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
Expected learning outcomes
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.
Lesson period: Second semester
(In case of multiple editions, please check the period, as it may vary)
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: complete intersections, double cover of P^2, elliptici surfaces and Abelian surfaces. with a particular emphasis on K3 and Abelian surfaces. Surfaces with Kodaira dimension equal to 1: elliptic surfaces covers of rational elliptic surfaces. Surfaces of general type: complete intersections, double covers of P^2, the Godeaux surface.
Prerequisites for admission
Basic knowledge of geometry and topology are required. The knowledge of the differential varieties, of the complex varieites and of the line bundles are strongly suggested.
Web pages on Ariel. The books mainly used are: - W. Barth, K. Hulek, C, Peters, A, van de Ven Compact complex surfaces. Second edition. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics 4. Springer-Verlag, Berlin, 2004. - A. Beauville, Complex Algebraic Surfaces, Second Edition, Cambridge Univ. Press, 1996. - M. Reid, Chapters on Algebraic Surfaces, in J. Kollár (ed.), Complex Algebraic Geometry, IAS/Park City Math. Ser., vol. 3, Amer. Math. Soc., Providence R.I., 1997.
Assessment methods and Criteria
The final examination consists of an oral exam.
In the oral exam, the student will be required to illustrate results an arguments which are complmentary to the ones of the course and which will be chosen from a list presented by the professor. If a student is interested in a specific argument which is not contained in the listed ones, but which is still related with the course, he can propose it to the Professor, who will decide if it is adeguate for the exam. Moreover, during the oral exam the student will be required to illustrate some of the results presented during the course and to answer to specific questions on the program of the course.
Final marks are given using the numerical range 0-30, and will be communicated immediately after the oral examination.