The aims of this course is to provide the notion of birational morphisms and of minimal model of surfaces, in order to obtain the classification of the algebraic surfaces and to study their geometric properties
The student will learn the basic results in birational geometry, in particular about the problem of the birational classification of the varieties. Moreover, one will acquire techniques for the construction and the study of projective varieties.
Background material: Complex manifolds. Subvarieties, divisors and holomorphic line bundles. The canonical bundle. Projective algebraic varieties. Ample line bundles and their properties. Basics of the theory of projective algebraic surfaces: Curves on a surface. Intersection theory on an algebraic surface. The Néron-Severi group and numerical equivalence. The Riemann-Roch theorem, Noether's formula. Genus formula. The cones of curves: The Hodge index theorem. The ample and the nef cones. Birational maps and minimal models: Rational maps and linear systems. Birational maps. Blowing-ups and their properties. Birational invariants. Minimal models. Kodaira dimension and classification by using birational invariants. Examples: surfaces with negative Kodaira dimension: blow ups of P^2 and del Pezzo surfaces. Surfaces with trivial Kodaira dimension, with a particular emphasis on K3 and Abelian surfaces. Surfaces with Kodaira dimension, equal to 1 and elliptic fibrations. Surfaces of general type and smooth cover of surfaces.