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Abstract
Let A be a noetherian local k-algebra with residue field k and associated graded ring G(A). Let 𝔟 be the conductor of A in its normalization , which we assume to be regular and finite over A. Assume also that Proj(G(A)) is reduced and
. Let r be the embedding dimension of A and
be the Hilbert function of
. We show that if the Hilbert function of Proj(G(A)) is equal to
, then G(A) is reduced. This allows us to compute, in the general case, the conductor of the local ring of singular isolated points on surfaces whose tangent cone is multiplanar; that is, it consists—as a set—of planes. We explicitly construct a class of such surfaces in the rational case.
ACKNOWLEDGMENT
The authors were partially supported by PRIN “Geometria sulle Varietà Algebriche,” Italian Ministry of Education, University, and Research.
Notes
1Of course, the situation would have been similar if we had used a symmetric power of a grassmannian, or a Hilbert scheme, instead of the rough (but handier) parameter space U.
2The multiplicity e(A) is e because the degree of Proj(G(A)) is e; the point x is non-normal, simply because A is nonregular and is regular.
3Indeed, tensoring the A f -module sequence 0 → A f → B f → B f /A f → 0 by B f we get that B f /A f ⊗ B f = 0. Since B f is finite over A f , if B f /A f ≠ 0, we could choose a submodule M/A f of B f /A f such that B f /M is generated by exactly one nonzero element. Hence B f /M≅ A f /𝔮 for some proper ideal 𝔮 and B f /M⊗ B f ≅ B f /𝔮B f would be 0, which is impossible because B f is integral over A f . An alternative argument, suggested by the geometric insight, may run as follows. It suffices to show that the conductor (A f :B f ) is A f . Suppose the contrary and choose a maximal ideal 𝔭 containing (A f :B f ). From B f /𝔭B f ⊗ A f /𝔭 B f /𝔭B f = B f /𝔭B f follows B f /𝔭B f = A f /𝔭, because A f /𝔭 is a field. Then the Nakayama lemma easily implies that the localizations of A f and B f at 𝔭 coincide (cf. the proof of Hartshorne, Citation1977, Chap. II, Lemma 7.4). Hence, for each element b of a finite system of generators of B f over A f , we could choose an a b ∈ A f − 𝔭 such that a b b ∈ A f . Then the product of the a b 's would be an element in (A f :B f ) lying outside 𝔭, and this contradicts the choice of 𝔭.
4Look at the congruences
Communicated by S. Goto.