Resolutions of singularities.

Yoshifumi Tsuchimoto

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

DEFINITION 07.1   Let $W$ be a scheme. The order of an ideal $I$ at $a\in W$ is defined as follows.

$$\operatorname{ord}_a I=\max \{k \vert \mathfrak{m}_a^k\supset I\}$$

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

$$\operatorname{top}(I)=\operatorname{top}(\operatorname{ord}(I))$$

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

$$J=M\cdot I$$

$M$ 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 $\operatorname{top}(i_a)$ of a certain function $i_\bullet=i_a$ .

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

LEMMA 07.2   For any point $a\in W$ , there exists a local hyper surface $V\subset W$ (``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 $\bullet_{-}$ may not commute. Therefore we need to decompose $J=M\cdot I$ and see how $M$ and $I$ change.

The function $i_a$ is then be defined (inductively) by:

$$i_a(J)=(\operatorname{ord}_a(I),\operatorname{ord}_a(Q),m_a,i_a(J_{-})).$$