Abstract. We answer a question by Niederreiter concerning the enumeration of a class of subspaces of finite dimensional vector spaces over finite fields by proving a conjecture by Ghorpade and Ram.
Tautological bundles of realizations of matroids were introduced in [BEST23] as a unifying geometric model for studying matroids. We compute the cohomologies of exterior and symmetric powers of these vector bundles, and show that they depend only on the matroid of the realization. As an application, we show that the log canonical bundle of a wonderful compactification of a hyperplane arrangement complement, in particular the moduli space of pointed rational curves M 0,n , has vanishing higher cohomologies.
For any matroid M , we compute the Tutte polynomial T M (x, y) using the mixed intersection numbers of certain tautological classes in the combinatorial Chow ring A • (M ) arising from Grassmannians. Using mixed Hodge-Riemann relations, we deduce a strengthening of the log-concavity of the h-vector of a matroid complex, improving on an old conjecture of Dawson whose proof was announced recently by Ardila, Denham and Huh.
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