ALGEBRAIC GEOMETRY
Academic year and teacher
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- Versione italiana
- Academic year
- 2022/2023
- Teacher
- ALEX MASSARENTI
- Credits
- 8
- Curriculum
- TEORICO
- Didactic period
- Secondo Semestre
- SSD
- MAT/03
Training objectives
- An introduction to the problems, concepts and methods of classical Algebraic Geometry. At the end of the course the student will have a clear understanding of the basics of Algebraic Geometry and will be able to continue with more advanced topics towards contemporary research.
Knowledge and skills: at the end of the course the student will have acquired the basic concepts and techniques of algebraic geometry, will be able to relate the different properties of algebraic varieties and use theoretical results in this regard, and will be able to solve problems and exercises in algebraic geometry. Prerequisites
- Algebra: rings, modules, polynomials, field extensions, algebraic and transcendent elements. Geometry I: affine spaces, quadrics.
Course programme
- 12 hours: Affine algebraic sets. Hilbert's basis theorem. Correspondence between ideals and algebraic sets. Hilbert's zero theorem.
Zariski topology. Irreducible sets. Decomposition into irreducible components.
Morphisms and rational applications. Regular functions. Dominant rational functions. Birational equivalence.
8 hours: Theory of dimension; dimension of the fiber.
12 hours: Zariski tangent space. Algebraic differential calculus. Regular points and singular points. Systems of local parameters.
16 hours: Projective and quasi-projective algebraic sets. Segre and Veronese varieties. Intersections in the projective space. Bertini's theorem.
8 hours: Divisors and linear systems. Differential forms and canonical class, birational invariants.
8 hours: Elements of the theory of algebraic curves. Riemann-Roch theorem and applications. Elliptic curves, hyperelliptic curves. Didactic methods
- Frontal lectures with exercises, examples, questions.
Learning assessment procedures
- Written exam with typical exercises and more theoretical exercises.
Reference texts
- R. Hartshorne, Algebraic Geometry, Berlin, Springer, 1977.
J. Harris, Algebraic Geometry - A First Course, Springer-Verlag, 1992.
I. Shafarevich, Basic Algebraic Geometry 1 - Varieties in projective spaces, Berlin, Springer-Verlag, 1974.
I. Dolgachev, Classical algebraic geometry - A modern view, Cambridge University Press, 2012.
