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Let $X\subset \mathbb C^n$ be a complex hypersurface (given by $F=0$ where $F$ is a polynomial). It is known then that $X$ admits a Whitney stratification. This is a decomposition of $X$ into smooth submanifolds (strata) that have some adjacency properties (Whitney conditions a and b).

Question. Does $X$ have a minimal stratification, i.e. such a stratification that for any other Whitney stratification of $X$ the strata of the minimal one are unions of the strata of the other one?

At least maybe this is known for varieties with certain type of singularities?

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The answer is yes for any reduced equidimensional analytic space. This is the proposition 3.2 (and remark after the proof) page 479 of Variétés polaires II by Bernard Teissier.

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  • $\begingroup$ Thanks a lot for the reference! Just to doublecheck, a reduced sub variety in $\mathbb C^n$ is equidimensional iff all its irreducible components have the same dimension? Also, a hypersuface $F=0$ is a reduced subvariety in $\mathbb C^n$ iff the polynomial $F$ is a product of distinct irreducible polynomials? $\endgroup$
    – aglearner
    Commented May 8, 2021 at 13:14
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    $\begingroup$ yes, these are standards definitions I believe... $\endgroup$
    – Libli
    Commented May 8, 2021 at 13:40

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