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For a subset $S$ of a smooth manifold $M$, a locally finite decomposition $$S = \bigsqcup_{\alpha} S_\alpha$$ into smooth submanifolds (strata) is called a Whitney stratification of $S$ if each pair $(S_\alpha, S_\beta)$ satisfies Whitney's condition (b). Whitney's condition (b) is usually defined using local coordinates and specifically the notion of convergence in the usual Euclidean topology.

It turns out that if $M$ is a nonsingular complex algebraic variety and $S$ is a subvariety, then there is a canonical Whitney stratification of $S$ whose strata are nonsingular algebraic sets.

My question is: in the complex algebraic category, can one define the notion of Whitney stratifications using only commutative algebra?

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  • $\begingroup$ Do you have in mind an example of a non-Whitney but algebraic stratification of something? $\endgroup$ Jan 12 at 21:50
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    $\begingroup$ @TomGoodwillie a classic example is given by Whitney's umbrella: the stratification by "regular locus" union "singular locus" is not Whitney. $\endgroup$ Jan 13 at 1:39
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    $\begingroup$ A closely related question that I've often wondered about: is it possible to define "topologically locally trivial fibration" in commutative algebra? Of course "smooth morphism" covers smooth fibres, but when the fibres are singular I'm not sure... $\endgroup$ Jan 13 at 1:42
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    $\begingroup$ @TomGoodwillie Given an algebraic variety, we can consider the set of its smooth points. Then for the set of singular points (which has strictly lower dimension) we can consider its set of smooth points again. This construction gives a stratification which is algebraic but not necessarily Whitney. $\endgroup$
    – UVIR
    Jan 13 at 1:49
  • $\begingroup$ @GeordieWilliamson I am a layman, but do universal local acyclicity maps work like that? $\endgroup$
    – Z. M
    Jan 13 at 7:21

2 Answers 2

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There is a purely algebraic characterisation of Condition (B) due to Le and Teissier, see Proposition 1.3.8 of the paper

Lê Dũng Tráng; Teissier, Bernard, Limites d’espaces tangents en géométrie analytique. (Limits of tangent spaces in analytic geometry), Comment. Math. Helv. 63, No. 4, 540-578 (1988). ZBL0658.32010..

Here's a summary. Assume that $X$ is a complex algebraic variety (embedded in $\mathbb{C}^n$ if affine or $\mathbb{P}^n$ if projective) and that $Y \subset X$ is a smooth quasiprojective subvariety. Then the pair $(X_\text{reg},Y)$ satisfies Condition (B) if and only if there is a containment of ideals $$I[\textbf{Con}(X) \cap \textbf{Con}(Y)] \subset \overline{I}[\kappa_X^{-1}(Y)]$$

Some explanation: here $I[Z]$ means the generating ideal of $Z$, while $\textbf{Con}(X)$ is the conormal variety of $X$ and $\kappa_X:\textbf{Con}(X) \to X$ is the conormal map. The bar on the right side here denotes integral closure. I learned about all this from Chapter 4 of Flores and Teissier's amazing survey

Flores, Arturo Giles; Teissier, Bernard, Local polar varieties in the geometric study of singularities, Ann. Fac. Sci. Toulouse, Math. (6) 27, No. 4, 679-775 (2018). ZBL1409.14002.

Martin Helmer and I have a recent paper which uses this Le-Teissier criterion to algorithmically construct Whitney stratifications of complex varieties. The hard part is bypassing the integral closure, which is computationally prohibitive. Martin even has a Macaulay2 implementation on his webpage which you can play around with if you have actual example varieties to stratify :)

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In the case that the variety is a hypersurface, it seems to me that an answer is provided by the Zariski equisingular stratification.

If $X = (f) \subset \Bbb C^{n+1}$ is a hypersurface, one defines the dimensionality type of $X$ at a point $x \in X$ inductively by $\mathrm{dt.}(X, x) = -1$ if $x \notin X$, and $\mathrm{dt.}(X, x) = 1 + \mathrm{dt.}(\Delta_{\pi}, 0)$ where $\pi : (X, x) \to (\Bbb C^n, 0)$ is a generic finite map provided by Noether normalization, and $\Delta_\pi$ is the discriminant, consisting of the hypersurface in $\Bbb C^n$ of points over which the map $\pi$ is not etale.

Hironaka proved that the function $\mathrm{dt.}$ is upper semicontinuous on $X$, so as a consequence the sets $\{x \in X : \mathrm{dt.}(x) \geq i\}$ are closed subvarieties of $X$. One defines a partition of $X$ by "strata" as the connected components of the fibers $X_i \setminus X_{i-1}$ of $\mathrm{dt.}$; Zariski showed that this is indeed a topological stratification, in the sense that boundary of a "stratum" is a collection "strata".

A proof of $(b)$-regularity may be found in Speder "Equisingularite et conditions de Whitney".

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