Spines for amoebas of rational curves (Q2024038)

Let \(V\) be an algebraic variety in the complex torus \((\mathbb{C}^{\times})^n = (\mathbb{C} \backslash \{0\})^n\), and consider the map \(\mathrm{Log} : (\mathbb{C}^{\times})^n \rightarrow \mathbb{R}^n\) defined by \(\mathrm{Log}(z_1, \dots, z_n) = (\log(|z_1|, \dots, \log|z_n|)\). Then, the image \(\mathrm{Log}(V)\) is called the amoeba of \(V\). In this paper, the authors consider a rational curve \(V\), and associate to it a tropical rational curve in \(\mathbb{R}^n\) called spine. The main results of the paper are described in Section 2. According to theorem 2.1, the amoeba of a complex curve of given toric degree \(\Delta\) can be approximated by a tropical rational curve of the same degree up to a constant which only depends on \(\Delta\) but not on the curve. Using that result, in Theorem 2.4 the tropical limits of families of rational complex curves of toric degree \(\Delta\) are described. The proofs of both results are fully developed in Sections 3 to 5 of the paper.