  We define the affine stratification number asn X of a scheme X. For X
equidimensional, it is the minimal number k such that there is a stratification
of X by locally closed affine subschemes of codimension at most k. We show that
the affine stratification number is well-behaved, and bounds many aspects of
the topological complexity of the scheme, such as vanishing of cohomology
groups of quasicoherent, constructible, and l-adic sheaves. We explain how to
bound asn X in practice. We give a series of conjectures (the first by E.
Looijenga) bounding the affine stratification number of various moduli spaces
of pointed curves. For example, the philosophy of [GV, Theorem *] yields: the
moduli space of genus g, n-pointed complex curves of compact type (resp. with
"rational tails") should have the homotopy type of a finite complex of
dimension at most 5g-6+2n (resp. 4g-5+2n). This investigation is based on work
and questions of Looijenga.
  One relevant example turns out to be a proper integral variety with no
embeddings in a smooth algebraic space. This one-paragraph construction appears
to be simpler and more elementary than the earlier examples, due to Horrocks
and Nori.
