AlgebraicGeometry: added graph_curve(::Graph)

docs/oscar_references.bib

@@ -185,6 +185,18 @@ @Article{BE23
   doi           = {10.1090/jag/810}
 }
 
+@Article{BE91,
+  author        = {Bayer, Dave and Eisenbud, David},
+  title         = {Graph Curves},
+  journal       = {Advances in Mathematics},
+  volume        = {86},
+  number        = {1},
+  pages         = {1--40},
+  year          = {1991},
+  month         = mar,
+  doi           = {10.1016/0001-8708(91)90034-5}
+}
+
 @Misc{BEO23,
   author        = {Besche, Hans Ulrich and Eick, Bettina and O'Brien, Eamonn},
   title         = {SmallGrp, The GAP Small Groups Library, Version 1.5.3},

docs/src/AlgebraicGeometry/Curves/ProjectiveCurves.md

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

src/AlgebraicGeometry/AlgebraicGeometry.jl

@@ -3,6 +3,7 @@ include("ToricVarieties/JToric.jl")
 include("Curves/AffinePlaneCurve.jl")
 include("Curves/ProjectivePlaneCurve.jl")
 include("Curves/ProjectiveCurve.jl")
+include("Curves/GraphCurve.jl")
 include("Curves/ParametrizationPlaneCurves.jl")
 include("Surfaces/K3Auto.jl")
 include("Surfaces/AdjunctionProcess/AdjunctionProcess.jl")

src/AlgebraicGeometry/Curves/GraphCurve.jl

@@ -0,0 +1,50 @@
+@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 = vertex_edge_graph(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 = vertex_edge_graph(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)
+    @req all(isequal(3),degree(G)) "G is not trivalent"
+    @req connectivity(G)==3 "G is not three-connected"
+
+    R,_ = graded_polynomial_ring(QQ,Int(nv(G)//2+1))
+    rowOfVariables = matrix(R,[gens(R)])
+
+    cycleMatrix = kernel(matrix(QQ,signed_incidence_matrix(G)); side=:right)
+    cycleMatrix = matrix(R,cycleMatrix) # converting to matrix over R for vcat below
+
+    vertexIdeals = MPolyIdeal[]
+    for v in 1:n_vertices(G)
+        edgesContainingV = findall(edge->(v in edge),collect(edges(G)))
+        cycleMatrix_v = cycleMatrix[edgesContainingV,:]
+        push!(vertexIdeals,ideal(minors(vcat(cycleMatrix_v,rowOfVariables),3)))
+    end
+    graphCurveIdeal = reduce(intersect,vertexIdeals)
+    graphCurve = projective_curve(graphCurveIdeal)
+
+    return graphCurve
+end

src/exports.jl

@@ -667,6 +667,7 @@ 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