Vanishing theorems for combinatorial geometries
Christopher Eur, Alex Fink, Matt Larson
TL;DR
The paper establishes strong vanishing theorems for line bundles on wonderful varieties tied to hyperplane arrangements, and shows that the resulting positivity of Euler characteristics extends to all matroids via tropical degenerations. By constructing ind_w M, a tropical initial degeneration, and proving it is a reduced, Cohen–Macaulay kindred subscheme, the authors connect cohomology vanishing, K-theory, and matroidal Hilbert functions, enabling a unified treatment across realizable and nonrealizable matroids. This approach yields a new proof of Speyer's f-vector conjecture and confirms Cohen–Macaulayness for higher order Orlik–Terao algebras, while also providing a framework that extends to building set and augmented wonderful varieties. The results unify vanishing, CM properties, and Hilbert function positivity in matroid theory, with implications for moduli spaces and related toric degenerations.
Abstract
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.