January 25th, 2000.

### Algebraic Geometry Lectures

#### Cortona, Italy, July 23rd - August 12th, 2000

## Rational curves
and higher dimensional geometry

#### Provisional Program

**1.** Introduction: existence of rational curves on manifolds whose
canonical divisor is not nef, cone theorem of Mori, extremal rays,
contractions, examples. * Readings *: [CKM] Lect. 1 - 4, [KM]
Chap. 1 and [K] II.4, III.1.

**2.** Deformations of curves and schemes parametrizing curves on
manifolds: definition and existence of Hom and Hilb, obstructions to
deformation, dimension and smoothness of Hom and Hilb, universal
families, evaluation morphism and its tangent. * Readings *: [K]
I.1, I.2 and [MP] I.1, I.2.

**3.** Proof of the existence of rational curves on manifolds with
K_X not nef, proof of Mori cone theorem. * Readings *: [CKM]
Lect. 1 - 4, [KM] Chap. 1, [K] II.4, III.1 and [MP] I.2

**4.** Locus of extremal rays and cohomological invariants related
to them. Extremal ray contractions are topologically elementary. *
Readings *: [K] IV.2, V.1 and [W1].

**5.** Uniruled and rationaly connected varieties: existence of good
curves, smoothening of curves, bound on the degree of Fano manifolds
with b_2=1. * Readings *: [K] II.3, IV.1 - IV.3, V.2 and [KMM].

**6.** Low degree rational curves on Fano manifolds; contact Fano
manifolds; classification results. * Readings *: [K] V.1, V.3,
[W2].

** Prerequisite **

The students are expected to have good general
background in algebra, topology, complex and algebraic geometry. The
latter should include the language of schemes and their morphisms as in
Hartshorne's texbook [H].

**Main Textbook**

[K] Kollar, * Rational curves on algebraic varieties*,
Ergebnisse der Mathematik u Grenzgebiete 32, Springer 1996.

**Additional Readings:**

[CKM] Clemens, Kollar, Mori, * Higher dimensional complex geometry*,
Asterisque 166, SMF 1988.

[DB] Debarre, *Introduction to methods in higher-dimensional
algebraic geometry*, lecture notes,
ps file from
Olivier Debarre's home page.

[H] Hartshorne, * Algebraic Geometry*, Graduate Text in Math 52,
Springer 1977.

[KM] Kollar, Mori, *Birational geometry of algebraic varieties*,
Cambridge Tracts in Math 134.

[KMM] Kollar, Miyaoka, Mori, * Rational curves on Fano varieties*,
in Springer Lect Notes in Math 1515.

[MP] Miyaoka, Peternell,
*Geometry of higher dimensional algebraic varieties*,
DMV Seminar 26, Birkhauser 1997.

[W1] Wisniewski, *Cohomological invariants of complex manifolds
coming from extremal rays*, in Asian J Math 2; ps file.

[W2] Wisniewski, *Lines and conics on contact Fano manifolds,
* preprint; ps file.

*Jaroslaw A. Wisniewski*