Algebraic Geometry - Selected Topics

Aleksandar T. Lipkovski

First version of the present text appeared as one-semester course lecture notes in algebraic geometry. The course for graduate students of mathematics, of geometrieal, topological and algebraic orientation, took place in the spring semester of 1994/95, and was organized on the initiative of Z. Markovic, head of Mathematical Institute in Belgrade, with great support from my colleagues from the Belgrade GTA Seminarl , especially R. Zivaljevic and S. Vrecica . I had a difficult task. In a short course one should have reached some relevant topics of algebraic geometry. Basics of algebraic geometry require an ample preliminary material, mostly from commutative algebra, homological algebra and topology. I tried to avoid this and to include only a minimal amount of such material. Consequently, the style of writing is laconic, with many references to existing (excellent) textbooks in algebraic geometry, but on the other side, it is consistent, in order to be readable, with some effort of course. The scope of the course should include some of the interesting and important results in algebraic geometry. Two such results are included, both classical but very important: the 27 lines on a cubic surface and the Riemann-Roch theorem for curves. I leave to the reader to judge, whether my task has been solved, and to which extent. The present text could serve different purposes. It could be used as an introduction for nonspecialists, who would like to understand what is going on in algebraic geometry, but are not willing to read long textbooks. It could also be used as a digest for students, who are preparing to take a serious course in algebraic geometry. Nowadays, algebraic geometry became an indispensable tool in many closely related or even far standing disciplines, such as theoretical physics, combinatorics and many others. Specialists in these fields may also find this text useful.