Symmetric powers

  1. Elemetary examples: symmetric powers of curves
    1. Smoothness
    2. Sphere, punctured sphere, elliptic curve
  2. Local algebra
    1. Weierstrass preparatory theorem
    2. The degree of a mapping is equal to the dimension of its local algebra. Example: quotient map f: Xn -> X(n).
  3. Mcdonald's formula for cohomology of a symmetric power.
  4. Resolution of singularities of the symmetric power of a surface.
    1. Hilbert scheme and Chow morphism. The ADHM model for a Hilbert scheme of points in C2.
    2. Smoothness of the Hilbert scheme (Fogarty theorem).
  5. Cohomology of the Hilbert scheme of points in a surface.
    1. Goettsche formula for Betti numbers (without proof).
    2. Action of the Heisenberg algebra on the cohomology (Nakajima operators).
    3. The Fock space of semiinfinite forms.