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Can one suggest some references where explicit calculations for blow up technique(along an ideal) and strict transformation is done in different examples?

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I don't know what tag this should be given, but I don't think it's "differential-geometry". And please don't SHOUT. I can hear you perfectly well from here. – Loop Space Nov 11 '09 at 9:44

Both Hartshorne and Fulton's Intersection theory contain many examples. Look for the terms "Blow up" and "Monoidal transform". There are also some "hands on" examples in Dolgachev's classical algebraic geometry online book . The standard example is describing a cubic surface as a plane with 6 blown up points; see Hartshorne V.4

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I think, you'd like to see explicit equations and if possible, a singularity completely solved by blowups. One can take a look at such thinks in Lectures on resolution of singularities by Janos Kollar ( The book starts with more than ten different methods of how to solve singularities on curves and then continues on resolution of singularities on surfaces. There you will see specific examples and explicit equations.

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I suggest Beauville's Complex Algebraic Surfaces.

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There are several explicit calculations given in Chapter IV.2 of Eisenbud and Harris' Geometry of Schemes

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