The goal of the course is to introduce the basic notions and techniques of algebraic geometry including the relevant parts of commutative algebra, and create a platform from which the students can take off towards more advance topics, both theoretical and applied, also in view of a master's thesis project. The fist part of the course provides some basic concepts in commutative algebra, such as localization, Noetherian property and prime ideals. The second part covers fundamental notions and results about algebraic and projective varieties over algebraically closed fields and develops the theory of algebraic curves from the viewpoint of modern algebraic Geometry. Finally, the student will be able to deal with some applications, as for instance Gröbner basis or cryptosystems on elliptic curves over finite fields.
Basic results in commutative algebra: rings an ideals, localization , Noetherian rings.
Affine and projective varieties.
|William Fulton||Algebraic Curves. An Introduction to Algebraic Geometry.||Addison-Wesley||2008|
|Robin Hartshorne||Algebraic Geometry||Springer-Verlag New York||1977||978-0-387-90244-9|
|Siegfried Bosch||Algebraic Geometry and Commutative Algebra||Springer-Verlag London||2013||978-1-4471-4828-9|
To pass the exam students must demonstrate that they have understood the fundamental concepts and demonstration techniques of algebraic geometry, and to be able to support their argumentation with mathematical rigor.
Attendance at lessons and preparation of a seminar on a topic agreed upon with the teacher is required.
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