Can we define Whitney stratification algebraically?

Question

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?

Answer 1

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 :)

Update (11th July 2023) There is an error in my paper with Martin mentioned above, which has been described and corrected in this erratum; the fix involves using primary decompositions rather than saturations. The arxiv version incorporates all of the corrections.

Answer 2

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".