Tropical Chow Hypersurfaces

TL;DR

This paper introduces the tropical Chow hypersurface, a new tropical geometric object derived from a given tropical variety, providing explicit construction methods and conjecturing its role in reconstructing the original variety.

Contribution

It defines the tropical Chow hypersurface, relates it explicitly to the tropical variety, and proves reconstruction for certain cases, advancing tropical geometry understanding.

Findings

Explicit method to obtain $ ext{Trop}(Z_X)$ from $ ext{Trop}(X)$

Geometric description of the tropical Chow hypersurface

Proof of reconstruction for curves in $ ext{P}^3$

Abstract

Given a projective variety $X$ of codimension $k+1$ in $\mathbb{P}^n$ the Chow hypersurface $Z_X$ is the hypersurface of the Grassmannian $\operatorname{Gr}(k, n)$ parametrizing projective linear spaces that intersect $X$. We introduce the tropical Chow hypersurface $\operatorname{Trop}(Z_X)$. This object only depends on the tropical variety $\operatorname{Trop}(X)$ and we provide an explicit way to obtain $\operatorname{Trop}(Z_X)$ from $\operatorname{Trop}(X)$. We also give a geometric description of $\operatorname{Trop}(Z_X)$. We conjecture that, as in the classical case, $\operatorname{Trop}(X)$ can be reconstructed from $\operatorname{Trop}(Z_X)$ and prove it for the case when $X$ is a curve in $\mathbb{P}^3$. This suggests that the tropical Chow hypersurface can be used to construct a tropical Chow variety.