[Paper Review] Tropical lines in planes and beyond

Philipp Jell, Hannah Markwig+2 more|arXiv (Cornell University)|Mar 5, 2020

Polynomial and algebraic computation14 references4 citations

TL;DR

This paper studies the tropicalization of spaces of lines and higher-dimensional linear subspaces in fixed linear varieties with constant coefficients. It proves these spaces are tropical linear subspaces whose structure is fully determined by matroids, specifically the matroid of lines in the associated hyperplane arrangement, which are generically Dilworth truncations, enabling a complete description of tropicalized Fano schemes and moduli spaces of degree-1 stable maps.

ABSTRACT

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.

Motivation & Objective

• To understand the tropicalization of spaces of d-dimensional linear subspaces in a fixed (d+1)-dimensional linear variety.
• To analyze the case where the ambient variety has constant coefficients, simplifying the tropicalization structure.
• To characterize the resulting tropical spaces as linear subspaces via their associated matroids.
• To identify the matroids arising in this context as those of lines in the hyperplane arrangement corresponding to the variety.
• To apply the results to tropicalized Fano schemes and moduli spaces of degree-1 stable maps to a plane.

Proposed method

• Restricting to varieties with constant coefficients to ensure well-behaved tropicalization.
• Proving that the space $ L(X) $ of $ d $-dimensional subspaces in a $ (d+1) $-dimensional variety $ X $ is itself a tropical linear subspace.
• Using matroid theory to describe the tropicalization of $ L(X) $, showing it is completely determined by the matroid of lines in the hyperplane arrangement of $ X $.
• Establishing that these matroids are generically Dilworth truncations of the underlying arrangement matroid.
• Applying the matroid-theoretic description to analyze tropicalized Fano schemes and moduli spaces of stable maps of degree 1.

Experimental results

Research questions

• RQ1 What is the tropicalization of the space of lines in a fixed plane with constant coefficients?
• RQ2 How can the tropical structure of spaces of higher-dimensional linear subspaces in a fixed ambient variety be characterized?
• RQ3 What is the relationship between the matroid of the tropicalized space $ L(X) $ and the hyperplane arrangement associated with $ X $?
• RQ4 In what way do Dilworth truncations arise naturally in this tropical geometric setting?
• RQ5 How do these results extend to the tropicalization of Fano schemes and moduli spaces of degree-1 stable maps?

Key findings

• The space $ L(X) $ of $ d $-dimensional linear subspaces in a $ (d+1) $-dimensional variety $ X $ with constant coefficients is a tropical linear subspace.
• The tropicalization of $ L(X) $ is completely determined by its associated matroid.
• This matroid is identified as the matroid of lines in the hyperplane arrangement corresponding to $ X $.
• Generically, this matroid arises as a Dilworth truncation of the arrangement's matroid.
• The results provide a complete description of the tropicalized Fano scheme parametrizing $ d $-dimensional subspaces in a $ (d+1) $-dimensional variety.
• The framework also yields a description of the tropicalization of the moduli space of stable maps of degree 1 to a plane.