Thursday, 9 November 2000

Day 19: Math 697R

Applicable Algebraic Geometry
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No Class 21 & 30 November

Definition.  Isomorphism, birational isomorphism
          (Peculiar feature of algebraic geometry)
          Isomorphic, Birationall Isomorphic.
            Classification of varieties.

Recall:  * maps P^1 --> P^1 & PGL_1

Theorem:  An Automorphism of P^1 fised 1, 2, or all points.

Example:  Elliptic Curve in P^2  (Characteristic not 2, 3)
          y |--> -y has 4 fixed points.

Theorem.   P^1 is not birationallly isomorphic to Elliptic Curve.

   Practical consequence.   Quadrics in Space.

Definition.  Projective & Affine varietites.

Theorem.  Quasi-projective varieties are covered by affine sets.
  (Like manifolds)

Products.  Segre embedding,    P^1 x P^1

Theorem. The image of a projective variety under a regular map 
         is closed.

Note not true for affine varieties.

Observe that a map of varieties factors as a closed embedding
    (the graph) followed by a projection

Definition.  Graph of a regular map.

Lemma.  The graph of a regular map is closed.

Theorem. If X is projective, then the projection X x Y --> Y
         takes closed varieties to closed varieties.

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Schubert Calculus

   Definition of Grassmannian.
   Parameterization by matrices, effect of changing matrices.
   Pl\"ucker coordinates.