4-element SubObjectIterator{RayVector{QQFieldElem}}:
 [1, 0]
 [-1, 0]
 [0, 1]
 [0, -1]

julia> maximal_cones(IncidenceMatrix, tv)
4×4 IncidenceMatrix
[1, 3]
[1, 4]
[2, 3]
[2, 4]


julia> Sigma = polyhedral_fan(tv)
Polyhedral fan in ambient dimension 2

4 Vanishing Sets from cohomCalg

4.1 The cohomCalg Algorithm

julia> P1 = projective_space(NormalToricVariety,1)
Normal toric variety

julia> l = toric_line_bundle(P1*P1, [-3,1])
Toric line bundle on a normal toric variety

julia> cohomology(l, 1)
4

4.2 Vanishing Sets

julia> P1 = projective_space(NormalToricVariety,1)
Normal toric variety

julia> v0, v1, v2 = vanishing_sets(P1*P1)
3-element Vector{ToricVanishingSet}:
 Toric vanishing set for cohomology indices [0]
 Toric vanishing set for cohomology indices [1]
 Toric vanishing set for cohomology indices [2]

julia> print_constraints(polyhedra(v0)[1])
-x_1 <= 0
-x_2 <= 0

julia> print_constraints(polyhedra(v1)[1])
-x_2 <= 0
x_1 <= -2

julia> print_constraints(polyhedra(v1)[2])
-x_1 <= 0
x_2 <= -2

julia> print_constraints(polyhedra(v2)[1])
x_2 <= -2
x_1 <= -2

5 Triangulations

julia> normal_toric_varieties_from_glsm([[1,1,1]])
1-element Vector{NormalToricVariety}:
 Normal toric variety

julia> normal_toric_varieties_from_glsm([[1,1,-1,-1]])
2-element Vector{NormalToricVariety}:
 Normal toric variety
 Normal toric variety

6 Blowup Resolution meets Star Subdivision

julia> rs = [1 1; -1 1];

julia> max_cones = IncidenceMatrix([[1,2]]);

julia> v = normal_toric_variety(IncidenceMatrix([[1,2]]), [1 1; -1 1]);

julia> set_coordinate_names(v, ["u1", "u2"])

julia> is_smooth(v)
false

julia> toric_ideal(v)
Ideal generated by
  -x1*x2 + x3^2

julia> rays(maximal_cones(v)[1])
2-element SubObjectIterator{RayVector{QQFieldElem}}:
 [1, 1]
 [-1, 1]

julia> bu = blow_up(v, [0,1])
Toric blowdown morphism

julia> v2 = domain(bu)
Normal toric variety

julia> is_smooth(v2)
true

julia> is_affine(v2)
false

julia> cox_ring(v2)
Multivariate polynomial ring in 3 variables over QQ graded by
  u1 -> [1]
  u2 -> [1]
  e -> [-2]

julia> stanley_reisner_ideal(v2)
Ideal generated by
  u1*u2
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