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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.

##### MSC:
 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.)
Gfan
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