[论文解读] Zariski equisingularity of surface singularities in $\mathbb C^3$ by a local invariant

Adam Parusiński, Laurenţiu Păunescu|arXiv (Cornell University)|Feb 16, 2026

Holomorphic and Operator Theory被引用 0

一句话总结

作者为解析曲面奇点在 ℂ³ 中定义了局部不变量 multiplicity sequence mult^*(V)，并证明一个族在泛化意义上的 Zariski 匀质性，当且仅当该不变量在族中保持常数。

ABSTRACT

We associate to every analytic surface singularity $(V,0)$ in $(\mathbb C^3,0)$, not necessarily isolated, an invariant $mult^* (V)$ and show that an analytic family of such singularities $(V_t,0)$, $t\in (\mathbb C^l,0)$, is generically Zariski equisingular if and only if $mult^* (V_t)$ is constant. The invariant, that we call the multiplicity sequence of $V$, takes into account the multiplicities of the successive discriminants of $V$ by generic corank one projections.

研究动机与目标

• Motivate the study of equisingularity for non-isolated surface singularities in ℂ³ and its relation to Teissier’s numerical invariants.
• Introduce the multiplicity sequence mult^*(V) as a local invariant built from discriminants of generic corank-one projections.
• Prove equivalence between generic Zariski equisingularity and constancy of mult^*(V_t) in analytic families.
• Relate mult^*(V) to Teissier’s numbers in the isolated case and establish analytic invariance and semicontinuity properties.

提出的方法

• Define the generalized discriminants and the ind(D_f) index for a Weierstrass polynomial representation of f.
• Construct the multiplicity sequence mult^*(V) from (mult_0(V), mult_0(D_f), i_0, mult_0(D_f^{i_0})) where i_0 = ind(D_f).
• Prove mult^*(V_t) is analytic-invariant and upper semicontinuous in ν-transverse Zariski equisingular families.
• Show that constancy of mult^*(V_t) is equivalent to ν-transverse Zariski equisingularity after generic linear changes of coordinates.
• Relate the sequence to Teissier’s μ^*(V_t), k(V_t), and φ(V_t) for isolated singularities via established formulas.

实验结果

研究问题

• RQ1 Can a local invariant control Zariski equisingularity for non-isolated surface singularities in ℂ³?
• RQ2 Is the multiplicity sequence mult^*(V_t) constant in a family if and only if the family is Zariski equisingular (up to ν-transverse changes)?
• RQ3 How does mult^*(V) relate to Teissier’s numbers in the isolated case?
• RQ4 Do generalized discriminants provide an analytic invariant and semicontinuity properties suitable for equisingularity analysis?

主要发现

• A new invariant mult^*(V) (the multiplicity sequence) is defined for analytic surface singularities in ℂ³.
• mult^*(V) is an analytic invariant and is upper semicontinuous in families.
• For isolated singularities, mult^*(V) can be expressed in terms of Teissier’s μ^*(V), k(V), and φ(V).
• An analytic family is ν-transverse Zariski equisingular if and only if mult^*(V_t) is constant.
• The main theorem yields equivalence among ν-transverse Zariski equisingularity, Zariski equisingularity (after generic linear change), and constancy of mult^*(V_t).