18/05/2006, 15:45 — 16:45 — Room P3.10, Mathematics Building
Igor Dolgachev, University of Michigan
Introduction to classical Cremona transformations II
The group structure of the group $Cr(2)$ of Cremona transformations of projective plane is unlike any other familiar group structures. Although it is generated by projective transformations and a single nonprojective transformation, its structure is very complicated. For example, the conjugacy classes of elements of given finite order are parametrized by an algebraic variety with finitely many irreducible components of different dimension. This is very different from the case of Lie groups. One of the oldest conjectures is that the group is simple as an abstract group.
In these lectures we will briefly discuss the now completed classification of conjugacy classes of finite groups of $Cr(2)$ which is equivalent to the classification of pairs $(S,G)$, where $S$ is a rational surface and $G$ is a finite group of its automorphisms, up to equivariant birational maps.
We will also discuss known examples of infinite subgroups of $Cr(2)$ which can be realized as automorphism groups of rational surfaces, and their relationship to complex dynamics of rational maps.
