Tropical Chow Hypersurfaces

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