Wednesday, April 13, 2016

For the last three weeks of the course, we will shift our attention from the general theory of algebraic geometry to specific examples coming from elliptic curves and linear algebraic groups.

For elliptic curves, we will use Milne’s book on elliptic curves.

For the theory of linear algebraic groups, we will use Humphreys book.

We will start with Humphrey’s book, moving quickly through a review of algebraic geometry from the first section of the book.  Skim through the first 12 pages, which should all be familiar.

  • Let k be a field. Check that in the category of affine k-schemes, the (categorical) product of two affine lines (A^1) over k is the plane A^2.
  • Read the construction of page 13 (Prop. 1.7) that identifies the product of two projective spaces with a closed subset of a larger projective space.  We can use this to give a provisional definition of the product of projective spaces.
  • page 14. Learn the definition of the Grassmannian and flag variety.
  • Skim to section 2.5.
  • We will discuss the “Hausdorff axiom” in Section 2.5.
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