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Belghitti-Gruson or Semple? (English) Zbl 1084.14505
Christensen, Chris (ed.) et al., Algebra, arithmetic and geometry with applications. Papers from Shreeram S. Ahhyankar’s 70th birthday conference, Purdue University, West Lafayette, IN, USA, July 19–26, 2000. Berlin: Springer (ISBN 3-540-00475-0/pbk). 259-262 (2003).
Summary: We want to make some advertising for two spaces, one defined J. G. Semple [Proc. Lond. Math. Soc., III. Ser. 4, 24–49 (1954; Zbl 0055.14505)], the other by Belghitti and Gruson [M. Belghitti, C. R. Acad. Sci., Paris, Sér. I 314, 541–545 (1992; Zbl 0815.14002)]. The \(n\)-th Semple’s tower over \(X\) is the smallest space where you can realize the set of all sequences of length \(n\) of Nash’s blowing-ups of curves embedded in \(X\), the \(n\)-th Belghitti-Gruson’s space parametrizes the set of sequences of length \(n\) of blowing-ups centered at near closed points of \(X\). The existence of such spaces is interesting by itself, and good descriptions (whatever it means) of them would be full of informations. The constructions of these spaces are a priori quite different. We show that, in fact, they are quite similar and closely related: they have the same Chow ring.
For the entire collection see [Zbl 1022.00011].

14E05 Rational and birational maps
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14C15 (Equivariant) Chow groups and rings; motives