Cotangent cohomology and quadratic obstructions of matroids

This is the authors' abstract. We don't add key points for this paper.

Let M be a finite matroid and let T²(M) denote the second cotangent cohomology of its Stanley–Reisner ring over an arbitrary field. We compute every multigraded component of T²(M) from the connected components of explicit graphs on flats in dual links. This gives the fine Hilbert series and proves that its coefficients are independent of the characteristic. We show that T²(M) vanishes if and only if every connected component of M has corank at most two, equivalently if and only if M has no U₁,₄-minor, proving the conjecture of Constantinescu, Klein, Nguyen, Singh, and Venturello. Explicit cup-product coordinates then produce a first-order deformation that fails to lift to second order whenever T²(M) is nonzero. Thus the affine deformation functor is formally smooth exactly when T²(M) vanishes, in every characteristic. The graph formula also yields sharp rank-two and rank-three bounds. A restricted labeled T² nonvanishing pattern determines every simple rank-three matroid on at least seven elements. Among simple rank-three matroids on n ≥ 8 elements, the near-pencil uniquely minimizes the standard-degree-zero obstruction dimension.