Tropical lines in planes and beyond
Philipp Jell, Hannah Markwig, Felipe Rincón, Benjamin Schröter · arXiv (Cornell University) · 2020
We study the tropicalization of the space $L(X)$ of lines contained in a fixed plane $X$, or, more generally, $d$-dimensional linear subspaces contained in a fixed $(d+1)$-dimensional linear variety. We restrict our attention to the case the variety $X$ has constant coefficients. We prove that these spaces $L(X)$ are linear subspaces themselves, and thus their tropicalization is completely determined by their associated matroids. We show that these matroids are equal to the matroid of lines of the hyperplane arrangement corresponding to $X$, which generically can be interpreted as Dilworth truncations. In this way, we can describe tropicalized Fano schemes parametrizing $d$-dimensional linear subspaces of a $(d+1)$-dimensional linear variety, and tropicalizations of moduli spaces of stable maps of degree $1$ to a plane.