Stacky Curves in Characteristic P
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Stack Completions and Morita Equivalence for Categories in a Topos Cahiers De Topologie Et Géométrie Différentielle Catégoriques, Tome 20, No 4 (1979), P
CAHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES MARTA BUNGE Stack completions and Morita equivalence for categories in a topos Cahiers de topologie et géométrie différentielle catégoriques, tome 20, no 4 (1979), p. 401-436 <http://www.numdam.org/item?id=CTGDC_1979__20_4_401_0> © Andrée C. Ehresmann et les auteurs, 1979, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ CAHIERS DE TOPOLOGIE Vol. XX-4 (1979) ET GEOMETRIE DIFFERENTIELLE STACK COMPLETIONS AND MORITA EQUIVALENCE FOR CATEGORIES IN A TOPOS by Marta BUNGE 1) 0. INTRODUCTION. The origin of this paper can be traced back to one of a series of lectures given by F.W. Lawvere [10, Lecture V]. In it, Lawvere dealt for the case of an arbitrary topos S , with the notion of stack, a notion which, for Grothendieck toposes, had been considered by J. Giraud [6], and which is given relative to a site. A topos S may always be regarded as a site with the regular epi- morphism topology, and the notion of stack over S is then defined with respect to this particular topology. Special as it may be, this notion of stack over a topos S plays an important role in the development of Category Theory over a base to- pos S . -
Vector Bundles As Generators on Schemes and Stacks
Vector Bundles as Generators on Schemes and Stacks Inaugural-Dissertation zur Erlangung des Doktorgrades der Mathematisch-Naturwissenschaftlichen Fakult¨at der Heinrich-Heine-Universit¨atD¨usseldorf vorgelegt von Philipp Gross aus D¨usseldorf D¨usseldorf,Mai 2010 Aus dem Mathematischen Institut der Heinrich-Heine-Universit¨atD¨usseldorf Gedruckt mit der Genehmigung der Mathematisch-Naturwissenschaftlichen Fakult¨atder Heinrich-Heine-Universit¨atD¨usseldorf Referent: Prof. Dr. Stefan Schr¨oer Koreferent: Prof. Dr. Holger Reich Acknowledgments The work on this dissertation has been one of the most significant academic challenges I have ever had to face. This study would not have been completed without the support, patience and guidance of the following people. It is to them that I owe my deepest gratitude. I am indebted to my advisor Stefan Schr¨oerfor his encouragement to pursue this project. He taught me algebraic geometry and how to write academic papers, made me a better mathematician, brought out the good ideas in me, and gave me the opportunity to attend many conferences and schools in Europe. I also thank Holger Reich, not only for agreeing to review the dissertation and to sit on my committee, but also for showing an interest in my research. Next, I thank the members of the local algebraic geometry research group for their time, energy and for the many inspiring discussions: Christian Liedtke, Sasa Novakovic, Holger Partsch and Felix Sch¨uller.I have had the pleasure of learning from them in many other ways as well. A special thanks goes to Holger for being a friend, helping me complete the writing of this dissertation as well as the challenging research that lies behind it. -
X → S Be a Proper Morphism of Locally Noetherian Schemes and Let F Be a Coherent Sheaf on X That Is flat Over S (E.G., F Is Smooth and F Is a Vector Bundle)
COHOMOLOGY AND BASE CHANGE FOR ALGEBRAIC STACKS JACK HALL Abstract. We prove that cohomology and base change holds for algebraic stacks, generalizing work of Brochard in the tame case. We also show that Hom-spaces on algebraic stacks are represented by abelian cones, generaliz- ing results of Grothendieck, Brochard, Olsson, Lieblich, and Roth{Starr. To accomplish all of this, we prove that a wide class of relative Ext-functors in algebraic geometry are coherent (in the sense of M. Auslander). Introduction Let f : X ! S be a proper morphism of locally noetherian schemes and let F be a coherent sheaf on X that is flat over S (e.g., f is smooth and F is a vector bundle). If s 2 S is a point, then define Xs to be the fiber of f over s. If s has residue field κ(s), then for each integer q there is a natural base change morphism of κ(s)-vector spaces q q q b (s):(R f∗F) ⊗OS κ(s) ! H (Xs; FXs ): Cohomology and Base Change originally appeared in [EGA, III.7.7.5] in a quite sophisticated form. Mumford [Mum70, xII.5] and Hartshorne [Har77, xIII.12], how- ever, were responsible for popularizing a version similar to the following. Let s 2 S and let q be an integer. (1) The following are equivalent. (a) The morphism bq(s) is surjective. (b) There exists an open neighbourhood U of s such that bq(u) is an iso- morphism for all u 2 U. (c) There exists an open neighbourhood U of s, a coherent OU -module Q, and an isomorphism of functors: Rq+1(f ) (F ⊗ f ∗ I) =∼ Hom (Q; I); U ∗ XU OXU U OU where fU : XU ! U is the pullback of f along U ⊆ S. -
Stacky Curves in Characteristic P
Stacky Curves in Characteristic p Andrew J. Kobin [email protected] Madison Moduli Weekend September 27, 2020 Introduction Root Stacks AS Root Stacks Classification Introduction Common problem: all sorts of information is lost when we consider quotient objects and/or singular objects. 2 Introduction Root Stacks AS Root Stacks Classification Introduction Solution: Keep track of lost information using orbifolds (topological and intuitive) or stacks (algebraic and fancy). 2 Introduction Root Stacks AS Root Stacks Classification Introduction Solution: Keep track of lost information using orbifolds (topological and intuitive) or stacks (algebraic and fancy). 2 Example: For the plane curve X : y2 − x = 0, stacks remember automorphisms like (x; y) $ (x; −y) using groupoids Introduction Root Stacks AS Root Stacks Classification Introduction Solution: Keep track of lost information using orbifolds (topological and intuitive) or stacks (algebraic and fancy). 2 Goal: Classify stacky curves (= orbifold curves) in char. p (First steps: “Artin–Schreier Root Stacks”, arXiv:1910.03146) Introduction Root Stacks AS Root Stacks Classification Complex Orbifolds Definition A complex orbifold is a topological space admitting an atlas fUig ∼ n where each Ui = C =Gi for a finite group Gi, satisfying compatibility conditions (think: manifold atlas but with extra info). Introduction Root Stacks AS Root Stacks Classification Algebraic Stacks There’s also a version of orbifold in algebraic geometry: an algebraic stack. Important class of examples we will focus on are Deligne–Mumford stacks ≈ smooth varieties or schemes with a finite automorphism group attached at each point. Introduction Root Stacks AS Root Stacks Classification Algebraic Stacks There’s also a version of orbifold in algebraic geometry: an algebraic stack. -
The Orbifold Chow Ring of Toric Deligne-Mumford Stacks
JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 18, Number 1, Pages 193–215 S 0894-0347(04)00471-0 Article electronically published on November 3, 2004 THE ORBIFOLD CHOW RING OF TORIC DELIGNE-MUMFORD STACKS LEVA.BORISOV,LINDACHEN,ANDGREGORYG.SMITH 1. Introduction The orbifold Chow ring of a Deligne-Mumford stack, defined by Abramovich, Graber and Vistoli [2], is the algebraic version of the orbifold cohomology ring in- troduced by W. Chen and Ruan [7], [8]. By design, this ring incorporates numerical invariants, such as the orbifold Euler characteristic and the orbifold Hodge num- bers, of the underlying variety. The product structure is induced by the degree zero part of the quantum product; in particular, it involves Gromov-Witten invariants. Inspired by string theory and results in Batyrev [3] and Yasuda [28], one expects that, in nice situations, the orbifold Chow ring coincides with the Chow ring of a resolution of singularities. Fantechi and G¨ottsche [14] and Uribe [25] verify this conjecture when the orbifold is Symn(S)whereS is a smooth projective surface n with KS = 0 and the resolution is Hilb (S). The initial motivation for this project was to compare the orbifold Chow ring of a simplicial toric variety with the Chow ring of a crepant resolution. To achieve this goal, we first develop the theory of toric Deligne-Mumford stacks. Modeled on simplicial toric varieties, a toric Deligne-Mumford stack corresponds to a combinatorial object called a stacky fan. As a first approximation, this object is a simplicial fan with a distinguished lattice point on each ray in the fan. -
Artin-Schreier Root Stacks
Artin–Schreier Root Stacks Andrew Kobin∗ Abstract We classify stacky curves in characteristic p > 0 with cyclic stabilizers of order p using higher ramification data. This approach replaces the local root stack structure of a tame stacky curve, similar to the local structure of a complex orbifold curve, with a more sensitive structure called an Artin–Schreier root stack, allowing us to incorporate this ramification data directly into the stack. As an application, we compute dimensions of Riemann–Roch spaces for some examples of stacky curves in positive characteristic and suggest a program for computing spaces of modular forms in this setting. 1 Introduction Over the complex numbers, Deligne–Mumford stacks with generically trivial stabilizer can be understood by studying their underlying complex orbifold structure. This approach leads one to the following classification of stacky curves: over C, or indeed any algebraically closed field of characteristic 0, a smooth stacky curve is uniquely determined up to isomorphism by its underlying complex curve and a finite list of numbers corresponding to the orders of the (always cyclic) stabilizer groups of the stacky points of the curve (cf. [GS]). Such a concise statement fails in positive characteristic since stabilizer groups may be nonabelian. Even in the case of stacky curves over an algebraically closed field k of charac- teristic p> 0, there exist infinitely many nonisomorphic stacky curves over k with the same underlying scheme and stacky point with stabilizer group abstractly isomorphic to Z/pZ. Thus any attempt at classifying stacky curves in characteristic p will require finer invariants arXiv:1910.03146v3 [math.AG] 14 Aug 2020 than the order of the stabilizer group. -
Moduli Stack of Oriented Formal Groups and Periodic Complex Bordism
MODULI STACK OF ORIENTED FORMAL GROUPS AND PERIODIC COMPLEX BORDISM ROK GREGORIC or Abstract. We study the non-connective spectral stack MFG, the moduli stack of ori- ented formal groups in the sense of Lurie. We show that its descent spectral sequence recovers the Adams-Novikov spectral sequence. For two E -forms of periodic complex bordism MP, the Thom spectrum and Snaith construction model, we describe the univer- ∞ or sal property of the cover Spec(MP) → MFG. We show that Quillen’s celebrated theorem on complex bordism is equivalent to the assertion that the underlying ordinary stack of or MFG is the classical stack of ordinary formal groups MFG. In order to carry out all of the above, we develop foundations of a functor of points approach♡ to non-connective spectral algebraic geometry. Introduction The goal of this paper is to show that the connection between the homotopy category of spectra and formal groups, which is at the heart of chromatic homotopy theory, may be interpreted in terms of spectral algebraic geometry. By formal group, we mean a smooth 1-dimensional commutative formal group, as is traditional in homotopy theory. These are classified by an algebro-geometric moduli stack, the moduli stack of formal groups, which ♡ we shall denote MFG. Here and in the rest of the paper, we will use the heart symbol ♡ in the subscript to emphasize the classical nature of algebro-geometric objects considered, in contrast to spectral algebro-geometric objects which will be our main focus. This notation is motivated by the fact that ordinary abelian groups sit inside spectra as the heart of the usual t- ♡ structure. -
Unbounded Quantifiers and Strong Axioms in Topos Theory
Unbounded quantifiers & strong axioms Michael Shulman The Question Two kinds of set theory Strong axioms Unbounded quantifiers and strong axioms Internalization The Answer in topos theory Stack semantics Logic in stack semantics Strong axioms Michael A. Shulman University of Chicago November 14, 2009 Unbounded quantifiers & strong axioms The motivating question Michael Shulman The Question Two kinds of set theory Strong axioms Internalization The Answer Stack semantics Logic in stack semantics What is the topos-theoretic counterpart of the Strong axioms strong set-theoretic axioms of Separation, Replacement, and Collection? Unbounded quantifiers & strong axioms Outline Michael Shulman The Question Two kinds of set theory Strong axioms 1 The Question Internalization The Answer Two kinds of set theory Stack semantics Logic in stack Strong axioms semantics Strong axioms Internalization 2 The Answer Stack semantics Logic in stack semantics Strong axioms Following Lawvere and others, I prefer to view this as an equivalence between two kinds of set theory. Unbounded quantifiers & strong axioms Basic fact Michael Shulman The Question Two kinds of set theory Strong axioms Internalization The Answer Stack semantics Elementary topos theory is equiconsistent with a Logic in stack semantics weak form of set theory. Strong axioms Unbounded quantifiers & strong axioms Basic fact Michael Shulman The Question Two kinds of set theory Strong axioms Internalization The Answer Stack semantics Elementary topos theory is equiconsistent with a Logic in stack semantics weak form of set theory. Strong axioms Following Lawvere and others, I prefer to view this as an equivalence between two kinds of set theory. Unbounded quantifiers & strong axioms “Membership-based” or Michael Shulman “material” set theory The Question Two kinds of set theory Strong axioms Data: Internalization The Answer • A collection of sets. -
Introduction to Algebraic Stacks
Introduction to Algebraic Stacks K. Behrend December 17, 2012 Abstract These are lecture notes based on a short course on stacks given at the Newton Institute in Cambridge in January 2011. They form a self-contained introduction to some of the basic ideas of stack theory. Contents Introduction 3 1 Topological stacks: Triangles 8 1.1 Families and their symmetry groupoids . 8 Various types of symmetry groupoids . 9 1.2 Continuous families . 12 Gluing families . 13 1.3 Classification . 15 Pulling back families . 17 The moduli map of a family . 18 Fine moduli spaces . 18 Coarse moduli spaces . 21 1.4 Scalene triangles . 22 1.5 Isosceles triangles . 23 1.6 Equilateral triangles . 25 1.7 Oriented triangles . 25 Non-equilateral oriented triangles . 30 1.8 Stacks . 31 1.9 Versal families . 33 Generalized moduli maps . 35 Reconstructing a family from its generalized moduli map . 37 Versal families: definition . 38 Isosceles triangles . 40 Equilateral triangles . 40 Oriented triangles . 41 1 1.10 Degenerate triangles . 42 Lengths of sides viewpoint . 43 Embedded viewpoint . 46 Complex viewpoint . 52 Oriented degenerate triangles . 54 1.11 Change of versal family . 57 Oriented triangles by projecting equilateral ones . 57 Comparison . 61 The comparison theorem . 61 1.12 Weierstrass compactification . 64 The j-plane . 69 2 Formalism 73 2.1 Objects in continuous families: Categories fibered in groupoids 73 Fibered products of groupoid fibrations . 77 2.2 Families characterized locally: Prestacks . 78 Versal families . 79 2.3 Families which can be glued: Stacks . 80 2.4 Topological stacks . 81 Topological groupoids . 81 Generalized moduli maps: Groupoid Torsors . -
Algebraic Stacks Many of the Geometric Properties That Are Defined for Schemes (Smoothness, Irreducibility, Separatedness, Properness, Etc...)
Algebraic stacks Tom´as L. G´omez Tata Institute of Fundamental Research Homi Bhabha road, Mumbai 400 005 (India) [email protected] 9 November 1999 Abstract This is an expository article on the theory of algebraic stacks. After intro- ducing the general theory, we concentrate in the example of the moduli stack of vector budles, giving a detailed comparison with the moduli scheme obtained via geometric invariant theory. 1 Introduction The concept of algebraic stack is a generalization of the concept of scheme, in the same sense that the concept of scheme is a generalization of the concept of projective variety. In many moduli problems, the functor that we want to study is not representable by a scheme. In other words, there is no fine moduli space. Usually this is because the objects that we want to parametrize have automorphisms. But if we enlarge the category of schemes (following ideas that go back to Grothendieck and Giraud, and were developed by Deligne, Mumford and Artin) and consider algebraic stacks, then we can construct the “moduli stack”, that captures all the information that we would like in a fine moduli space. The idea of enlarging the category of algebraic varieties to study moduli problems is not new. In fact A. Weil invented the concept of abstract variety to give an algebraic arXiv:math/9911199v1 [math.AG] 25 Nov 1999 construction of the Jacobian of a curve. These notes are an introduction to the theory of algebraic stacks. I have tried to emphasize ideas and concepts through examples instead of detailed proofs (I give references where these can be found). -
The Canonical Ring of a Stacky Curve John Voight David Zureick-Brown
The canonical ring of a stacky curve John Voight David Zureick-Brown Author address: Department of Mathematics, Dartmouth College, 6188 Kemeny Hall, Hanover, NH 03755, USA E-mail address: [email protected] Department of Mathematics and Computer Science, Emory Univer- sity, Atlanta, GA 30322 USA E-mail address: [email protected] Contents Chapter 1. Introduction 1 1.1. Motivation: Petri's theorem 1 1.2. Orbifold canonical rings 1 1.3. Rings of modular forms 2 1.4. Main result 3 1.5. Hassett-Keel program 4 1.6. Generalizations 5 1.7. Organization 5 1.8. Acknowledgements 6 Chapter 2. Canonical rings of curves 7 2.1. Setup 7 2.2. Terminology 8 2.3. Low genus 11 2.4. Pointed gin: High genus and nonhyperelliptic 12 2.5. Gin and pointed gin: Rational normal curve 15 2.6. Pointed gin: Hyperelliptic 16 2.7. Gin: Nonhyperelliptic and hyperelliptic 19 2.8. Summary 22 Chapter 3. A generalized Max Noether's theorem for curves 23 3.1. Max Noether's theorem in genus at most 1 23 3.2. Generalized Max Noether's theorem (GMNT) 24 3.3. Failure of surjectivity 25 3.4. GMNT: nonhyperelliptic curves 26 3.5. GMNT: hyperelliptic curves 27 Chapter 4. Canonical rings of classical log curves 29 4.1. Main result: classical log curves 29 4.2. Log curves: Genus 0 30 4.3. Log curves: Genus 1 30 4.4. Log degree 1: hyperelliptic 31 4.5. Log degree 1: nonhyperelliptic 33 4.6. Exceptional log cases 35 4.7. Log degree 2 36 4.8. -
Wild Ramification and Stacky Curves
Wild Ramification and Stacky Curves Andrew Kobin Cedarburg, Wisconsin Bachelor of Science, Wake Forest University, 2013 Master of Arts, Wake Forest University, 2015 A Dissertation Presented to the Graduate Faculty of the University of Virginia in Candidacy for the Degree of Doctor of Philosophy Department of Mathematics University of Virginia May, 2020 ©Copyright by Andrew Kobin 2020 All Rights Reserved Abstract The local structure of Deligne–Mumford stacks has been studied for decades, but most results require a tameness hypothesis that avoids certain phenom- ena in positive characteristic. We tackle this problem directly and classify stacky curves in characteristic p > 0 with cyclic stabilizers of order p using higher ramification data. Our approach replaces the local root stack struc- ture of a tame stacky curve, similar to the local structure of a complex orb- ifold curve, with a more sensitive structure called an Artin–Schreier root stack, allowing us to incorporate the ramification data directly into the stack. A com- plete classification of the local structure of stacky curves, and more generally Deligne–Mumford stacks, will require a broader understanding of root struc- tures, and we begin this program by introducing a higher-order version of the Artin–Schreier root stack. Finally, as an application, we compute dimensions of Riemann–Roch spaces for some examples of stacky curves in positive char- acteristic and suggest a program for computing spaces of modular forms using the theory of stacky modular curves. Acknowledgments I always knew this would be the last part of my thesis I would write – even in the final days of my journey, there are still people supporting me and helping me across the finish line.