Intersection Theory on Linear Subvarieties of Toric Varieties

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 $Σ$ is supported on the tropicalization of $Z$. It turns out that cocycles on $\overline Z$ canonically correspond to Minkowski weights on $Σ$ and that the cup product is described by the intersection product on the tropical matroid variety $\operatorname{Trop}(Z)$.

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