We study the isospectral Hilbert scheme
X
n
X_{n}
, defined as the reduced fiber product of
(
C
2
)
n
(\mathbb {C}^{2})^{n}
with the Hilbert scheme
H
n
H_{n}
of points in the plane
C
2
\mathbb {C}^{2}
, over the symmetric power
S
n
C
2
=
(
C
2
)
n
/
S
n
S^{n}\mathbb {C}^{2} = (\mathbb {C}^{2})^{n}/S_{n}
. By a theorem of Fogarty,
H
n
H_{n}
is smooth. We prove that
X
n
X_{n}
is normal, Cohen-Macaulay and Gorenstein, and hence flat over
H
n
H_{n}
. We derive two important consequences. (1) We prove the strong form of the
n
!
n!
conjecture of Garsia and the author, giving a representation-theoretic interpretation of the Kostka-Macdonald coefficients
K
λ
μ
(
q
,
t
)
K_{\lambda \mu }(q,t)
. This establishes the Macdonald positivity conjecture, namely that
K
λ
μ
(
q
,
t
)
∈
N
[
q
,
t
]
K_{\lambda \mu }(q,t)\in {\mathbb N} [q,t]
. (2) We show that the Hilbert scheme
H
n
H_{n}
is isomorphic to the
G
G
-Hilbert scheme
(
C
2
)
n
/
/
S
n
(\mathbb {C}^{2})^{n}{/\!\!/}S_n
of Nakamura, in such a way that
X
n
X_{n}
is identified with the universal family over
(
C
2
)
n
/
/
S
n
({\mathbb C}^2)^n{/\!\!/}S_n
. From this point of view,
K
λ
μ
(
q
,
t
)
K_{\lambda \mu }(q,t)
describes the fiber of a character sheaf
C
λ
C_{\lambda }
at a torus-fixed point of
(
C
2
)
n
/
/
S
n
({\mathbb C}^2)^n{/\!\!/}S_n
corresponding to
μ
\mu
. The proofs rely on a study of certain subspace arrangements
Z
(
n
,
l
)
⊆
(
C
2
)
n
+
l
Z(n,l)\subseteq (\mathbb {C}^{2})^{n+l}
, called polygraphs, whose coordinate rings
R
(
n
,
l
)
R(n,l)
carry geometric information about
X
n
X_{n}
. The key result is that
R
(
n
,
l
)
R(n,l)
is a free module over the polynomial ring in one set of coordinates on
(
C
2
)
n
(\mathbb {C}^{2})^{n}
. This is proven by an intricate inductive argument based on elementary commutative algebra.