Tropical Fano Schemes

TL;DR

This paper introduces a tropical analogue of the Fano Scheme for projective varieties, showing it forms a polyhedral complex and exploring its properties, especially in relation to classical Fano schemes and toric varieties.

Contribution

It defines the tropical Fano Scheme, proves it is a polyhedral complex, and investigates its relationship with classical Fano schemes, including cases where they coincide.

Findings

Tropical Fano Scheme is a polyhedral complex.

In general, tropical and classical Fano schemes differ, but they coincide for toric varieties.

Explicit examples show the inclusion can be strict for certain linear spaces.

Abstract

We define a tropical version $\F_d(\trop X)$ of the Fano Scheme $\F_d(X)$ of a projective variety $X\subseteq \mathbb P^n$ and prove that $\F_d(\trop X)$ is the support of a polyhedral complex contained in $\trop \Grp(d,n)$. In general $\trop \F_d(X)\subseteq \F_d(\trop X)$ but we construct linear spaces $L$ such that $\trop \F_1(X)\subsetneq \F_1(\trop X)$ and show that for a toric variety $\trop \F_d(X)=\F_d(\trop X)$.