Wednesday 8 November 2000

Day 18: Math 697R

Applicable Algebraic Geometry
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 Definition: Quasi-Projective Variety
                 (Locally closed subvariety)

 Extensions of tangent space, smooth & singular points,
               dimension, irreducible/reducible
        to Projective varieties
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 *Maps of Projective Varieties

 Only get reasonable functions of allow F/G F,G forms of
       the same degree.
 - Field of rational functions on an irreducible X
 - Domain of regularity; regular function 
    (need algebraic closure)
 - Rational maps (tuples of forms of the same degree)
   When two are equal.
   Domain of regularity; regular map.
 - Rational functions = rational maps to P^1

 - Example:  Linear projection
             Twisted Cubic
 - Isomorphism, Birational Isomorphism, function fields

 - Example:   Classify Birational Automorphisms of P^1
       - All rational maps are regular!
       - If biregular, given by quotient of linear forms
       - GL_2(k) -->> Aut P^1
         Quotient = scalars, called PSL_2(k).

       - One or two fixed points of such a map

 - Example Elliptic curve, automorphism of order 2 with
        4 fixed points.
   Theorem.  Function field of P^1 is not equal to that of
             Elliptic curve

 - Definition: Projective & Affine varietites
 - Quasi-projective varieties have open covers
 - Manifold-like.

 Products.  Segre Embedding.  P^1xP^1.