Resolution of Singularities - Examples - Minimal Resolutions Need Not Exist

Minimal Resolutions Need Not Exist

Minimal resolutions (resolutions such that every resolution factors through them) exist in dimensions 1 and 2, but not always in higher dimensions. The Atiyah flop gives an example in 3 dimensions of a singularity with no minimal resolution. Let Y be the zeros of xy = zw in A4, and let V be the blowup of Y at the origin. The exceptional locus of this blowup is isomorphic to PP1, and can be blown down to P1 in 2 different ways, giving two small resolutions X1 and X2 of Y, neither of which can be blown down any further.

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