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 -adic sheaves. We explain how to bound asn X in practice. We give a series

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... 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 (Example 4.9) 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 [Ho] and Nori [N].