Reading seminar 'Invariant theory'
Contact. Olivier Benoist. Office T10.
Schedule. The reading seminar will take place in Salle Verdier on Tuesdays, sometimes in the afternoon (16h15-17h45), sometimes in the morning (10h15-11h45), sometimes both. The precise schedule is below, and will be regularly updated.
- September 17th: Introductory meeting.
- October 2nd (16h15-17h45): Invariant theory of finite groups (Vincent Bouis).
- October 9th (16h15-17h45): The Cohen-Macaulay property (Yitong Wang).
- October 16th (16h15-17h45): Molien's formula (Colin Davalo).
- October 23rd (16h15-17h45): The Chevalley-Shephard-Todd theorem (Andréa Negro).
- November 6th (10h15-11h45): Gröbner bases (Keyu Wang).
- November 13th (10h15-11h45): Nagata's counterexample I: elliptic curves (Misha Popov).
- November 13th (16h15-17h45): Nagata's counterexample II: a ring of invariants that is not finitely generated (Haohao Liu).
- November 20th (16h15-17h45): Brauer groups (Yitong Wang).
- November 27th (10h15-11h45): Group cohomology (Colin Davalo).
- November 27th (16h15-17h45): Infinite Galois theory (Andréa Negro).
- December 4th (16h15-17h45): Faddeev's theorem (Haohao Liu).
- December 11th (10h15-11h45): The unramified Brauer group (Misha Popov).
- December 11th (16h15-17h45): The Bogomolov-Saltman example: a field of invariants that is not purely transcendental (Keyu Wang).
- A few books on invariant theory are [Sturmfels, Algorithms in invariant theory], [Mukai, An introduction to invariants and moduli] and [Dolgachev, Lectures on invariant theory]. Useful references for commutative algebra are [Eisenbud, Commutative algebra with a view toward algebraic geometry] or [Lang, Algebra].
- We will follow [Sturmfels, Chapter 2] for the first five talks, concerning the invariant theory of finite groups. An introductory reference for Gröbner bases is [Cox, Little, O'Shea, Ideals, varieties, algorithms].
- The next two talks are devoted to the exposition of Nagata's counterexample to Hilbert's 14th problem given in [Mukai, §2.5] (another reference with a different proof is [Dolgachev, Chapter 4]). Additional material concerning elliptic curves, and useful to read the proof in Mukai's book may be found in [Silverman, The arithmetic of elliptic curves, Chapter VI].
- In the last six talks, we will present Saltman and Bogomolov's negative answer to the Noether problem. A short reference is [Shafarevich, On Lüroth's problem]. We will rather use the book [Gille, Szamuely, Central simple algebras and Galois cohomology] as a background reference for Brauer groups (Chapter 2), for Galois cohomology of finite and profinite groups (Chapters 3 and 4) as well as for the computation of the Brauer groups of k((t)) and k(t) (§6.3-6.4). The application to Noether's problem is given in §6.6-6.7. The construction of the relevant finite group is also detailed in [Gorchinskiy, Shramov, Unramified Brauer group and its applications, §5.2].