Math 620: Algebraic Geometry

Instructor: Frank Sottile
Lectures: TΘ 9:35–10:50 Blocker 148
Course webpage: www.math.tamu.edu/~sottile/teaching/14.1/620.html
Grading: Based on regular homework and class participation.
Homework: One or two problems each week. Found here.

Prerequisites: Graduate algebra or permission of instructor. This may be taken concurrently with Math 654.
Make up classes Wednesday 30 April 3-4 PM, and Thursday 1 May 9-12:00. CE 007.
Material on applications of algebraic geometry:
A gentle introduction to algebraic geometry in applications.
The SIAM activity group on Algebraic Geometry.
Schedule
  • Affine algebraic sets
    • Ideals and affine varieties
    • Irreducibility
    • Nullstellensatz
    • First step towards Bézout's Theorem
  • Projective algebraic sets
    • Projective spaces
    • Ideal of a projective algebraic set
  • Sheaves and varieties
    • Structural sheaf of affine algebraic sets
    • Algebraic varieties
    • Local rings
    • Sheaves of modules on varieties
  • Dimension
    • Topological definition and the link with algebra
    • Dimension and counting equations
    • Morphisms and dimension
  • Tangent spaces and singular points
    • Singular points
    • Regular local rings
    • Curves
  • Bézout's Theorem
    • Intersection multiplicities
    • Bézout's Theorem
  • Sheaf cohomology
  • Arithmetic genus of curves and the weak Riemann-Roch theorem
    • Euler-Poincaré characteristic
    • Degree and genus of projective curves, Riemann-Roch 1
    • Divisors on a curve and Riemann-Roch 2
  
Last modified: Tue Jan 28 08:07:46 CST 2014