## Birational geometry of algebraic varieties (Math 290)

**Instructor:**Dori Bejleri (bejleri [at] math [dot] university [dot] edu)

**Course description:**The classification of algebraic varieties up to birational equivalence is one of the major questions of higher dimensional algebraic geometry. This course will serve as an introduction to the subject, focusing on the minimal model program (MMP). If you are interested in participating in this course but are not registered, please fill out

**this form**.

**Discussion platform:**We will use the birational geometry Discord server for class discussions. Please use

**this link**to join.

**Canvas page**

**Time and place:**Monday + Wednesday 12:00pm - 1:15pm ET on Zoom (see Canvas or fill out the above form). Lectures will be available both synchronously and asynchronously.

**Office hours:**Thursday 9:00am - 11:00am ET on Zoom (same link as class).

**Syllabus**

**Lectures:**

1. Overview - 09/02/2020. (Notes, video.)

2. Surfaces + Castelnuovo's contraction theorem [H V.1 + V.5, M 1.1] - 09/09/2020. (Notes, video.)

3. The cone of curves [L 1.5, M 1.2] - 09/14/2020. (Notes, video.)

4. Surfaces with canonical bundle not nef + Castelnuovo's rationality criteria [M 1.3 - 1.4] - 09/16/2020. (Notes, video.)

5. Big line bundles, the Albanese morphism, minimal models and existence of plurcanonical sections in dimension 2 [L 2.2, B V, M 1.5] - 09/21/2020. (Notes, video.)

6. Abundance in dimension 2 [M 1.5 - 1.6] - 09/23/2020. (Notes, video.)

7. Classification of surfaces [M 1.7, B] - 09/28/2020. (Notes, video.)

8. Classification of surfaces (cont.), cone of curves in higher dimensions [M 1.7, KM 1] - 09/30/2020. (Notes, video.)

9. Extremal contractions, positivity criteria [KM 2.1, KM 1.5] - 10/05/2020. (Notes, video.)

10. The relative setting, divisors on singular schemes, canonical and terminal singularities [KM 1.5, KM 2.1-2.2] - 10/07/2020. (Notes, video.)

11. Log pairs and discrepencies [KM 2.3] - 10/14/2020. (Notes, video.)

12. Singularities of pairs [KM 2.3] - 10/19/2020. (Notes, video.)

13. Singularities (cont.), Kodaira vanishing [KM 2.3 - 2.4] - 10/21/2020. (Notes, video.)

14. Kawamata-Viehweg vanishing [KM 2.5] - 10/26/2020. (Notes, video.)

15. Kawamata-Viehweg vanishing (cont.), cone theorems [KM 2.5 + 3.1] - 10/28/2020. (Notes, video.)

16. Proof of the basepoint free theorem [KM 3.1 - 3.2] - 11/02/2020. (Notes, video.)

17. Proof of the non-vanishing theorem [KM 3.5] - 11/09/2020. (Notes, video.)

18. Proof of the rationality theorem [KM 3.4] - 11/11/2020. (Notes, video.)

19. The cone theorem I [KM 3.3] - 11/16/2020. (Notes, video.)

20. The cone theorem II [KM 3.3 + 3.6] - 11/18/2020. (Notes, video.)

21. The log minimal model program and the relative case [KM 3.6 - 3.7] - 11/23/2020. (Notes, video.)

22. Types of models, adjunction and the different [KM 3.8, K1 4.1] - 11/30/2020. (Notes, video.)

23. Inversion of adjunction, survey of BCHM [KM 5.4, K1 4.1, BCHM] - 12/02/2020. (Notes, video.)

24. Survey of BCHM (cont.), semi-log canonical singularities [BCHM, K1 5.1 - 5.2] - 12/07/2020. (Notes, video.)

25. Moduli of stable pairs [K1 5.2, K2] - 12/09/2020. (Notes, video.)

**Problem sets**

Homework 1 - due 11/23/2020.

**Texts**

[KM] Kollár-Mori,

*Birational geometry of algebraic varieties.*Cambridge University Press, 1998.

[H] Hartshorne,

*Algebraic Geometry.*Springer.

[M] Matsuki,

*Introduction to the Mori program.*Springer, 2002.

[L] Lazarsfeld,

*Positivity in algebraic geometry, I.*Springer, 2003.

[B] Beauville,

*Complex Algebraic Surfaces.*Cambridge University Press, 1996.

[K1] Kollár,

*Singularities of the Minimal Model Program.*Cambridge University Press, 2013.

[K2] Kollár,

*Families of varieties of general type.*Manuscript available on authors website.

[BCHM] Bikar-Hacon-Cascini-McKernan,

*Existence of minimal models for varieties of log general type.*JAMS, 2010.