Birational transformations of small degree (Q2873888)

The Cremona group \(\text{Bir}(\mathbb{P}^2)\) of birational self-maps of the projective plane has attracted much attention in the recent years. This group stands at the confluence of algebraic geometry, group theory and dynamical systems. This is one of the richness of the subject but also makes it difficult for the beginner to penetrate the increasingly technical recent literature. This book aims at being a bridge between the classical literature (Hudson, Godeaux, Semple and Roth\dots) and more recent results. The authors took the radical step to focus almost exclusively on maps of small degree, where small means at most 3: In fact most of the book is concerned with the study of quadratic maps. This restriction allows the authors to introduce many aspects of the general theory in a very concrete way, by the careful study of many examples. Another bias of the book is towards the particular interests of the authors: many links with the theory of foliations are pointed out which are not easy to find elsewhere. Each chapter ends with a few open problems, which are often both simple-looking and challenging. In conclusion this is a pleasant introduction to a classical subject in algebraic geometry, which makes a good companion to the more systematic exposition of [\textit{M. Alberich-Carramiñana}, Geometry of the plane Cremona maps. Lecture Notes in Mathematics. 1769. Berlin: Springer (2002; Zbl 0991.14008)].NEWLINENEWLINEHere follows a more precise description of the content of each chapter.NEWLINENEWLINEChapter 1 introduces the main objects under study: quadratic birational maps of \(\mathbb{P}^2\) up to left-right equivalence, that is up to pre and post-composition by elements of \(\text{PGL}_3\). There are three such orbits of respective dimension 12, 13 and 14 sitting in the projective space \(\mathbb{P}^{17}\) of triple of homogeneous quadratic polynomial in three variables. Some criterion are given to decide whether a quadratic rational map is birational.NEWLINENEWLINEChapter 2 focus on the classification of quadratic flows, that is holomorphic families \(\phi_t\) of quadratic birational maps such that \(\phi_t \circ \phi_s = \phi_{t+s}\).NEWLINENEWLINEIn Chapter 3 the authors propose a way to associate a foliation \(\mathcal{F}(f)\) to a rational map \(f: \mathbb{P}^2 \to \mathbb{P}^2\). In particular when \(f\) is a generic quadratic birational map, \(f\) admits 3 indeterminacy points and 4 fixed points, and these correspond to the 7 singular points of the associated foliation. As an example of application of this circle of ideas from foliation theory, the problem of classifying quadratic birational maps up to birational conjugacy is studied.NEWLINENEWLINEIn Chapter 4 the focus is on dynamical properties, and basic notions such as algebraic stability and dynamical degree are introduced. After a study of birational quadratic maps admitting an invariant curve, an invariant fibration, or an invariant foliation, the authors turn to the theme of periodic points. For a map of the form \(g = A\sigma\) with \(A\) an automorphism with coefficients algebraically independent over \(\mathbb Q\), they show for instance that the union of indeterminacy points for all \(g^{-n}\), \(n >0\), is Zariski dense. The same property is true for the set of periodic points of \(g\). The chapter ends with a classification of quadratic birational maps \(f\) such that \(f^2\) is also quadratic: it turns out that there are interesting examples besides quadratic flows.NEWLINENEWLINEChapter 5 is concerned with properties of \(\text{Bir}(\mathbb{P}^2)\) as an abstract group, and starts with a nice short proof of the fact that the Cremona group is not a linear group. Then it is shown that the centralizer of a generic quadratic birational transformation is an infinite cyclic group, and that the group generated by the standard quadratic involution and \(n\) generic linear automorphisms is isomorphic to a free product \(\mathbb{Z} * \dots * \mathbb{Z} * \mathbb{Z}/2\). Finally, it is shown that the normal subgroup generated by a quadratic or Jonquières map is equal to the whole Cremona group.NEWLINENEWLINEChapter 6 and Appendix A are devoted to a discussion of cubic birational maps of \(\mathbb{P}^2\). The authors obtain 32 different classes of left-right equivalence classes of such maps. The main invariant is the geometry of the exceptional locus, which is a certain configuration of lines and conics. There are 15 admissible configurations, and the configurations of the exceptional set of \(f\) and \(f^{-1}\) are invariant under left-right composition by automorphism, but the classification is finer than that: for instance there are 4 distinct classes where the configuration of both exceptional sets of \(f\) and \(f^{-1}\) is a union of two lines.