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Resolutions of Quotient Singularities and Their Cox Rings Phd Dissertation

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Resolutions of Quotient Singularities and Their Cox Rings Phd Dissertation University of Warsaw Faculty of Mathematics, Informatics and Mechanics Maksymilian Grab Resolutions of quotient singularities and their Cox rings PhD dissertation Supervisor prof. dr hab. Jaros law Wi´sniewski Institute of Mathematics University of Warsaw Auxiliary Supervisor dr Maria Donten-Bury Institute of Mathematics University of Warsaw Author's declaration: I hereby declare that I have written this dissertation myself and all the contents of the dissertation have been obtained by legal means. March 11, 2019 ................................................... Maksymilian Grab Supervisors' declaration: The dissertation is ready to be reviewed. March 11, 2019 ................................................... prof. dr hab. Jaros law Wi´sniewski March 11, 2019 ................................................... dr Maria Donten-Bury ABSTRACT 3 Abstract The aim of this dissertation is to investigate the geometry of resolutions of quotient sin- n gularities C =G for G ⊂ SLn(C) with use of an associated algebraic object { the Cox ring. We are interested in the construction of all crepant resolutions and a combinatorial description of birational relations among them. This information can be read off from the structure of the Cox ring. We give a method to construct certain finitely generated subalgebras of the Cox ring R(X) n of a crepant resolution X ! C =G and present three methods to verify when such a subring is actually the whole Cox ring. The construction relies on the embedding, investigated by Donten-Bury and Wi´sniewski, of the Cox ring into the Laurent polynomial ring over an invariant ring of the commutator subgroup [G; G] ⊂ G. The first method to verify whether a constructed subalgebra is equal to R(X) relies on a n criterion involving valuations of crepant divisors over the singularity C =G. Such valua- tions can be computed using the McKay correspondence of Ito and Reid as restrictions of certain monomial valuations on the field of rational functions C(x1; : : : ; xn). We apply this method to the family of three-dimensional quotient singularities given by groups acting via 3 reducible representations on C . We obtain a presentation of R(X) in terms of generators and relations, which is then used to study the geometry of resolutions for several examples of quotient singularities. One example is the infinite series of quotient singularities given by dihedral groups, investigated previously by Nolla de Cellis and Sekiya with different methods. We provide an alternative treatment for such quotients. We also investigate the 3 geometry of crepant resolutions in the simplest examples of the quotient C =G when such a resolution contracts a divisor to a point. Due to limitations of methods used prior to our work examples exhibiting such a phenomenon were not studied earlier even though they are typical among resolutions of three-dimensional quotient singularities. Two examples we present belong to the family of reducible representations and one belongs to the family of irreducible representations which is substantially harder to analyze. The second method is based on the characterization theorem for the Cox ring in terms of Geometric Invariant Theory. We present an example of its use when we give another proof in the case of dihedral groups. We also give a general method to bound degrees of generators of the Cox ring by use of the Kawamata-Viehweg vanishing and multigraded Castelnuovo-Mumford regularity. We use this method in the study of three examples of symplectic quotient singularities in dimension four. In this study we use another technique to verify that a constructed subalgebra is equal to the Cox ring, based on the algebraic torus action on the resolution. The action allows us, via the Lefschetz-Riemann-Roch theorem, to compute the important part of the Hilbert series of the Cox ring. We expect that this idea may be generalized and used to study other examples. Keywords: finite group action, quotient singularity, resolution of singularities, crepant resolution, symplectic resolution, Cox ring, algebraic torus action. AMS MSC 2010 classification: 14E15, 14E30, 14E16, 14L30, 14L24, 14C35, 14C40, 14C20, 14Q15 4 Streszczenie n Celem niniejszej rozprawy jest zbadanie geometrii rozwiaza´nosobliwo´sci, ilorazowych C =G dla G ⊂ SLn(C) przy u_zyciu stowarzyszonego obiektu algebraicznego { pier´scienia Coxa. Interesuje nas skonstruowanie wszystkich rozwiaza´nkrepantnych, i kombinatoryczny opis relacji biwymiernych pomiedzy, nimi. Opis taki mo_zna odczyta´cze struktury pier´scienia Coxa. Podajemy metode, konstrukcji pewnych sko´nczenie generowanych podalgebr pier´scienia n Coxa R(X) rozwiazania, krepantnego X ! C =G i trzy metody pozwalajace, rozstrzyga´c,, kiedy taki podpier´scie´njest ca lym pier´scieniem Coxa. Konstrukcje, przeprowadzamy w oparciu o badane przez Donten-Bury i Wi´sniewskiego zanurzenie pier´scienia Coxa w pier´s- cie´nwielomian´owLaurenta nad pier´scieniem niezmiennik´owkomutanta [G; G] ⊂ G. Pierwszy spos´obsprawdzenia, czy skonstruowana algebra jest r´owna R(X), opiera sie, na n kryterium zwiazanym, z waluacjami dywizor´owkrepantnych nad osobliwo´scia, C =G. Walu- acje te mo_zna obliczy´cprzy u_zyciu odpowiednio´sci McKaya w sensie Ito i Reida jako za- we_zenia, pewnych waluacji jednomianowych na pier´scieniu funkcji wymiernych C(x1; : : : ; xn). Stosujemy to kryterium do rodziny tr´ojwymiarowych osobliwo´sci ilorazowych zwiazanych, z reprezentacjami rozk ladalnymi. Otrzymujemy prezentacje, R(X) poprzez generatory i relacje, kt´orej nastepnie, u_zywamy, aby zbada´cgeometrie, rozwiaza´nkilku, przyk lad´ow osobliwo´sci ilorazowych. Jednym z nich jest niesko´nczona seria osobliwo´sci ilorazowych zadanych przez grupy dihedralne, badana wcze´sniej przez Nolle, de Cellis i Sekiye, przy u_zyciu innych metod. Analizujemy r´ownie_zgeometrie, rozwiaza´nkrepantnych, w najprost- 3 szych przypadkach osobliwo´sci C =G o rozwiazaniu, ´sciagaj, acym, dywizor do punktu. Ze wzgledu, na ograniczenia stosowanych uprzednio metod przyk lady takich rozwiaza´nnie, by ly dotychczas badane pomimo ich powszechno´sci w´sr´odrozwiaza´ntr´ojwymiarowych, os- obliwo´sci ilorazowych. Dwa przyk lady nale_za, do rodziny reprezentacji rozk ladalnych a jeden do rodziny reprezentacji nieprzywiedlnych, kt´oresa, istotnie trudniejsze do przeanal- izowania. Drugi spos´objest oparty o twierdzenie charakteryzujace, pier´scienie Coxa w terminach geometrycznej teorii niezmiennik´ow(GIT). Przedstawiamy jego przyk ladowe zastosowanie, przeprowadzajac, alternatywny dow´odw przypadku grup dihedralnych. Podajemy r´ownie_zog´olna, metode, ograniczania stopni generator´owpier´scienia Coxa przy u_zyciu twierdzenia Kawamaty-Viehwega o znikaniu i wielogradowanej wersji regularno´sci Castelnuovo-Mumforda. Stosujemy te, metode, do zbadania trzech przyk lad´owsymplekty- cznych osobliwo´sci ilorazowych w wymiarze cztery. Stosujemy tutaj inna, technike, do rozstrzygniecia,, czy skonstruowana podalgebra jest r´owna ca lemu pier´scieniowi Coxa, opierajac, sie, na dzia laniu algebraicznego torusa na rozwiazaniu., To dzia lanie pozwala nam, dzieki, twierdzeniu Lefschetza-Riemanna-Rocha, obliczy´cwa_zna, cze´s´cfunkcji, Hilberta pier´s- cienia Coxa. Spodziewamy sie,, _zeprzedstawione w tej cze´sci, pracy metody zwiazane, z dzia laniem torusa moga, zosta´cuog´olnione i wykorzystane do badania innych przyk lad´ow. S lowa kluczowe: dzia lanie grupy sko´nczonej, osobliwo´s´cilorazowa, rozwiazanie, osobli- wo´sci, rozwiazanie, krepantne, rozwiazanie, symplektyczne, pier´scie´nCoxa, dzia lanie torusa algebraicznego Klasyfikacja AMS MSC 2010: 14E15, 14E30, 14E16, 14L30, 14L24, 14C35, 14C40, 14C20, 14Q15 ACKNOWLEDGEMENTS 5 Acknowledgements I would like to thank both my advisors { Jaros law Wi´sniewski and Maria Donten-Bury, for helping me at this beginning of work in mathematics. For sharing many valuable ideas in countless discussions, nurturing growth of my independent point of view on the research problems. For the uncountable amount of their time and support that I got from them during my studies. And for all the patience, especially Maria, who is my coauthor. I am grateful to professors Jaros law Buczy´nski, Mariusz Koras, Adrian Langer, Andrzej Weber for the opportunity to learn from them during courses they gave and seminars they led. Many thanks to my senior colleagues Piotr Achinger, Agnieszka Bodzenta-Skibi´nska, Weronika Buczy´nska, Robin Guilbot, Grzegorz Kapustka, Oskar Kedzierski,, Karol Palka, Piotr Pokora for inspiring conversations on mathematics and research, and for many oc- casions to learn from them while I was preparing seminar talks and attending conferences and seminars they organized. In the course of my studies I have an opportunity to meet and discuss with my colleagues and friends Adam Czapli´nski, Maciek Ga lazka,, Joachim Jelisiejew, Tomek Pe lka, Kuba Pawlikowski, Eleonora Romano,Lukasz Sienkiewicz, Robert Smiech,´ Kuba Witaszek, Ma- ciek Zdanowicz, Magda Zielenkiewicz { big thanks to each one of you. I am also grateful to people who introduced me to the world of algebraic geometry and commutative algebra: Kuba Byszewski, S lawomir Rams, Kamil Rusek and, especially, S lawomir Cynk, who supervised my master thesis. I also owe the big thank you to all the people outside my research circles who were sup- porting me during the period of my PhD studies. Colleagues from the academia doing work in other fields, my considerate flatmates, caring friends, people with whom I spent a lot of time talking, playing go, climbing, improvising and doing other fantastic
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