On Cremona transformations and quadratic complexes (Q1032574)

A cubo-cubic transformation is a birational map \(\phi : \mathbb{P}^3 \dasharrow \mathbb{P}^3\) of degree 3. According to the classification by H. Hudson, there are three kinds of them: Determinantal, de jonquiéres and ruled. The paper under review gives a new construction of some cubo-cubic transformations by using quadratic line complexes. It is shown that any cubo-cubic transformation \(\phi\) whose base locus contains a smooth quintic curve of genus 2 is constructed in the following way. Let \(X=Q_1 \cap Q_2 \subset \mathbb{P}^5\) a smooth complete intersection of two smooth quadric hypersurfaces. Let \(L_1, L_2 \subset X\) two distinct lines and \(E \subset \mathbb{P}^5\) a 3-plane disjoint from the two lines. Let \(\pi_i : X \dasharrow E\), \(i=1,2\), be the projection from the lines \(L_1\) and \(L_2\) to \(E\cong \mathbb{P}^3\), respectively. The authors show that \(\pi_i\) are birational and hence \(\phi=\pi_1\pi_2^{-1}\) gives a birational automorphism of \(\mathbb{P}^3\) which is cubo-cubic. Moreover, if \(L_1\cap L_2 = \emptyset\), then \(\phi\) is determinantal. Otherwise it is of de jonquiére type.