Abstract
In this short note, we want to make some advertising for two spaces one defined by Semple in [S], the other by Belghitti and Gruson in [B]. The 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 blowingups 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.
In this note, X is a regular irreducible scheme over an algebraic closed field k. A curve in an ambient space is a closed reduced irreducible subscheme of dimension 1 of the ambient space.
This paper has been written after many long and fruitful conversations with L. Gruson, M. Lejeune-Jalabert and O. Piltant
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Belghitti M., Variétés des points infiniment voisins d’ordre n du plan. C.R.A.S. t. 314, Série I, 541–545 (1992).
Colley S.J., Kennedy G., A higher-order contact formula for plane curves. Common. Algebra 19, No.2, 479–508 (1991).
Demailly J.P. Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials. Kollár, János (ed.) et al., Algebraic geometry. Proceedings of the Summer Research Institute, Santa Cruz, CA, USA, July 9-29, 1995. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 62(pt.2), 285–360 (1997).
Halphen G.H. Sur la recherche des points d’une courbe algébrique plane qui satisfont à une condition exprimée par une équation différentielle algébrique et sur les questions analogues dans l’espace. uvres, I, Gauthier-Villars, p. 475–543.
Lejeune-Jalabert M. Infinitely near points: a differential approach. In preparation.
Semple J.G. Some investigations in the geometry of curve and surface elements. Proc. London Math. Soc. (3) 4 24–29 (1954).
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Cossart, V. (2004). Belghitti-Gruson or Semple?. In: Christensen, C., Sathaye, A., Sundaram, G., Bajaj, C. (eds) Algebra, Arithmetic and Geometry with Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18487-1_14
Download citation
DOI: https://doi.org/10.1007/978-3-642-18487-1_14
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-00475-2
Online ISBN: 978-3-642-18487-1
eBook Packages: Springer Book Archive