The general reference has been . See also an article (arXiv:math/0211423)
The top locus of an upper semicontinuous function on is a set of points of where attains its maximum. We put:
We consider an variety contained in a regular variety . We let the defining ideal of . We decompose:
is the ``resolved part'', whereas is the ``unresolved part'', of the ideal .
Objective: By blowing up several times, reduce the order of the ideal .
We need to find the center of the blowing up. It is given as a top locus of a certain function .
We now introduce a result which is specific to the characteristic zero case.
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 in .
There are two problems:
The function is then be defined (inductively) by: