Vanishing theorems for combinatorial geometries

We establish strong vanishing theorems for line bundles on wonderful varieties of hyperplane arrangements, and we show that the resulting positivity properties of Euler characteristics extend to all matroids. We achieve this by showing that every degeneration of a wonderful variety within the permutohedral toric variety is reduced and Cohen--Macaulay. The same holds for a larger class of subschemes in products of projective lines that we call "kindred," which are characterized by matroidal Hilbert polynomials. Our results give a new proof of the 20-year-old f-vector conjecture of Speyer and resolve the conjecture of Tohaneanu that higher order Orlik--Terao algebras are Cohen--Macaulay.