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06086


Physik Sommersemester 2007
06086 Seminar zur Einführung in die algebraische Geometrie [S]
   
Dozent Wendland K., Leschke K., Mencattini I., Gray O.
Dauer 2 SWS
Studiensemester 6
Schein Ja (Vortrag)
Termin Mi, 12:15-13:45, 2006 L1
Inhalt The goal this seminar is to provide an introduction to algebraic geometry.

To define algebraic geometry, we could say that it is the study of the solutions of systems of polynomial equations in an affine or projective space of dimension $n$. The solutions of such a set of equations is what is called an algebraic variety. Every branch of mathematics has some guiding problems. In algebraic geometry, the most important of those is the classification problem. In its strongest form, this problem could be stated as follows: classify all algebraic varieties up to isomorphism. This is a very difficult problem, that can be divided in easier subproblems. In the first place we could ask about the classification of algebraic varieties up to birational equivalence. Then, after having identified a reasonable subset of a birational equivalence class, we could proceed to the classification, up to isomorphism, of its members. Finally, we could ask how far an arbitrary variety is from one of the good ones considered before (e.g we could pretend to classify up to isomomorphism all the birational equivalent non singular projective varieties and then, given a general projective variety, ask how far it is from being non singular).
Very often a classification problem in algebraic geometry consists of a discrete part and a continuous part (e.g classification of non singular algebraic curves is given in terms of the genus, the discrete part, and then the study of the moduli space, the continuous part). In the attempt to follow such a project, we are forced to introduce numerical invariants, with the goal of distinguishing between non isomorphic varieties.
Algebraic geometry is probabily the oldest and one of the more technical branches of mathematics. This seminar is meant to be only an introduction to this beautiful subject.

Vorkenntnisse minimale Kenntnisse commutative algebra
Literatur Reid, Miles: Undergraduate Algebraic Geometry,
Cambridge Univ. Press, LMS Student Texts 12
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