On Cremona transformations of \(\mathbb P^{3}\) which factorize in a minimal form (Q2871815)

Any Cremona transformation \(\phi : \mathbb P^3 \dashrightarrow \mathbb P^3\) can be factored as a composition of elementary links, according to the Sarkisov program. However, in higher dimensions the geometric content of this factorization is somewhat less transparent than in dimension two.NEWLINENEWLINEThis paper undertakes the classification of Sarkisov links of a special type in which the geometry can be understood explicitly. If \(\phi_1 : X_1 \to Y_1\) and \(\phi_2 : X_2 \to Y_2\) are two Mori fiber spaces, a Sarkisov link of Type II is given by a divisorial contraction \(\tau_1 : Z_1 \to X_1\), a sequence of log flips \(\chi : Z_1 \dashrightarrow Z_2\), and a divisorial contraction \(\tau_2 : Z_2 \to X_2\). This paper is concerned with \textit{special} links of Type II, defined to be those for which \(Z_1 = Z_2\) (so that there are no flips), and \(X_2\) is smooth. In other words, one blows up a smooth subvariety of \(X_1\), and then contracts a divisor to obtain a smooth \(X_2\).NEWLINENEWLINEThe first result of this paper is a classification of special links of Type II from \(\mathbb P^3\) to a Fano variety of Picard rank \(1\). These fall into five classes, which are described explicitly and have quite nice geometry. Three of the five classes have been previously studied and are introduced briefly; the other two are new and described in detail.NEWLINENEWLINEWe mention one example to indicate the flavor of the constructions: let \(\Gamma \subset \mathbb P^3\) be a smooth quintic curve of genus \(2\). The linear system of cubics vanishing on \(\Gamma\) determines a map \(\mathbb P^3 \dashrightarrow \mathbb P^5\), with image \(X_2\) a complete intersection of two quadrics. The inverse of this map can be realized by blowing up a line in \(X_2\) and then contracting the strict transforms of the other lines in \(X_2\) which meet \(L\).NEWLINENEWLINEThe paper next considers Cremona transformations \(\phi : \mathbb P^3 \dashrightarrow \mathbb P^3\) which can be factored as a composition of two special Type II links, i.e.\ decomposed in the form \(\mathbb P^3 \dashrightarrow X \dashrightarrow \mathbb P^3\), where \(X\) is Fano of Picard rank \(1\) and in which both maps are special Type II links; these are the factorizations to which the title alludes. There are \(8\) classes of such transformations, which are again described in detail; these maps can have degrees \(2\), \(3\), \(4\), \(6\), or \(9\).