Math 620: Algebraic Geometry

• 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