Organizers: Dori Bejleri, Junyan Zhao
Overview
Artin stacks provide a flexible framework for studying moduli spaces that incorporate objects with automorphisms, allowing for a richer and more accurate geometric interpretation of parameter spaces. The theory of good moduli spaces, introduced by Alper, offers a way to extract meaningful "coarse" moduli spaces from stacks in a way that retains essential geometric and categorical information.
This seminar aims to develop a working understanding of Artin stacks and good moduli spaces, focusing on both foundational aspects and recent developments. We will begin with the basic theory of stacks, emphasizing examples and moduli problems that naturally lead to Artin stacks. Then we will study the theory of good moduli spaces, highlighting its contrast with coarse moduli spaces in the Deligne–Mumford setting and exploring key results, techniques, and applications.
Schedules
We meet every Wednesday from 2 pm to 4 pm with a 10 minutes break.
(06/04/2025) Definition and basic properties of good moduli spaces [Speaker: Junyan Zhao]
2. (06/11/2025) Local structure of good moduli spaces [Speaker: Myeong Jae Jeon]
3. (06/25/2025) Good moduli gerbes, root stacks, and valuative criteria [Speaker: Dori Bejleri]
4. (??/??/2025) Existence of good moduli spaces, criterion by AHLH [Speaker: ]
5. (??/??/2025) Theta stratification and Langton's algorithm [Speaker: ]
6. (??/??/2025) Existence of good moduli spaces, criterion by AFS [Speaker: ]
7. (??/??/2025) Examples: moduli of vector bundles, moduli of Fano varieties [Speaker: ]
References
1. Alper, Jarod, Good moduli spaces for Artin stacks. Annales de l'Institut Fourier, Volume 63 (2013) no. 6, pp. 2349-2402.
2. Alper, Jarod, Local properties of good moduli spaces. Tohoku Math. J. (2) 64 (2012), no. 1, 105–123.
3. Alper, Jarod; Hall, Jack; Rydh, David A Luna \étale slice theorem for algebraic stacks. Ann. of Math. (2) 191 (2020), no. 3, 675–738.
4. Alper, Jarod, Jack Hall, and David Rydh. "The \'etale local structure of algebraic stacks." arXiv preprint arXiv:1912.06162 (2019).
5. Alper, Jarod; Halpern-Leistner, Daniel; Heinloth, Jochen Existence of moduli spaces for algebraic stacks. Invent. Math. 234 (2023), no. 3, 949–1038.
6. Alper, Jarod; Fedorchuk, Maksym; Smyth, David Ishii Second flip in the Hassett-Keel program: existence of good moduli spaces Compos. Math. 153 (2017), no. 8, 1584–1609.
7. Alper, Jarod; Blum, Harold; Halpern-Leistner, Daniel; Xu, Chenyang Reductivity of the automorphism group of K-polystable Fano varieties. Invent. Math. 222 (2020), no. 3, 995–1032.
8. Halpern-Leistner, Daniel. "On the structure of instability in moduli theory." arXiv preprint arXiv:1411.0627 (2014).
The picture was stolen from Alper's personal website