M390C, Algebraic Geometry

" 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!'' (D. Mumford, 1975)

This is the webpage of the Math 390C Algebraic Geometry Course. The class meets MWF 1:00-2:00 in RLM 12.166.

  • Professor: Gavril Farkas, e-mail: gfarkas@math.utexas.edu
     Office hours: Monday and Friday 2-3, Tuesday 3-4.
  • Algebraic geometry occupies a central role in modern mathematics interacting with fields like theoretical physics, number theory, topology and differential geometry. For instance, 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 varieties of dimension three or more (Mori theory) and the understanding of moduli spaces.
    This course aims to introduce the basic notions and techniques of modern algebraic geometry. A tentative sampling of topics to be discussed include among other things the classical theory of affine and projective varieties, rational and regular maps, the algebraic notion of dimension and the Hilbert polynomial, syzygies. Very early in the course we'll start talking about sheaves and schemes and pursue the study of algebraic varieties using this 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 varieties like Grassmannians and scrolls.
  • References: We will not follow any particular reference too closely but it will be useful to own Hartshorne's "Algebraic Geometry" and Mumford's "Red Book of Varieties and Schemes".
  • Homework problems:
    Problem set nr. 1
    Problem set nr. 2
    Problem set nr. 3
    Problem set nr. 4
    Problem set nr. 5
    Problem set nr. 6
    Problem set nr. 7
    Problem set nr. 8
    Problem set nr. 9
  • These are the course notes in dvi and pdf format as taken and typed up by the students enrolled in this class. I have only done minimal editing. Please let me know if you find any typos or inaccuracies:
    January 21, 2005 dvi pdf
    January 26, 2005 dvi pdf
    January 28, 2005 dvi pdf
    January 31, 2005 dvi pdf
    February 2, 2005 dvi pdf
    February 4, 2005 dvi pdf
    February 7, 2005 dvi pdf
    February 14, 2005 dvi pdf
    February 17, 2005 dvi pdf
    February 23, 2005 dvi pdf
    February 25, 2005 dvi pdf
    February 28, 2005 dvi pdf
    March 2, 2005 dvi pdf
    March 4, 2005 dvi pdf
    March 7, 2005 dvi pdf
    April 4, 2005 dvi pdf
    April 6, 2005 dvi pdf
    April 18, 2005 dvi pdf
    April 20, 2005 dvi pdf
    April 25, 2005 dvi pdf
    Algebraic Geometry Notes (This is a slightly edited and compressed version of the course notes taken by all students who attended this class).