RESOLUTIONS OF SINGULARITIES.

YOSHIFUMI TSUCHIMOTO

07. A quick overview The general reference has been [2]. See also an article [1](arXiv:math/0211423)

Definition 7.1. Let W be a scheme. The order of an ideal I at a ∈ W is defined as follows.

ord_{a} I = max{k|𝔪_{a}^{k} ⊃ I}

The top locus top(t) of an upper semicontinuous function t on V is a set of points of V where t attains its maximum. We put:

top(I) = top(ord(I))

We consider an variety X contained in a regular variety W. We let J = J_{X} the defining ideal of X. We decompose:

J = M · I

M is the “resolved part”, whereas I is the “unresolved part”, of the ideal J. Objective: By blowing up several times, reduce the order o of the ideal I.

We need to find the center Z of the blowing up. It is given as a top locus top(i_{a}) of a certain function i_{•} = i_{a}.

We now introduce a result which is specific to the characteristic zero case.

Lemma 7.2. For any point a ∈ W, there exists a local hyper surface V ⊂ W (“a hyper surface of maximal contact”) such that “blow ups in the center in V conatins all the equiconstant points.”

By using the lemma above, we develop an inductive argument on the dimension. Namely, by using the theory of “coefficient ideals”, we define an ideal J_{−} in V.

There are two problems:

(1) Z_{−} may not be transversal to the exceptional locus F. Additional “small” blow ups (along with, say, Q) are needed.

(2) “Blow ups” and •_{−} may not commute. Therefore we need to decompose J = M · I and see how M and I change.

The function i_{a} is then be defined (inductively) by:

i_{a}(J) = (ord_{a}(I), ord_{a}(Q), m_{a}, i_{a}(J_{−})).

References

[1] Herwig Encinas, S.; Hauser, Strong resolution of singularities in characteristic zero, Comment. Math. Helv. 77 (2002), no. 4, 821–.

[2] Herwig Hauser, The Hironaka theorem on resolution of singularities (or: A proof we always wanted to understand)., Bull. Am. Math. Soc., New Ser. 40 (2003), no. 3, 323–403 (English).