2004/06/19 by Mike Roth, Ravi Vakil, Roth, Mike +1
Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Meromorphic and Entire Functions #Primary 14A15 #Secondary 14H10 #math.AG #msc:14A15 #msc:14H10
paper · pdf · doi:10.48550/arxiv.math/0406384
17 pages, to appear in Proceedings of "Workshop on algebraic structures and moduli spaces", July 14-20, 2003, Centre de Recherches Mathematiques, Universite de Montreal
arxiv created 2004/06/19 · openalex publication_date 2004/06/19 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.