Constructions of rationality of some four-dimensional Fano varieties of index 2 (Q1311492)

Fano four-dimensional varieties \(V\) of index 2 with \(\text{Pic} V = \mathbb{Z}\) are considered (over \(\mathbb{C})\). The only discrete invariant for such \(V\) is its genus \(g\), for which the only possibilities are \(2\leq g\leq 10\). What it is shown is that for each \(g\), with \(5 \leq g \leq 8\), it exists a rational four-dimensional Fano variety \(V = V_{2g - 2} \subseteq \mathbb{P}^{g + 2}\) as above. -- All the constructions of rationality are essentially based on finding a Fano variety \(V\) containing a special plane \(P\), blowing up \(V\) along \(P\) and determining a linear system on the blow-up \(\widetilde V\) which determines a morphism on \(\widetilde V\) whose image is rational.