Zariski's multiplicity question and aligned singularities
Zariski's famous multiplicity conjecture [cf. \textit{O. Zariski}, Bull. Am. Math. Soc. 77, 481--491 (1971; Zbl 0236.14002)] can be briefly formulated as follows. Let \(f_t:({\mathbb C}^n, 0)\to (\mathbb C, 0)\) define a family of hypersurface singularities. Assume the family is topologically constant (i.e., in case of isolated singularities the Milnor number \(\mu(f_t)=\dim_\mathbb C \mathbb Cx_1, \ldots, x_n/\langle \frac{\partial f_t}{\partial x_1}, \ldots \frac{\partial f_t}{\partial x_n}\rangle\) is constant). Does this imply that the multiplicity \(m(f_t)=\text{ord}_x f_t\) is constant? This conjecture is proved only for very special cases. The author considers Zariski's question for a special class of non-isolated singularities and gives a proof.
scientific article; zbMATH DE number 5013362