ALGEBRAIC GEOMETRY
Academic year and teacher
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- Versione italiana
- Academic year
- 2015/2016
- Teacher
- PALTIN IONESCU
- Credits
- 9
- Didactic period
- Secondo Semestre
- SSD
- MAT/03
Training objectives
- Introduction to the problems, concepts and methods of classic Algebraic Geometry. When finishing the course, the student will posses a comprehensive image about the subject of Algebraic Geometry and will be prepared to read more advanced text books, leading him towards contemporary research.
Prerequisites
- General facts from the Algebra course: rings, modules, polynomials, field extensions, algebraic and transcendental elements. General facts from the Geometry course: affine and projective spaces, quadrics.
The Commutative Algebra course from the first semester of the Master programme: localisation, finiteness properties, tensor product, integral elements, prime and maximal ideals. Course programme
- 12 hours: Affine algebraic sets. The Hilbert basis theorem. Correspondence between ideals and algebraic sets. The Hilbert Nullstellensatz. Zariski topology. Irreducible sets. Decomposition into irreducible components.
Morphisms and rational maps. Regular maps. Dominant rational maps. Birational equivalence.
9 hours: Dimension Theory. Dimension of fibres.
9 hours: Zariski tangent space. Embedded and abstract tangent space. Algebraic differential calculus. Smooth points and singular points. Systems of local parameters.
12 hours: Projective and quasi-projective algebraic sets. Segre and Veronese varieties. Intersections in projective space. The fundamental theorem (the image is closed). Applications, Bertini's Theorem and the principle of counting constants.
9 hours: Divisors and linear systems.
Differential forms and canonical class, birational invariants.
12 hours: Elements of the theory of algebraic curves. Riemann-Roch theorem and applications. Elliptic and hiperelliptic curves. Didactic methods
- Oral lectures including: exercises, examples, questions, dialog with the class.
Learning assessment procedures
- Oral examination, based on a given list of possible subjects. The first subject concerns the general theory, while the second one is about the theory of algebraic curves. Every subject which is part of the oral examination has the same weight in the final grade.
Reference texts
- Ph. Ellia, Un'introduzione alla geometria algebrica, dispense del corso, Ferrara, 2005.
I. R. Shafarevich, Basic Algebraic Geometry I, Springer 1994.
