Tuesday, 7 November 2000

Day 17: Math 697R

Applicable Algebraic Geometry
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8 November:   Make up lecture

21 & 30 No class.

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Projective Algebraic Geometry

   Definition of projective space: n+1 tuples modulo scalars

   P^n is the space of lines P(V).

   Example:  P^1.  P^1_R is S^1, P^1_C is S^2,
             (Do this explicitly)

 Polynomials on projective space.
   Making sense of f(x) ---> homogeneous forms, & can 
    only make sense of f(x)=0.

 Definition of Projective variety.

 Examples of hyperplanes and linear subspaces.

 Definition.  The ideal of a subset of P^n

 Definition.  Homogeneous ideal.

 Observe:  S-polynomials, reduction, Buchberger algorithm, all
           preserve homogeneity.
           [Get: Finite homogeneous Gr\"obner basis]

 Algebraic-Geometric Dictionary is not so exact: 
   m_0.
 Theorem on homogeneous ideals.

 Discuss affine cone over a projective variety.

 Definition.  Zariski topology on P^n

 Relation of Z-topology on P^n to Z-topology on A^n
   - Coordinate hyperplanes
   - Principal affine pieces  U_i
   - local coordinates for the pieces.

 Theorem.   The map A^n --> U_i is a homeomorphism.

 Corollary.  X in P^n is closed iff X\cap U_i is closed, all i.

 Definition.  Locally closed subset

--> Most concepts concerning affine varieties carry over verbatim
    * Tangent spaces, smooth & Singular points.
    * Reducible, Irreducible,
    * Dimension.