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Stable reduction of curves and tame ramification
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
2007 (English)Doctoral thesis, comprehensive summary (Other scientific)
Abstract [en]

This thesis treats various aspects of stable reduction of curves, and consists of two separate papers. In Paper I of this thesis, we study stable reduction of curves in the case where a tamely ramified base extension is sufficient. If X is a smooth curve defined over the fraction field of a strictly henselian discrete valuation ring, there is a criterion, due to T. Saito, that describes precisely, in terms of the geometry of the minimal model with strict normal crossings of X, when a tamely ramified extension suffices in order for X to obtain stable reduction. For such curves we construct an explicit extension that realizes the stable reduction, and we furthermore show that this extension is minimal. We also obtain a new proof of Saito's criterion, avoiding the use of adic cohomology and vanishing cycles. In Paper II, we study group actions on regular models of curves. If X is a smooth curve defined over the fraction field K of a complete discrete valuation ring R, every tamely ramified field extension K0=K with Galois group G induces a G-action on the extension XK0 of X to K0. We study the extension of this G-action to certain regular models of XK0 . In particular, we are interested in the induced action on the cohomology groups of the structure sheaf of the special fiber of such a regular model. We obtain a formula for the Brauer trace of the endomorphism induced by a group element on the alternating sum of the cohomology groups. Inspired by this global study, we also consider similar group actions on the cohomology of the structure sheaf of the exceptional locus of a tame cyclic quotient singularity, and obtain an explicit polynomial formula for the Brauer trace of the endomorphism induced by a group element on the alternating sum of the cohomology groups. We apply these results to study a natural filtration of the special fiber of the Néronmodel of the Jacobian of X by closed, unipotent subgroup schemes. We show that the jumps in this filtration only depend on the fiber type of the special fiber of the minimal regular model with strict normal crossings for X over Spec(R), and in particular are independent of the residue characteristic. Furthermore, we obtain information about where these jumps can occur. We also compute the actual jumps for each of the finitely many possible fiber types for curves of genus 1 and 2.

Place, publisher, year, edition, pages
Stockholm: Matematik , 2007. , 11, 19, 69 p.
Series
Trita-MAT, ISSN 1401-2286 ; 07:09
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-4494ISBN: 978-91-7178-764-4 (print)OAI: oai:DiVA.org:kth-4494DiVA: diva2:12534
Public defence
2007-10-12, Sal F3, KTH, Lindstedtsvägen 26, Stockholm, 13:00
Opponent
Supervisors
Note
QC 20100712Available from: 2007-09-21 Created: 2007-09-21 Last updated: 2010-08-26Bibliographically approved
List of papers
1. Stable reduction of curves and tame ramification
Open this publication in new window or tab >>Stable reduction of curves and tame ramification
2010 (English)In: Mathematische Zeitschrift, ISSN 0025-5874, E-ISSN 1432-1823, Vol. 265, no 3, 529-550 p.Article in journal (Refereed) Published
Abstract [en]

We study stable reduction of curves in the case where a tamely ramified base extension is sufficient. If X is a smooth curve defined over the fraction field of a strictly henselian discrete valuation ring, there is a criterion, due to Saito, that describes precisely, in terms of the geometry of the minimal model with strict normal crossings of X, when a tamely ramified extension suffices in order for X to obtain stable reduction. For such curves we construct an explicit extension that realizes the stable reduction, and we furthermore show that this extension is minimal. We also obtain a new proof of Saito's criterion, avoiding the use of l-adic cohomology and vanishing cycles.

Keyword
Stable reduction; Tame cyclic quotient singularities; Tame ramification
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-24226 (URN)10.1007/s00209-009-0528-5 (DOI)000277603500003 ()
Note
QC 20100826Available from: 2010-08-26 Created: 2010-08-26 Last updated: 2017-12-12Bibliographically approved
2. Galois actions on models of curves
Open this publication in new window or tab >>Galois actions on models of curves
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We study group actions on regular models of curves. If $X$ is a smooth curve defined over the fraction field $K$ of a complete d.v.r. $R$, every tamely ramified extension $K'/K$ with Galois group $G$ induces a $G$-action on $X_{K'}$. In this paper we study the extension of this $G$-action to certain regular models of $X_{K'}$. In particular, we obtain a formula for the Brauer trace of the endomorphism induced by a group element on the alternating sum of the cohomology groups of the structure sheaf of the special fiber of such a regular model. Inspired by this global study, we also consider similar questions for Galois actions on the exceptional locus of a tame cyclic quotient singularity. We apply these results to study a natural filtration of the special fiber of the N\'eron model of the Jacobian of $X$ by closed, unipotent subgroup schemes. We show that the jumps in this filtration only depend on the fiber type of the special fiber of the minimal regular model with strict normal crossings for $X$ over $\Spec(R)$, and in particular are independent of the residue characteristic. Furthermore, we obtain information about where these jumps occur. We also compute the jumps for each of the finitely many possible fiber type for curves of genus 1 and 2.

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-24227 (URN)
Note
QC 20100826Available from: 2010-08-26 Created: 2010-08-26 Last updated: 2010-08-26Bibliographically approved

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Citation style
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More styles
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