Second Level Degree in Mathematics
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ALGEBRAIC GEOMETRY 1 - 8 CFU
Teacher
Orsola Tommasi
Scheduled Period
I Year - 2 Semester | 02/03/2020 - 12/06/2020
Hours: 64 (32 esercitazione, 32 lezione)
Prerequisites
Many results are based on results from commutative algebra. Basic knowledge of commutative algebra (corresponding to roughly the first half of the commutative algebra course) is recommended.
Target skills and knowledge
Knowledge of the basic concepts, constructions and techniques of algebraic geometry. Competence in relating the different properties of algebraic varieties and the main theoretical results about them. Problem solving skills in algebraic geometry.
Examination methods
Written exam, possibly taking homework assignments into account.
Assessment criteria
Mastering the key techniques and concepts of algebric geometry.
Competence in applying the theoretical results on algebraic varieties and their properties in specific examples, for instance in the solution of exercises.
Problem solving skills in algebraic geometry.
Course contents
This course is intended as a foundational course in algebraic geometry, starting from the basics of the subject and progressing to more avanced techniques such as the study of sheaves and schemes.
Contents:
Affine varieties.
The Zariski topology.
The sheaf of regular functions on a variety.
Morphisms of varieties.
Projective varieties.
Dimension of a variety.
Introduction to schemes.
Planned learning activities and teaching methods
Lectures. Homework, in the form of weekly exercise sheets. The weekly exercise sheets are discussed during problem sessions.
Additional notes about suggested reading
The course is based on Andreas Gathmann's lecture notes at TU Kaiserslautern, available online at
https://www.mathematik.uni-kl.de/~gathmann/de/alggeom.php
The structure of the course will follow the 2003 version of Gathmann's notes, with some material added or substituted from the version of 2014.
There are weekly exercise sheets available on the Moodle page of the course.
Additional references:
- for the parts about affine varieties and varieties in projective space, a good complementary reference is
I. R. Shafarevich, Basic algebraic geometry. 1. Varieties in projective space. Second edition. Translated from the 1988 Russian edition and with notes by Miles Reid. Springer-Verlag, Berlin, 1994. xx+303 pp. ISBN: 3-540-54812-2
- for the part about schemes and sheaves one may refer to
I. R. Shafarevich, Basic algebraic geometry. 2.Schemes and complex manifolds. Second edition. Translated from the 1988 Russian edition and with notes by Miles Reid. Springer-Verlag, Berlin, 1994. xiv+269 pp. ISBN: 3-540-57554-5
I. G. Macdonald, Algebraic geometry. Introduction to schemes. W. A. Benjamin, Inc., New York-Amsterdam 1968 vii+113 pp.