A remark on isolated complex hypersurface singularities

Abstract: Let $(X, \bf 0)$ be the germ of a hypersurface in $(\mathbb C^n,\bf 0)$ with an ordinary singularity of multiplicity $m$ at the origin $\bf 0$. A natural question to ask is whether $X$ and its tangent cone at the origin are analytically isomorphic. The answer is negative in general, in view of a theorem of Kioji Saito. However there is an integer $D(n,m)&gt;m$ such that, given a \emph{regular} homogeneous polynomial $f(x_1,\ldots, x_n)$ of degree $m$ (this means that ${ f=0}$ is a smooth hypersurface in $\PP<sup>{n-1}$) then, for all $d\geq D(n,m)$, any convergent power series of the form $g=f+ o(d)$ (here, as usual, $o(d)$ stays for a power series of order at least $d$), defines a germ ${ g=0}$ which is analytically equivalent to the germ ${ f=0}$. In this note we compute $D(n,m)$ explicitly. We also give an extension to the case in which $f$ is a quasihomogeneous polynomial.

• The paper provides an explicit determinacy bound D(n, m) that guarantees analytic equivalence of a complex singularity to its tangent cone after high order perturbations.
• It utilizes the graded structure of the Milnor algebra and combinatorial kernel computations to establish precise stabilization conditions.
• The results extend to quasi-homogeneous hypersurfaces and offer practical insights for deformation theory and computational algebraic geometry.

Analytical Determinacy of Isolated Complex Hypersurface Singularities

Overview

This work investigates the analytic classification of isolated singularities of complex hypersurfaces, with a focus on the relation between a germ $(X, 0) \subset (\mathbb{C}^n, 0)$ exhibiting an ordinary singularity of multiplicity $m$ and its tangent cone at the origin. The main contribution is the explicit computation of a determinacy bound $D(n, m)$ such that for sufficiently high order perturbations, the analytic equivalence class of the germ stabilizes to that of the tangent cone. The results apply to both homogeneous and quasi-homogeneous cases, extending and sharpening prior results, including a negative answer in general to the analytic equivalence question posed by Saito's theorem.

Technical Background

The classical question is to determine whether a complex hypersurface germ with an ordinary singularity is analytically isomorphic to its tangent cone. This is known to hold for $m=2$, some low-dimensional cases, and degree/multiplicity constraints, but fails in general as established by Saito. The analytic equivalence is formalized via right-equivalence of defining equations, utilizing the algebro-geometric structure of the Jacobian ideal, the Milnor and Tyurina algebras, and finite determinacy theory as developed by Greuel, Lossen, and Shustin. The tangent cone is the zero set of the degree-$m$ homogeneous part $f$ of the defining power series $g$. A germ is $k$-determined if any perturbation coinciding up to order $k$ yields an analytically equivalent singularity.

Main Theorems and Explicit Determinacy Bound

The central achievement is the explicit bound:

$$D(n,m) = \begin{cases} 3 & \text{if } m=2 \ 4 & \text{if } n=2, m=3 \ n(m-2) + 1 & \text{otherwise} \end{cases}$$

For any regular homogeneous polynomial $m$0 of degree $m$1 in $m$2 variables, any germ $m$3 with $m$4 defines a singularity analytically equivalent to the cone defined by $m$5. The proof relies on the structure of the Milnor algebra as a graded Gorenstein algebra with socle in degree $m$6. Surjectivity of the relevant multiplication maps and the combinatorial computation of the kernel dimensions, via Lemma 2.8, provide precise control.

Sharpness is established: counterexamples are constructed using the Fermat polynomial and explicit higher-order terms, demonstrating that for $m$7, analytic equivalence can fail. The stabilizing property thus only holds strict for $m$8.

Extension to Quasi-Homogeneous Hypersurfaces

The result is generalized to regular quasi-homogeneous polynomials of isobaric type $m$9. In this case, the determinacy bound adapts to the weighted degree of the Hessian:

$D(n, m)$0

where regularity is defined in terms of partial derivatives forming a regular sequence.

Theoretical and Practical Implications

This analysis conclusively delineates when the local analytic type of an isolated ordinary hypersurface singularity is determined by the tangent cone. The explicit bounds for determinacy have direct implications in deformation theory and singularity theory, informing when analytic moduli coincide with algebraic data. In the quasi-homogeneous context, the weighted determinacy is essential for computations in weighted projective spaces and for understanding singularity adjacencies and their versal deformations.

The explicit bounds on determinacy have applications in computational algebraic geometry, especially for algorithms determining moduli of singularities or implementing equivalence checks in computer algebra systems.

Directions for Future Research

Potential future advances include:

• Extension to non-isolated and more general singularity types (e.g., complete intersections, Cohen-Macaulay singularities).
• Algorithmic implementation of analytic equivalence testing using the provided determinacy bounds.
• Investigation of determinacy in positive characteristic or other valuation fields.
• Connections to the study of the locus of non-finite determinacy within parameter spaces.

Conclusion

The paper gives a precise answer to when the germ of a complex hypersurface with an ordinary isolated singularity is determined, up to analytic equivalence, by small perturbations of its tangent cone. This resolves an explicit instance of the finite determinacy problem, sharpening previous qualitative results with concrete bounds, and advances the understanding of analytic moduli for complex hypersurface singularities.