Wild quotient singularities of arithmetic surfaces and their regular models [Elektronische Ressource] / vorgelegt von Franz Johannes Király

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• Mathematics

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• Publié par: universitat_ulm
• Publié le: 01 janvier 2010
• Nombre de lectures: 30

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Wild quotient singularities of
arithmetic surfaces and their
regular models
Dissertation zur Erlangung des Doktorgrades Dr. rer. nat.
der Fakult¨at fur¨ Mathematik und Wirtschaftswissenschaften
der Universit¨at Ulm
Vorgelegt von Franz Johannes Kir´aly aus Ulm
Ulm, 2010Dekan: Prof. Dr. Werner Kratz
Gutachter: Prof. Dr. Werner Lu¨tkebohmert
Datum der Promotion: 02. November 2010Contents
1 Introduction 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 About notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Tame quotient singularities 4
2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Basic facts on invariant rings . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Excellent rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.4 Tame local actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.5 Toric schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.6 Extension to mixed characteristic . . . . . . . . . . . . . . . . . . . . 17
2.7 Desingularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.8 A characterization of toric surface singularities . . . . . . . . . . . . 24
3 p-cyclic group actions on local normal rings 26
3.1 Motivating examples in dimension 2 . . . . . . . . . . . . . . . . . . 26
3.2 A criterion for monogeneity . . . . . . . . . . . . . . . . . . . . . . . 32
3.3 Application to models of curves . . . . . . . . . . . . . . . . . . . . . 38
4 Wild quotient singularities of surfaces 46
4.1 Brief overview on chapter 4 . . . . . . . . . . . . . . . . . . . . . . . 46
4.2 Models of local rings . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2.1 Definition of models . . . . . . . . . . . . . . . . . . . . . . . 50
4.2.2 Zariski valuations . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2.3 Introducing the G -action . . . . . . . . . . . . . . . . . . . . 56
4.2.4 The graph structure of models and the G -action . . . . . . . 60
4.2.5 Minimal regular realization of a valuation . . . . . . . . . . . 63
4.2.6 Rings of components and parameters . . . . . . . . . . . . . . 66
I4.2.7 Application to components with K -rational center . . . . . 67
4.3 Resolution of wild quotient singularities . . . . . . . . . . . . . . . . 70
4.3.1 Examining the naive approach . . . . . . . . . . . . . . . . . 70
4.3.2 Critical components and correspondence of models . . . . . . 73
4.3.3 The augmentation ideal on critical components . . . . . . . . 78
4.3.4 Application to components with K -rational center . . . . . 79
4.4 Tame descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.4.1 Galois extension and tame descent . . . . . . . . . . . . . . . 85
4.4.2 Tame invariance of critical components . . . . . . . . . . . . 89
4.4.3 Quotients of stable models. . . . . . . . . . . . . . . . . . . . 90
Bibliography 93
IIChapter 1
Introduction
1.1 Introduction
Resolutionofsingularitiesisoneoftheoldestandyetstilloneofthemostintriguing
topics in mathematics. Ever since Isaac Newton’s calculations on Puiseux series
expansionsofcurves,ithasbeenofmajorinteresttodescribeorremovesingularities
on varieties. The first rigorous proof for desingularization of surfaces over the
complexnumberswasgiven1934byWalker[Wal34]. Afewyearslater, Zariskigave
proofsfordesingularizationofsurfaces[Zar39]andthree-folds[Zar44]overarbitrary
fields of characteristic zero. In 1964, Hironaka gave his famous proof for reduced
schemes of finite type over fields of characteristic zero [Hir64], Abhyankar presented
hisresultforresolutionoverfieldsofcharacteristicsevenorgreaterin1966[Abh66].
In 1978, Lipman proved desingularization for quasi-excellent surfaces which also
include arithmetic surfaces [Lip78]. For the remaining cases, desingularization is
still an open problem despite many efforts; however, Grothendieck has proved that
quasi-excellency is a necessary condition for a locally Noetherian scheme to have
a desingularization [Gro65, 7.9]. By Lipman’s result, we know that it is also a
sufficient condition for dimension two and less.
This thesis is motivated by the study of quotient singularities, occurring on the
scheme of orbits by a finite group action on a regular scheme, the so-called quotient
scheme. Quotient singularities are an important class of singularities in any char-
acteristic which canonically arise in context of actions on schemes. The problem
of quotient singularities was first studied by Jung [Jun08] and Hirzebruch [Hir53]
for a cyclic group acting on a regular complex surface. Here the quotient surface
exhibits the famous Hirzebruch-Jung singularities; their resolution combinatorics
can be related directly to the action of the group G. General groups acting on
complex varieties were later examined by Cartan [Car57] and Brieskorn [Bri68].
When one generalizes the problem to arbitrary (locally Noetherian quasi-excellent)
schemes, one has to note that the problem of quotient singularities is essentially of
local nature, so it is merely a question in commutative algebra. One can reduce to
the following situation:
Let B a Noetherian local regular ring, and G,→ AutB a finite group action on
B. Denote by k the residue field of B, by p = chark its characteristic. The
Gring of invariants A =B is the subring of B consisting exactly of the elements
invariant under G and it can happen that A is not regular. Here B plays the
role of a local germ of the scheme in consideration, taken at a closed point; and A
1is the corresponding germ on the quotient scheme. The main goal in the theory of
quotient singularities is to relate the structure of A , e.g. the combinatorics of its
minimal desingularization, to the group action of G on B.
The case where p is coprime to #G, the so-called tame case, has been extensively
studied in literature. In this context, the Hirzebruch-Jung singularities over the
complex numbers can be generalized with minor efforts to the case of a cyclic group
acting on varieties over arbitrary fields of characteristic zero. The same can be
done for arbitrary group actions. There have been also some ad-hoc adaptations of
theseresultstomoregeneral B mostlyindimension 2 , e.g. thecaseofarithmetic
surfacesin[Vie77]. Moreover,inthetamecase,acelebratedtheoremofSerre[Ser68]
classifiesexactlywhenthering A isregularintermsof G, namelyifandonlyif G
is generated by so-called pseudo-reflections. There seems to be consensus that the
tame case is relatively well understood, and that those results can be generalized to
arbitraryringsinthetamecase. However, asof2010, thereisnostandardreference
in literature which goes beyond particular applications.
The case where p divides #G is called the wild case, since the classical methods
from the tame case fail by elementary means. To the author’s knowledge, the only
results concern the simplest non-trivial case where G =Z/pZ, and B is regular
of very specific form: M. Artin [Art75] has obtained a few results for p = 2 and
B =k[[X ,X ]]. InPeskin’sthesis, [Pes83]severalbasicresultsaboutthewildcase1 2
are collected and the results of Artin are generalized for specific group actions with
p≥ 3.
There seems to have been little progress after that until Lorenzini’s unpublished
paper [Lor06], in which some structural results on quotient singularities with focus
on quotients of stable models of curves by prime cyclic actions are obtained. The
biggest part of those results uses the global machinery of N´eron models and many
ad-hoccombinatorialconstructionswhichunfortunatelygivesnoinsightonhowthe
G -action relates to the structure of the singularities. However, in the context of
the local results of Artin these results suggest that it might be possible to obtain
similar structural results only with local methods.
The goal of this thesis is to understand the relation between B and A in terms
of the group action G on B, in the simplest non-trivial case where G =hσi is
prime cyclic.
In chapter 2, we collect and extend classical results on tame quotient singularities
and discuss them in the context of toric geometry.
In chapter 3, we prove an algebraic result about the invariant morphism. We prove
that B is a monogenous A -algebra if and only if the augmentation ideal
I ={(σ−id)b ; b∈B}G
of B isprincipal. Ifinparticular B isregular, thisimpliesthat A isalsoregular.
In chapter 4, we assume that B is a local regular normal crossings germ of an
arithmeticsurfaceoverthespectrumofacompletediscretevaluationring R. Using
birationalgeometry,combinatorialmethodsandtheresultsfromchapter3,werelate
the structure of the minimal normal crossings desingularization of A to the group
action of G on B. Furthermore, we examine the behavior of the singularity of A
with respect to tame base extension.
A more detailed overview on the content and the utilized methods can be found at
the beginning of each chapter.
21.2 About notation
Inthisthesiswewilltrytousecommonsymbolsandnotations. However,wewantto
pointoutseveralthingswhichmightbesourcesofconfusionsincethecorresponding
notation is not uniform worldwide. At most occasions, this will be also said in the
text.
For a ring C, we will denote by Q(C)