Functions with isomorphic Jacobian ideals

Shoshitaishvili, A. N.

doi:10.1007/bf01077939

Cited by 13 publications

(4 citation statements)

References 2 publications

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“…We have that the left side of (5) is spanned over r by vl(w), ..., VN(W ). (l), and the left side of (5) equals the right side when w = 0 or 1, by (4). It follows that the left side of (5) is contained in the right side, for all w. Let d denote the dimension of the right side of (5), considered as a C-vector space.…”

Section: (Zr ]mentioning

confidence: 99%

Classification of isolated hypersurface singularities by their moduli algebras

1982

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We will prove that two germs of complex analytic hypersurfaces with isolated singularities are biholomorphically equivalent if and only if they have the same dimension and their moduli algebras are isomorphic. This result was announced in [1]. Examples given in [1] show that the analogous statements over the real numbers and over all fields of characteristic p > 0 are false. is finite dimensional. The germ at 0 of A(V) is the base space for the miniversal deformation of (V, 0), so it seems natural to call A(V), provided with the obvious C-algebra structure, the moduli algebra of V. Our theorem will state that V is determined by A(V) and also by the following C-algebra: Theorem. Suppose (V, O) and (W, O) are germs of hypersurfaces in 117 "+ 1, and V-O is non-singular. Then the following conditions are equivalent. (i) (V, O) is biholomorphically equivalent to (W, 0). (ii) A(V) is isomorphic to A(W) as a C-algebra. (iii) B(V) is isomorphic to B(W) as a i12-algebra.

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Section: (Zr ]mentioning

confidence: 99%

Classification of isolated hypersurface singularities by their moduli algebras

1982

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“…Take f = y 2 + x 3 y and g = f + x 5 in characteristic 2, then T k (f ) = T k (g) for k = 0, 1 but f c ∼ g (f has two branches and g is irreducible as can be verified by using Singular [2]). (4) It was proved in [7], see also [8], that for an isolated quasi-homogeneous singularity f ∈ m ⊂ C{x} and any g ∈ m, M (f ) ∼ = M (g) as C-algebras implies that f r ∼ g. This theorem does not have an analogue in characteristic p, even if we use the higher Milnor algebras. For example, for the homogeneous polynomials f = x p+1 + y p+1 and g = f + x p , we have M k (f ) = M k (g) as K-algebras for all k, but f is not right equivalent to g.…”

Section: Resultsmentioning

confidence: 99%

Mather-Yau theorem in positive characteristic

Greuel

Pham

2016

J. Algebraic Geom.

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The well-known Mather-Yau theorem says that the isomorphism type of the local ring of an isolated complex hypersurface singularity is determined by its Tjurina algebra. It is also well known that this result is wrong as stated for power series f in K [[x]] over fields K of positive characteristic. In this note we show that, however, also in positive characteristic the isomorphism type of an isolated hypersurface singularity f is determined by an Artinian algebra, namely by a "higher Tjurina algebra" for sufficiently high index, for which we give an effective bound. We prove also a similar version for the "higher Milnor algebra" considered as K[[f ]]-algebra.

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“…Theorem 2.6 [Shoshitaishvili 1976;Benson and Yau 1990]. If f and g are germs of isolated weighted homogeneous singularities at the origin in ‫ރ‬ n , then f and g are right-equivalent if and only if f and g are contact-equivalent.…”

Section: Then the Geometric Genus Pmentioning

confidence: 99%

Complete characterization of isolated homogeneous hypersurface singularities

Yau

Zuo

2015

Pacific J. Math.

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Dedicated to Professor Michael Artin on the occasion of his 80th birthday Let X be a nonsingular projective variety in ‫ސރ‬ n−1. Then the cone over X in ‫ރ‬ n is an affine variety V with an isolated singularity at the origin. It is a very natural and important question to ask when an affine variety with an isolated singularity at the origin is a cone over nonsingular projective variety. This problem is very hard in general. In this paper we shall treat the hypersurface case. Given a function f with an isolated singularity at the origin, we can ask whether f is a weighted homogeneous polynomial or a homogeneous polynomial after a biholomorphic change of coordinates. The former question was answered in a celebrated 1971 paper by Saito. However, the latter question had remained open for 40 years until Xu and Yau solved it for f with three variables. Recently, Yau and Zuo solved it for f with up to six variables. However, the methods they used are hard to generalize. In this paper, we solve the latter question for general n completely; i.e., we show that f is a homogeneous polynomial after a biholomorphic change of coordinates if and only if µ = τ = (ν − 1) n , where µ, τ and ν are the Milnor number, Tjurina number and multiplicity of the singularity respectively. We also prove that there are at most µ 1/n + 1 multiplicities within the same topological type of the isolated hypersurface singularity, while the famous Zariski multiplicity problem asserts that there is only one multiplicity.

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Functions with isomorphic Jacobian ideals

Cited by 13 publications

References 2 publications

Classification of isolated hypersurface singularities by their moduli algebras

Classification of isolated hypersurface singularities by their moduli algebras

Mather-Yau theorem in positive characteristic

Complete characterization of isolated homogeneous hypersurface singularities

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