Families of curves and alterations. (English) Zbl 0868.14012

Summary: 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 than the author’s previous results [ “Smoothness, semi-stability and alterations”, Publ. Math., Inst. Hautes Étud. Sci. 83, 51-93 (1996)]. In addition we deal with situations where a finite group acts.


14H10 Families, moduli of curves (algebraic)
Full Text: DOI Numdam EuDML


[1] [1] , Smoothness, semi-stability and alterations, Publications Mathématiques I.H.E.S. · Zbl 0916.14005
[2] [2] and , On extending families of curves, to appear in Journal of Algebraic Geometry. · Zbl 0922.14017
[3] [3] and , Arithmetic moduli of elliptic curves, Annals of Mathematics Studies 108, Princeton University Press (1985). · Zbl 0576.14026
[4] [4] , Desingularization of two-dimensional schemes, Annals of Mathematics, 107(1978), 151-207. · Zbl 0369.14005
[5] [5] and , Geometric invariant theory, Second Enlarged Edition, Ergebnisse der Mathematik und ihrer Grenzgebiete 34, Springer Verlag (1982). · Zbl 0504.14008
[6] [6] and , Critères de platitude et de projectivité, Techniques de “platification” d’un module, Inventiones Mathematicae, 13 (1971), 1-89. · Zbl 0227.14010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.