Birational Geometry Reading Seminar

This is my plan of the reading program of birational geometry for the beginner of this area! Aiming to read the basic aspect in the birational geometry, both lower dimensional ($\dim X=2$) and higher dimensional ($\dim X\geq 3$) in algebraic geometry.

-1. Our Notes

Here is our notes: BirGeo23, update at 2023/03/07.

0. Preliminaries

Including the basic theory of schemes and cohomology of coherent sheaves at the level of Hartshorne’s book chapter 1-4 and some reults in chapter 5.

1. The basic things in birational geometry

• The chapter 1-5 in Kollar/Mori’s Birational Geometry of Algebraic Varieties. These things including the most basic knowledge of birational geometry, including the minimal model program and its singularity theory.

Here are my remarks in the book:

1. Some Remarks of the Kollar and Mori’s Birational Geometry of Algebraic Varieties Chapter 1-2;

2. Some Remarks of the Kollar and Mori’s Birational Geometry of Algebraic Varieties Chapter 3;

3. Some Remarks of the Kollar and Mori’s Birational Geometry of Algebraic Varieties Chapter 4-5.

• Chapter 1-10 in Classification of Higher Dimensional Algebraic Varieties by Hacon/Kovács. More recent book and MMP in Birational Geometry.

More references: (1) Higher-Dimensional Algebraic Geometry by Olivier Debarre;

(2) Rational Curves on Algebraic Varieties by J. Kollár;

(3) Singularities of the Minimal Model Program by J. Kollár;

(4) Introduction to the Mori Program by Kenji Matsuki;

(5) Exercises in the birational geometry of algebraic varieties (Kol08) by J. Kollár;

(6) Introduction to the log minimal model program for log canonical pairs by O. Fujino；

(7) Positivity in Algebraic Geometry I/II by R. Lazarfeld;

(8) Foundations of the Minimal Model Program by O. Fujino.

2. The BCHM

Existence of minimal models for varieties of log general type by Birkar/Cascini/Hacon/McKernan. The classical paper which shows that the canonical ring of a smooth projective variety is finitely generated.

More references: (1) Classification of Higher Dimensional Algebraic Varieties by Hacon/Kovács chapter 11-16;

(2) The Minimal Model Program for Varieties of Log General Type (MMP) by Hacon;

(3) Foundations of the Minimal Model Program by Fujino.

3. Moduli of verieties of general type

Families of varieties of general type by J. Kollár. The moduli theory of KSBA-stable varieties of general type has been extended to higher dimension in full generality by contributions of many people, culminating in this book (modbook-final). This theory heavily relies on developments in birational geometry, in particular, on existence of minimal models (BCHM) and boundedness of varieties of general type (Hacon/McKernan/Xu).

More references: (1) Singularities of the Minimal Model Program by J. Kollár;

(2) C. D. Hacon, J. McKernan and C. Xu, Boundedness of moduli of varieties of general type, J. Eur. Math. Soc. 20 (2018), 865–901;

(3) Moduli of algebraic varieties by Caucher Birkar, arxiv2211.11237.

4. More things need to add

Such as BAB-conjecture or moduli of algebraic varieties by Caucher Birkar, arxiv2211.11237.

$\infty$. Some websites of learning birational geometry

(1) Minimal Model Program Learning Seminar;

(2) Birational geometry of algebraic varieties (Math 290);

(3) Minimal Model Program Learning Seminar, Spring 2021;