  We use the geometry of the stellahedral toric variety to study matroids. We
identify the valuative group of matroids with the cohomology ring of the
stellahedral toric variety, and show that valuative, homological, and numerical
equivalence relations for matroids coincide. We establish a new log-concavity
result for the Tutte polynomial of a matroid, answering a question of Wagner
and Shapiro-Smirnov-Vaintrob on Postnikov-Shapiro algebras, and calculate the
Chern-Schwartz-MacPherson classes of matroid Schubert cells. The central
construction is the "augmented tautological classes of matroids," modeled after
certain vector bundles on the stellahedral toric variety.
