[Paper Review] Whitney equisingularity of families of surfaces in $\mathbb{C}^3$

María Aparecida Soares Ruas, O. N. Silva|arXiv (Cornell University)|Aug 30, 2016

Algebraic Geometry and Number Theory12 references3 citations

TL;DR

This paper investigates Ruas's long-standing conjecture that topological triviality of 1-parameter unfoldings of finitely determined map germs $f:(\mathbb{C}^2,0)\to(\mathbb{C}^3,0)$ implies Whitney equisingularity. Using counterexamples, the authors show the conjecture is false in general, demonstrating multiple ways in which constancy of the Milnor number $\mu(D(f_t))$ of the double point curve does not guarantee Whitney equisingularity. However, they identify a class of homogeneous map germs with $\gcd(n,m)\neq2$ for which the conjecture holds, providing a complete answer to the problem after 25 years.

ABSTRACT

In this work, we study families of singular surfaces in $\mathbb{C}^3$ parametrized by $\mathcal{A}$-finitely determined map germs. We consider the topological triviality and Whitney equisingularity of an unfolding $F$ of a finitely determined map germ $f:(\mathbb{C}^2,0) ightarrow(\mathbb{C}^3,0)$. We investigate the following conjecture: topological triviality implies Whitney equisingularity of the unfolding $F$? We provide a complete answer to this conjecture, given counterexamples showing how the conjecture can be false.

Motivation & Objective

• To resolve Ruas's 1994 conjecture that topological triviality of 1-parameter unfoldings of finitely determined map germs $f:(\mathbb{C}^2,0)\to(\mathbb{C}^3,0)$ implies Whitney equisingularity.
• To investigate whether constancy of the Milnor number $\mu(D(f_t))$ of the double point curve is sufficient for Whitney equisingularity.
• To provide a complete answer to the conjecture by constructing counterexamples and identifying conditions under which it holds.
• To clarify the role of polar multiplicities and Hilbert-Samuel multiplicities in characterizing Whitney equisingularity for surface families in $\mathbb{C}^3$.

Proposed method

• Construction of explicit counterexamples to Ruas's conjecture using families of map germs with constant $\mu(D(f_t))$ but non-constant higher invariants.
• Application of Gaffney's characterization of Whitney equisingularity via constancy of $\mu(D(f_t))$, $m_1(f_t(\mathbb{C}^2),0)$, and $m_0(f_t(D(f_t)))$.
• Analysis of the double point curve $D(f)$ and its lifting $D^2(f)$ using the ideal $\mathcal{I}^2(f)$ generated by $ (f\times f)^*\mathcal{I}_3 $ and the $2\times2$ minors of the Jacobian matrix $\alpha_{ij}$.
• Computation of invariants such as $\mu(D(f_t))$, $m_0(f_t(D(f_t)))$, and $\mu_1(f_t(\mathbb{C}^2))$ via parametrization and degree analysis of irreducible components of $D(f)$.
• Use of the formula $m_0(f(D(f))) = \frac{dn}{2}$ for homogeneous map germs $f(x,y) = (x^n, y^m, (x+y)^k)$ with $n,m,k$ coprime in pairs and $n < m < k$, where $d = nmk - n - m - k + 2$.
• Application of Lemma 7.1 and Theorem 7.2 to prove that when all components of $f(D(f))$ are smooth, topological triviality implies Whitney equisingularity.

Experimental results

Research questions

• RQ1 Does topological triviality of a 1-parameter unfolding $F$ of a finitely determined map germ $f:(\mathbb{C}^2,0)\to(\mathbb{C}^3,0)$ imply Whitney equisingularity?
• RQ2 Is the constancy of the Milnor number $\mu(D(f_t))$ of the double point curve sufficient for Whitney equisingularity of the unfolding $F$?
• RQ3 Under what conditions does topological triviality imply Whitney equisingularity for families of surfaces in $\mathbb{C}^3$?
• RQ4 Can the conjecture be true for specific classes of map germs, such as homogeneous ones with $\gcd(n,m)\neq2$?
• RQ5 What role do the Hilbert-Samuel multiplicity $m_0(f_t(D(f_t)))$ and first polar multiplicity $m_1(f_t(\mathbb{C}^2),0)$ play in characterizing Whitney equisingularity?

Key findings

• The conjecture that topological triviality implies Whitney equisingularity for 1-parameter unfoldings of finitely determined map germs $f:(\mathbb{C}^2,0)\to(\mathbb{C}^3,0)$ is false in general, as demonstrated by explicit counterexamples.
• The Milnor number $\mu(D(f_t))$ alone is insufficient to guarantee Whitney equisingularity, as shown by examples where $\mu(D(f_t))$ is constant but $m_0(f_t(D(f_t)))$ or $m_1(f_t(\mathbb{C}^2),0)$ vary.
• For homogeneous map germs $f(x,y) = (x^n, y^m, (x+y)^k)$ with $n,m,k$ coprime in pairs and $n < m < k$, the double point curve $D(f)$ has $d = nmk - n - m - k + 2$ smooth irreducible components, all of which are identification components.
• In this class of homogeneous map germs, if $\gcd(n,m) \neq 2$, then topological triviality implies Whitney equisingularity, as all components of $f(D(f))$ are smooth, so $m_0(f(D(f))) = 1$ for each component.
• The formula $m_0(f(D(f))) = \frac{dn}{2}$ holds for such homogeneous map germs, where $d$ is the number of irreducible components of $D(f)$ and $n$ is the degree of the first component.
• The first polar multiplicity $\mu_1(f(\mathbb{C}^2))$ is given by $(n-1)(m-1) + dn$ for these homogeneous map germs, with $d = nmk - n - m - k + 2$.