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Resolutions of singularities.

Yoshifumi Tsuchimoto

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\fbox{07. A quick overview}

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.

$\displaystyle \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:

$\displaystyle \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:

$\displaystyle 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:

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

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