Let
L
L
be an ample line bundle on a smooth complex projective variety
X
X
of dimension three such that there exists a smooth member
Z
Z
of
|
L
|
\vert L\vert
. When the restriction
L
Z
L_{Z}
of
L
L
to
Z
Z
is very ample and
(
Z
,
L
Z
)
(Z,L_{Z})
is a Bordiga surface, it is proved that there exists an ample vector bundle
E
\mathcal {E}
of rank two on
P
2
\mathbb {P}^{2}
with
c
1
(
E
)
=
4
c_{1}(\mathcal {E}) = 4
and
3
≤
c
2
(
E
)
≤
10
3 \leq c_{2}(\mathcal {E}) \leq 10
such that
(
X
,
L
)
=
(
P
P
2
(
E
)
,
H
(
E
)
)
(X,L) = (\mathbb {P}_{\mathbb {P}^{2}}(\mathcal {E}),H(\mathcal {E}))
, where
H
(
E
)
H(\mathcal {E})
is the tautological line bundle on the projective space bundle
P
P
2
(
E
)
\mathbb {P}_{\mathbb {P}^{2}}(\mathcal {E})
associated to
E
\mathcal {E}
.