Projective Curves

ProjectiveCurve

A reduced projective curve, defined as the vanishing locus of a homogeneous (but not necessarily radical) ideal.

Examples

julia> R, (w, x, y, z) = graded_polynomial_ring(QQ, [:w, :x, :y, :z]);

julia> M = matrix(R, 2, 3, [w x y; x y z])
[w   x   y]
[x   y   z]

julia> V = minors(M, 2)
3-element Vector{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}:
 w*y - x^2
 w*z - x*y
 x*z - y^2

julia> I = ideal(R, V);

julia> TC = projective_curve(I)
Projective curve
  in projective 3-space over QQ with coordinates [w, x, y, z]
defined by ideal (w*y - x^2, w*z - x*y, x*z - y^2)

invert_birational_map(phi::Vector{T}, C::ProjectiveCurve) where {T <: MPolyRingElem}

Return a dictionary where image represents the image of the birational map given by phi, and inverse represents its inverse, where phi is a birational map of the projective curve C to its image in the projective space of dimension size(phi) - 1. Note that the entries of inverse should be considered as representatives of elements in R/image, where R is the basering.

invert_birational_map(phi::Vector{T}, C::ProjectivePlaneCurve) where {T <: MPolyRingElem}

Return a dictionary where image represents the image of the birational map given by phi, and inverse represents its inverse, where phi is a birational map of the projective plane curve C to its image in the projective space of dimension size(phi) - 1. Note that the entries of inverse should be considered as representatives of elements in R/image, where R is the basering.

geometric_genus(X::AbsProjectiveScheme{<:Field}; algorithm::Symbol=:default) -> Int

Given a projective curve X return the genus of X, i.e. the integer $p_g = p_a - \delta$ where $p_a$ is the arithmetic genus and \delta the $\delta$-invariant of the curve.

The algorithm keyword can be specified to

• :normalization to compute $\delta$ a normalization
• :primary_decomposition to proceed with a primary decomposition

Normalization is usually slower, but not always.

Examples

julia> R, (x,y,z) = graded_polynomial_ring(QQ, [:x, :y, :z]);

julia> C = plane_curve(z*x^2-y^3)
Projective plane curve
  defined by 0 = x^2*z - y^3

julia> geometric_genus(C)
0

graph_curve(G::Graph)

Return the graph curve of G, i.e., a union of lines whose dual graph is G, see [BE91].

Assumes that G is trivalent and 3-connected.

Examples

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