Course description: This is a year-long basic course in Algebraic Geometry for master's students, complemented by necessary facts from Commutative Algebra.
Books. In my exposition I will mostly follow the book:
Algebraic Varieties by G.R.Kempf,
Cambridge University Press
(London Math. Society, Lecture Notes Series, v.172).
Sometimes for exercises I will use the book:
Introduction to Commutative Algebra
by M.F. Atiyah and I.G. MacDonald.
Home assignments. I will give problem assignments weekly. These problem assignments are the integral part of the course - they will contain many important points for which there is not enough time in the course itself. Submission of solutions is obligatory. The students should submit solutions a week after they get the the assignments.
Exam. In each semester there will be a midterm exam in class and a final take-home exam.
Syllabus of Part I (Winter semester 2013):
Affine algebraic varieties
Zariski topology
Noether's normalization lemma
Hilbert's basis theorem and Nullstellensatz
Projective varieties and general algebraic varieties
Products of algebraic varieties
Separated and complete varieties
Decomposition into irreducible components
Dimension - different definitions and properties
Principal ideal theorem
Smooth points and tangent spaces
Degree of a projective variety
Classical examples of algebraic varieties
Elements of theory of schemes
Syllabus of Part II. (Spring semester 2014):
Algebraic curves and their non-singular models
Riemann-Roch theorem - elementary approach
Sheaves
Coherent sheaves and localization. Serre's lemma
Cohomologies and elements of homological algebra
Higher cohomological operations with sheaves. Base change
Different versions of Riemann-Roch theorem
and its applications
Jacobians of curves
Weil's proof of Riemann hypothesis
for curves over finite fields