The closure of a linear space in a product of lines

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