Day 11
Applicable Algebraic Geometry



Wednesday 25 5:00 1530  LGRT


 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.

 - Definition Smooth & Singular points

 - Intrinsic Definition of Tangent Space;
     The Map d_x, restriction to m_x, 
     Theorem d_x : m_x/m_x^2 iso to dual of tangent space
    
 - Functoriality of tangent Space

 - Isomorphic = isomorphic tangent spaces

 - Algebraic Groups are smooth


 - Definition of dimension.

 - Give alterate definitions.