Theorem Resoltion Singularities by Hironaka

By Hironaka

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This gives a rough idea how a mobile transforms under blowup. , not on the flags chosen for M), because the components of the invariant are independent. In conclusion, mobiles are a somewhat heavy and slowly moving vehicle, but they run and run and run . . until they are resolved. 6. Setups. To motivate the definition of setups of mobiles, let us assume that we have already blown up several times and have thus arrived at an ideal J in W . It will be the controlled transform of the ideal we started with — with respect to a given control c (you may just as well think of J as being the total transform).

So there is just one string of ideals (Jn , . . , J1 ) appearing in the flag Wn ⊃ . . ⊃ W1 at a, and Ji−1 is the coefficient ideal of Ji in Wi−1 . In the general case, we shall denote a setup also by (Jn , . . , J1 ), though the flag and the other ideals are part of the structure. It can be checked that the flag determines all the remaining ingredients of a setup. Not all mobiles admit punctual setups (because the factorization Ji = Mi · Ii need not hold if Di is not appropriate), but those arising under blowup as the transforms of a mobile with initially empty combinatorial handicap will do.

For ideals I which are not principal, it is required that there is at least one element f in I of order o = orda I with this property. We have seen before that osculating to I implies adjacent to I. As we have just shown that osculating persists at equiconstant points under blowup, we conclude that osculating also implies permanent maximal contact, and that’s what we were looking for. This is one of the rare instances in mathematical research where reality produces a truly favorable coincidence.

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