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Daily Overview |
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AG3: Algebraic Geometry Location: A701 Session Chair: Mateusz Michalek Session Chair: Botong Wang | |
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The omega invariant of a matroid via h* vectors of section rings 1: Carnegie Mellon University; 2: Queen Mary University of London; 3: Princeton University Speyer's 2005 f-vector conjecture asserted the nonnegativity of the coefficients of a matroid invariant he defined, notably the leading coefficient omega(M). When M is represented by a hyperplane arrangement, Larson identified omega(M) as the Euler characteristic of a certain anti-nef line bundle L^-1 on the wonderful compactification of the arrangement, up to a predictable sign. Eur and Larson used this to show that omega(M) is the leading coefficient in the "h* polynomial" of L, i.e. the numerator of the Hilbert series of the total coordinate ring of the image of the map to projective space defined by L. If the image is sufficiently nice (arithmetically Cohen-Macaulay), then the h* polynomial necessarily has nonnegative coefficients. The work I'll be talking about completes the argument, giving a proof that omega is positive for all matroids. We show that wonderful varieties degenerate inside the permutahedral toric variety to a Cohen--Macaulay union of torus orbits controlled by a second matroid. This union of torus orbits can be defined when M is not representable, and the needed Euler characteristic can be computed on the degeneration. | |
