16/05/2006, 14:30 — 15:30 — Room P3.10, Mathematics Building
Sandro Verra, Universitá di Roma III
Introduction to classical Cremona transformations I
A Cremona transformation is a birational automorphism of a complex projective space $\mathbb{P}^r$. The study of such transformations and the group which they generate was a popular subject of classical algebraic geometry flourishing more than one century ago. Although much progress has been made in the two-dimensional case, in spite of the efforts of many classical and modern algebraic geometers, most of the fundamental problems in the higher-dimensional case remain unsolved. The aim of these lectures is to report on the classical techniques for studying Cremona transformations and specifically on the rich legacy of classical examples. The plan is to present a series of examples in modern terms and to use these examples to introduce basic constructions and techniques, as well as more recent results and open problems. The program will include some of the following topics:
- Base locus and numerical characters of Cremona transformations.
- Transformations defined by quadrics.
- The cubo-cubic transformation of $\mathbb{P}^3$ and related topics.
- Birational involutions of $\mathbb{P}^2$ and $\mathbb{P}^3$.
- Homaloidal linear system of surfaces with finite base locus.
- Cremona transformations with smooth and connected base locus.
- Classification problems.
