most recent 30 from http://mathoverflow.net 2013-06-18T21:33:23Z http://mathoverflow.net/feeds/user/12840 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/54860/when-is-the-quotient-of-a-tropical-curve-also-a-tropical-curve Max Kutler 2011-02-09T08:05:54Z 2011-02-10T06:23:54Z

<p>A plane tropical curve $\Gamma$ is the corner locus of a tropical polynomial in two variables. That is, it is the set of points at which the tropical polynomial, which is a piecewise-linear concave function, fails to be linear. Equivalently, $\Gamma$ is a weighted graph in the plane satisfying certain, fairly rigid conditions. (For instance, each edge has rational slope, and the weighted sum of the primitive vectors of the edges around a given vertex is zero. See <a href="http://arxiv.org/abs/math/0601322v1" rel="nofollow">Gathmann</a> for a more thorough introduction to tropical plane curves.) Of particular importance to my question is the fact that $\Gamma$ has no finite points of valence 1.</p> <p>When the actual polynomial defining $\Gamma$ is not of particular importance to the question at hand, we can shift our attention to more abstract structures. Such an approach was recently used successfully by <a href="http://arxiv.org/abs/1006.4869" rel="nofollow">Joyner, Ksir, and Melles</a>. They first define a <em>star-shaped set</em> to be any set of the form <code>\[ S(n,r) = \{ z \in \mathbb{C} \mid z = t \exp(2\pi i/n) \ \text{for some $t \in [0,r)$ and $k \in \mathbb{Z}$}\}, \]</code> where $n$ is a positive integer and $r$ is a positive real number. The set $S(n,r)$ is given the path metric and the metric topology. An <em>abstract tropical curve</em> is then defined to be a compact connected topological space with the property that every point has a neighborhood homeomorphic and isometric to a star-shaped set. Further, informally speaking, each edge is given a positive integer weight and no finite leaves are allowed.</p> <p>By shifting our focus to abstract tropical curves, we lose the rigid constraints on tropical plane curves, while we retain their topological and metric structure. Moreover, this approach is more general: we are no longer restricted to plane curves, and can consider tropical curves in larger-dimensional spaces.</p> <hr> <p>Let $\Gamma$ be a tropical plane curve, and let $G$ be a nontrivial subgroup of $\text{Aut}(\Gamma)$. Under what conditions is the quotient $\Gamma/G$ is also a tropical curve. Because I fear this may be too restrictive a question---I see no obvious examples---I would like to include abstract tropical curves in my question. If $\Gamma$ is an (abstract) tropical curve, when is $\Gamma/G$ an abstract tropical curve? In this case, it is relatively easy to cook up examples of genus zero abstract tropical curves for which this works. A rotationally symmetric genus one curve also works, with $G$ the group of rotations. However, it fails in other cases. For instance, if $G$ contains a reflection, it is possible that $\Gamma/G$ has finite leaves. Can anyone supply a more illuminating example?</p>