Cell decompositions have been used in topology for a long time as a tool in computing cohomology, but the notion in algebraic geometry and arithmetic geometry of paving by affine spaces (or "affine paving" for short) seems to be a little more recent. An early occurrence is found in the survey Barry Mazur gave in 1974: Eigenvalues of Frobenius acting on algebraic varieties over finite fields, Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974), pp. 231–261, Amer. Math. Soc., Providence, R.I., 1975. He considers only complete varieties (where the most natural examples arise), but the definition makes sense for any variety $X$: require $X$ to have a filtration $0=X_0 \subset X_1 \subset \dots \subset X_d = X$ with each $X_i$ closed and with $X_i - X_{i-1}$ a finite disjoint union of copies of affine space $\mathbb{A}^i$. (Though not stated explicitly, the union here might be empty.)

This definition was used by Hotta and Springer in their influential 1977 paper here. But as time went on, a broader definition seems to have taken over: given a filtration of $X$ by closed subvarieties as above, one only requires that each $X_i - X_{i-1}$ be a finite disjoint union of copies of a single affine space (of some dimension).

It's not clear to me whether these two definitions (narrower and broader) are actually interchangeable, but the underlying notion of affine paving is itself quite natural. The prototype is $X=\mathbb{P}^1$, with a closed point and a complementary affine line. More generally, a flag variety $G/B$ has a suitable filtration with differences given by unions of Bruhat cells $BwB/B$, each isomorphic to affine space of dimension $\ell(w)$ as $w$ ranges over the Weyl group. One can then try to form an affine paving of a Springer fiber (or more general Hessenberg variety) inside $G/B$ by simply intersecting with a Bruhat-type filtration. Work in this direction occurs in a number of places, including $\S11$ of J.C. Jantzen's monograph Nilpotent orbits in representation theory, contained in the 2004 Birkhauser volume Lie Theory (Progress in Mathematics 228). There are also research papers by T. Haines (on Kazhdan-Lusztig "purity"), J. Tymoczko, E. Sommers, M. Precup (on Hessenberg varieties), etc.

Given the range of applications in the literature, should the broader or narrower definition of "covering by affine spaces" be regarded now as standard?

To me it seems awkward to leave ambiguity in a widely used mathematical definition. But it also seems unnatural to use a broader-than-needed definition.

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    $\begingroup$ To muddy the water further, Fulton, Intersection Theory, 1.9.1 uses an even broader definition: $X_i-X_{i-1}$ is a union of affine spaces of possibly varying dimensions. He refers to this as 'a scheme with a "cellular decomposition"'. He goes on to establish some desirable properties, such as the Chow group maps onto homology with a basis given by closures of the affine spaces. I suspect that this may be sufficient for most applications. $\endgroup$ – Donu Arapura Nov 11 '14 at 15:38
  • $\begingroup$ @Donu: Thanks for the reference. I thought I had seen this variant somewhere but couldn't recall exactly. Fulton does give it a new name, though the transition from variety to scheme probably isn't significant here. Broader definitions may be more flexible, but do they still carry all the consequences of the earlier ones? I'm also unsure what natural examples are covered by a broader version that aren't already covered by a narrower one. $\endgroup$ – Jim Humphreys Nov 11 '14 at 17:07

The definitions are not equivalent: consider $$\mathbb{A}^2 \times 0 \cup 0 \times \mathbb{P}^1 \subset \mathbb{A}^2 \times \mathbb{P}^1$$

Then you cannot see the line before you see the plane.

That is, you can't take any subvariety out of $\mathbb{A}^2$ first, since you can't stratify $\mathbb{A}^2$-minus-anything by cells. You can't take $0 \times (\mathbb{P}^1 \setminus 0)$ out first, because it's not closed. So there's no dimension one closed subvariety you can remove, leaving the complement a union of affines, so there's no Barry Mazur stratification. But you can take

$$\mathbb{A}^2 \times 0 \subset \mathbb{A}^2 \times 0 \cup 0 \times \mathbb{P}^1$$

which is a stratification in the other sense.

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  • $\begingroup$ Thanks very much for the example (though it's probably not something one stumbles across "in nature"). I suspected the earlier definitions weren't equivalent but wasn't quite sure. I'm used to seeing well-behaved varieties. $\endgroup$ – Jim Humphreys Nov 11 '14 at 21:06

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