Durham Workshop 2024 (Yue's group)

docs/oscar_references.bib

@@ -1,3 +1,15 @@
+@article{BE91,
+  title = {Graph Curves},
+  author = {Bayer, Dave and Eisenbud, David},
+  year = {1991},
+  month = mar,
+  journal = {Advances in Mathematics},
+  volume = {86},
+  number = {1},
+  pages = {1--40},
+  issn = {0001-8708},
+  doi = {10.1016/0001-8708(91)90034-5},
+}
 
 @Article{ABGJ18,
   author        = {Allamigeon, Xavier and Benchimol, Pascal and Gaubert, Stéphane and Joswig, Michael},

docs/src/AlgebraicGeometry/Curves/ProjectiveCurves.md

@@ -8,4 +8,5 @@ CurrentModule = Oscar
 ProjectiveCurve
 invert_birational_map
 geometric_genus
+graph_curve
 ```

docs/src/PolyhedralGeometry/Polyhedra/constructions.md

@@ -190,6 +190,7 @@ rss_associahedron
 signed_permutahedron
 stable_set_polytope
 transportation_polytope
+tutte_lifting
 zonotope
 zonotope_vertices_fukuda_matrix
 ```

docs/src/PolyhedralGeometry/intro.md

@@ -182,6 +182,7 @@ Lower dimensional polyhedral objects can be visualized through polymake's backen
 
 ```@docs
 visualize(P::Union{Polyhedron{<:Union{Float64,FieldElem}}, Cone{<:Union{Float64,FieldElem}}, PolyhedralFan{<:Union{Float64,FieldElem}}, PolyhedralComplex{<:Union{Float64,FieldElem}}, SubdivisionOfPoints{<:Union{Float64,FieldElem}}, Graph, SimplicialComplex}; kwargs...)
+visualize(::Vector)
 ```
 
 

src/AlgebraicGeometry/Curves/ProjectiveCurve.jl

@@ -85,3 +85,62 @@ end
 ################################################################################
 # The union of two curves is a curve
 Base.union(C::T, D::T) where T <: AbsProjectiveCurve = ProjectiveCurve(union(underlying_scheme(C),underlying_scheme(D)))
+
+
+
+function edge_matrix(G::Graph)
+    M = zero_matrix(ZZ, nv(G), ne(G))
+    for (i,e) in enumerate(edges(G))
+        M[src(e),i] = 1
+        M[dst(e),i] = -1
+    end
+    return M
+end
+
+@doc raw"""
+    graph_curve(G::Graph)
+
+Return the graph curve of `G`, i.e., a union of lines whose dual graph is `G`, see [BE91](@cite).
+
+Assumes that `G` is trivalent and 3-connected.
+
+# Examples
+```jldoctest
+julia> G1 = edgegraph(simplex(3))
+Undirected graph with 4 nodes and the following edges:
+(2, 1)(3, 1)(3, 2)(4, 1)(4, 2)(4, 3)
+
+julia> C1 = graph_curve(G1)
+Projective curve
+  in projective 2-space over QQ with coordinates [x1, x2, x3]
+defined by ideal (x1^2*x2*x3 - x1*x2^2*x3 + x1*x2*x3^2)
+
+julia> G2 = edgegraph(cube(3))
+Undirected graph with 8 nodes and the following edges:
+(2, 1)(3, 1)(4, 2)(4, 3)(5, 1)(6, 2)(6, 5)(7, 3)(7, 5)(8, 4)(8, 6)(8, 7)
+
+julia> C2 = graph_curve(G2)
+Projective curve
+  in projective 4-space over QQ with coordinates [x1, x2, x3, x4, x5]
+defined by ideal with 5 generators
+
+```
+"""
+function graph_curve(G::Graph; check::Bool=true)
+    if check
+        @req all(isequal(3),degree(G)) "G is not trivalent"
+    end
+
+    R,_ = graded_polynomial_ring(QQ,Int(nv(G)//2+1))
+    EG = kernel(Oscar.edge_matrix(G);side=:right)
+
+    edgesG = Vector{Int}.(collect(edges(G)))
+    lineIdeals = MPolyIdeal[]
+    for v in vertices(G)
+        EGv = matrix(R,EG[findall(edge->(v in edge),edgesG),:])
+        push!(lineIdeals,ideal(minors(vcat(EGv,matrix(R,[gens(R)])),3)))
+    end
+    graphCurveIdeal = reduce(intersect,lineIdeals)
+
+    return projective_curve(graphCurveIdeal)
+end

src/PolyhedralGeometry/Polyhedron/standard_constructions.jl

@@ -2455,3 +2455,40 @@ function vertex_figure(P::Polyhedron{T}, n::Int; cutoff=nothing) where {T<:scala
     Polymake.polytope.vertex_figure(pm_object(P), n - 1; opts...), coefficient_field(P)
   )
 end
+
+@doc raw"""
+    tutte_lifting(G::Graph{Undirected})
+
+Compute a realization of `G` in $\mathbb{R}^3$, i.e., a polyhedron whose edge graph is `G`.  Assumes that `G` is planar, 3-connected, and that is has a triangular facet.
+
+# Examples
+```jldoctest
+julia> G = edgegraph(simplex(3))
+Undirected graph with 4 nodes and the following edges:
+(2, 1)(3, 1)(3, 2)(4, 1)(4, 2)(4, 3)
+
+julia> pG = tutte_lifting(G)
+Polytope in ambient dimension 3
+
+julia> vertices(pG)
+4-element SubObjectIterator{PointVector{QQFieldElem}}:
+ [0, 0, 0]
+ [1, 0, 1//3]
+ [0, 1, 0]
+ [1//3, 1//3, 0]
+
+julia> faces(IncidenceMatrix,pG,1)
+6×4 IncidenceMatrix
+[1, 2]
+[1, 3]
+[1, 4]
+[2, 3]
+[2, 4]
+[3, 4]
+
+```
+"""
+function tutte_lifting(G::Graph{Undirected})
+  pmG = Polymake.graph.Graph{Undirected}(; ADJACENCY=G)
+  return polyhedron(Polymake.polytope.tutte_lifting(pmG))
+end

src/PolyhedralGeometry/visualization.jl

@@ -70,6 +70,27 @@ function visualize(
   Polymake.visual(pmo; kwargs...)
 end
 
+@doc raw"""
+    visualize(P::Vector; kwargs...)
+
+Visualize a vector of polyhedral objects `P`, i.e., all polyhedral objects in `P` in one visualization.  See [`visualize`](@ref Oscar.visualize(::Union{SimplicialComplex, Cone{<:Union{Float64, FieldElem}}, Graph, PolyhedralComplex{<:Union{Float64, FieldElem}}, PolyhedralFan{<:Union{Float64, FieldElem}}, Polyhedron, SubdivisionOfPoints{<:Union{Float64, FieldElem}}})) for which polyhedral objects and keyword arguments are possible.
+
+# Examples
+```julia
+julia> p = simplex(3)
+Polytope in ambient dimension 3
+
+julia> P = [p,p+[3,0,0],p+[0,3,0],p+[0,0,3]]
+4-element Vector{Polyhedron{QQFieldElem}}:
+ Polytope in ambient dimension 3
+ Polytope in ambient dimension 3
+ Polytope in ambient dimension 3
+ Polytope in ambient dimension 3
+
+julia> visualize(P)
+
+```
+"""
 function visualize(P::Vector; kwargs::Dict=Dict{Int,Nothing}())
   for p in P
     @req p isa visual_supported_types "Can not visualize objects of type $(typeof(P))"

src/exports.jl

@@ -648,6 +648,7 @@ export graded_map
 export graded_polynomial_ring
 export grading_group
 export graph
+export graph_curve
 export graph_from_adjacency_matrix
 export graph_from_edges
 export grassmann_pluecker_ideal
@@ -1566,6 +1567,7 @@ export tropical_variety
 export truncate
 export turn_denominator_into_polyhedron
 export tutte_connectivity
+export tutte_lifting
 export tutte_polynomial
 export twist
 export twisting_sheaf