RESOLUTIONS OF SINGULARITIES.

YOSHIFUMI TSUCHIMOTO

08. Coefficient ideal From the paper of Encinas and Hauser:

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The coefficient ideal of an ideal I of W at a with respect to V is an ideal in V which is built from the coefficients of the Taylor expansion of the elements of I with respect to the equations defining V. Let x, y and y be regular systems of parameters of 𝒪_{W,a} and 𝒪_{V,a} so that x = 0 defines V in W. For f in I denote by a_{f,α} the elements of 𝒪_{V,a} so that f = ∑_{α} a_{f,α} · x^{α} holds after passage to the completion. Then we set

coeff_{V} I = ∑_{|α|<c}(a_{f,α}; f ∈ I)^{\frac{c!}{c−|α|}} .

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Example 8.1. Let us consider a curve C = {(t^{3}, t^{5}, t^{7})} ∈ 𝔸^{3}. By using a theory of groebner basis, we may eliminate the variable t and obtain an ideal

I = (y^{7} − z^{5}, xz − y^{2}, xy^{5} − z^{4}, x^{2}y^{3} − z^{3}, x^{3}y − z^{2}, x^{4} − zy)

Let us choose V = {z = 0} as the hypersurface. Then:

(1) When c = 1, we have

coeff_{V} (I) = (y^{2}, x^{3}y, x^{4})

(2) When c = 2, we have

coeff_{V} (I) = (y^{2}, x^{3}y, x^{4}) + (x, y)^{2} = (x^{2}, xy, y^{2})

(3) When c = 3, we have

coeff_{V} (I) = (y^{2}, x^{3}y, x^{4})^{2} + (x, y)^{3} + (1)^{6} = (1).

Problem 8.2. To compute the coefficient ideal of a given ideal I, Is it sufficient to consider only coefficients of generators of I?

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