Birational geometry

Winter 2020/2021

Jaroslaw Wisniewski

Summary: The course will provide an introduction to birational geometry methods in higher dimensional complex algebraic geometry. Prerequisites: courses in commutative algebra, and algebraic geometry, and algebraic methods in geometry and topology. Official page at USOS GIF: stability condition for quotiens of a lagrangian grassmanian by the Cartan torus, or the Mori chamber structure on the respective Cox ring (joint project with Joachim Jelisiejew).

Topics, provisional list, to be modified.

1. Introduction, case of surfaces intersection on surfaces, blow-ups, factorization of birational morphisms, resolution of rational maps of surfaces.
2. Classical methods: Cartier and Weil divisors, differentials, canonical divisors, adjunction, classification of surfaces.
3. Birational maps and singularities: rational singularities on surfaces, quotients singularities, resolution of singularities.
4. Rational curves on algebraic varieties: existence, Mori cone theorem, contractions, main trivialities of Mori theory.
5. Minimal model program: rudiments, flips and flops, minimal models, canonical models.
6. Algebraic group actions and quotients: alebraic tori, 1-parameter subgroups and characters, toric varieties and their combinatorics, linearization.
7. Mumford's GIT: invariants, stable and semistable sets, geometric and good quotients, quotient by torus action.
8. Cox rings, Mori Dream Spaces: variations of GIT, birational geometry of quotients, case of toric varieties.
9. Cobordism and bordism: birational geometry associated to C* action, fixed points, Bialynicki-Birula decomposition, Morelli-Wlodarczyk cobordism, factorisation theorem.

Problem and exercise sheets:

1. Set I, surfaces, blow-ups, resolving rational maps.
2. Set II, differential forms, canonical divisor.
3. Set IIa, vector bundles, Euler sequence.
4. Set III, ruled surfaces, minimal surfaces.
5. Set IV, Frobenius, Mori, and Hironaka
6. Set V, del Pezzo surfaces, Gosset polytopes.
7. Set Va, automorphisms of degree 5 del Pezzo.
8. Set VI, Atiyah flop, different views.
9. Set VII, finite, C* quotients of affine space.
10. Set VIII, stability, geometric quotients.
11. Set IX, (C*)^r quotients, resolution of abelian singularities
12. Set X, toric varieties as quotients
13. Set XI, torus quotient of a Grassmannian
14. Set XII, torus orbits and quotients of quadrics.

Assessment methods and assessment criteria: An active participation in problem solving sessions is necessary to get the credit, the students are expected to prepare solutions to problems and prepare examples. Midterm and final, both take home. Oral exam in the end of the course.

Readings:

• Arzhantsev, Derenthal, Hause, Laface, Cox Rings, part I
• Clemens, Kollar, Mori, Higher dimensional complex geometry
• Dearre, Higher dimensional algebraic geometry.
• Durfee, Fifteen characterizations of rational double points
• Hu, Keel, Mori dream spaces and GIT
• Kolar, The structure of algebraic threefolds: an introduction to Mori's program
• Kollar, Mori, Birational geometry of algebraic varieties
• Laface, Velasco, A survey on Cox rings
• Matsuki, Introduction to the Mori program.
• Reid, What is a flip?