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arXiv · 1902.06507

Matroid connectivity and singularities of configuration hypersurfaces

Graham Denham·Mathias Schulze·Uli Walther
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Abstract

Consider a linear realization of a matroid over a field. One associates with it a configuration polynomial and a symmetric bilinear form with linear homogeneous coefficients. The corresponding configuration hypersurface and its non-smooth locus support the respective first and second degeneracy scheme of the bilinear form. We show that these schemes are reduced and describe the effect of matroid connectivity: for (2-)connected matroids, the configuration hypersurface is integral, and the second degeneracy scheme is reduced Cohen-Macaulay of codimension 3. If the matroid is 3-connected, then also the second degeneracy scheme is integral. In the process, we describe the behavior of configuration polynomials, forms and schemes with respect to various matroid constructions.

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Graham Denham, Mathias Schulze, Uli Walther. 2019-02-18. Matroid connectivity and singularities of configuration hypersurfaces. https://doi.org/10.1007/s11005-020-01352-3

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