Tropical Geometry also has the concept of “space of (tropical) curves passing through some fixed points”. The article under review deals with the space of tropical plane curves with a given support set of \(n\) points given, passing through \(n-2\) points in general position.
To be more precise, let \[ {\mathcal A}=\{a_1,\dots, a_n\}\subset\{(r,s,t)\in\big(\mathbb{Z}_{\geq0}\big)^3:\,r+s+t=d\} \] for some integer \(d\). If \(C\subset{\mathbb T}{\mathbb P}^2\) is a configuration of \(n-2\) points, Denote with \(L_C\) the set of tropical plane curves with support \({\mathcal A}\) and passing through \(C\). \(C\) is said to be general with respect to \({\mathcal A}\) if the set does not lie on a tropical projective curve with support \({\mathcal A}\setminus\{a_i,\,a_j\}\) for any pair \(i, j\). If \(C\) is general, then \(L_C\) is a tropical line. A tropical line \(L\) is said to be compatible with \({\mathcal A}\) if \(L\) is an \(n\)-tree and the following conditions holds: if a trivalent quartet \((i j| k l)\) is a subtree of \(L\), then the convex hull of \(a_i,\,a_j,\,a_k,\,a_l\) has at least one of the segments \(\text{conv}(a_i,\,a_j)\) or \(\text{conv}(a_k,\,a_l)\) as an edge.
In [J. Richter-Gebert, B. Sturmfels and T. Theobald, Contemp. Math. 377, 289–317 (2005; Zbl 1093.14080)], it is proven that for each configuration \(C\) of \(n-2\) points, the linear pencil \(L_C\) is compatible with \({\mathcal A},\) and it was left as an open question whether the converse was true.
The main result of this paper is a positive answer to this open question. The main result reads as follows: Let \(L\) be a tropical projective line in \({\mathbb T}{\mathbb P}^2\) which is compatible with \({\mathcal A}\). Assume that \(L\) is trivalent and that each trivalent vertex of \(L\) corresponds to a maximal triangulation of \(\text{conv}({\mathcal A})\). Then there exists a general configuration \(C\) of \(n-2\) points in \({\mathbb T}{\mathbb P}^2\) such that \(L=L_C\).