Characteristic Polynomial of Arrangements and Multiarrangements

123 Pages·2017·0.8 MB·English

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Mehdi Garrousian, The University of Western Ontario
Supervisor: Dr. Graham Denham, The University of Western Ontario
A thesis submitted in partial fulfillment of the requirements for the Doctor of Philosophy degree
in Mathematics
© Mehdi Garrousian 2011
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Characteristic Polynomial of
Arrangements and Multiarrangements
(Spine title: Characteristic Polynomial of Arrangements)
(Thesis format: Monograph)
by
Mehdi Garrousian
Graduate Program
in
Mathematics
A thesis submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
School of Graduate and Postdoctoral Studies
The University of Western Ontario
London, Ontario, Canada
(cid:13)c Mehdi Garrousian 2011
Certificate of Examination
THE UNIVERSITY OF WESTERN ONTARIO
SCHOOL OF GRADUATE AND POSTDOCTORAL STUDIES
Chief Adviser: Examining Board:
Professor Graham Denham Professor Ajneet Dhillon
Advisory Committee: Professor Marc Moreno Maza
Professor Ajneet Dhillon Professor Lex Renner
Professor Lex Renner Professor Hal Schenck
The thesis by
Mehdi Garrousian
entitled:
Characteristic Polynomial of Arrangements and Multiarrangements
is accepted in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Date:
Chair of Examining Board
Professor Margaret McNay
ii
Abstract
This thesis is on algebraic and algebraic geometry aspects of complex hyperplane
arrangements and multiarrangements. We start by examining the basic properties
of the logarithmic modules of all orders such as their freeness, the cdga structure,
the local properties and close the first chapter with a multiarrangement version of a
theorem due to M. Musta¸ta˘ and H. Schenck.
In the next chapter, we obtain long exact sequences of the logarithmic modules
of an arrangement and its deletion-restriction under the tame conditions. We observe
howthetameconditionstransferbetweenanarrangementanditsdeletion-restriction.
In chapter 3, we use some tools from the intersection theory and show that the
intersection cycle of a certain projective variety has a closed answer in terms of the
characteristic polynomial. This result is used to compute the leading parts of the
Hilbert polynomial and Hilbert series of the logarithmic ideal. As a consequence, we
recover some of the classical results of the theory such as the Solomon-Terao formula
for tame arrangements. This is done by computing the Hilbert series in two different
ways. We also introduce the notion of logarithmic Orlik-Terao ideal and show that
the intersection lattice parametrizes a primary decomposition. The chapter is closed
by a generalization of logarithmic ideals to higher orders. It is shown that these ideals
detect the freeness of the corresponding logarithmic modules.
The last chapter is a generalization of the notion of logarithmic ideal to mul-
tiarrangements. Some of the basic properties of these ideals are investigated. It is
shown that one obtains a natural resolution of this ideal by logarithmic modules un-
der the tame condition. In the final section it is shown that the intersection cycle of
the logarithmic ideal of a free multiarrangement is obtained from its characteristic
polynomial, similar to simple arrangements.
Keywords: arrangement/multiarrangement,derivationmodule,logarithmicideal,char-
acteristic polynomial, Tutte polynomial, intersection cycle, Chow ring.
iii
Co-Authorship
Chapter three is a joint work with my supervisor, Dr. Graham Denham, and Dr.
Mathias Schulze. I made contributions to the geometric deletion-restriction formula
which leads to the computation of the intersection cycle of the logarithmic variety.
In particular, I gave a formulation of the main formula of Section 3.2 by giving an
algebraic defining ideal for the component of the restriction. The appendix to chapter
three is independent from this joint work.
iv
Acknowledgements
It is my pleasure to express my deep gratitude to my advisor, professor Graham
Denham, for introducing me to the fascinating area of the current thesis and his
generous sharing of several beautiful ideas that led to the present work. He patiently
taught me to be rigorous and articulate in my math and look for the smallest hints to
the general facts in concrete examples. These are qualities that I need to foster in the
years to come. Moreover, I am grateful to my advisor for his financial support that
facilitated the completion of this thesis. I would also like to thank professor Mathias
Schulze who coauthored the material of chapter 3 and made insightful comments on
an earlier draft.
My warm appreciation also goes to my advisory committee, professor Ajneet
Dhillon and professor Lex Renner as well as my external examiners, professor Marc
Moreno Maza and professor Hal Schenck, for their careful reading and invaluable
corrections and suggestions. I would also like to cherish the memory of professor
Richard Kane who would have been my examiner if he was among us.
During my PhD studies I had the opportunity of attending conferences on hy-
perplane arrangements and met the leading researchers of this area. I thank all of
the organizers of the following conferences for providing excellent academic enviro-
ment and financial support: The Algebraic Geometry and Topology of Hyperplane
Arrangements at Northeastern University in 2011; Intensive Research Program on
Configuration Spaces at Centro di Ricerca Matematica Ennio De Giorgi, in Pisa in
Summer 2010; Mathematical Society of Japan (SI) 2009 on Arrangements of Hyper-
planes at Hokkaido Univeristy in 2009; Conference in Honour of Peter Orlik at Fields
Institute in 2008. Parts of this thesis were carried out during my two months visit in
Pisa. I also thank professor Alex Suciu and professor Nicole Lemire for helping with
the travel expenses.
This is also an opportunity to thank my teachers during the past years who all
have indirectly contributed to this thesis. In particular I feel indebted to my master’s
advisor at the University of Tehran, professor Rahim Zaare-Nahandi, who was an
inspiring mathematician and caring teacher. I am also very grateful for the attention
and support of the current and past graduate chairs of the mathematics department
at Western, professor Dan Christensen and professor Andr´e Boivin. The mathematics
v
department was a much more live place due to the presence of professor Ja´n Min´aˇc
who inspired me to follow research with more passion and energy.
I would also like to mention some of my friends here. Mona was kind and
supportive during the difficult times of writing this thesis and provided the much
needed company. Ali and Farzad were wonderful officemates, Arash always shared
his great advices with me and proof read some of my documents and Javad gave very
useful tips about tweaking the Latex file.
At last but not least, my thank goes to my family for their love and support.
My father mentored me at times when computing a common denominator seemed the
hardest task in the world and my mother believed in my determination for studying
math when I was only a high school student.
vi
To my parents and my brothers
vii
Table of Contents
Certificate of Examination . . . . . . . . . . . . . . . . . . . . . . . . . ii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Co-Authorship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
List of Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Basic Definitions and Operations . . . . . . . . . . . . . . . . . . . . 1
1.2 Derivations and Ka¨hler Forms . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Background and Motivations . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.1 Classical Results . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.2 Transition to Multiarrangements . . . . . . . . . . . . . . . . 16
1.4 Multiarrangements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.4.1 D and Ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.4.2 Saito’s Criterion for Freeness . . . . . . . . . . . . . . . . . . . 26
1.4.3 cdga Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.4.4 Homological Dimensions and Local Properties . . . . . . . . . 33
1.5 Characteristic and Poincar´e Polynomials . . . . . . . . . . . . . . . . 37
2 Long Exact Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.1 LES for D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
p
2.2 LES for Ωp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.3 D -exactness vs. Ωp-exactness . . . . . . . . . . . . . . . . . . . . . . 51
p
2.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
viii
3 Intersection Cycle, Recurrence and the Characteristic Polynomial 54
3.1 Critical Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.1.1 A Concrete Example . . . . . . . . . . . . . . . . . . . . . . . 57
3.2 A Geometric Deletion-Restriction Formula . . . . . . . . . . . . . . . 58
3.2.1 case 1: Localization . . . . . . . . . . . . . . . . . . . . . . . . 61
3.2.2 case 2: Complement of Restriction . . . . . . . . . . . . . . . 62
3.2.3 case 3: Blow Up . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2.4 Intersection Classes and Multiplicities . . . . . . . . . . . . . . 64
3.3 Tutte Polynomial and Recursion . . . . . . . . . . . . . . . . . . . . . 69
3.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.5 Appendix to Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.5.1 Logarithmic OT Ideals . . . . . . . . . . . . . . . . . . . . . . 83
3.5.2 Higher Order Logarithmic Ideals . . . . . . . . . . . . . . . . 86
4 Logarithmic Ideals of Multiarrangements . . . . . . . . . . . . . . . 90
4.1 Freeness via Log Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.3 Tameness, Resolutions and Hilbert Series . . . . . . . . . . . . . . . . 98
4.4 Intersection Class with Multiplicity . . . . . . . . . . . . . . . . . . . 103
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Curriculum Vitae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
ix