Cremona Book Project

March 2024

Project

The project is to write a book on the Cremona group with the following scope in mind:

• Dimension 2 only;
• Rational surfaces only;
• Algebraic setting, over any base field;
• Full proofs (not a survey);
• As self-contained and elementary as possible.

The book covers the "Dynamical story" (algebraic stability, dynamical degree), the "Hyperboloid story" (blowing-up everything, consequences of negative curvature), and the "Sarkisov story" (factorization, generators and relations).

Comments are very welcome, including minor typos!

Download

Here you will find some preliminary versions, that should be updated regularly.

Latest version:

• 2024, July 27: Some corrections in chapter "Presentations", and first draft of genral introduction.

Older versions:

• 2024, June 19: Some (slow) progress on chapters "Halphen" and "Presentations".
• 2024, April 18: First draft of new (and last?) chapter "Open surfaces".
• 2024, March 31: Many corrections in part III.
• 2024, March 13: Almost done with chapter "Real Cremona Group".
• 2024, March 1: Corrections in chapter "Perfect Fields" and appendix "Pieces".
• 2024, February 15: Many corrections, mostly in part III.
• 2023, August 18: Many corrections in part II.
• 2023, July 6: Corrections chapters 1 to 8 + new tables in appendix H.
• 2023, April 18: Corrections in last two chapters (normal subgroups and Tits alternative).
• 2023, March 26: Corrections up to Chapter 8.
• 2023, March 3: Corrections in chapter Topologies.
• 2023, February 5: Some cleaning up, up to Chapter 9.
• 2023, January 8: Some cleaning up.
• 2022, November 1: First draft of five new chapters (characteristic, presentations, Halphen twists, finite subgroups, real Cremona group).
• 2022, September 25: Tits alternative over any field.
• 2022, August 9: Some cleaning up, and first batch of epigraphs!
• 2022, July 20: Added first draft of chapter about realizing dynamical degrees.
• 2022, July 2: Added chapter on rank r fibrations over perfect fields.
• 2022, February 16: Some cleaning up.
• 2022, February 7: Added draft of chapter on group automorphisms of the Cremona group.
• 2022, January 21: Back to work after a semester of teaching! Added draft of chapter about Jonquières length.
• 2021, August 14: Added draft of chapter about Tits alternative.
• 2021, August 7: Some improvements in first five chapters.
• 2021, May 1st: Some cleaning-up, and added first draft of chapter about normal subgroups.
• 2021, April 25: Some cleaning-up, and added first draft about the cube complex.
• 2021, April 2: Added draft of chapter on G-Sarkisov program.
• 2021, February 2: Added draft of chapters on Polynomial Automorphisms, and on Del Pezzo Surfaces.
• 2020, December 28: Preliminary version of first 6 chapters.