# 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:

• Talks should be accessible to first year graduate students. The audience will be quite diverse, and you should not assume any familiarity with schemes, or with complex surfaces, or with hyperbolic geometry, or with geometric group theory. Please use the language of complex projective varieties or manifolds instead of schemes.
• Include examples and pictures!
• Complete proofs are not required. If you think a proof is too hard, you can just give the intuition or skip it entirely.

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.