Blowup 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) nonlectures (see the Disclaimer) which are guaranteed to extend beyond the Judgement Day January 13, 2008. Follow the new postings!

Trigonometric blowup and its effect on the simplest singularities. Advantages and drawbacks.

Algebraic blowup in the real and complex domain. Exceptional divisor and its exceptionality. Complex Möbius band.

Blowup of analytic curves and singular foliations. Dicritical and nondicritical cases. Computation in the local coordinates.

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 blowup. Selfintersection index (of the exceptional divisor).

Multiplicity of foliation and its blowup (principal formula)


Proof of the Desingularization theorem: the main part.

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:

D. Mumford, Algebraic Geometry I. Complex projective manifolds. Springer, 1976, §2(AB).

V. Arnold, S. GuseinZade, A. Varchenko, Singularities of differentiable mappings, vol. 1, §5 (Multiplicity of holomorphic maps).
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