Let
L
^
\widehat {L}
be a very ample line bundle on an
n
n
-dimensional projective manifold
X
^
\widehat {X}
, i.e., assume that
L
^
≈
i
∗
O
P
N
(
1
)
\widehat {L}\approx i^*\mathcal {O}_{\mathbb {P}_ N}(1)
for some embedding
i
:
X
^
↪
P
N
i:\widehat {X}\hookrightarrow \mathbb {P}_ N
. In this article, a study is made of the meromorphic map,
φ
^
:
X
^
→
Σ
\widehat {\varphi } : \widehat {X}\to \Sigma
, associated to
|
K
X
^
+
(
n
−
2
)
L
^
|
|K_{\widehat {X}}+(n-2)\widehat {L}|
in the case when the Kodaira dimension of
K
X
^
+
(
n
−
2
)
L
^
K_{\widehat {X}}+(n-2)\widehat {L}
is
≥
3
\ge 3
, and
φ
^
\widehat {\varphi }
has a
1
1
-dimensional image. Assume for simplicity that
n
=
3
n=3
. The first main result of the paper shows that
φ
^
\widehat \varphi
is a morphism if either
h
0
(
K
X
^
+
L
^
)
≥
7
h^0(K_{\widehat X}+\widehat L)\geq 7
or
κ
(
X
^
)
≥
0
\kappa (\widehat {X})\geq 0
. The second main result of this paper shows that if
κ
(
X
^
)
≥
0
\kappa (\widehat X)\ge 0
, then the genus,
g
(
f
)
g(f)
, of a fiber,
f
f
, of the map induced by
φ
^
\widehat \varphi
on hyperplane sections is
≤
6
\leq 6
. Moreover, if
h
0
(
K
X
^
+
L
^
)
≥
21
h^0(K_{\widehat X}+\widehat L)\ge 21
then
g
(
f
)
≤
5
g(f)\leq 5
, a connected component
F
F
of a general fiber of
φ
^
\widehat \varphi
is either a
K
3
K3
surface or the blowing up at one point of a
K
3
K3
surface, and
h
1
(
O
X
^
)
≤
1
h^1(\mathcal {O}_{\widehat X})\le 1
. Finally the structure of the finite to one part of the Remmert-Stein factorization of
φ
^
\widehat \varphi
is worked out.