Algebraic Geometry (Wintersemester 2019-20)
4 SWS VL pro Woche, Mittwoch 9-11 und Donnerstag 9-11 Uhr, RUD 25, 1.115.
Vorlesender: Prof. Dr. Gavril Farkas
Büro: Humboldt Universität, Institut für Mathematik, Adlershof, Zimmer 1.401
Übungen: Do 11 - 13 Uhr, RUD25, 1.115
" As it turned out, the field seems to have acquired the reputation of being esoteric, exclusive and very abstract with adherents who are secretly plotting to take over all the rest of mathematics!'' (David Mumford, 1975)
Algebraic geometry occupies a central role in modern mathematics interacting with fields like theoretical physics, number theory, topology and differential geometry. Startling advances in the study of parameter (moduli) spaces have been inspired by ideas from physics, elliptic curves play a crucial role in arithmetic, while the study of real 4-manifolds is very much connected to the classical theory of algebraic surfaces. Within algebraic geometry, there has been great progress over the last three decades especially in the study of classification of varieties of dimension three or more (Minimal Model Program) and the understanding of moduli spaces.
This course aims to introduce the basic notions and techniques of modern algebraic geometry. Topics to be discussed include among other things the classical theory of affine and projective varieties, sheaves and schemes, the algebraic notion of dimension and the Hilbert polynomial, syzygies. Very early in the course we will introduce sheaves and schemes and pursue the study of algebraic varieties using this modern language. Since algebraic geometry may sometimes seem to be abstract, a special emphasis will be placed on examples and we will describe in detail explicit algebraic varieties.
References: We will not follow any particular reference too closely but it will be useful to consult Hartshorne's book "Algebraic Geometry", Harris's "Algebraic geometry: a first course" and Mumford's book "Red Book of Varieties and Schemes".
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