Geodesic
Stellahedral geometry of matroids
Forum of Mathematics, Pi, Tome 11 (2023)

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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 toric vector bundles on the stellahedral toric variety. 
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Christopher Eur; June Huh; Matt Larson. Stellahedral geometry of matroids. Forum of Mathematics, Pi, Tome 11 (2023). doi: 10.1017/fmp.2023.24

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