On P¹-bundles on P³ that are small modifications of a Fano fourfold

We find necessary conditions for the projectivization of a rank two vector bundle E on P³C to be related to a smooth Fano 4-fold by a small Q-factorial modification (SQM). Furthermore, we fully describe the birational geometry of PP³(E) when E is stable, with c₁=-1, c₂=2. In this case, we prove that there exists a flip from X⁺ to PP³(E), where X⁺ is smooth and Fano. We also determine the birational geometry and numerical invariants of X⁺.