Course: Algebraic Surfaces
Prerequisites: basic algebraic geometry, algebraic curves.
Programme:
- Basics of minimal model programme
will be introduced as soon as they will be required.
Canonical bundle, adjunction formula.
Divisors and curves, Picard and Neron--Severi groups, K\"ahler and Mori cones.
Criteria of ampleness. Hodge structure of surfaces, Hodge index theorem.
Vanishing theorems, Serre duality.
Kodaira dimension. Finite generation of canonical and Cox rings.
Blowups and exceptional curves. Castelnuovo contraction theorem.
Cone theorem, extremal rays, contraction theorem.
Minimal models. Canonical models and canonical singularities.
- Uniruled surfaces.
Rational surfaces. Castelnuovo's rationality criterium.
Hesse pencil and other interesting pencils.
Del Pezzo surfaces, lines on them, exceptional root systems.
Rational Jacobian elliptic surfaces.
Du Val singularities.
Rational elliptic surfaces without a section. Halphen pencils.
Conic bundles, ruled surfaces, Tsen theorem, projectivization of vector bundles. Hirzebruch surfaces. Scrolls.
Coble surfaces. Severi--Brauer varieties.
- Kodaira dimension zero.
Abelian surfaces.
Bielliptic surfaces.
Kummer surfaces.
$K3$ surfaces.
Torelli theorem.
Enriques surfaces. Reye and Cayley models. Connection with Coble surfaces.
- Kodaira dimension one, elliptic surfaces.
Neron--Kodaira--Tate classification of minimal models of elliptic curves over a local field.
Mordell--Weil group, Shioda--Tate formula.
Theory of Jacobian elliptic surfaces.
Ogg--Shafarevich theory, principal homogeneous spaces over elliptic curves.
- Surfaces of general type.
Surfaces with $p_g=q=0$. Example: Godeaux surface.
Campedelli surfaces.
Barlow surface. Determinantal quintics and Catanese surfaces.
Beauville surfaces. Bidisc quotients.
Bogomolov--Miyaoka--Yau inequality. Ball quotients. Mostow rigidity.
Fake projective planes.
Rigid configurations of lines on a plane, and other rigid configurations.
Fiberations by higher genus curves, Shafarevich and Mordell conjectures.
A proof of Shafarevich conjecture.
Parshin's trick and a proof of Mordell conjecture.
Last time read: 2015-2016, HSE/IUM.
Some videos:
1,
2a,
2b,
2c,
3 (cubic)
4 (seminar).
More could appear at IUM videos page.
EXAM #1