**Programme:**

- 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.

Last time read: Fall 2015, IUM.

Some videos are available online: 1, 2, 3, 4 (Abugaliev) 'Castelnuovo's criterion'.

More videos could appear at IUM videos page.