Rational curves on manifolds and Mori theory

Freie Universitaet, Berlin, October 17th - December 19th , 2011

Jaroslaw Wisniewski, Warsaw University

Course webpage at Freie Unveristaet
picture
ruled surface
source: Wikipedia
Summary: The course will give an easy introduction to Mori theory which emerged in 1980's as a part of a birational classification scheme of complex varieties, known as the minimal model program.

Prerequisite: The course will be aimed at students who have good working knowledge of commutative algebra, topology and basic algebraic geometry, including the language of schemes, as in Hartshorne's textbook. Apart of taking part in lectures, the students are expected to solve homework problems and participate actively in exercise sessions.

Task: The students will learn fundamental results as well as basic tools of Mori theory.


Contents of lectures:

  1. Weil divisors and Cartier divisors. Principal divisors and class group, and Picard group. Invertible sheaves and line bundles.
  2. Pulling back divisors, linear systems and maps into projective space, coherent sheaves generated by global sections.
  3. Ample and very ample line bundles, cohomology, theorems A and B of Serre.
  4. Intersection of curves and divisors, selfintersection of divisors on subvarieties, theorem of Nakai.
  5. Numerical equivalence. Kleiman theorem. Nef divisors. Cone of curves and cone of nef divisors. Big divisors. Kodaira lemma.
  6. Stein factorisation and contractions. Fundamental triviality of the Mori program: faces of cones vs contractions. Surfaces: Castenuovo contraction theorem.
  7. Parameter sapaces for morphisms of curves, dimension estimate via Riemann-Roch. Existence of rational curves, K_X not nef, char(k)>0, Frobenius morphism trick.
  8. Rational curves on Fano manifolds, coming back to char(k)=0. The cone theorem of Mori. Contractions of extremal rays. Case of surfaces.

Problem sheets (will be posted regularly every week):

  1. First problem set, Oct 18th, review on Cech cohomology.
  2. Second problem set, Oct 25th, cohomology of invertible sheaves on projective space.
  3. Third problem set, Nov 1st, spanedness, ampleness and degree of divisors on curves.
  4. Fourth problem set, Nov 8th, graded modules vs. coherent sheaves on P^n.
  5. Fifth problem set, Nov 15th, saturated ideals, Euler sequence.
  6. Sixth problem set, Nov 22nd, push-forwad vs. pull-back, canonical divisor, adjunction.
  7. Seventh problem set, Nov 29nd, maps of surfaces, resolving indeterminancy of a rational map, first breaking lemma.
  8. Eigth problem set, Dec 6th, ruled surfaces, second breaking.

Readings:

Algebraic Geometry textbooks: Additional readings: