Moduli and Stacks Learning Seminar

The aim of this seminar is to learn about algebraic stacks and how they are applied to moduli theory in algebraic geometry.

More concretely, we will prove the following results:

The moduli space $\overline{\mathcal{M}}_g$ of stable curves of genus $g > 1$ is a smooth, proper, irreducible Deligne–Mumford stack of dimension $3g - 3$ which admits a projective coarse moduli space $\overline{M}_g$.
The moduli space $\mathcal{M}^{\mathrm{ss}}_X(r, d)$ of semistable vector bundles of rank $r$ and degree $d$ over a fixed smooth connected projective curve $X$ of genus $g > 1$ is a smooth, universally closed, irreducible algebraic stack of dimension $r^2(g - 1)$ which admits a projective good moduli space $M^{\mathrm{ss}}_X(r, d)$.

Talks

Introduction to moduli and stacks ([A] (0)) Speaker: Myeonhu Kim (Princeton University) Date: February 10, 2023 Notes
Sites, sheaves, prestacks ([A] (2.1), (2.2), (2.3.1-4)) Speaker: Myeonhu Kim (Princeton University) Date: February 17, 2023 Notes
Fiber products of prestacks and stacks ([A] (2.3.5-6), (2.4)) Speaker: Myeonhu Kim (Princeton University) Date: February 24, 2023 Notes
Algebraic spaces, Deligne–Mumford Stacks, algebraic stacks ([A] (3.1)) Speaker: Myeonhu Kim (Princeton University) Date: March 3, 2023 Notes
Examples, representability of the diagonal ([A] (3.9), (3.2)) Speaker: Junyan Zhao (University of Illinois at Chicago) Date: March 10, 2023 Notes
First properties, equivalence relations and groupoids ([A] (3.3-4)) Speaker: Zhengze Xin (Stony Brook University) Date: March 17, 2023 Notes
Dimension, tangent spaces, residual gerbes ([A] (3.5)) Speaker: Lu Qi (Princeton University) Date: March 24, 2023 Notes
Characterization of Deligne–Mumford stacks, lifting/valuative criteria ([A] (3.6-8)) Speaker: Junyao Peng (Princeton University) Date: March 31, 2023
Quasi-coherent sheaves and cohomology ([A] (4.1)) Speaker: Neelima Borade (Princeton University) Date: April 9, 2023
Quotients by finite groups, local structure theorem ([A] (4.2)) Speaker: Saket Shah (University of Michigan) Date: April 14, 2023 Notes
Keel-Mori Theorem (existence of coarse moduli spaces) ([A] (4.3)) Speaker: Myeonhu Kim (Princeton University) Date: April 23, 2023 Notes
Results on structures of algebraic spaces and Deligne–Mumford stacks ([A] (4.4-5)) Speaker: Liyuan Shang (Rutgers University) Date: April 30, 2023
Families of smooth curves, nodal curves ([A] (5.1.4), (5.2)) Speaker: Hyunsuk Kim (University of Michigan) Date: May 5, 2023 Notes
Stable curves ([A] (5.3)) Speaker: Supravat Sarkar (Princeton University) Date: May 14, 2023 Notes
The stack of all curves ([A] (5.4)) Speaker: Yueqiao Wu (University of Michigan) Date: May 19, 2023 Notes
Stable reduction of curves ([A'] (5.5.1-4)) Speaker: Junyao Peng (Princeton University) Date: May 26, 2023 Notes
Explicit stable reduction, gluing and forgetful morphisms ([A'] (5.5.5-6), (5.6)) Speaker: Myeonhu Kim (Princeton University) Date: June 5, 2023 Notes
Irreducibility of $\overline{\mathcal{M}}_{g, n}$ ([A'] (5.7)) Speaker: Myeonhu Kim (Princeton University) Date: June 9, 2023 Notes
Projectivity of $\overline{M}_{g, n}$ ([A'] (5.8)) Speaker: Zhiyuan Chen (Princeton University) Date: June 16, 2023 Notes
No seminar this week
Quasi-coherent sheaves on algebraic stacks ([A'] (6.1.1-6)) Speaker: Benjamin Church (Stanford University) Date: June 30, 2023 Notes
Good moduli spaces of algebraic stacks ([A'] (6.3.1-7)) Speaker: Lu Qi (Princeton University) Date: July 8, 2023 Notes
Local structure of algebraic stacks ([A'] (6.5.1-5)) Speaker: Myeonhu Kim (Princeton University) Date: July 23, 2023 Notes
$\Theta$- and $S$-completeness ([A'] (6.7.1-4)) Speaker: Zhengze Xin (Stonybrook University) Date: July 27, 2023
Proof of existence theorem of good moduli spaces ([A'] (6.7.5-7)) Speaker: Zhengze Xin (Stonybrook University) Date: July 29, 2023
Moduli stacks of vector bundles, semistability ([ABBLT] (1, 2)) Speaker: Myeonhu Kim (Princeton University) Date: August 4, 2023 Notes
Good moduli spaces of stacks of semistable vector bundles ([ABBLT] (3)) Speaker: Lu Qi (Princeton University) Date: August 13, 2023
Determinantal line bundles ([ABBLT] (4)) Speaker: Benjamin Church (Stanford University) Date: August 18, 2023
Projectivity of $M^{\mathrm{ss}}_X(r, d)$ and $M^{\mathrm{ss}}_X(r, L)$ ([ABBLT] (5)) Speaker: Junyan Zhao (University of Illinois at Chicago) Date: August 25, 2023 Notes
An introduction to the Emerton–Gee stack Speaker: Neelima Borade (Princeton University) Date: August 28, 2023 Notes

References

[A] Stacks and Moduli, Alper (January 18, 2023 ver.)
[A'] Stacks and Moduli, Alper (February 23, 2023 ver.)
[ABBLT] Projectivity of the moduli space of vector bundles on a curve, Alper etc.
[H] Moduli Theory, Halpern-Leistner
[K] Projectivity of complete moduli, Kollár
[O] Algebraic Spaces and Stacks, Olsson
[SP] The Stacks Project