Last class!

Plucker embedding, Veronese map, Segre embedding.
Line bundles and sections. Sections of O(m) on projective space. Contravariance in spaces of sections.
Associated bundle construction.
Statement of the Borel-Weil theorem.

Extra class Monday 12:30 PM

We'll have an extra class Monday 12:30-2 PM, location still to be determined.

Thursday June 8

Some puzzle stuff: through four generic lines in space, two pass.

Proj of a graded algebra over C. The special case that the ring is generated in degree 1.

Lemma we didn't prove.
1) If G an algebraic group acts on R, hence on Proj R, the stabilizers have finitely many connected components.
2) If a point x is not G-fixed, then the G-orbit through x isn't closed, and the extra stuff in the closure is of lower dimension.

Borel's theorem. If B is the group of upper triangular matrices, acting on R hence on Proj R, then B has a fixed point in Proj R.

Proposition. Let V be a rep of GL_n. Then each closed orbit X of GL_n on the projective space PV is a GL_n-equivariant image of GL_n/B. If V is irreducible, then it is the linear span of the affine cone over X.

GL_n/B is a flag manifold, which embeds in a product of Grassmannians. The Plucker embedding of a Grassmannian.

Tuesday 6/6/6

Grassmannians. The Stiefel manifold. Orbits of N on the Grassmannian. Schubert varieties, which give a basis for homology. An intersection calculation in the cohomology ring of a Grassmannian. Factoid: the cohomology ring of the Grassmannian, with the Schubert basis, is isomorphic to the puzzle ring with its statuatory basis.