Counting tropical rational space curves with cross-ratio constraints (Q2148103)

Consider the enumerative problem of determining the number \(N_d(\mathbb{P}^m, \mathcal{C})\) of rational curves in \(\mathbb{P}^m\) of fixed degree \(d\) that satisfy a list \(\mathcal{C}\) of geometric conditions. Possible conditions in \(\mathcal{C}\) are point incidences and cross-ratio conditions. \textit{I. Tyomkin} [Adv. Math. 305, 1356--1383 (2017; Zbl 1401.14242)] provides a correspondence theorem which states that the number \(N_d(\mathbb{P}^m, \mathcal{C})\) is equal to the tropical number \(N_\Delta^{\mathrm{trop}}(\mathbb{R}^m, \mathcal{C}^{\mathrm{trop}})\), i.e. the weighted count of rational tropical curves in \(\mathbb{R}^m\) with fixed degree \(\Delta\) that satisfy tropical analogs \(\mathcal{C}^{\mathrm{trop}}\) of the conditions \(\mathcal{C}\). This motivates the search for a formula computing the tropical count. In the present article, the author provides such a formula for \(m = 3\). More precisely, the tropical enumerative problem under consideration in this article allows point and line incidence, point and line tangency, and (non-)degenerate tropical cross-ratio conditions. Using evaluation and forgetful maps, all of these conditions become tropical cycles in the moduli space \(M_{0,n}(\mathbb{R}^m, \Delta)\) of tropical rational stable maps to \(\mathbb{R}^m\) of degree \(\Delta\). If the conditions are in general position and in the correct number, then the tropical count of interest \(N_\Delta^{\mathrm{trop}}(\mathbb{R}^m, \mathcal{C}^{\mathrm{trop}})\) is the degree of the zero dimensional intersection of these tropical cycles, i.e. a weighted count of the tropical objects satisfying \(\mathcal{C}^{\mathrm{trop}}\). Specializing to \(m = 3\), the computation of \(N_\Delta^{\mathrm{trop}}(\mathbb{R}^3, \mathcal{C}^{\mathrm{trop}})\) is reduced to computing the weighted count of \textit{floor decomposed} tropical space curves. The remaining problem is approached via so-called \textit{cross-ratio floor diagrams}, which contain only the information on the floor structure of a floor decomposed tropical curve. Using some notion of multiplicity for these diagrams, the main theorem of the article expresses the weighted count of (floor decomposed) tropical space curves satisfying \(\mathcal{C}^{\mathrm{trop}}\) as a weighted count of floor diagrams. Crucially, the multiplicity of a diagram can be computed vertex-by-vertex and moreover the local multiplicity at a vertex is an enumerative number of tropical curves in \(\mathbb{R}^2\). The latter of which were computed by the author in previous work thus giving an explicit formula.