Positivity in Algebraic Geometry

Lecture 1, March 04.

Weil divisors, Cartier divisors and invertible sheaves (see Hartshorne, Section II.6).

Lecture 2, March 05.

Morphisms to projective space and linear systems (see Hartshorne, Section II.7).

Ample and very ample line bundles: definitions and Serre vanishing (see Hartshorne, Sections II.7 and III.5).

Lecture 3, March 06.

Intersection numbers (see Debarre, Section 1.2).

Nakai-Moishezon criterion of ampleness (see Debarre, Section 1.5).

Lecture 4, March 07.

Nef divisors (see Debarre, Section 1.6)

Kleinman's criterion for ampleness (see Debarre, Section 1.7).

Lecture 5, March 14.

Relative cone of curves and rigidity Lemma (See Debarre, Section 1.3).

Big divisors (see Lazarsfeld, Section 2.2A).

Basic examples.

Lecture 6, March 15.

Mumford's example of a non-closed NE(X) (see Lazarsfeld, 1.4.A).

Overview of Mori theory.

Negativity lemma (Kollár-Mori, Lemma 3.39).

Lecture 7, March 19.

Terminal and canonical singularities.

Lecture 8, March 20.

Running an MMP: types of contractions and their properties. Flips.

Lecture 9, March 27.

Bend and Break lemmas (see Debarre).

Lecture 10, March 28.

Fano maifolds are uniruled (see Debarre)

Cone theorem in the smooth case via B&B (see Debarre).

References:

• O. Debarre, Higher-dimensional algebraic geometry. Springer, 2001.

• R. Hartshorne, Algebraic Geometry, Springer, 1977.

• R. Lazarsfeld, Positivity in Algebraic Geometry, Vol. I, II. Springer, 2004.