Courses

Topics in Algebraic Geometry: the theory of complements and its application (2019 Fall, Prof. Shokurov)

This graduate course will begin with an overview of the theory of complements and will talk about the recent progress by Birkar and Shokurov in this field. Some unpublished results of Shokurov will also be discussed in the course.

Algebraic Geometry 1: minimal model program for surfaces (2019 Fall, Jingjun Han)

This is an introduction course on the minimal model program. For surfaces, we will cover the cone theorem, the non-vanishing for pairs and generalized polarized pairs, the abundance theorem, Sarkisov program, Zariski decomposition, the ACC for MLD, the ACC for a-LCT, the theory of complements, Birkar-Borisov-Alexeev-Borisov theorem, boundedness of LCT of linear systems, etc.

Syllabus

1. Overview and the basepoint free theorem

2. Rationality theorem

3. Cone theorem (1)

4. Cone theorem (2)

5. MMP for surfaces

6. ACC for MLDs (1)

7. ACC for MLDs (2)

8. Epsilon-complements (1)

9. Epsilon-complements (2)

10. BBAB theorem for surfaces

11. Complements (1)

12. Complements (2)

Algebraic Geometry 2: minimal model program for varieties of log general type (2020 Spring, Jingjun Han)

This course aims to introduce the celebrated paper ``Existence of Minimal Models for Varieties of Log General Type'' by Cacuher Birkar, Paolo Cascini, Christopher D. Hacon, and James Mckernan.

Syllabus (Week)

1. Introduction to the minimal model program in high dimensions.

2. Shokurov type polytope and the minimal model program with scaling

3. Special termination+existence of plt flips imply the existence of minimal models (1)

4. Special termination+existence of plt flips imply the existence of minimal models (2)

5. Finiteness of models and the termination of MMP with scaling (1)

6. Finiteness of models and the termination of MMP with scaling (2)

7. Sarkisov links (1)

8. Sarkisov links (2)

9. Length of extremal rays and bend and break

10. Special termination

11. Dlt models

12. Generalized negativity lemma and varieties fibered by good minimal models

13. Compactify MMPs

14. Termination of flips for threefolds

15. Existence of minimal models for fourfolds

16. Termination of flips for fourfolds

17. Nonvanishing (1)

18. Nonvanishing (2)