Università Roma Tre

1) Linear systems of hypersurfaces in Pn Generalities and basic notions, examples in plane and space geometry.
2) Corrispondences in Pn x Pn Intersection theory, Segre classes, numerical characters of a correspondence. Examples.
2) Cremina transformations Linear system of rational hypersurfaces. Homaloidal linear systems Basic Cremona transformations of Pn.
4) In P2 Quadratic transformations, birational involutions, de Jonquiéres trasformations. Structure of the group. Some finite subgroups.
5) In P3: Linear system of rational surfaces. Transformations defined by quadrics or cubicss. Their inverse transformations.
6) Factorization in P3 Birational maps of P3 and Mori theory. Classification of elementary links with applications. Noether-Fano inequalities. I
teoremi di Max Noether per P2 e di Hilda Hudson per P3. Cenni storici e risultati recenti sul gruppo di Cremona.

Testi di riferimento
- I. Dolgachev ‘Lectures on Cremona Transformations, Ann Arbor-Rome’ 2010 -11 http://www.math.lsa.umich.edu/~idolga/cremonalect.pdf
- I. Dolgachev ‘Classical Algebraic Geometry’, Cambridge University Press 2012
Altri riferimenti didattici o bibliografici utili:
- I. Cheltsov, C. Shramov, ‘Cremona Groups and the Icosahedron’ CRC Press New York 2016
- I. Cheltsov, C. Shramov, Three embeddings of the Klein simple group into the Cremona group of rank three, Transformation Groups
- A. Verra ‘Lectures on Cremona Transformations’, School and Workshop, Torino 2006, pdf.
- Si vedano inoltre le voci ‘Cremona group’ e ‘Cremona transformation’ in Encyclopedia of Mathematics. European Math. Society