Projective Bundles and Blowing-Ups II

We study the blowing-up [Formula: see text] of a smooth projective variety [Formula: see text] along a smooth center [Formula: see text] that is equipped with a projective bundle structure over a variety [Formula: see text]. If the Picard number [Formula: see text] is 1 and [Formula: see text] is at most 4, we classify all such pairs [Formula: see text]. If [Formula: see text] is a projective space [Formula: see text] ([Formula: see text]) and [Formula: see text] is 2, we show that [Formula: see text] is a linear subspace in [Formula: see text].