# Stable reduction of curves and tame ramification

Abstract (Summary)

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 K?/K with Galois group G induces a G-action on the
extension XK? of X to K?. We study the extension of this G-action to certain regular
models of XK?. 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 Néron
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 can occur. We also compute the actual jumps for each of the finitely
many possible fiber types for curves of genus 1 and 2.
Bibliographical Information:

Advisor:

School:Kungliga Tekniska högskolan

School Location:Sweden

Source Type:Doctoral Dissertation

Keywords:MATHEMATICS

ISBN:978-91-7178-764-4

Date of Publication:01/01/2007