  Our main theorems provide a single geometric setting in which polynomial
representatives for Schubert classes in the integral cohomology ring of the
flag manifold are determined uniquely, and have positive coefficients for
geometric reasons. This results in a geometric explanation for the naturality
of Schubert polynomials and their associated combinatorics.
  Given a permutation w in S_n, we consider a determinantal ideal I_w whose
generators are certain minors in the generic n x n matrix (filled with
independent variables). Using `multidegrees' as simple algebraic substitutes
for torus-equivariant cohomology classes on vector spaces, our main theorems
describe, for each ideal I_w:
  - variously graded multidegrees and Hilbert series in terms of ordinary and
double Schubert and Grothendieck polynomials;
  - a Gr\"obner basis consisting of minors in the generic n x n matrix;
  - the Stanley-Reisner complex of the initial ideal in terms of known
combinatorial diagrams associated to permutations in S_n; and
  - a procedure inductive on weak Bruhat order for listing the facets of this
complex, thereby generating the coefficients of Schubert polynomials by a
positive recursion on combinatorial diagrams.
  We show that the initial ideal is Cohen-Macaulay, by identifying the
Stanley-Reisner complex as a special kind of ``subword complex in S_n'', which
we define generally for arbitrary Coxeter groups, and prove to be shellable by
giving an explicit vertex decomposition. We also prove geometrically a general
positivity statement for multidegrees of subschemes.
