The exam will start asking you to explain one of the following theorems:
-Nakai-Moishezon's criterion.
-Kleinman's criterion for ampleness.
-Non-vanishing theorem.
-Base point free theorem.
-Cone theorem.
Lecture 1, Oct. 10.
Overview of Mori theory.
Weil, Cartier divisors and invertible sheaves (see Hartshorne, Section II.6).
Lecture 2, Oct. 17.
Intersection numbers (see Debarre, Section 1.2).
Linear systems and rational maps (see Lazarsfeld I, Section 1.1.B).
Lecture 3, Oct. 24.
Nakai-Moishezon criterion of ampleness and nef divisors (see Debarre, Sections 1.5, 1.6).
Lecture 4, Nov. 7.
Kleinman's criterion for ampleness (see Debarre, Section 1.7).
Relative cone of curves and rigidity Lemma (See Debarre, Section 1.3).
Mumford's example of a non-closed NE(X) (see Lazarsfeld, 1.4.A).
Lecture 5, Nov. 14.
Exceptional locus of birational morphism (cf. Debarre, Section 1.10): negativity lemma (Kollár-Mori, Lemma 3.39), purity theorem and ramification formula for smooth varieties .
Lecture 6, Nov. 21.
Terminal and canonical singularities, examples and dimension 2 case.
Lecture 7, Nov. 28.
Big divisors (see Lazarsfeld, Section 2.2A).
Canonical models (see Debarre, Section 7.2).
Lecture 8, Dec. 12.
Singularities of pairs (see Kollár-Mori, Section 2.3).
Multiplier ideal sheaves (see Lazarsfeld, Sections 9.2 and 9.3.G).
Lecture 9, Dec. 19.
Vanishing theorems (without proof): Kodaira, Kawamata-Viehweg Nadel (for proofs see for example Matsuki, Chapter 5).
Local vanishing and Nadel vanishing (Lazarsfeld, Theorems 9.4.1 and 9.4.8).
Shokurov non-vanishing theorem (see CKL, Sections 2.3, 2.4 and Theorem 3.11).
Lecture 10, Jan. 16.
Base point free theorem (see CKL, Theorem 3.4).
Rationality theorem (see CKL, Theorem 3.5).
Lecture 11, Jan. 23.
Cone theorem and contraction theorem (see Kollár-Mori, Theorem 3.7).
Lecture 12, Jan. 30.
Running an MMP: type of contractions, minimal models and Mori fibre spaces. Flips.
The results of BCHM.
References:
• A. Corti, A-S. Kaloghiros, and V. Lazić, Introduction to the minimal model program and the existence of flips. Bull. Lond. Math. Soc. 43 (2011), no. 3, 415–418.
• O. Debarre, Higher-dimensional algebraic geometry. Springer, 2001.
• R. Hartshorne, Algebraic Geometry, Springer, 1977.
• J. Kollár and S. Mori, Birational geometry of algebraic varieties. Cambridge University Press,
1998.
• R. Lazarsfeld, Positivity in Algebraic Geometry, Vol. I, II. Springer, 2004.
• K. Matsuki, Introduction to Mori theory. Springer, 2002.