Tropical Fano Schemes

We define a tropical version ${\text{F}}_d(\text{trop}X)$ of the Fano Scheme ${\text{F}}_d(X)$ of a projective variety $X\subseteq {\mathbb{P}}^n$ and prove that ${\text{F}}_d(\text{trop}X)$ is the support of a polyhedral complex contained in $\text{trop}\text{Grp}(d,n)$. In general $\text{trop}{\text{F}}_d(X)\subseteq {\text{F}}_d(\text{trop}X)$ but we construct linear spaces $L$ such that $\text{trop}{\text{F}}_1(X)⊊{\text{F}}_1(\text{trop}X)$ and show that for a toric variety $\text{trop}{\text{F}}_d(X)={\text{F}}_d(\text{trop}X)$.