Tuesday, 17 October 2000
Day 10:
Applicable Algebraic Geometry


*  Schedule the next couple of weeks
    19, 24, *25*, 26


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 - Algebraic-Geometric Dictionary II
     Category equivalence, Affine Varieties, f.g reduced K-algebras

 - Principal affine open set
     Definition, Coordinate ring

 - Examples: K^\times, GL_n

 - Define: Affine algebraic group

 - Mention Chevalley's Theorem about affine algeberaic groups are
                           all linear

 - X irreducible, I(X) is prime, so K[X] has no zero divisors (integral domain)
    quotient field is the function field of X

 - Example:  V(x^2 + y^2 + 2y)  -x/y = (y+2)/x

 - Definition: Regular point of a rational function.
               Set of regular points is non-empty & open

 - Theorem.   When K is closed, rational functions that are everyehere
             regular are regular functions on X

 - Rational maps, Domains.

 - Image of a rational map; dominant maps correspond to field inclusions.

 - Composition of dominant maps; birational equivalence.

 Section 5: Smooth & Singular Points, Dimension.

 - Taylor expansion of f in K[A^n]
 - Differential

 - Definition.   Zariski Tangent Space
 - Example of cubics
 - Example of Special Linear Group

 - Theorem.  An affine variety has a non empty open subset of points
             where the tangent space has minimal dimension.