# Sergei Yakovenko's blog: on Math and Teaching

## 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.