A birational construction of projective compactifications of \(\mathbb{C}^{3}\) with second Betti number equal to one (Q1409672)

The author gives an explicit construction of all projective compactifications of \(\mathbb{C}^{3}\) with second Betti number equal to one. He constructs explicitly birational mappings \[ V_{22}\rightarrow V_{5}\rightarrow Q_{3}\rightarrow\mathbb{P}^{3}, \] where \(Q_{3}\) is the complex quadric in \(\mathbb{P}^{4}\), \(V_{5}\) the Fano threefold of degree \(5\) in \(\mathbb{P}^{6}\), and \(V_{22}\) a Fano threefold of genus \(12\) in \(\mathbb{P}^{13}\).