  We study the isospectral Hilbert scheme X_n, defined as the reduced fiber
product of C^2n with the Hilbert scheme H_n of points in the plane, over the
symmetric power S^n C^2. We prove that X_n is normal, Cohen-Macaulay, and
Gorenstein, and hence flat over H_n. We derive two important consequences.
  (1) We prove the strong form of the "n! conjecture" of Garsia and the author,
giving a representation-theoretic interpretation of the Kostka-Macdonald
coefficients K_{lambda,mu}(q,t). This establishes the Macdonald positivity
conjecture, that K_{lambda,mu}(q,t) is always a polynomial with non-negative
integer coefficients.
  (2) We show that the Hilbert scheme H_n is isomorphic to the Hilbert scheme
of orbits C^2n//S_n, in such a way that X_n is identified with the universal
family over C^2n//S_n.
