DOCSLIB.ORG

Resolutions of Quotient Singularities and Their Cox Rings Phd Dissertation

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

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
Recommended publications
  • Homomorphism and Isomorphism Theorems Generalized from a Relational Perspective

    Homomorphism and Isomorphism Theorems Generalized from a Relational Perspective

    Homomorphism and Isomorphism Theorems Generalized from a Relational Perspective Gunther Schmidt Institute for Software Technology, Department of Computing Science Universit¨at der Bundeswehr M¨unchen, 85577 Neubiberg, Germany [email protected] Abstract. The homomorphism and isomorphism theorems traditionally taught to students in a group theory or linear algebra lecture are by no means theorems of group theory. They are for a long time seen as general concepts of universal algebra. This article goes even further and identifies them as relational properties which to study does not even require the concept of an algebra. In addition it is shown how the homomorphism and isomorphism theorems generalize to not necessarily algebraic and thus relational structures. Keywords: homomorphism theorem, isomorphism theorem, relation al- gebra, congruence, multi-covering. 1 Introduction Relation algebra has received increasing interest during the last years. Many areas have been reconsidered from the relational point of view, which often pro- vided additional insight. Here, the classical homomorphism and isomorphism theorems (see [1], e.g.) are reviewed from a relational perspective, thereby sim- plifying and slightly generalizing them. The paper is organized as follows. First we recall the notion of a heterogeneous relation algebra and some of the very basic rules governing work with relations. With these, function and equivalence properties may be formulated concisely. The relational concept of homomorphism is defined as well as the concept of a congruence which is related with the concept of a multi-covering, which have connections with topology, complex analysis, and with the equivalence problem for flow-chart programs. We deal with the relationship between mappings and equivalence relations.
  • PROJECTIVE LINEAR CONFIGURATIONS VIA NON-REDUCTIVE ACTIONS 3 Do Even for the Study of Low-Dimensional Smooth Projective Geometry

    PROJECTIVE LINEAR CONFIGURATIONS VIA NON-REDUCTIVE ACTIONS 3 Do Even for the Study of Low-Dimensional Smooth Projective Geometry

    PROJECTIVE LINEAR CONFIGURATIONS VIA NON-REDUCTIVE ACTIONS BRENT DORAN AND NOAH GIANSIRACUSA Abstract. We study the iterated blow-up X of projective space along an arbitrary collection of linear subspaces. By replacing the universal torsor with an A1-homotopy equivalent model, built from A1-fiber bundles not just algebraic line bundles, we construct an “algebraic uniformization”: X is a quotient of affine space by a solvable group action. This provides a clean dictionary, using a single coordinate system, between the algebra and geometry of hypersurfaces: effective divisors are characterized via toric and invariant-theoretic techniques. In particular, the Cox ring is an invariant subring of a Pic(X)-graded polynomial ring and it is an intersection of two explicit finitely generated rings. When all linear subspaces are points, this recovers a theorem of Mukai while also giving it a geometric proof and topological intuition. Consequently, it is algorithmic to describe Cox(X) up to any degree, and when Cox(X) is finitely generated it is algorithmic to verify finite generation and compute an explicit presentation. We consider in detail the special case of M 0,n. Here the algebraic uniformization is defined over Spec Z. It admits a natural modular interpretation, and it yields a precise sense in which M 0,n is “one non-linearizable Ga away” from being a toric variety. The Hu-Keel question becomes a special case of Hilbert’s 14th problem for Ga. 1. Introduction 1.1. The main theorem and its context. Much of late nineteenth-century geometry was pred- icated on the presence of a single global coordinate system endowed with a group action.
  • THE DERIVED CATEGORY of a GIT QUOTIENT Contents 1. Introduction 871 1.1. Author's Note 874 1.2. Notation 874 2. the Main Theor

    THE DERIVED CATEGORY of a GIT QUOTIENT Contents 1. Introduction 871 1.1. Author's Note 874 1.2. Notation 874 2. the Main Theor

    JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 28, Number 3, July 2015, Pages 871–912 S 0894-0347(2014)00815-8 Article electronically published on October 31, 2014 THE DERIVED CATEGORY OF A GIT QUOTIENT DANIEL HALPERN-LEISTNER Dedicated to Ernst Halpern, who inspired my scientific pursuits Contents 1. Introduction 871 1.1. Author’s note 874 1.2. Notation 874 2. The main theorem 875 2.1. Equivariant stratifications in GIT 875 2.2. Statement and proof of the main theorem 878 2.3. Explicit constructions of the splitting and integral kernels 882 3. Homological structures on the unstable strata 883 3.1. Quasicoherent sheaves on S 884 3.2. The cotangent complex and Property (L+) 891 3.3. Koszul systems and cohomology with supports 893 3.4. Quasicoherent sheaves with support on S, and the quantization theorem 894 b 3.5. Alternative characterizations of D (X)<w 896 3.6. Semiorthogonal decomposition of Db(X) 898 4. Derived equivalences and variation of GIT 901 4.1. General variation of the GIT quotient 903 5. Applications to complete intersections: Matrix factorizations and hyperk¨ahler reductions 906 5.1. A criterion for Property (L+) and nonabelian hyperk¨ahler reduction 906 5.2. Applications to derived categories of singularities and abelian hyperk¨ahler reductions, via Morita theory 909 References 911 1. Introduction We describe a relationship between the derived category of equivariant coherent sheaves on a smooth projective-over-affine variety, X, with a linearizable action of a reductive group, G, and the derived category of coherent sheaves on a GIT quotient, X//G, of that action.
  • Moduli Spaces and Invariant Theory

    Moduli Spaces and Invariant Theory

    MODULI SPACES AND INVARIANT THEORY JENIA TEVELEV CONTENTS §1. Syllabus 3 §1.1. Prerequisites 3 §1.2. Course description 3 §1.3. Course grading and expectations 4 §1.4. Tentative topics 4 §1.5. Textbooks 4 References 4 §2. Geometry of lines 5 §2.1. Grassmannian as a complex manifold. 5 §2.2. Moduli space or a parameter space? 7 §2.3. Stiefel coordinates. 8 §2.4. Complete system of (semi-)invariants. 8 §2.5. Plücker coordinates. 9 §2.6. First Fundamental Theorem 10 §2.7. Equations of the Grassmannian 11 §2.8. Homogeneous ideal 13 §2.9. Hilbert polynomial 15 §2.10. Enumerative geometry 17 §2.11. Transversality. 19 §2.12. Homework 1 21 §3. Fine moduli spaces 23 §3.1. Categories 23 §3.2. Functors 25 §3.3. Equivalence of Categories 26 §3.4. Representable Functors 28 §3.5. Natural Transformations 28 §3.6. Yoneda’s Lemma 29 §3.7. Grassmannian as a fine moduli space 31 §4. Algebraic curves and Riemann surfaces 37 §4.1. Elliptic and Abelian integrals 37 §4.2. Finitely generated fields of transcendence degree 1 38 §4.3. Analytic approach 40 §4.4. Genus and meromorphic forms 41 §4.5. Divisors and linear equivalence 42 §4.6. Branched covers and Riemann–Hurwitz formula 43 §4.7. Riemann–Roch formula 45 §4.8. Linear systems 45 §5. Moduli of elliptic curves 47 1 2 JENIA TEVELEV §5.1. Curves of genus 1. 47 §5.2. J-invariant 50 §5.3. Monstrous Moonshine 52 §5.4. Families of elliptic curves 53 §5.5. The j-line is a coarse moduli space 54 §5.6.
  • Mirror Symmetry for GIT Quotients and Their Subvarieties

    Mirror Symmetry for GIT Quotients and Their Subvarieties

    Mirror symmetry for GIT quotients and their subvarieties Elana Kalashnikov September 6, 2020 Abstract These notes are based on an invited mini-course delivered at the 2019 PIMS-Fields Summer School on Algebraic Geometry in High-Energy Physics at the University of Saskatchewan. They give an introduction to mirror constructions for Fano GIT quotients and their subvarieties, es- pecially as relates to the Fano classification program. They are aimed at beginning graduate students. We begin with an introduction to GIT, then construct toric varieties via GIT, outlining some basic properties that can be read off the GIT data. We describe how to produce a Laurent 2 polynomial mirror for a Fano toric complete intersection, and explain the proof in the case of P . We then describe conjectural mirror constructions for some non-Abelian GIT quotients. There are no original results in these notes. 1 Introduction The classification of Fano varieties up to deformation is a major open problem in mathematics. One motivation comes from the minimal model program, which seeks to study and classify varieties up to birational morphisms. Running the minimal model program on varieties breaks them up into fundamental building blocks. These building blocks come in three types, one of which is Fano varieties. The classification of Fano varieties would thus give insight into the broader structure of varieties. In this lecture series, we consider smooth Fano varieties over C. There is only one Fano variety in 1 dimension 1: P . This is quite easy to see, as the degree of a genus g curve is 2 2g.
  • Introduction to Abstract Algebra (Math 113)

    Introduction to Abstract Algebra (Math 113)

    Introduction to Abstract Algebra (Math 113) Alexander Paulin, with edits by David Corwin FOR FALL 2019 MATH 113 002 ONLY Contents 1 Introduction 4 1.1 What is Algebra? . 4 1.2 Sets . 6 1.3 Functions . 9 1.4 Equivalence Relations . 12 2 The Structure of + and × on Z 15 2.1 Basic Observations . 15 2.2 Factorization and the Fundamental Theorem of Arithmetic . 17 2.3 Congruences . 20 3 Groups 23 1 3.1 Basic Definitions . 23 3.1.1 Cayley Tables for Binary Operations and Groups . 28 3.2 Subgroups, Cosets and Lagrange's Theorem . 30 3.3 Generating Sets for Groups . 35 3.4 Permutation Groups and Finite Symmetric Groups . 40 3.4.1 Active vs. Passive Notation for Permutations . 40 3.4.2 The Symmetric Group Sym3 . 43 3.4.3 Symmetric Groups in General . 44 3.5 Group Actions . 52 3.5.1 The Orbit-Stabiliser Theorem . 55 3.5.2 Centralizers and Conjugacy Classes . 59 3.5.3 Sylow's Theorem . 66 3.6 Symmetry of Sets with Extra Structure . 68 3.7 Normal Subgroups and Isomorphism Theorems . 73 3.8 Direct Products and Direct Sums . 83 3.9 Finitely Generated Abelian Groups . 85 3.10 Finite Abelian Groups . 90 3.11 The Classification of Finite Groups (Proofs Omitted) . 95 4 Rings, Ideals, and Homomorphisms 100 2 4.1 Basic Definitions . 100 4.2 Ideals, Quotient Rings and the First Isomorphism Theorem for Rings . 105 4.3 Properties of Elements of Rings . 109 4.4 Polynomial Rings . 112 4.5 Ring Extensions . 115 4.6 Field of Fractions .
  • Impulse Response Sequences and Construction of Number Sequence Identities

    Impulse Response Sequences and Construction of Number Sequence Identities

    1 2 Journal of Integer Sequences, Vol. 16 (2013), 3 Article 13.8.2 47 6 23 11 Impulse Response Sequences and Construction of Number Sequence Identities Tian-Xiao He Department of Mathematics Illinois Wesleyan University Bloomington, IL 61702-2900 USA [email protected] Abstract In this paper, we investigate impulse response sequences over the integers by pre- senting their generating functions and expressions. We also establish some of the corre- sponding identities. In addition, we give the relationship between an impulse response sequence and all linear recurring sequences satisfying the same linear recurrence rela- tion, which can be used to transfer the identities among different sequences. Finally, we discuss some applications of impulse response sequences to the structure of Stirling numbers of the second kind, the Wythoff array, and the Boustrophedon transform. 1 Introduction Many number and polynomial sequences can be defined, characterized, evaluated, and clas- sified by linear recurrence relations with certain orders. A number sequence an is called a sequence of order r if it satisfies the linear recurrence relation of order r { } r an = pjan j, n r, (1) − ≥ j=1 X for some constants pj, j = 1, 2,...,r, pr = 0, and initial conditions aj (j = 0, 1,...,r 1). Linear recurrence relations with constant6 coefficients are important in subjects including− 1 combinatorics, pseudo-random number generation, circuit design, and cryptography, and they have been studied extensively. To construct an explicit formula of the general term of a number sequence of order r, one may use a generating function, a characteristic equation, or a matrix method (See Comtet [4], Niven, Zuckerman, and Montgomery [12], Strang [15], Wilf [16], etc.) Let Ar be the set of all linear recurring sequences defined by the homogeneous linear recurrence relation (1) with coefficient set E = p ,p ,...,p .
  • Moduli Spaces of Stable Maps to Projective Space Via

    Moduli Spaces of Stable Maps to Projective Space Via

    MODULI SPACE OF STABLE MAPS TO PROJECTIVE SPACE VIA GIT YOUNG-HOON KIEM AND HAN-BOM MOON n-1 Abstract. We compare the Kontsevich moduli space M0;0(P ; d) of stable d 2 n maps to projective space with the quasi-map space P(Sym (C )⊗C )==SL(2). Consider the birational map ¯ d 2 n n-1 : P(Sym (C ) ⊗ C )==SL(2) 99K M0;0(P ; d) which assigns to an n-tuple of degree d homogeneous polynomials f1; ··· ; fn 1 n-1 in two variables, the map f = (f1 : ··· : fn): P P . In this paper, for d = 3, we prove that ¯ is the composition of three blow-ups followed by two blow-downs. Furthermore, we identify the blow-up/down! centers explicitly n-1 in terms of the moduli spaces M0;0(P ; d) with d = 1; 2. In particular, n-1 M0;0(P ; 3) is the SL(2)-quotient of a smooth rational projective variety. n-1 2 2 The degree two case M0;0(P ; 2), which is the blow-up of P(Sym C ⊗ n n-1 C )==SL(2) along P , is worked out as a warm-up. 1. Introduction The space of smooth rational curves of given degree d in projective space admits various natural moduli theoretic compactifications via geometric invariant theory (GIT), stable maps, Hilbert scheme or Chow scheme. Since these compactifications give us important but different enumerative invariants, it is an interesting problem to compare the compactifications by a sequence of explicit blow-ups and -downs. We expect all the blow-up centers to be some natural moduli spaces (for lower degrees).
  • Good Moduli Spaces for Artin Stacks Tome 63, No 6 (2013), P

    Good Moduli Spaces for Artin Stacks Tome 63, No 6 (2013), P

    R AN IE N R A U L E O S F D T E U L T I ’ I T N S ANNALES DE L’INSTITUT FOURIER Jarod ALPER Good moduli spaces for Artin stacks Tome 63, no 6 (2013), p. 2349-2402. <http://aif.cedram.org/item?id=AIF_2013__63_6_2349_0> © Association des Annales de l’institut Fourier, 2013, tous droits réservés. L’accès aux articles de la revue « Annales de l’institut Fourier » (http://aif.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://aif.cedram.org/legal/). Toute re- production en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement per- sonnelle du copiste est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Ann. Inst. Fourier, Grenoble 63, 6 (2013) 2349-2402 GOOD MODULI SPACES FOR ARTIN STACKS by Jarod ALPER Abstract. — We develop the theory of associating moduli spaces with nice geometric properties to arbitrary Artin stacks generalizing Mumford’s geometric invariant theory and tame stacks. Résumé. — Nous développons une théorie qui associe des espaces de modules ayant de bonnes propriétés géométriques des champs d’Artin arbitraires, généra- lisant ainsi la théorie géométrique des invariants de Mumford et les « champs modérés ». 1. Introduction 1.1. Background David Mumford developed geometric invariant theory (GIT) ([30]) as a means to construct moduli spaces.
  • Arxiv:1203.6643V4 [Math.AG] 12 May 2014 Olwn Conditions: Following Group, Hog GT U Ae Sfcsdo Hs Htw N Spleasant Perspect Is Let New find a We Categories

    Arxiv:1203.6643V4 [Math.AG] 12 May 2014 Olwn Conditions: Following Group, Hog GT U Ae Sfcsdo Hs Htw N Spleasant Perspect Is Let New find a We Categories

    VARIATION OF GEOMETRIC INVARIANT THEORY QUOTIENTS AND DERIVED CATEGORIES MATTHEW BALLARD, DAVID FAVERO, AND LUDMIL KATZARKOV Abstract. We study the relationship between derived categories of factorizations on gauged Landau-Ginzburg models related by variations of the linearization in Geometric Invariant Theory. Under assumptions on the variation, we show the derived categories are compara- ble by semi-orthogonal decompositions and describe the complementary components. We also verify a question posed by Kawamata: we show that D-equivalence and K-equivalence coincide for such variations. The results are applied to obtain a simple inductive description of derived categories of coherent sheaves on projective toric Deligne-Mumford stacks. This recovers Kawamata’s theorem that all projective toric Deligne-Mumford stacks have full exceptional collections. Using similar methods, we prove that the Hassett moduli spaces of stable symmetrically-weighted rational curves also possess full exceptional collections. As a final application, we show how our results recover Orlov’s σ-model/Landau-Ginzburg model correspondence. 1. Introduction Geometric Invariant Theory (GIT) and birational geometry are closely tied. The lack of a canonical choice of linearization of the action, which may once have been viewed as a bug in the theory, has now been firmly established as a marvelous feature for constructing new birational models of a GIT quotient. As studied by M. Brion-C. Procesi [BP90], M. Thaddeus [Tha96], I. Dolgachev-Y. Hu [DH98], and others, changing the linearization of the action leads to birational transformations between the different GIT quotients (VGIT). Conversely, any birational map between smooth and projective varieties can be obtained through such GIT variations [W lo00, HK99].
  • Introduction to Algebraic Theory of Linear Systems of Differential

    Introduction to Algebraic Theory of Linear Systems of Differential

    Introduction to algebraic theory of linear systems of differential equations Claude Sabbah Introduction In this set of notes we shall study properties of linear differential equations of a complex variable with coefficients in either the ring convergent (or formal) power series or in the polynomial ring. The point of view that we have chosen is intentionally algebraic. This explains why we shall consider the D-module associated with such equations. In order to solve an equation (or a system) we shall begin by finding a “normal form” for the corresponding D-module, that is by expressing it as a direct sum of elementary D-modules. It is then easy to solve the corresponding equation (in a similar way, one can solve a system of linear equations on a vector space by first putting the matrix in a simple form, e.g. triangular form or (better) Jordan canonical form). This lecture is supposed to be an introduction to D-module theory. That is why we have tried to set out this classical subject in a way that can be generalized in the case of many variables. For instance we have introduced the notion of holonomy (which is not difficult in dimension one), we have emphazised the connection between D-modules and meromorphic connections. Because the notion of a normal form does not extend easily in higher dimension, we have also stressed upon the notions of nearby and vanishing cycles. The first chapter consists of a local algebraic study of D-modules. The coefficient ring is either C{x} (convergent power series) or C[[x]] (formal power series).
  • The Derived Category of a GIT Quotient

    The Derived Category of a GIT Quotient

    Geometric invariant theory Derived categories Derived Kirwan Surjectivity The derived category of a GIT quotient Daniel Halpern-Leistner September 28, 2012 Daniel Halpern-Leistner The derived category of a GIT quotient Geometric invariant theory Derived categories Derived Kirwan Surjectivity Table of contents 1 Geometric invariant theory 2 Derived categories 3 Derived Kirwan Surjectivity Daniel Halpern-Leistner The derived category of a GIT quotient Geometric invariant theory Derived categories Derived Kirwan Surjectivity What is geometric invariant theory (GIT)? Let a reductive group G act on a smooth quasiprojective (preferably projective-over-affine) variety X . Problem Often X =G does not have a well-behaved quotient: e.q. CN =C∗. Grothendieck's solution: Consider the stack X =G, i.e. the equivariant geometry of X . Mumford's solution: the Hilbert-Mumford numerical criterion identifies unstable points in X , along with one parameter subgroups λ which destabilize these points X ss = X − funstable pointsg, and (hopefully) X ss =G has a well-behaved quotient Daniel Halpern-Leistner The derived category of a GIT quotient Geometric invariant theory Derived categories Derived Kirwan Surjectivity Example: Grassmannian The Grassmannian G(2; N) is a GIT quotient of V = Hom(C2; CN ) by GL2. The unstable locus breaks into strata Si . Maximal Destabilizer Fixed Locus Unstable (one param. subgroup) (column vectors) Stratum t 0 λ = [0; 0] S = fthe 0 matrixg 0 0 t 0 t 0 λ = [0; ∗], ∗= 6 0 S = frank 1 matricesg 1 0 1 1 λi chosen to maximize the numerical invariant hλi ; deti=jλi j. Daniel Halpern-Leistner The derived category of a GIT quotient The topology of GIT quotients Theorem (M.