Spring 2023 Informal AG Seminar on MMP
This is the website for the informal algebraic geometry seminar for Spring 2023. We will meet weekly on Fridays at 1:30pm in Room 622.

The organizers are Simon Felten, Andres Fernandez Herrero and Akash Sengupta.

## Meetings and lectures

1. 01/27/23 Organizational meeting.
2. We will decide on the topics and speakers for the seminar.

3. 01/27/23 Overview of the Minimal Model Program
4. Lecture by Simon Felten.

The minimal model program aims at exhibiting a 'nice' representative within any birational equivalence class of algebraic varieties -- these are the so-called minimal models.

In this introductory talk, we will motivate the study of minimal models and give an overview over the algorithm that is used to construct them..

5. 02/03/23 Bend and break, rational curves
6. Lecture by Alex Scheffelin.

State deformation theory results for rational curves.
Prove bend and break.
Existence of rational curves when K_X not nef.

References: [D] or [KM] Chapter 1.

7. 02/10/23 Cone of curves and cone theorem for smooth varieties
8. Lecture by Nicolas Vilches Reyes.

Basic notions work with normal projective varieties.
Review Weil and Cartier divisors with coefficients in Z, Q, and R.
Recall ampleness and nef-ness for these divisors.
Recall numerical equivalence.
Finite-dimensionality of N (X) , N1(X).
Cone of effective curves.
Explain Q-factoriality.
Make the audience comfortable with considering a pair (X, D) with D = ∑i ai Di a Q-Weil divisor as a basic object of study.
Prove Cone theorem.

Reference: [KM] Chapter 1.

9. 02/17/23 Resolution of singularities and singularities of MMP
10. Lecture by Simon Felten.

Recall existence of resolution of singularities.
For a resolution f : Y -- > X, compare divisors on V and X.
Define terminal and canonical singularities.
Explain independence of choice of resolution.
Define log canonical, log terminal, Kawamata log terminal, purely log terminal etc. singularities.
Discuss independence of choice of resolution.
Maybe also discuss rational singularities.
Give examples.
Mention du Val singularities as the canonical singularities in dimension 2.
No terminal singularities in dimension 2.

Reference: [KM] Chapter 2.

11. 02/24/23 MMP in dimension 2
12. Lecture by Morena Porzio.

Discuss the general MMP process.
Go through the steps of MMP in dimension 2 (for nonsingular projective surfaces).
Focus should be on the big picture, rather explain what a theorem means than giving a full proof
Start by asking whether KX is nef.
Cone theorem.
Extremal contractions and contraction theorem.
Explain the two types of extremal contractions: either it is a contraction of a (−1)-curve, or it is a fibration over a curve, or it is P2 mapping to a point.
Introduce the hard dichotomy theorem.
Explain that the process always terminates.
Give an overview of the possible outcomes for κ(X) = −∞.

References: [KM] Chapter 1, [M] Chapter 1.

13. 03/03/23 Basepoint-free Theorem and the Cone Theorem
14. Lecture by Akash Sengupta.

TBA.

Reference: [KM].

15. 03/10/23 The Rationality Theorem and the Non-vanishing Theorem
16. Lecture by Anna Abasheva.

TBA.

Reference: [KM] Chapter 3.

17. 03/24/23 The relative cone theorem, running the minimal model program
18. Lecture by Amal Mattoo.

Three Types of Contractions and Flips: introduce the possible three types of contractions.
Briefly discuss that we can continue our program easily in the case of a divisorial contraction.
Recall that the algorithm stops after a fibering contraction.
Explain flips.
Discuss that the existence of flips is equivalent to the finite generation of the canonical ring.
Discuss termination of flips.

References: [KM] Chapter 3, [M] Chapter 11.

19. 03/31/23 Basic Properties of Minimal Models and Mori Fiber Spaces
20. Lecture by Nathan Chen.

Discuss uniruledness and ruledness of Mori fiber spaces.
For minimal models, discuss existence of an effective pluricanonical divisor
Abundance conjecture, hard dichotomy.
Discuss the Iitaka fibration.
The canonical model.
Discuss the canonical singularities occurring on the canonical model.
Finite generation of the canonical ring.
Xcan = Proj(R).
If time permits, give an overview over the Sarkisov program, introduce flops as the connections between different possible minimal models of the same space.

Reference: [M] Chapter 12,13.

21. 04/07/23 Surface Singularities
22. Lecture by Andres Fernandez Herrero.

Discuss the classification of surface singularities and examples.

References: [KM] Chapter 4, [M] Chapter 4.

23. 03/14/23 Multiplier ideals and Nadel vanishing
24. Lecture by TBA.

(possibly skip or find speaker later).

25. 04/21/23 BCHM I
26. Lecture by Kuan-Wen Chen / Akash Sengupta.

Overview and summary of results.

Reference: [BCHM].

27. 04/28/23 BCHM II
28. Lecture by Kuan-Wen Chen / Akash Sengupta.

Some ideas and key steps.

Reference: [BCHM].

## References

1. [KM] Kollar and Mori. "Birational geometry of algebraic varieties".
2. [Mat] Matsuki. "Introduction to the Mori program".
3. [D] Debarre. "Higher-Dimensional Algebraic Geometry".
4. [BCHM] Birkar, Cascini, Hacon and McKernan. "Existence of minimal models for varieties of log general type".