Sergei Yakovenko's blog: on Math and Teaching

Wednesday, January 2, 2008

“Auxiliary Lessons” שעור עזר) 11-13) January {3, 10, 17}, 2008

Blow-up and Desingularization Theorem(s)

We will start a long discussion of one of the most useful results in the local analysis of complex analytic objects. It will take at least two (perhaps, three) non-lectures (see the Disclaimer) which are guaranteed to extend beyond the Judgement Day January 13, 2008. Follow the new postings!

  1. Trigonometric blow-up and its effect on the simplest singularities. Advantages and drawbacks.
  2. Algebraic blow-up in the real and complex domain. Exceptional divisor and its exceptionality. Complex Möbius band.
  3. Blow-up of analytic curves and singular foliations. Dicritical and nondicritical cases. Computation in the local coordinates.
  4. Intersections and their multiplicity.
    • Divisors, their local representations and transformations by holomorphic maps.
    • Divisors and cocycles. Effective divisors
    • Isolated intersection between effective divisors. Intersection multiplicity (algebraic, geometric and deformational  construction).  
    • Principal theorem of Singularity theory. An idea of the proof.
    • Intersection index and its property as a bilinear form.
    • Intersection index and blow-up. Self-intersection index (of the exceptional divisor). 
    • Multiplicity of foliation and its blow-up (principal formula)
  5. Proof of the Desingularization theorem: the main part.
  6. Cleaning the field: desingulation of cuspidal points, elimination of resonant nodes and tangencies with the exceptional divisor.

 Recommended reading: Section 8 from the book (printing disabled).Auxiliary reading:

  1. D. Mumford, Algebraic Geometry I. Complex projective manifolds. Springer, 1976, §2(A-B).
  2. V. Arnold, S. Gusein-Zade, A. Varchenko, Singularities of differentiable mappings, vol. 1, §5 (Multiplicity of holomorphic maps).
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