The moduli space of tropical curves with fixed Newton polygon (Q2075161)

Let \(\Delta\) be a lattice polygon. It is said to be \emph{hyperelliptic} if all interior lattice points are colinear. Denote by \(\mathbb{M}_\Delta\) the moduli space parametrizing plane tropical curves (i.e. weighted balanced one-dimensional polyhedral complexes in \(\mathbb{R}^2\)) with Newton polygon \(\Delta\). This is a locus inside the moduli space of plane tropical curves of fixed genus as introduced by [\textit{S. Brodsky} et al., Res. Math. Sci. 2, Paper No. 4, 31 p. (2015; Zbl 1349.14043)] which in turn can be embedded into the moduli space of tropical curves (i.e. metric graphs) of that genus. The authors show that if \(\Delta\) is nonhyperelliptic or if \(\Delta\) is hyperelliptic and maximal among the lattice polygons with the same set of interior lattice points as \(\Delta\), then \(\dim \mathbb{M}_\Delta\) is equal to the dimension of the moduli space of nondegenerate algebraic curves with Newton polygon \(\Delta\) as described in [\textit{W. Castryck} and \textit{J. Voight}, Algebra Number Theory 3, No. 3, 255--281 (2009; Zbl 1177.14089)]. This generalizes results of [\textit{S. Brodsky} et al., Res. Math. Sci. 2, Paper No. 4, 31 p. (2015; Zbl 1349.14043)]. The strategy of the proof is to compute \(\dim \mathbb{M}_\Delta\) explicitly. To do so, write \(\mathbb{M}_\Delta = \bigcup \mathbb{M}_\mathcal{T}\), where the union runs over all unimodular triangulations \(\mathcal{T}\) of \(\Delta\) and \(\mathbb{M}_\mathcal{T}\) denotes the moduli space of plane tropical curves dual to \(\mathcal{T}\). The authors give a formula for \(\dim \mathbb{M}_\mathcal{T}\) that holds for regular \(\mathcal{T}\) and nonhyperelliptic \(\Delta\). This then reduces the computation of \(\dim \mathbb{M}_\Delta\) to finding a triangulation \(\mathcal{T}\) that maximizes \(\dim \mathbb{M}_\mathcal{T}\) and that is additionally regular. The authors call a maximizing \(\mathcal{T}\) \emph{beehive triangulation}. Existence of regular beehive triangulations for maximal nonhyperelliptic as well as nonmaximal nonhyperelliptic \(\Delta\) is shown explicitly. The result is a formula for \(\dim \mathbb{M}_\Delta\) purely in terms of the combinatorics of \(\Delta\). The remaining case of \(\Delta\) maximal and hyperelliptic is reduced to computing \(\dim \mathbb{M}_\mathcal{T}\) for a specific \(\mathcal{T}\) and \(\Delta\). This done explicitly and the claim follows. The case of nonmaximal hyperelliptic \(\Delta\) remains open.