Course: Proofs of Irrationality
Lectures by: Sergey Galkin
Where: Independent University of Moscow. Room 303.
When: every Saturday. Fall 2015.
This page: https://mccme.ru/~galkin/rat.html.
Prerequisites: basic algebraic geometry, algebraic curves.
Videos of first 4 lectures are available online:
4 (Абугалиев) 'Castelnuovo's criterion'.
- Rationality, stable rationality, retract-rationality, unirationality, rational connectedness.
- Examples of rational varieties. Rationality of intersection of two quadrics.
- Birational invariants.
Resolution of singularities and weak decomposition theorem.
Holomorphic contravariant tensors.
- Castelnuovo's rationality criterium.
- Rationality of surfaces over non-algebraically-closed fields.
Del Pezzo fibrations.
- Conic bundles, discriminant. Double covers and Prym varieties.
Intermediate Jacobian of a conic bundle.
- Artin-Mumford's example of stably irrational unirational threefold.
Torsion in homology. Brauer group.
- Clemens-Griffiths's proof of irrationality of a smooth cubic threefold.
Variety of lines on a cubic hypersurface.
Weil's intermediate Jacobian and Griffiths's component.
- Iskovskikh-Manin's proof of irrationality of a smooth quartic threefold.
Method of maximal singularities. Birational rigidity.
- Kollár's method: holomorphic forms in finite characteristic.
- Voisin's degeneration method. Stable irrationality of a very general double cover of three-space branched in a quartic.
- Beauville's proofs using Voisin's degeneration method.
- Work of Colliot-Thélène and Pirutka. Stable irrationality of a very general quartic threefold.
- Stable irrationality of a very general quartic fourfold after Totaro.
More videos should appear at IUM videos page.