  A set of polynomials G in a polynomial ring S over a field is said to be a
universal Groebner basis, if G is a Groebner basis with respect to every term
order on S. Twenty years ago Bernstein, Sturmfels, and Zelevinsky proved that
the set of the maximal minors of a matrix X of variables is a universal
Groebner basis. Boocher recently proved that any initial ideal of the ideal of
maximal minors of X has a linear resolution.
  In this paper we give a quick proof of the results mentioned above. Our proof
is based on a specialization argument. Then we show that similar statements
hold in a more general setting, for matrices of linear forms satisfying certain
homogeneity conditions. More precisely, we show that the set of maximal minors
of a matrix L of linear forms is a universal Groebner basis for the ideal I
that it generates, provided that L is column-graded. Under the same assumption
we show that every initial ideal of I has a linear resolution. Furthermore, the
projective dimension of I and of its initial ideals is n-m, unless I=0 or a
column of L is identically 0. Here L is a matrix of size m times n, and m is
smaller than or equal to n. If instead L is row-graded, then we prove that I
has a universal Groebner basis of elements of degree m and that every initial
ideal of I has a linear resolution, provided that I has the expected
codimension.
  The proofs are based on a rigidity property of radical Borel fixed ideals in
a multigraded setting: We prove that if two Borel fixed ideals I and J have the
same multigraded Hilbert series and I is radical, then I = J. We also discuss
some of the consequences of this rigidity property.
