Content: The aim of the course is to familiarise PhD students with the concepts and the methods of modern birational geometry, especially from the point of view of Mori theory. After an introduction to the notions of positivity in higher–dimensional algebraic geometry, we will review the birational geometry of algebraic surfaces over arbitrary fields. This will be the occasion to illustrate the Minimal Model Program. We will prove Mori’s bend and break technique to produce rational curves and deduce the cone theorem for smooth projective varieties over perfect fields. As an application, we establish the MMP for smooth algebraic surfaces. Time permitting, we can either prove the Castelnuovo–Enriques–Bombieri–Mumford classification theorem, or study the birational geometry of complex 3-folds.
Diary
26th February: Sheaves, stalks, short exact sequences. Vector bundles and line bundles. Tautological line bundle on the projective space.
4th March: Vector bundles and locally free sheaves. Examples. Weil divisors and class group.
15th April: Computation of class groups for some curves. Cartier divisors, invertible sheaves and Picard group.
22nd April: Maps to projective space, ample line bundles and Serre's theorems on ampleness and finiteness.
29th April: Differential, canonical class, Serre duality. Riemann--Roch for curves and geometric version.
13th May: Intersection form on the Picard group of algebraic surfaces, blow-ups and examples.
20th May: Ample and nef cone, duality with the Mori cone.
27th May: Numerical criteria for ample and nef. Asymptotic Riemann--Roch.
3 June: Deformation of morphisms and curves. Bend and Break.
10 June: The Cone theorem for smooth varieties and the Minimal Model Program for surfaces.
17th June: Divisorial contractions and flips in higher dimension. The Minimal Model Program for complex projective varieties.
24th June: Survey on the recent developments of the MMP (after BCHM).
References:
Lazarsfeld, R. Positivity in algebraic geometry. I.
Bădescu, Lucian. Algebraic surfaces.
Beauville, Arnaud. Surfaces algébriques complexes.
Birkar, Cascini, Hacon, McKernan: Existence of minimal models for varieties of log general type.
Debarre, Olivier. Higher-dimensional algebraic geometry.
Kollár, János; Mori, Shigefumi. Birational geometry of algebraic varieties.
Kawakita, Masayuki. Complex algebraic threefolds.