Yu.Burman

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.