Tropical Fano Schemes

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 $\trop \F_d(X)=\F_d(\trop X)$.

16 pages, 3 figures. Version 2 contains generalised version of Theorems 2.3 and 4.2 where not just tropicalized lines are considered but any tropicalized linear space