The closure of a linear space in a product of lines

Abstract

Given a linear space L in affine space $${\mathbb {A}}^n$$An, we study its closure $$\widetilde{L}$$L~ in the product of projective lines $$({\mathbb {P}}^1)^n$$(P1)n. We show that the degree, multigraded Betti numbers, defining equations, and universal Grobner basis of its defining ideal $$I(\widetilde{L})$$I(L~) are all combinatorially determined by the matroid M of L. We also prove that $$I(\widetilde{L})$$I(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 $$\widetilde{L}$$L~ also gives rise to two new objects with interesting properties: the cocircuit polytope and the external activity complex of a matroid.