Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Stable modification of relative curves

Author: Michael Temkin
Journal: J. Algebraic Geom. 19 (2010), 603-677
Published electronically: June 9, 2010
MathSciNet review: 2669727
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Abstract | References | Additional Information

Abstract: We generalize theorems of Deligne-Mumford and de Jong on semi-stable modifications of families of proper curves. The main result states that after a generically étale alteration of the base any (not necessarily proper) family of multipointed curves with semi-stable generic fiber admits a minimal semi-stable modification. The latter can also be characterized by the property that its geometric fibers have no certain exceptional components. The main step of our proof is uniformization of one-dimensional extensions of valued fields. Riemann-Zariski spaces are then used to obtain the result over any integral base.

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Additional Information

Michael Temkin
Affiliation: School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, New Jersey 08540
Address at time of publication: Institute of Mathematics, Hebrew University, Giv$’$at-Ram, 91904 Jerusalem, Israel
MR Author ID: 332870

Received by editor(s): July 26, 2007
Received by editor(s) in revised form: February 24, 2010
Published electronically: June 9, 2010
Additional Notes: This article is based on a portion of my Ph.D. thesis; I want to thank my advisor Professor V. Berkovich. I am absolutely indebted to B. Conrad and I owe a lot to A. Ducros for pointing out various mistakes and inaccuracies, and for many suggestions that led to two revisions of the paper that improved the exposition. I express my deep gratitude to the Israel Clore Foundation for supporting my doctoral studies and to the Max Planck Institute for Mathematics, where a portion of this paper was written. A final revision was made when the author was staying at the IAS; the author was supported by NFS grant DMS-0635607.