Tuesday,  14  November 2000

Day 20: Math 697R

Applicable Algebraic Geometry
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No class 21 & 30 November
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Definition:  Graph of a regular map.
Lemma:  The graph is closed.
   Suffices to do this when X, Y are affine, then easy.

Theorem: The image of a projective variety under a regular
         map is closed.
(Give example of V(xy-1) --> A^1)

Proof: Since X \subset P^n is closed, we can assume that X=P^n.
 Since Y is covered by affines, we can assume that Y is affine space.

 Claim:  Z \subset P^n x A^m  ==> Projection \pi(Z) of Z is closed.

 Z is defined by polynomials g_1,...g_l with 
  g_i a homogeneous polynomial in s with coefficients polynomials in t.

 For y\in A^m, \pi^{-1}(y)\cap Z are solutions to g_i(s;y)=0.
 Non-empty only if m_0^a\not\subset the ideal, all a>0

 Set T_a := those y\in A^m with m_0^a not a subset of the ideal.
 \pi(Z) = intersection T_a.

 Claim: T_a is closed:

  m_0^a is in ideal if and only if some linear map is onto.  \QED
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Schubert Calculus
  Grass(p,V) or Grass(p,n).  Grass(1,V) = P(V).

  A p-plane is represented as row space of full-rank matrix,
indeterminacy = GL_p;  Grass is the space of orbits.

  Pl\"ucker coordinates for Grassmannian.  Different reps multiply
  them by a scalar ----> Pl\"ucker map to projective space.

 [ Give intrinsic definition ]

Theorem.  The Pl\"ucker map is an injection whose image is a smooth,
  irreducible subvariety of Projective space of dimension pm.

Proof:  Irreducible, as the image of an irreducible variety under a 
regular map.

Show the intersection with U_alpha is smooth, irreducible, and 
isomorphic to Mat_{pxm}.

\/\/\/-> Get equations for the Grassmannian.