Let V be an analytic subvariety of an open subset
Ω
\Omega
of
C
n
{{\text {C}}^n}
of pure dimension r; for any
p
∈
V
p \in V
, there exists an
n
−
r
n - r
dim plane T such that
π
T
:
V
→
C
r
{\pi _T}:V \to {{\text {C}}^r}
, the projection along T to
C
r
{{\text {C}}^r}
, is a branched covering of finite sheeting order
μ
(
V
,
p
,
T
)
\mu (V,p,T)
in some neighborhood of V about p.
π
T
{\pi _T}
is called a global parametrization of V if
π
T
{\pi _T}
has all discrete fibers, e.g.
dim
p
V
∩
(
T
+
p
)
=
0
{\dim _p}V \cap (T + p) = 0
for all
p
∈
V
p \in V
. Theorem.
B
=
{
(
p
,
T
)
∈
V
×
G
(
n
−
r
,
n
)
|
dim
p
V
∩
(
T
+
p
)
>
0
}
B = \{ (p,T) \in V \times G(n - r,n)|{\dim _p}V \cap (T + p) > 0\}
is an analytic set. If
π
2
:
V
×
G
→
G
{\pi _2}:V \times G \to G
is the natural projection, then
π
2
(
B
)
{\pi _2}(B)
is a negligible set in G. Theorem.
{
(
p
,
T
)
∈
V
×
G
|
μ
(
V
,
p
,
T
)
≥
k
}
\{ (p,T) \in V \times G|\mu (V,p,T) \geq k\}
is an analytic set. For each
p
∈
V
p \in V
, there is a least
μ
(
V
,
p
)
\mu (V,p)
and greatest
m
(
V
,
p
)
m(V,p)
sheeting multiplicity over all
T
∈
G
T \in G
. If
Ω
\Omega
is Stein, V is the locus of finitely many holomorphic functions but its ideal in
O
(
Ω
)
\mathcal {O}(\Omega )
is not necessarily finitely generated. Theorem. If
μ
(
V
,
p
)
\mu (V,p)
is bounded on V, then its ideal is finitely generated.