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For a tropical curve $Z$, let us call $Z_0$ this curve with its 1-valent points removed.

(Def [5] of Joyner-Ksir-Grant Melles) Let the automorphism group of a tropical curve $Z$ be a map $g: Z \to Z$ such that $g$ is a homeomorphism of on the underlying topological space of $Z$, $g$ is an isometry on $Z_0$, and $g$ preserves multiplicities. (If $Z$ is not homeomorphic to a circle, then $g$ will be a graph automorphism on the minimal graph of $Z$, taking vertices to vertices and edges to edges.)

Let's say $X$ is a curve over a field $k$. We may fix a valuation and look at either its tropicalization $\text{trop}(X)$ or its corresponding tropical hypersurface $Y := \text{trop}(I(X))$.

My question is: How are the automorphism groups of $Y$ and $X$ related? Is $\text{Aut}(Y)$ contained in $\text{Aut}(X)$? How are the automorphism groups of $Y$ and $\text{Jac}(Y)$ related?

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  • $\begingroup$ What do you mean by $V(X)$? $\endgroup$
    – user2520938
    Commented Sep 24, 2020 at 9:30
  • $\begingroup$ $V(X)$ suggests "variety of $X$." But maybe the OP meant the opposite, $I(X)$; i.e., the tropical hypersurface is the tropicalization of the equations defining $X$. $\endgroup$
    – Sam Hopkins
    Commented Sep 24, 2020 at 14:13
  • $\begingroup$ Apologies, I indeed meant $I(X)$, edited to reflect that. $\endgroup$
    – Catherine Ray
    Commented Sep 24, 2020 at 17:30
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    $\begingroup$ Possibly the right notion of automorphisms for tropical curves is the one in: arxiv.org/abs/1006.0446 (I guess there is an issue of graphs vs. metric graphs...) $\endgroup$
    – Sam Hopkins
    Commented Sep 24, 2020 at 22:44
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    $\begingroup$ I believe the case I want to consider is that of automorphism groups of a metric topological graph. As defined in: citeseerx.ist.psu.edu/viewdoc/…. Edited to reflect that. $\endgroup$
    – Catherine Ray
    Commented Sep 26, 2020 at 23:36

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