Tropical curves

Introduction

A tropical curve is a graph with multiplicities on its edges. If embedded, it is a polyhedral complex of dimension (at most) one.

Construction

In addition to converting from TropicalVariety, objects of type TropicalCurve can be constructed from:

Properties

In addition to the properties inherited from TropicalVariety, objects of type TropicalCurve have the following exclusive properties:

graph — Method

graph(f::AbsAffineSchemeMor)

Return the graph of $f : X → Y$ as a subscheme of $X×Y$ as well as the two projections to $X$ and $Y$.

Examples

julia> Y = affine_space(QQ,3)
Affine space of dimension 3
  over rational field
with coordinates [x1, x2, x3]

julia> R = OO(Y)
Multivariate polynomial ring in 3 variables x1, x2, x3
  over rational field

julia> (x1,x2,x3) = gens(R)
3-element Vector{QQMPolyRingElem}:
 x1
 x2
 x3

julia> X = subscheme(Y, x1)
Spectrum
  of quotient
    of multivariate polynomial ring in 3 variables x1, x2, x3
      over rational field
    by ideal (x1)

julia> f = inclusion_morphism(X, Y)
Affine scheme morphism
  from [x1, x2, x3]  scheme(x1)
  to   [x1, x2, x3]  affine 3-space over QQ
given by the pullback function
  x1 -> 0
  x2 -> x2
  x3 -> x3

julia> graph(f)
(scheme(x1, -x1, x2 - x2, x3 - x3), Hom: scheme(x1, -x1, x2 - x2, x3 - x3) -> scheme(x1), Hom: scheme(x1, -x1, x2 - x2, x3 - x3) -> affine 3-space over QQ with coordinates [x1, x2, x3])

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graph(TropC::TropicalCurve{minOrMax,false})

Return the graph of an abstract tropical curve TropC. Same as polyhedral_complex(tc).

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