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# Resolutions of singularities.

Yoshifumi Tsuchimoto

From the paper of Encinas and Hauser:

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The coefficient ideal of an ideal of at a with respect to is an ideal in which is built from the coefficients of the Taylor expansion of the elements of with respect to the equations defining . Let and be regular systems of parameters of and so that defines in . For in denote by the elements of so that holds after passage to the completion. Then we set

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EXAMPLE 08.1   Let us consider a curve . By using a theory of groebner basis, we may eliminate the variable and obtain an ideal

Let us choose as the hypersurface. Then:
1. When , we have

2. When , we have

3. When , we have

PROBLEM 08.2   To compute the coefficient ideal of a given ideal , Is it sufficient to consider only coefficients of generators of ?

Let be a closed subscheme of a regular scheme . We want to resolve the singularity of . If there exists an regular hypersurface such that , then we may replace by . So we may (have to) assume that is not contained in such hypersurfaces. Instead, we have for each point a hypersurface of maximal contact'' . is not canonical, but is good enough to define the invariant and then (afterwards) determine the center of blow up.

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2014-07-24