Ample bundles and introduction to Mori theory, 2013/2014

Jaroslaw Wisniewski

Fall: Wednedays, 8:30 - 10:00 (lecture), 10:15-11:45 (problem session), room 1780, Banacha 2

Spring: Wednesydays: 16:15-17:45 (lecture only), room 5420, Banacha 2.

Course webpage at USOS
picture
ruled surface
(a water-tower in Ciechanow)
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.

Topics:

  1. Weil divisors and Cartier divisors. Class group, and Picard group. Invertible sheaves and line bundles.
  2. Pulling back divisors, linear systems and maps into projective space,
  3. Coherent sheaves, generation by global sections.
  4. Ample and very ample line bundles, cohomology, theorems A and B of Serre.
  5. Intersection of curves and divisors, selfintersection of divisors on subvarieties, theorem of Nakai.
  6. Numerical equivalence. Kleiman theorem. Nef divisors. Cone of curves and cone of nef divisors.
  7. Big divisors. Kodaira lemma.
  8. Stein factorisation and contractions. Fundamental triviality of the Mori program: faces of cones vs contractions.
  9. Surfaces: Castenuovo contraction theorem. Hodge index theorem.
  10. Parameter spaces for morphisms of curves, dimension estimate via Riemann-Roch.
  11. Existence of rational curves, K_X not nef, char(k)>0, Frobenius morphism trick.
  12. Rational curves on Fano manifolds, coming back to char(k)=0.
  13. Mori's proof of Harstshorne-Frankel conjecture. Lazarsfeld's proof of Remmert- Van de Ven conjecture.
  14. The cone theorem of Mori. Contractions of extremal rays. Case of surfaces.
  15. Locus of extremal rays, types of extremal contractions, length of a ray.
  16. Minimal model program. How it works in case of surfaces. Examples.
  17. Log resolutions. Types of singularities in the MMP.
  18. Kodaira-Kawamata-Viehweg vanishing theorem (according to Koll'ar).
  19. Applications of KKV vanishing theorem, ideas on proving base-point freeness.
  20. Zariski decomposition on surfaces (according to Bauer); case of Del Pezzo surfaces.
  21. Total coordinate (Cox) ring of a variety, GIT quotient of Spec of a f.g. Cox ring.

Problem sheets:

  1. 1st problem set, for Oct 2nd, review on Cech cohomology and the case of P^n.
  2. 2nd problem set, for Oct 9th, trivialities on gbgs and ample, degree of divisors on curves.
  3. 3rd problem set, Oct 23rd, spanedness of vector bundles, splitting over P^1.
  4. 4th problem set, Oct30th, intersection on surfaces, adjunction.
  5. 5th problem set, Nov13th, Euler sequence.
  6. 6th problem set, Nov20th, resolution of indeterminacy of a rational map of surfaces, Mori's breaking.
  7. 7th problem set, Nov27th, geometry of surfaces and birational morphisms.
  8. 8th problem set, Dec4th, ruled surfaces, minimal surfaces.
  9. 9th problem set, Dec18th, Euler-Atiyah extension.
  10. 10th problem set, Jan15th-22nd, singularities and resolutions (under construction, there will be more)
  11. problem set at Miles Reid homepage, Du Val singularities.

Readings:

Algebraic Geometry textbooks: Additional readings: