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Edit: I realize the mathematics below is lacking a precise phrasing. I hope that the intuitiion behind the question is clear enough that a reader will understand the question and provide guidance. The question is essentially to what extent a degeneration of tropical curves reflects an actual degeneration of complex tropical curves in $(\mathbb{C}^*)^2$ More precisely, On page 10 of his paper,

http://arxiv.org/pdf/math/0312530v4.pdf

Mikhalkin discusses the degeneration of a smooth tropical curve to a nodal tropical curve. Given such a local degeneration, which we assume occurs in a one dimensional real family paramaterized by $t$, via some reconstruction process, we can associate for each $t$ an actual hypersurface in $(\mathbb{C}^*)^2$ (either a curve in some degenerated complex structure or a complex tropical curve), whose tropicalization is the tropical curve $\Pi_t$. Is it true that the limiting curve over the nodal tropical curve is nodal?

To be more demanding, can one associate to this degeneration in a canonical way a fibration of curves in $(\mathbb{C}^*)^2$,

$H_t \to Spec(\mathbb{C}[\tau,\tau^{-1}])$

such that one fiber has a nodal curve and the rest of the fibers are smooth? The following example makes me believe this may be possible: In the local model corresponding to Mikhalkin's example, we may consider the family of hypersurfaces:

$$\tau+x+y+xy $$

when $\tau$ is not $0,1$, the smooth tropical curve is a smooth deformation retract of the amoebaas of curves in this family. The singular tropical is the (tropical) amoeba of:

$$ 1+x+y+xy=(1+x)(1+y) $$

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1 Answer 1

If you ask is it true that a tropical nodal curve can be presented such a way as a limit of nodal curves then the answer is no.

What is true:

Curves in a family which tropicalises to tropical curve of genus g has genus at least g.

So, tropical nonsingular curve may be presented as a limit of nonsingular curves and a tropical limit of singular curves is singular too.

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