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The closure of a linear space in a product of lines. (English) Zbl 1331.05051
Summary: Given a linear space \(L\) in affine space \({\mathbb {A}}^n\), we study its closure \(\widetilde{L}\) in the product of projective lines \(({\mathbb {P}}^1)^n\). We show that the degree, multigraded Betti numbers, defining equations, and universal Gröbner basis of its defining ideal \(I(\widetilde{L})\) are all combinatorially determined by the matroid \(M\) of \(L\). We also prove that \(I(\widetilde{L})\) and all of its initial ideals are Cohen-Macaulay with the same Betti numbers, and can be used to compute the \(h\)-vector of \(M\). This variety \(\widetilde{L}\) also gives rise to two new objects with interesting properties: the cocircuit polytope and the external activity complex of a matroid.

05B35 Combinatorial aspects of matroids and geometric lattices
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
52B40 Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.)
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