Families of curves and alterations
Annales de l'Institut Fourier, Volume 47 (1997) no. 2, pp. 599-621.

In this article it is shown that any family of curves can be altered into a semi-stable family. This implies that if S is an excellent scheme of dimension at most 2 and X is a separated integral scheme of finite type over S, then X can be altered into a regular scheme. This result is stronger then the results of [ Smoothness, semi-stability and alterations to appear in Publ. Math. IHES]. In addition we deal with situations where a finite group acts.

Dans l’article on prouve que toute famille de courbes peut être altérée en une famille semi-stable. Soit S un schéma excellent de dimension 0, 1 ou 2 et soit X un schéma séparé de type fini sur S. Alors le résultat implique qu’on peut altérer X en un schéma régulier. C’est un résultat plus fort que ceux de [Smoothness, semi-stability and alterations à paraître dans Publ. Math. IHES]. De plus, on considère des situations où un groupe fini agit, et on obtient des résultats analogues.

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Jong, A. Johan de. Families of curves and alterations. Annales de l'Institut Fourier, Volume 47 (1997) no. 2, pp. 599-621. doi : 10.5802/aif.1575. http://archive.numdam.org/articles/10.5802/aif.1575/

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[3] N.M. Katz and B. Mazur, Arithmetic moduli of elliptic curves, Annals of Mathematics Studies 108, Princeton University Press (1985). | MR | Zbl

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[5] D. Mumford and J. Fogarty, Geometric invariant theory, Second Enlarged Edition, Ergebnisse der Mathematik und ihrer Grenzgebiete 34, Springer Verlag (1982). | MR | Zbl

[6] M. Raynaud and L. Gruson, Critères de platitude et de projectivité, Techniques de "platification" d'un module, Inventiones Mathematicae, 13 (1971), 1-89. | Zbl

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