Semester: Fall
Instructor: Chris Brav
Course description: The seminar will illustrate the general theory of algebraic geom-
etry and commutative algebra by focusing on the special case of varieties over a field,
using concrete techniques, particularly Groebner bases, resultants, and elimination
theory.
Prerequisites: Some familiarity with rings, ideals, and modules.
Curriculum:
Division algorithm for polynomials in many variables.
Groebner bases.
Hilbert's Basissatz
Ideal membership problem.
Computing solutions of polynomial systems of equations.
Computing the closure of the image of a map of affine varieties (elimination
theory).
Resultants and elimination theory.
Hilbert's Nullstellensatz
Irreducible components. Dimension.
Projective elimination theory.
Textbooks:
Cox, Little, O'Shea. Ideals, Varieties, and Algorithms. Springer.
Hassett. Introduction to Algebraic Geometry. Cambridge Univ. Press.