Cremona reading group $\def\Z{\mathbb Z} \def\Q{\mathbb Q} \def\Torelli{\mathcal I}$

The reading group will meet Tuesday and Fridays from 3:30-4:30pm in 384H to discuss the group of birational transformations of CP^2.

General references include Cantat's survey and Deserti's survey. There is also a shorter survey by Cantat and lecture notes by Dolgachev. Deserti maintains a large list of publications on the Cremona group, which includes other surveys. The book Geometry of the plane Cremona maps is available online through the Stanford library.

Guidelines for speakers:

The following is a tentative schedule of talks. Topics and speakers are subject to change.

Tuesday June 28 (Alex). Overview. References: Cantat's shorter survey, Favre's lecture notes. See Lemma 2.1 of Blanc-Furter (and Theorem 1.5 of Bass et al.) for a description of the set of degree d birational maps. Alex's lecture notes.

Friday July 1 (Tony). Individual Birational automorphisms of CP^2: Blowing up, Zarski's theorem, and examples. References: Capter 1 of Deserti's survey, and possibly also some of Chapter 4 on quadratic maps.

Tuesday July 5 (Bena). Algebraic structure and important subgroups of the Cremona group. References: Chapter 2 of Deserti's survey. The results are also discussed in Section 3 of Cantat's survey.

Friday July 8 (Volunteer?). TBD. (Finite subgroups of the Cremona group? Groups of birational automorphisms of varieties other than CP^2? Hodge theory and the Hodge-Riemann bilinear relations?)

Tuesday July 12 (Evan). Topology, closed normal subgroups. References: Section 2.1 of Cantat's survey, Blanc, Blanc-Zimmermann. In a recent Annals paper Blanc-Furter showed that the Cremona group is not an infinite dimensional algebraic variety (ind-variety), addressing questions of Shafarevich, Mumford and Serre.

Friday July 15 (Francois). Dynamical degree and algebraic stability. References: Theorem 0.1 (proven on page 11) of Diller-Favre, and also Chapter 3 of Deserti's survey. Statement 1.5 of Cheltsov.

Tuesday July 19 (Laura). Linear and quadratic degree growth, and automorphisms preserving a fibration. References: Theorem 0.2 (proven in Section 4) of Diller-Favre. See Theorem 4.6 in Cantat's survey for more references.

Friday July 22 (Volunteer?). Picard-Manin space, action is linear and faithful. Theorem 3.3.1 of Deserti's survey, Section 4.1 of Blanc-Cantat, Manin.

Tuesday July 26 (Weston). Hyperbolic space, classification of isometries, overview of Neilson-Thurston classification of mapping classes. References: Section 4.1 of Cantat's survey.

Friday July 29 (Alex). Tit's Alternative. References: Cantat's Annals paper.