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, in particular:
- I try to "simplify" proofs, meaning that I take the risk to transform a perfectly correct but rather long argument into a very short mistake. If you spot such a problem, please tell me!
- If you feel that I missed something important in my "might be included list", you should convince me to add an item.
Download
Here you will find some preliminary versions, that should be updated regularly.
Latest version:
Older versions:
- 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.