From Discrete Algorithms to Real Algebraic Geometry
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Simple Infinite Presentations for the Mapping Class Group of a Compact
SIMPLE INFINITE PRESENTATIONS FOR THE MAPPING CLASS GROUP OF A COMPACT NON-ORIENTABLE SURFACE RYOMA KOBAYASHI Abstract. Omori and the author [6] have given an infinite presentation for the mapping class group of a compact non-orientable surface. In this paper, we give more simple infinite presentations for this group. 1. Introduction For g ≥ 1 and n ≥ 0, we denote by Ng,n the closure of a surface obtained by removing disjoint n disks from a connected sum of g real projective planes, and call this surface a compact non-orientable surface of genus g with n boundary components. We can regard Ng,n as a surface obtained by attaching g M¨obius bands to g boundary components of a sphere with g + n boundary components, as shown in Figure 1. We call these attached M¨obius bands crosscaps. Figure 1. A model of a non-orientable surface Ng,n. The mapping class group M(Ng,n) of Ng,n is defined as the group consisting of isotopy classes of all diffeomorphisms of Ng,n which fix the boundary point- wise. M(N1,0) and M(N1,1) are trivial (see [2]). Finite presentations for M(N2,0), M(N2,1), M(N3,0) and M(N4,0) ware given by [9], [1], [14] and [16] respectively. Paris-Szepietowski [13] gave a finite presentation of M(Ng,n) with Dehn twists and arXiv:2009.02843v1 [math.GT] 7 Sep 2020 crosscap transpositions for g + n > 3 with n ≤ 1. Stukow [15] gave another finite presentation of M(Ng,n) with Dehn twists and one crosscap slide for g + n > 3 with n ≤ 1, applying Tietze transformations for the presentation of M(Ng,n) given in [13]. -
Algebraic Curves and Surfaces
Notes for Curves and Surfaces Instructor: Robert Freidman Henry Liu April 25, 2017 Abstract These are my live-texed notes for the Spring 2017 offering of MATH GR8293 Algebraic Curves & Surfaces . Let me know when you find errors or typos. I'm sure there are plenty. 1 Curves on a surface 1 1.1 Topological invariants . 1 1.2 Holomorphic invariants . 2 1.3 Divisors . 3 1.4 Algebraic intersection theory . 4 1.5 Arithmetic genus . 6 1.6 Riemann{Roch formula . 7 1.7 Hodge index theorem . 7 1.8 Ample and nef divisors . 8 1.9 Ample cone and its closure . 11 1.10 Closure of the ample cone . 13 1.11 Div and Num as functors . 15 2 Birational geometry 17 2.1 Blowing up and down . 17 2.2 Numerical invariants of X~ ...................................... 18 2.3 Embedded resolutions for curves on a surface . 19 2.4 Minimal models of surfaces . 23 2.5 More general contractions . 24 2.6 Rational singularities . 26 2.7 Fundamental cycles . 28 2.8 Surface singularities . 31 2.9 Gorenstein condition for normal surface singularities . 33 3 Examples of surfaces 36 3.1 Rational ruled surfaces . 36 3.2 More general ruled surfaces . 39 3.3 Numerical invariants . 41 3.4 The invariant e(V ).......................................... 42 3.5 Ample and nef cones . 44 3.6 del Pezzo surfaces . 44 3.7 Lines on a cubic and del Pezzos . 47 3.8 Characterization of del Pezzo surfaces . 50 3.9 K3 surfaces . 51 3.10 Period map . 54 a 3.11 Elliptic surfaces . -
Recognizing Surfaces
RECOGNIZING SURFACES Ivo Nikolov and Alexandru I. Suciu Mathematics Department College of Arts and Sciences Northeastern University Abstract The subject of this poster is the interplay between the topology and the combinatorics of surfaces. The main problem of Topology is to classify spaces up to continuous deformations, known as homeomorphisms. Under certain conditions, topological invariants that capture qualitative and quantitative properties of spaces lead to the enumeration of homeomorphism types. Surfaces are some of the simplest, yet most interesting topological objects. The poster focuses on the main topological invariants of two-dimensional manifolds—orientability, number of boundary components, genus, and Euler characteristic—and how these invariants solve the classification problem for compact surfaces. The poster introduces a Java applet that was written in Fall, 1998 as a class project for a Topology I course. It implements an algorithm that determines the homeomorphism type of a closed surface from a combinatorial description as a polygon with edges identified in pairs. The input for the applet is a string of integers, encoding the edge identifications. The output of the applet consists of three topological invariants that completely classify the resulting surface. Topology of Surfaces Topology is the abstraction of certain geometrical ideas, such as continuity and closeness. Roughly speaking, topol- ogy is the exploration of manifolds, and of the properties that remain invariant under continuous, invertible transforma- tions, known as homeomorphisms. The basic problem is to classify manifolds according to homeomorphism type. In higher dimensions, this is an impossible task, but, in low di- mensions, it can be done. Surfaces are some of the simplest, yet most interesting topological objects. -
Calculus Terminology
AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential -
THE DISJOINT ANNULUS PROPERTY 1. History And
THE DISJOINT ANNULUS PROPERTY SAUL SCHLEIMER Abstract. A Heegaard splitting of a closed, orientable three-manifold satis- ¯es the Disjoint Annulus Property if each handlebody contains an essential annulus and these are disjoint. This paper proves that, for a ¯xed three- manifold, all but ¯nitely many splittings have the disjoint annulus property. As a corollary, all but ¯nitely many splittings have distance three or less, as de¯ned by Hempel. 1. History and overview Great e®ort has been spent on the classi¯cation problem for Heegaard splittings of three-manifolds. Haken's lemma [2], that all splittings of a reducible manifold are themselves reducible, could be considered one of the ¯rst results in this direction. Weak reducibility was introduced by Casson and Gordon [1] as a generalization of reducibility. They concluded that a weakly reducible splitting is either itself reducible or the manifold in question contains an incompressible surface. Thomp- son [16] later de¯ned the disjoint curve property as a further generalization of weak reducibility. She deduced that all splittings of a toroidal three-manifold have the disjoint curve property. Hempel [5] generalized these ideas to obtain the distance of a splitting, de¯ned in terms of the curve complex. He then adapted an argument of Kobayashi [9] to produce examples of splittings of arbitrarily large distance. Hartshorn [3], also following the ideas of [9], proved that Hempel's distance is bounded by twice the genus of any incompressible surface embedded in the given manifold. Here we introduce the twin annulus property (TAP) for Heegaard splittings as well as the weaker notion of the disjoint annulus property (DAP). -
K3 Surfaces and String Duality
RU-96-98 hep-th/9611137 November 1996 K3 Surfaces and String Duality Paul S. Aspinwall Dept. of Physics and Astronomy, Rutgers University, Piscataway, NJ 08855 ABSTRACT The primary purpose of these lecture notes is to explore the moduli space of type IIA, type IIB, and heterotic string compactified on a K3 surface. The main tool which is invoked is that of string duality. K3 surfaces provide a fascinating arena for string compactification as they are not trivial spaces but are sufficiently simple for one to be able to analyze most of their properties in detail. They also make an almost ubiquitous appearance in the common statements concerning string duality. We arXiv:hep-th/9611137v5 7 Sep 1999 review the necessary facts concerning the classical geometry of K3 surfaces that will be needed and then we review “old string theory” on K3 surfaces in terms of conformal field theory. The type IIA string, the type IIB string, the E E heterotic string, 8 × 8 and Spin(32)/Z2 heterotic string on a K3 surface are then each analyzed in turn. The discussion is biased in favour of purely geometric notions concerning the K3 surface itself. Contents 1 Introduction 2 2 Classical Geometry 4 2.1 Definition ..................................... 4 2.2 Holonomy ..................................... 7 2.3 Moduli space of complex structures . ..... 9 2.4 Einsteinmetrics................................. 12 2.5 AlgebraicK3Surfaces ............................. 15 2.6 Orbifoldsandblow-ups. .. 17 3 The World-Sheet Perspective 25 3.1 TheNonlinearSigmaModel . 25 3.2 TheTeichm¨ullerspace . .. 27 3.3 Thegeometricsymmetries . .. 29 3.4 Mirrorsymmetry ................................. 31 3.5 Conformalfieldtheoryonatorus . ... 33 4 Type II String Theory 37 4.1 Target space supergravity and compactification . -
Intersection Theory on Algebraic Stacks and Q-Varieties
Journal of Pure and Applied Algebra 34 (1984) 193-240 193 North-Holland INTERSECTION THEORY ON ALGEBRAIC STACKS AND Q-VARIETIES Henri GILLET Department of Mathematics, Princeton University, Princeton, NJ 08544, USA Communicated by E.M. Friedlander Received 5 February 1984 Revised 18 May 1984 Contents Introduction ................................................................ 193 1. Preliminaries ................................................................ 196 2. Algebraic groupoids .......................................................... 198 3. Algebraic stacks ............................................................. 207 4. Chow groups of stacks ....................................................... 211 5. Descent theorems for rational K-theory. ........................................ 2 16 6. Intersection theory on stacks .................................................. 219 7. K-theory of stacks ........................................................... 229 8. Chcrn classes and the Riemann-Roth theorem for algebraic spaces ................ 233 9. comparison with Mumford’s product .......................................... 235 References .................................................................. 239 0. Introduction In his article 1161, Mumford constructed an intersection product on the Chow groups (with rational coefficients) of the moduli space . fg of stable curves of genus g over a field k of characteristic zero. Having such a product is important in study- ing the enumerative geometry of curves, and it is reasonable -
Physics 7D Makeup Midterm Solutions
Physics 7D Makeup Midterm Solutions Problem 1 A hollow, conducting sphere with an outer radius of 0:253 m and an inner radius of 0:194 m has a uniform surface charge density of 6:96 × 10−6 C/m2. A charge of −0:510 µC is now introduced into the cavity inside the sphere. (a) What is the new charge density on the outside of the sphere? (b) Calculate the strength of the electric field just outside the sphere. (c) What is the electric flux through a spherical surface just inside the inner surface of the sphere? Solution (a) Because there is no electric field inside a conductor the sphere must acquire a charge of +0:510 µC on its inner surface. Because there is no net flow of charge onto or off of the conductor this charge must come from the outer surface, so that the total charge of the outer 2 surface is now Qi − 0:510µC = σ(4πrout) − 0:510µC = 5:09µC. Now we divide again by the surface area to obtain the new charge density. 5:09 × 10−6C σ = = 6:32 × 10−6C/m2 (1) new 4π(0:253m)2 (b) The electric field just outside the sphere can be determined by Gauss' law once we determine the total charge on the sphere. We've already accounted for both the initial surface charge and the newly introduced point charge in our expression above, which tells us that −6 Qenc = 5:09 × 10 C. Gauss' law then tells us that Qenc 5 E = 2 = 7:15 × 10 N/C (2) 4π0rout (c) Finally we want to find the electric flux through a spherical surface just inside the sphere's cavity. -
Intersection Theory
William Fulton Intersection Theory Second Edition Springer Contents Introduction. Chapter 1. Rational Equivalence 6 Summary 6 1.1 Notation and Conventions 6 .2 Orders of Zeros and Poles 8 .3 Cycles and Rational Equivalence 10 .4 Push-forward of Cycles 11 .5 Cycles of Subschemes 15 .6 Alternate Definition of Rational Equivalence 15 .7 Flat Pull-back of Cycles 18 .8 An Exact Sequence 21 1.9 Affine Bundles 22 1.10 Exterior Products 24 Notes and References 25 Chapter 2. Divisors 28 Summary 28 2.1 Cartier Divisors and Weil Divisors 29 2.2 Line Bundles and Pseudo-divisors 31 2.3 Intersecting with Divisors • . 33 2.4 Commutativity of Intersection Classes 35 2.5 Chern Class of a Line Bundle 41 2.6 Gysin Map for Divisors 43 Notes and References 45 Chapter 3. Vector Bundles and Chern Classes 47 Summary 47 3.1 Segre Classes of Vector Bundles 47 3.2 Chern Classes . ' 50 • 3.3 Rational Equivalence on Bundles 64 Notes and References 68 Chapter 4. Cones and Segre Classes 70 Summary . 70 . 4.1 Segre Class of a Cone . 70 X Contents 4.2 Segre Class of a Subscheme 73 4.3 Multiplicity Along a Subvariety 79 4.4 Linear Systems 82 Notes and References 85 Chapter 5. Deformation to the Normal Cone 86 Summary 86 5.1 The Deformation 86 5.2 Specialization to the Normal Cone 89 Notes and References 90 Chapter 6. Intersection Products 92 Summary 92 6.1 The Basic Construction 93 6.2 Refined Gysin Homomorphisms 97 6.3 Excess Intersection Formula 102 6.4 Commutativity 106 6.5 Functoriality 108 6.6 Local Complete Intersection Morphisms 112 6.7 Monoidal Transforms 114 Notes and References 117 Chapter 7. -
'Cycle Groups for Artin Stacks'
Cycle groups for Artin stacks Andrew Kresch1 28 October 1998 Contents 1 Introduction 2 2 Definition and first properties 3 2.1 Thehomologyfunctor .......................... 3 2.2 BasicoperationsonChowgroups . 8 2.3 Results on sheaves and vector bundles . 10 2.4 Theexcisionsequence .......................... 11 3 Equivalence on bundles 12 3.1 Trivialbundles .............................. 12 3.2 ThetopChernclassoperation . 14 3.3 Collapsing cycles to the zero section . ... 15 3.4 Intersecting cycles with the zero section . ..... 17 3.5 Affinebundles............................... 19 3.6 Segre and Chern classes and the projective bundle theorem...... 20 4 Elementary intersection theory 22 4.1 Fulton-MacPherson construction for local immersions . ........ 22 arXiv:math/9810166v1 [math.AG] 28 Oct 1998 4.2 Exteriorproduct ............................. 24 4.3 Intersections on Deligne-Mumford stacks . .... 24 4.4 Boundedness by dimension . 25 4.5 Stratificationsbyquotientstacks . .. 26 5 Extended excision axiom 29 5.1 AfirsthigherChowtheory. 29 5.2 The connecting homomorphism . 34 5.3 Homotopy invariance for vector bundle stacks . .... 36 1Funded by a Fannie and John Hertz Foundation Fellowship for Graduate Study and an Alfred P. Sloan Foundation Dissertation Fellowship 1 6 Intersection theory 37 6.1 IntersectionsonArtinstacks . 37 6.2 Virtualfundamentalclass . 39 6.3 Localizationformula ........................... 39 1 Introduction We define a Chow homology functor A∗ for Artin stacks and prove that it satisfies some of the basic properties expected from intersection theory. Consequences in- clude an integer-valued intersection product on smooth Deligne-Mumford stacks, an affirmative answer to the conjecture that any smooth stack with finite but possibly nonreduced point stabilizers should possess an intersection product (this provides a positive answer to Conjecture 6.6 of [V2]), and more generally an intersection prod- uct (also integer-valued) on smooth Artin stacks which admit stratifications by global quotient stacks. -
UC Berkeley UC Berkeley Electronic Theses and Dissertations
UC Berkeley UC Berkeley Electronic Theses and Dissertations Title The arithmetic Hodge-index theorem and rigidity of algebraic dynamical systems over function fields Permalink https://escholarship.org/uc/item/4xb7w3f8 Author Carney, Alexander Publication Date 2019 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California The arithmetic Hodge-index theorem and rigidity of algebraic dynamical systems over function fields by Alexander Carney Adissertationsubmittedinpartialsatisfactionofthe requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Associate Professor Xinyi Yuan, Chair Associate Professor Burkhard Militzer Associate Professor Sug Woo Shin Spring 2019 The arithmetic Hodge-index theorem and rigidity of algebraic dynamical systems over function fields Copyright 2019 by Alexander Carney 1 Abstract The arithmetic Hodge-index theorem and rigidity of algebraic dynamical systems over function fields by Alexander Carney Doctor of Philosophy in Mathematics University of California, Berkeley Associate Professor Xinyi Yuan, Chair In one of the fundamental results of Arakelov’s arithmetic intersection theory, Faltings and Hriljac (independently) proved the Hodge-index theorem for arithmetic surfaces by relating the intersection pairing to the negative of the Neron-Tate height pairing. More recently, Moriwaki and Yuan–Zhang generalized this to higher dimension. In this work, we extend these results to projective varieties over transcendence degree one function fields. The new challenge is dealing with non-constant but numerically trivial line bundles coming from the constant field via Chow’s K/k-image functor. As an application of the Hodge-index theorem to heights defined by intersections of adelic metrized line bundles, we also prove a rigidity theorem for the set height zero points of polarized algebraic dynamical systems over function fields. -
Intersection Theory
APPENDIX A Intersection Theory In this appendix we will outline the generalization of intersection theory and the Riemann-Roch theorem to nonsingular projective varieties of any dimension. To motivate the discussion, let us look at the case of curves and surfaces, and then see what needs to be generalized. For a divisor D on a curve X, leaving out the contribution of Serre duality, we can write the Riemann-Roch theorem (IV, 1.3) as x(.!Z'(D)) = deg D + 1 - g, where xis the Euler characteristic (III, Ex. 5.1). On a surface, we can write the Riemann-Roch theorem (V, 1.6) as 1 x(!l'(D)) = 2 D.(D - K) + 1 + Pa· In each case, on the left-hand side we have something involving cohomol ogy groups of the sheaf !l'(D), while on the right-hand side we have some numerical data involving the divisor D, the canonical divisor K, and some invariants of the variety X. Of course the ultimate aim of a Riemann-Roch type theorem is to compute the dimension of the linear system IDI or of lnDI for large n (II, Ex. 7.6). This is achieved by combining a formula for x(!l'(D)) with some vanishing theorems for Hi(X,!l'(D)) fori > 0, such as the theorems of Serre (III, 5.2) or Kodaira (III, 7.15). We will now generalize these results so as to give an expression for x(!l'(D)) on a nonsingular projective variety X of any dimension. And while we are at it, with no extra effort we get a formula for x(t&"), where @" is any coherent locally free sheaf.