Birational geometry of rational quartic surfaces (Q2193429)

Two subvarieties \(X,Y\) in \(\mathbb{P}^{3}\) are Cremona equivalent (CE) if there is a birational map of \(\mathbb{P}^{3}\) that maps \(X\) to \(Y\). The author studies the very interesting and classical problem of finding divisor which are CE. He proves that all rational quartic \(S\) in \(\mathbb{P}^{3}\) are CE to a plane. The proof is a case-by-case analysis. The first relevant case is when \(S\) is singular along a twisted cubic. After that, he proves that if \(S\) has isolated singularities, then \(S\) is CE to a quartic surface with non isolated singularities. Finally, he concludes the proof by resolving that last case. In particular, if \(\mathrm{Sing}(S)\) is a conic, the author produces a good model of \((\mathbb{P}^{3},S)\) with threshold between \(0\) and \(1\). The author proves that the existence of a good model with such a threshold is a necessary and sufficient condition for \(S\) to be CE to a plane.