  Given a linear space L in affine space A^n, we study its closure L' in the
product of projective lines (P^1)^n. We show that the degree, multigraded Betti
numbers, defining equations, and universal Grobner basis of its defining ideal
I(L') are all combinatorially determined by the matroid M of L. We also prove
I(L') and all of its initial ideals are Cohen-Macaulay with the same Betti
numbers. In so doing, we prove that the initial ideals of I(L') are the
Stanley-Reisner ideals of an interesting family of simplicial complexes related
to the basis activities of M. We also describe the state polytope of I(L'),
which is related to the matroid basis polytope of M.
