Fall 2020

Fall 2020: This semester I am teaching an Advanced Graduate Course intended for 3rd year or above graduate students, postdocs and interested faculty members. It will be an online course taught over Zoom. Meeting ID and Passwords will be sent every week via emails. An outline of the course is given below.

Title: Singularities of the Minimal Model Program (MMP).

Lecture Schedule: Monday and Wednesday, 3:00PM-5:00PM. First lecture will be held on August 17th (Monday).

Main Text: We will mainly follow the book ``Birational Geometry of Algebraic Varieties'' by Kollár and Mori.

Course Content: Chapter 2, Parts of Chapter 4, and most of Chapter 5 of the above book. Chapter 2 is about the definitions, examples and basic properties of various MMP singularities. Chapter 4 is about Classification of log terminal surface singularities via dual graphs of its minimal resolution. Chapter 5 is about the Adjunction and Inversion of Adjunction properties of MMP singularities. In this chapter we will also discuss rational singularities and show that most of the MMP singularities are rational. If there is more time towards the end of the course, then I will talk about F-singularities (coming from the Tight closure theory in commutative algebra) and their relation to MMP singularities in positive characteristic.

Prerequisites: Chapter II and III materials of Hartshorne’s Algebraic Geometry book or its equivalent is required for this course. Specially, the concept of blow-ups, Birational morphism, divisors, line bundles, ample and very ample line bundles, Linear System, Bartini’s theorem, etc. are required.

Additional References: Following texts maybe used for additional references.

1. Paper: ``Singularities of Pairs'' by János Kollár.

2. Book: ``Singularities of the Minimal Model Program'' by János Kollár.

3. Book: ``Introduction to the Mori Program'' by Kenji Matsuki. Warning: This particular book has lots of mistakes, sometime even in the definitions, so one needs to be careful while reading this book.

Lecture Notes: Lecture 1 (Aug 17), Lecture 2 (Aug 19), Lecture 2 Complementary Notes, Lecture 3 (Aug 24), Lecture 4 (Aug 26), Lecture 5 (Aug 31), Lecture 6 (Sep 02), Lecture 7 (Sep 07), Lecture 8 (Sep 09), Lecture 9 (Sep 14), Lecture 10 (Sep 16), Lecture 11 (Sep 21), Lecture 12 (Sep 23), Lecture 13 (Sep 28), Lecture 14 (Sep 30), Lecture 15 (Oct 05), Lecture 16 (Oct 07), Lecture 17 (Oct 12), Lecture 18 (Oct 14), Lecture 19 (Oct 19), Lecture 20 (Oct 26), Lecture 21 (Oct 28), Lecture 22 (Nov 04), Lecture 23 (Nov 09), Lecture 24 (Nov 11).

Video Links of Lectures: Lecture 1, Lecture 2, Lecture 3, Lecture 4, Lecture 5, Lecture 6, Lecture 7, Lecture 8, Lecture 9, Lecture 10, Lecture 11, Lecture 12, Lecture 13, Lecture 14, Lecture 15, Lecture 16, Lecture 17, Lecture 18, Lecture 19, Lecture 20, Lecture 21, Lecture 22, Lecture 23, Lecture 24.

Exercise Sets: Exercises 1