Thursday, 16 November 2000

Day 21: Math 697R

Applicable Algebraic Geometry
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No Class on 21 and 30 November
Exercises:  Pass out & Discuss what to do
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Discuss the binomial coefficient

* Intrinsic definition of the Pl\"ucker map

Theorem.  The Pl\"ucker embedding is an injection and the image
          is a smooth, irreducible subvariety of dimension mp

  - Irreducible : image of matrices
  - Show intersection with U_alpha has this property

Remark.  Identifies G_alpha with A^mp, giving coordinate charts 
     Exercise: compute the transition functions.
     --- Grassmannian is a compactification of matrices ---

   T_H = Mat_pxm

   Coordinate-free description of open neighborhood of H

  /\/\-> T_H = Hom(H,K)
  
   Even more coordinate-free

Remark.  Equations for the Grassmannian from the proof

  Define the ideal of the Grassmannian  I_m,p

  Example m=p=2. The equations from the Theorem

        Do m=3, p=2, and then generalize.

  Show these are a Gr\"obner basis for I_m,p,

  Do the poset of minors