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.

Note:

The type TropicalCurve can be thought of as subtype of TropicalVariety in the sense that it should have all properties and features of the latter.

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 -> x1
  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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