We prove that if
M
M
is a closed smooth manifold and
M
i
{M_i}
,
i
=
1
,
…
,
k
i = 1, \ldots ,k
, are transversally intersecting closed smooth submanifolds of
M
M
, then there exist a nonsingular algebraic set
Z
Z
and nonsingular algebraic subsets
Z
i
{Z_i}
,
i
=
1
,
…
,
k
i = 1, \ldots ,k
, of
Z
Z
such that
(
M
;
M
1
,
…
,
M
k
)
(M;{M_1}, \ldots ,{M_k})
is diffeomorphic to
(
Z
;
Z
1
,
…
,
Z
k
)
(Z;{Z_1}, \ldots ,{Z_k})
. We discuss a generalization and the consequences of this result.