Intersection theory on linear subvarieties of toric varieties

Abstract

We give a complete description of the cohomology ring $$A^*(\overline{Z})$$ of a compactification of a linear subvariety $$Z$$ of a torus in a smooth toric variety whose fan $$\Sigma $$ is supported on the tropicalization of $$Z$$ . It turns out that cocycles on $$\overline{Z}$$ canonically correspond to Minkowski weights on $$\Sigma $$ and that the cup product is described by the intersection product on the tropical matroid variety $${{\mathrm{Trop}}}(Z)$$ .