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.

中文翻译：

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构型超曲面的拟阵连通性和奇异性

考虑一个拟阵在一个域上的线性实现。人们将其与配置多项式和具有线性齐次系数的对称双线性形式相关联。相应的配置超曲面及其非光滑轨迹支持双线性形式的相应第一和第二简并方案。我们证明了这些方案被简化并描述了拟阵连通性的影响：对于（2-）连通拟阵，构型超曲面是积分的，第二个简并方案被简化为 codimension 3 的 Cohen-Macaulay。如果拟阵是 3-连通的，那么第二个简并方案也是积分的。在这个过程中，我们描述了关于各种拟阵构造的配置多项式、形式和方案的行为。