Characteristic Polynomial of Arrangements and Multiarrangements

Western University Scholarship@Western Electronic Thesis and Dissertation Repository 4-21-2011 12:00 AM Characteristic Polynomial of Arrangements and Multiarrangements Mehdi Garrousian The University of Western Ontario Supervisor Dr. Graham Denham The University of Western Ontario Graduate Program in Mathematics A thesis submitted in partial fulfillment of the equirr ements for the degree in Doctor of Philosophy © Mehdi Garrousian 2011 Follow this and additional works at: https://ir.lib.uwo.ca/etd Part of the Algebra Commons, Algebraic Geometry Commons, and the Discrete Mathematics and Combinatorics Commons Recommended Citation Garrousian, Mehdi, "Characteristic Polynomial of Arrangements and Multiarrangements" (2011). Electronic Thesis and Dissertation Repository. 142. https://ir.lib.uwo.ca/etd/142 This Dissertation/Thesis is brought to you for free and open access by Scholarship@Western. It has been accepted for inclusion in Electronic Thesis and Dissertation Repository by an authorized administrator of Scholarship@Western. For more information, please contact [email protected]. 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 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. Mustat¸˘aand 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 how the tame conditions transfer between an arrangement and its deletion-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 multiarrangements. 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 under 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, derivation module, logarithmic ideal, characteristic 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 hyperplane 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 AndréBoivin. The mathematics v department was a much more live place due to the presence of professor JánMináˇc 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 KählerForms......................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 PoincaréPolynomials................ 37 2 Long Exact Sequences .......................... 45 2.1 LES for Dp ................................ 45 2.2 LES for Ωp ................................ 49 p 2.3 Dp-exactness vs. Ω -exactness...................... 51 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.........................

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