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Arxiv:0907.0298V3 [Math.AG] 9 Jul 2010 E Oefrmn Rtmtcadgoerccnieain.Wieth While Considerations
ELLIPTIC SURFACES MATTHIAS SCHUTT¨ AND TETSUJI SHIODA Abstract. This survey paper concerns elliptic surfaces with section. We give a detailed overview of the theory including many examples. Emphasis is placed on rational elliptic surfaces and elliptic K3 surfaces. To this end, we particularly review the theory of Mordell-Weil lattices and address arithmetic questions. Contents 1. Introduction 1 2. Elliptic curves 3 3. Elliptic surfaces 7 4. Singular fibres 10 5. Base change and quadratic twisting 17 6. Mordell-Weil group and N´eron-Severi lattice 22 7. Torsion sections 32 8. Rational elliptic surfaces 38 9. Extremal rational elliptic surfaces 42 10. Semi-stable elliptic surfaces 45 11. Mordell-Weil lattice 48 12. Elliptic K3 surfaces 58 13. Arithmetic of elliptic K3 surfaces 72 14. Ranks of elliptic curves 84 References 88 arXiv:0907.0298v3 [math.AG] 9 Jul 2010 1. Introduction Elliptic surfaces are ubiquitous in the theory of algebraic surfaces. They play a key role for many arithmetic and geometric considerations. While this feature Date: February 22, 2010. 2000 Mathematics Subject Classification. Primary 14J27; Secondary 06B05, 11G05, 11G07, 11G50, 14J20, 14J26, 14J28. Key words and phrases. Elliptic surface, elliptic curve, N´eron-Severi group, singular fibre, Mordell-Weil group, Mordell-Weil lattice, Tate algorithm, K3 surface. Partial funding from DFG under grant Schu 2266/2-2 and JSPS under Grant-in-Aid for Scientific Research (C) No. 20540051 is gratefully acknowledged. 1 2 MATTHIAS SCHUTT¨ AND TETSUJI SHIODA has become ever more clear during the last two decades, extensive survey papers and monographs seem to date back exclusively to the 80’s and early 90’s. -
TASI Lectures on F-Theory
CERN-TH-2018-126 TASI Lectures on F-theory Timo Weigand1,2 1CERN, Theory Division, CH-1211 Geneva 23, Switzerland 2Institut f¨ur Theoretische Physik, Ruprecht-Karls-Universit¨at, Philosophenweg 19, 69120, Heidelberg, Germany Abstract F-theory is perhaps the most general currently available approach to study non-perturbative string compactifications in their geometric, large radius regime. It opens up a wide and ever- growing range of applications and connections to string model building, quantum gravity, (non- perturbative) quantum field theories in various dimensions and mathematics. Its computational power derives from the geometrisation of physical reasoning, establishing a deep correspondence between fundamental concepts in gauge theory and beautiful structures of elliptic fibrations. These lecture notes, which are an extended version of my lectures given at TASI 2017, introduce some of the main concepts underlying the recent technical advances in F-theory compactifications and their various applications. The main focus is put on explaining the F-theory dictionary between the local and global data of an elliptic fibration and the physics of 7-branes in Type IIB compactifications to arXiv:1806.01854v1 [hep-th] 5 Jun 2018 various dimensions via duality with M-theory. The geometric concepts underlying this dictionary include the behaviour of elliptic fibrations in codimension one, two, three and four, the Mordell-Weil group of rational sections, and the Deligne cohomology group specifying gauge backgrounds. Contents 1 Introduction 1 2 Setting the stage for F-theory 3 2.1 Type IIB string theory, SL(2, Z) duality and [p,q]7-branes ............... 3 2.2 Elliptic curves, SL(2, Z)bundleandellipticfibrations . -
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. -
MINIMAL MODEL THEORY of NUMERICAL KODAIRA DIMENSION ZERO Contents 1. Introduction 1 2. Preliminaries 3 3. Log Minimal Model Prog
MINIMAL MODEL THEORY OF NUMERICAL KODAIRA DIMENSION ZERO YOSHINORI GONGYO Abstract. We prove the existence of good minimal models of numerical Kodaira dimension 0. Contents 1. Introduction 1 2. Preliminaries 3 3. Log minimal model program with scaling 7 4. Zariski decomposition in the sense of Nakayama 9 5. Existence of minimal model in the case where º = 0 10 6. Abundance theorem in the case where º = 0 12 References 14 1. Introduction Throughout this paper, we work over C, the complex number ¯eld. We will make use of the standard notation and de¯nitions as in [KM] and [KMM]. The minimal model conjecture for smooth varieties is the following: Conjecture 1.1 (Minimal model conjecture). Let X be a smooth pro- jective variety. Then there exists a minimal model or a Mori ¯ber space of X. This conjecture is true in dimension 3 and 4 by Kawamata, Koll¶ar, Mori, Shokurov and Reid (cf. [KMM], [KM] and [Sho2]). In the case where KX is numerically equivalent to some e®ective divisor in dimen- sion 5, this conjecture is proved by Birkar (cf. [Bi1]). When X is of general type or KX is not pseudo-e®ective, Birkar, Cascini, Hacon Date: 2010/9/21, version 4.03. 2000 Mathematics Subject Classi¯cation. 14E30. Key words and phrases. minimal model, the abundance conjecture, numerical Kodaira dimension. 1 2 YOSHINORI GONGYO and McKernan prove Conjecture 1.1 for arbitrary dimension ([BCHM]). Moreover if X has maximal Albanese dimension, Conjecture 1.1 is true by [F2]. In this paper, among other things, we show Conjecture 1.1 in the case where º(KX ) = 0 (for the de¯nition of º, see De¯nition 2.6): Theorem 1.2. -
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. -
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 . -
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. -
Stacky Curves in Characteristic P
Stacky Curves in Characteristic p Andrew J. Kobin [email protected] Connecticut Summer School in Number Theory June 12, 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 (preprint available!) 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. One important class of examples can be viewed as 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. One important class of examples can be viewed as smooth varieties or schemes with a finite automorphism group attached at each point. -
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. -
Arxiv:2105.04352V1 [Math.AG] 10 May 2021
A NON-ARCHIMEDEAN ANALOGUE OF CAMPANA’S NOTION OF SPECIALNESS JACKSON S. MORROW AND GIOVANNI ROSSO ABSTRACT. Let K be an algebraically closed, complete, non-Archimedean valued field of charac- teristic zero, and let X be a K-analytic space (in the sense of Huber). In this work, we pursue a non-Archimedean characterization of Campana’s notion of specialness. We say X is K-analytically special if there exists a connected, finite type algebraic group G/K, a dense open subset U ⊂ Gan with codim(Gan \ U) > 2, and an analytic morphism U → X which is Zariski dense. With this definition, we prove several results which illustrate that this definition correctly captures Campana’s notion of specialness in the non-Archimedean setting. These results inspire us to make non-Archimedean counterparts to conjectures of Campana. As preparation for our proofs, we prove auxiliary results concerning the indeterminacy locus of a meromorphic mapping between K-analytic spaces, the notion of pseudo-K-analytically Brody hyperbolic, and extensions of meromorphic maps from smooth, irreducible K-analytic spaces to the analytification of a semi-abelian variety. 1. Introduction The study of rational points on varieties over number fields is one of the fundamental questions in arithmetic geometry. For curves, we have a good understanding of the behaviour of the rational points, and this behaviour is governed by the genus of the curve. If the curve has genus g 6 1, then the rational points on the curve become infinite after a finite extension of the base number field; in this case, we say the curve is arithmetically special. -
Signature Redacted Department of Physics May 3, 2018
Classification of base geometries in F-theory by Yinan Wang B. S., Peking University (2013) Submitted to the Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2018 Massachusetts Institute of Technology 2018. All rights reserved. A uth or ............................... Signature redacted Department of Physics May 3, 2018 Signature redacted Certified by.........................Signature red ated *ashington Taylor Professor Thesis Supervisor Signature redacted Accepted by ..................... MASSACHUSETTS INSTITUTE Scott A. Hughes Interim ssociate Department Head JUN 2 5 2018 LIBRARIES ARCHIVES 2 Classification of base geometries in F-theory by Yinan Wang Submitted to the Department of Physics on May 3, 2018, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics Abstract F-theory is a powerful geometric framework to describe strongly coupled type JIB supcrstring theory. After we compactify F-theory on elliptically fibercd Calabi-Yau manifolds of various dimensions, we produce a large number of minimal supergravity models in six or four spacetime dimensions. In this thesis, I will describe a current classification program of these elliptic Calabi-Yau manifolds. Specifically, I will be focusing on the part of classifying complex base manifolds of these elliptic fibrations. Besides the usual algebraic geometric description of these base manifolds, F-theory provides a physical language to characterize them as well. One of the most important physical feature of the bases is called the "non-Higgsable gauge groups", which is the minimal gauge group in the low energy supergravity model for any elliptic fibration on a specific base. -
[Math.AG] 26 Jan 2003
NEF DIMENSION OF MINIMAL MODELS FLORIN AMBRO Abstract. We reduce the Abundance Conjecture in dimension 4 to the following numerical statement: if the canonical divisor K is nef and has maximal nef dimension, then K is big. From this point of view, we “classify” in dimension 2 nef divisors which have maximal nef dimension, but which are not big. 0. Introduction A minimal model is a complex projective variety X with at most ter- minal singularities, whose canonical divisor K is numerically effective (nef): K · C ≥ 0 for every curve C ⊂ X. Up to dimension three, min- imal models have a geometrical characterization (Kawamata [9, 12], Miyaoka [14, 15, 16]) : Abundance Conjecture . [11] Let X be a minimal model. Then the linear system |kK| is base point free, for some positive integer k. In dimension four, it is enough to show that X has positive Kodaira dimension if K is not numerically trivial (Kawamata [9], Mori [17]). A direct approach is to first construct the morphism associated to the expected base point free pluricanonical linear systems: 0 f : X → Proj(⊕k≥0H (X,kK)). Since K is nef, f is the unique morphism with connected fibers which contracts exactly the curves C ⊂ X with K · C = 0. Tsuji [22] and Bauer et al [2] have recently solved this existence problem birationally: arXiv:math/0301305v1 [math.AG] 26 Jan 2003 for any nef divisor D on X, there exists a rational dominant map f : X− → Y such that f is regular over the generic point of Y and a very general curve C is contracted by f if and only if D · C = 0.