On the untwisting of general de Jonquières and cubo-cubic Cremona transformations of \(\mathbb P^{3}\)
In his article [J. Algebr. Geom. 4, No. 2, 223--254, appendix 248--254 (1995; Zbl 0866.14007)], \textit{A. Corti} gave a rigorous proof of the Sarkisov program for threefolds, proving that every birational map between two Mori fiber spaces (of dimension three) factorizes into a product of four types of elementary links. To any such a map \(\Phi: X\dashrightarrow X'\), one associates a suitable triple of rational numbers \((\mu, \lambda, \epsilon)\), called the Sarkisov degree of \(\Phi\). It is possible to construct an elementary link \(\Phi':X\dashrightarrow X_1\) such that the Sarkisov degree of \(\Phi\circ(\Phi')^{-1}\) is strictly less than the Sarkisov degree of \(\Phi\) (with respect to the lexicographic order), so to start an induction process. Such a \(\Phi'\) is said to untwist \(\Phi\). In the article under review, the author studies this situation for three types of Cremona transformations of \(\mathbb P^3\): general de Jonquières transformations, both of degree \(2\) and of degree greater than \(2\), and special cubo-cubic transformations. In all the three cases he gets the precise form of the elementary links appearing in the Sarkisov decomposition, and proves that this form provides a characterization of the three types of Cremona transformations considered.
scientific article; zbMATH DE number 6010045
scientific article