Study-unit ALGEBRAIC GEOMETRY
| Course name | Mathematics |
|---|---|
| Study-unit Code | 55A00044 |
| Curriculum | Matematica per la crittografia |
| Lecturer | Alessandro Tancredi |
| Lecturers |
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| Hours |
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| CFU | 6 |
| Course Regulation | Coorte 2022 |
| Supplied | 2023/24 |
| Supplied other course regulation | |
| Learning activities | Caratterizzante |
| Area | Formazione teorica avanzata |
| Sector | MAT/03 |
| Type of study-unit | Obbligatorio (Required) |
| Type of learning activities | Attività formativa monodisciplinare |
| Language of instruction | Italian |
| Contents | Algebraic varieties as ringed spaces. Affine and projective varieties. Dimension of an algebraic variety. Regular and singular points of an algebraic varieries. |
| Reference texts | J. Bochnak, M. Coste, M. F. Roy, Real algebraic geometry. Springer 1998 D. Munford, The red book of varieties and schemes. Springer LNM 1358, 1988 I. R. Shafarevich, Basic Algebraic Geometry. Springer 1974 Further notes and references will be supplied by the lecturer |
| Educational objectives | The course introduces to the theory of algebraic varieties as ringed spaces. Its goal is to familiarize the students with the tools they will need in order to use algebraic varieties, also with regard to other fields of the geometry. |
| Prerequisites | Elements of commutative algebra and fields theory, wich are stated as needed, and some elementary topology. |
| Teaching methods | Face-to-face, office hours, usage of the platform “Unistudium” (https://www.unistudium.unipg.it) |
| Learning verification modality | The final exam consists in an oral discussion of about an hour on the subjects developped during the course. A detailed list of the subjects is provided at the end of the lectures. The aim of the exam is to evaluate the level and the quality of the knowledge the students have acquired and to check their ability in the exposition. |
| Extended program | Noetherian topological spaces. Sheaves and ringed spaces. Algebraic sets. Zariski topology. Polynomial and regular functions on algebraic sets. Affine varieties. Prevarieties and their morphisms: products of prevarieties. Algebraic varieties. Rational morphisms. Dimension of a variety. The local ring of a point of an algebraic variety: tangent and cotangent spaces. Regular and singular points of an algebraic variety. Algebraic varieties over an algebraic closed and over a really closed field. Projective varieties. Complexification af an affine and projective real agebraic set. Analytic structure of real and complex varieties. |
| Obiettivi Agenda 2030 per lo sviluppo sostenibile | 04 |