Tuesday, 10 October 2000
Day 8
Math 697R - Applicable Algebraic Goemetry


 * 15 October 5:30 PM

 Discuss make up sessions/ lectures
   -  Week of October 24-27
   -  Week of November 15
   -  One week in December?

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Generic properties of varieties

 - Theorem relating Zariski topology to usual (Euclidean) topology
    * Z-closed/open ==> E-closed/open
    * Non-empty & E-open ==> Z dense
    * Z closed, not all space ==> nowhere E dense
    * R^n is Z_closed in C^n   

 - Example of A^1
 - Definition of generic subset
 - Generic matrix is invertible
 - Generic polynomial has n distinct roots
 - Discriminant

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Unique Factorization for Varieties

 - Recall unique factorization for polynomials
 - Hilbert Basis Theorem
 - Def.  reducible/ireducible
 - Def.  prime ideal
 - Hypersurface V(f) is irreducible if and only if f is irreducble
 - An affine variety is irreducible if and only if its ideal is prime
 
 - Theorem:  Affine varieties are finite unions of irreducibles

 - Irredundant decomposition  (Compare to polynomials)
 -  Fact: irreducible complex varieties are connected

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Regular and Rational functions

 - Def.  Regular function
 -       Coordinate ring

 - Theorem.  For an algebraically closed field,
      Coordinate rings are precisely the finitely generated
      algebras without nilpotents.