A Surface of Maximal Canonical Degree 11
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
ON the CONSTRUCTION of NEW TOPOLOGICAL SPACES from EXISTING ONES Consider a Function F
ON THE CONSTRUCTION OF NEW TOPOLOGICAL SPACES FROM EXISTING ONES EMILY RIEHL Abstract. In this note, we introduce a guiding principle to define topologies for a wide variety of spaces built from existing topological spaces. The topolo- gies so-constructed will have a universal property taking one of two forms. If the topology is the coarsest so that a certain condition holds, we will give an elementary characterization of all continuous functions taking values in this new space. Alternatively, if the topology is the finest so that a certain condi- tion holds, we will characterize all continuous functions whose domain is the new space. Consider a function f : X ! Y between a pair of sets. If Y is a topological space, we could define a topology on X by asking that it is the coarsest topology so that f is continuous. (The finest topology making f continuous is the discrete topology.) Explicitly, a subbasis of open sets of X is given by the preimages of open sets of Y . With this definition, a function W ! X, where W is some other space, is continuous if and only if the composite function W ! Y is continuous. On the other hand, if X is assumed to be a topological space, we could define a topology on Y by asking that it is the finest topology so that f is continuous. (The coarsest topology making f continuous is the indiscrete topology.) Explicitly, a subset of Y is open if and only if its preimage in X is open. With this definition, a function Y ! Z, where Z is some other space, is continuous if and only if the composite function X ! Z is continuous. -
The Birational Isomorphism Types of Smooth Real Elliptic Curves
Rose-Hulman Institute of Technology Rose-Hulman Scholar Mathematical Sciences Technical Reports (MSTR) Mathematics 8-17-2004 The Birational Isomorphism Types of Smooth Real Elliptic Curves Sean A. Broughton Rose-Hulman Institute of Technology, [email protected] Follow this and additional works at: https://scholar.rose-hulman.edu/math_mstr Part of the Algebraic Geometry Commons, and the Geometry and Topology Commons Recommended Citation Broughton, Sean A., "The Birational Isomorphism Types of Smooth Real Elliptic Curves" (2004). Mathematical Sciences Technical Reports (MSTR). 44. https://scholar.rose-hulman.edu/math_mstr/44 This Article is brought to you for free and open access by the Mathematics at Rose-Hulman Scholar. It has been accepted for inclusion in Mathematical Sciences Technical Reports (MSTR) by an authorized administrator of Rose-Hulman Scholar. For more information, please contact [email protected]. The Birational Isomorphism Types of Smooth Real Elliptic Curves S.A. Broughton Mathematical Sciences Technical Report Series MSTR 04-05 August 17, 2004 Department of Mathematics Rose-Hulman Institute of Technology http://www.rose-hulman.edu/math Fax (812)-877-8333 Phone (812)-877-8193 The Birational Isomorphism Types of Smooth Real Elliptic Curves S. Allen Broughton Department of Mathtematics Rose-Hulman Institute of Technology August 17, 2004 Contents 1 Introduction 1 2Definitions and background information 2 3 Weierstrass form, double covers, and invariants 4 4 Reduction to a Weierstrass normal form 5 5 The real moduli space in complex moduli space 8 6 Real forms and real birational isomorphism 9 7 Real forms and symmetries of elliptic curves 12 8 The birational isomorphism types of real elliptic curves 13 1Introduction In this note we determine all birational isomorphism types of real elliptic curves and show that it is the same as the orbit space of smooth cubic real curves in P 2(R) under linear projective equivalence. -
Canonical Maps
Canonical maps Jean-Pierre Marquis∗ D´epartement de philosophie Universit´ede Montr´eal Montr´eal,Canada [email protected] Abstract Categorical foundations and set-theoretical foundations are sometimes presented as alternative foundational schemes. So far, the literature has mostly focused on the weaknesses of the categorical foundations. We want here to concentrate on what we take to be one of its strengths: the explicit identification of so-called canonical maps and their role in mathematics. Canonical maps play a central role in contemporary mathematics and although some are easily defined by set-theoretical tools, they all appear systematically in a categorical framework. The key element here is the systematic nature of these maps in a categorical framework and I suggest that, from that point of view, one can see an architectonic of mathematics emerging clearly. Moreover, they force us to reconsider the nature of mathematical knowledge itself. Thus, to understand certain fundamental aspects of mathematics, category theory is necessary (at least, in the present state of mathematics). 1 Introduction The foundational status of category theory has been challenged as soon as it has been proposed as such1. The literature on the subject is roughly split in two camps: those who argue against category theory by exhibiting some of its shortcomings and those who argue that it does not fall prey to these shortcom- ings2. Detractors argue that it supposedly falls short of some basic desiderata that any foundational framework ought to satisfy: either logical, epistemologi- cal, ontological or psychological. To put it bluntly, it is sometimes claimed that ∗The author gratefully acknowledge the financial support of the SSHRC of Canada while this work was done. -
On the Group of Automorphisms of a Quasi-Affine Variety 3
ON THE GROUP OF AUTOMORPHISMS OF A QUASI-AFFINE VARIETY ZBIGNIEW JELONEK Abstract. Let K be an algebraically closed field of characteristic zero. We show that if the automorphisms group of a quasi-affine variety X is infinite, then X is uniruled. 1. Introduction. Automorphism groups of an open varieties have always attracted a lot of attention, but the nature of this groups is still not well-known. For example the group of automorphisms of Kn is understood only in the case n = 2 (and n = 1, of course). Let Y be a an open variety. It is natural to ask when the group of automorphisms of Y is finite. We gave a partial answer to this questions in our papers [Jel1], [Jel2], [Jel3] and [Jel4]. In [Iit2] Iitaka proved that Aut(Y ) is finite if Y has a maximal logarithmic Kodaira dimension. Here we focus on the group of automorphisms of an affine or more generally quasi-affine variety over an algebraically closed field of characteristic zero. Let us recall that a quasi-affine variety is an open subvariety of some affine variety. We prove the following: Theorem 1.1. Let X be a quasi-affine (in particular affine) variety over an algebraically closed field of characteristic zero. If Aut(X) is infinite, then X is uniruled, i.e., X is covered by rational curves. This generalizes our old results from [Jel3] and [Jel4]. Our proof uses in a significant way a recent progress in a minimal model program ( see [Bir], [BCHK], [P-S]) and is based on our old ideas from [Jel1], [Jel2], [Jel3] and [Jel4]. -
Positivity in Algebraic Geometry I
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 48 Positivity in Algebraic Geometry I Classical Setting: Line Bundles and Linear Series Bearbeitet von R.K. Lazarsfeld 1. Auflage 2004. Buch. xviii, 387 S. Hardcover ISBN 978 3 540 22533 1 Format (B x L): 15,5 x 23,5 cm Gewicht: 1650 g Weitere Fachgebiete > Mathematik > Geometrie > Elementare Geometrie: Allgemeines Zu Inhaltsverzeichnis schnell und portofrei erhältlich bei Die Online-Fachbuchhandlung beck-shop.de ist spezialisiert auf Fachbücher, insbesondere Recht, Steuern und Wirtschaft. Im Sortiment finden Sie alle Medien (Bücher, Zeitschriften, CDs, eBooks, etc.) aller Verlage. Ergänzt wird das Programm durch Services wie Neuerscheinungsdienst oder Zusammenstellungen von Büchern zu Sonderpreisen. Der Shop führt mehr als 8 Millionen Produkte. Introduction to Part One Linear series have long stood at the center of algebraic geometry. Systems of divisors were employed classically to study and define invariants of pro- jective varieties, and it was recognized that varieties share many properties with their hyperplane sections. The classical picture was greatly clarified by the revolutionary new ideas that entered the field starting in the 1950s. To begin with, Serre’s great paper [530], along with the work of Kodaira (e.g. [353]), brought into focus the importance of amplitude for line bundles. By the mid 1960s a very beautiful theory was in place, showing that one could recognize positivity geometrically, cohomologically, or numerically. During the same years, Zariski and others began to investigate the more complicated be- havior of linear series defined by line bundles that may not be ample. -
University of Cape Town Department of Mathematics and Applied Mathematics Faculty of Science
The copyright of this thesis vests in the author. No quotation from it or information derived from it is to be published without full acknowledgementTown of the source. The thesis is to be used for private study or non- commercial research purposes only. Cape Published by the University ofof Cape Town (UCT) in terms of the non-exclusive license granted to UCT by the author. University University of Cape Town Department of Mathematics and Applied Mathematics Faculty of Science Convexity in quasi-metricTown spaces Cape ofby Olivier Olela Otafudu May 2012 University A thesis presented for the degree of Doctor of Philosophy prepared under the supervision of Professor Hans-Peter Albert KÄunzi. Abstract Over the last ¯fty years much progress has been made in the investigation of the hyperconvex hull of a metric space. In particular, Dress, Espinola, Isbell, Jawhari, Khamsi, Kirk, Misane, Pouzet published several articles concerning hyperconvex metric spaces. The principal aim of this thesis is to investigateTown the existence of an injective hull in the categories of T0-quasi-metric spaces and of T0-ultra-quasi- metric spaces with nonexpansive maps. Here several results obtained by others for the hyperconvex hull of a metric space haveCape been generalized by us in the case of quasi-metric spaces. In particular we haveof obtained some original results for the q-hyperconvex hull of a T0-quasi-metric space; for instance the q-hyperconvex hull of any totally bounded T0-quasi-metric space is joincompact. Also a construction of the ultra-quasi-metrically injective (= u-injective) hull of a T0-ultra-quasi- metric space is provided. -
Algebra & Number Theory
Algebra & Number Theory Volume 4 2010 No. 2 mathematical sciences publishers Algebra & Number Theory www.jant.org EDITORS MANAGING EDITOR EDITORIAL BOARD CHAIR Bjorn Poonen David Eisenbud Massachusetts Institute of Technology University of California Cambridge, USA Berkeley, USA BOARD OF EDITORS Georgia Benkart University of Wisconsin, Madison, USA Susan Montgomery University of Southern California, USA Dave Benson University of Aberdeen, Scotland Shigefumi Mori RIMS, Kyoto University, Japan Richard E. Borcherds University of California, Berkeley, USA Andrei Okounkov Princeton University, USA John H. Coates University of Cambridge, UK Raman Parimala Emory University, USA J-L. Colliot-Thel´ ene` CNRS, Universite´ Paris-Sud, France Victor Reiner University of Minnesota, USA Brian D. Conrad University of Michigan, USA Karl Rubin University of California, Irvine, USA Hel´ ene` Esnault Universitat¨ Duisburg-Essen, Germany Peter Sarnak Princeton University, USA Hubert Flenner Ruhr-Universitat,¨ Germany Michael Singer North Carolina State University, USA Edward Frenkel University of California, Berkeley, USA Ronald Solomon Ohio State University, USA Andrew Granville Universite´ de Montreal,´ Canada Vasudevan Srinivas Tata Inst. of Fund. Research, India Joseph Gubeladze San Francisco State University, USA J. Toby Stafford University of Michigan, USA Ehud Hrushovski Hebrew University, Israel Bernd Sturmfels University of California, Berkeley, USA Craig Huneke University of Kansas, USA Richard Taylor Harvard University, USA Mikhail Kapranov Yale -
The Canonical Map and the Canonical Ring of Algebraic Curves
The canonical map and the canonical ring of algebraic curves. Vicente Lorenzo Garc´ıa The canonical map and the canonical ring of algebraic curves. Introduction. LisMath Seminar. Basic tools. Examples. Max Vicente Lorenzo Garc´ıa Noether's Theorem. Enriques- Petri's Theorem. November 8th, 2017 Hyperelliptic case. 1 / 34 Outline The canonical map and the canonical ring of algebraic curves. Vicente Introduction. Lorenzo Garc´ıa Introduction. Basic tools. Basic tools. Examples. Examples. Max Noether's Theorem. Max Noether's Theorem. Enriques- Petri's Theorem. Hyperelliptic Enriques-Petri's Theorem. case. Hyperelliptic case. 2 / 34 Outline The canonical map and the canonical ring of algebraic curves. Vicente Introduction. Lorenzo Garc´ıa Introduction. Basic tools. Basic tools. Examples. Examples. Max Noether's Theorem. Max Noether's Theorem. Enriques- Petri's Theorem. Hyperelliptic Enriques-Petri's Theorem. case. Hyperelliptic case. 3 / 34 The canonical map and the canonical ring of algebraic curves. Vicente Lorenzo Garc´ıa Definition Introduction. We will use the word curve to mean a complete, nonsingular, one dimensional scheme C Basic tools. over the complex numbers C. Examples. Max Noether's Definition Theorem. The canonical sheaf ΩC of a curve C is the sheaf of sections of its cotangent bundle. Enriques- Petri's Theorem. Hyperelliptic case. 4 / 34 The canonical map and the canonical ring of algebraic curves. Vicente Theorem (Serre's Duality) Lorenzo Garc´ıa Let L be an invertible sheaf on a curve C. Then there is a perfect pairing 0 −1 1 Introduction. H (C; ΩC ⊗ L ) × H (C; L) ! K: Basic tools. 1 0 −1 Examples. -
The Topology of Path Component Spaces
The topology of path component spaces Jeremy Brazas October 26, 2012 Abstract The path component space of a topological space X is the quotient space of X whose points are the path components of X. This paper contains a general study of the topological properties of path component spaces including their relationship to the zeroth dimensional shape group. Path component spaces are simple-to-describe and well-known objects but only recently have recieved more attention. This is primarily due to increased qtop interest and application of the quasitopological fundamental group π1 (X; x0) of a space X with basepoint x0 and its variants; See e.g. [2, 3, 5, 7, 8, 9]. Recall qtop π1 (X; x0) is the path component space of the space Ω(X; x0) of loops based at x0 X with the compact-open topology. The author claims little originality here but2 knows of no general treatment of path component spaces. 1 Path component spaces Definition 1. The path component space of a topological space X, is the quotient space π0(X) obtained by identifying each path component of X to a point. If x X, let [x] denote the path component of x in X. Let qX : X π0(X), q(x) = [x2] denote the canonical quotient map. We also write [A] for the! image qX(A) of a set A X. Note that if f : X Y is a map, then f ([x]) [ f (x)]. Thus f determines a well-defined⊂ function f !: π (X) π (Y) given by⊆f ([x]) = [ f (x)]. -
Mathematische Annalen
Math. Ann. DOI 10.1007/s00208-014-1133-4 Mathematische Annalen On the group of automorphisms of a quasi-affine variety Zbigniew Jelonek Received: 8 March 2014 / Revised: 19 July 2014 © The Author(s) 2014. This article is published with open access at Springerlink.com Abstract Let K be an algebraically closed field of characteristic zero. We show that if the automorphisms group of a quasi-affine variety X over K is infinite, then X is uniruled. Mathematics Subject Classification 14 R 10 1 Introduction Automorphism groups of open varieties have always attracted a lot of attention, but the nature of these groups is still not well-known. For example the group of automorphisms of Kn is understood only in the case n = 2 (and n = 1, of course). Let Y be an open variety. It is natural to ask when the group Aut(Y ) of automorphisms of Y is finite. A partial answer to this question is given in our papers [7–9] and [10]. In [6], Iitaka proved that Aut(Y ) is finite if Y has a maximal logarithmic Kodaira dimension. Here we focus on the group of automorphisms of an affine or, more generally, quasi-affine variety over an algebraically closed field of characteristic zero. Let us recall that a quasi-affine variety is an open subvariety of some affine variety. We prove the following: Theorem 1.1 Let X be a quasi-affine (in particular affine) variety over an alge- braically closed field of characteristic zero. If the automorphism group Aut(X) is infinite, then X is uniruled, i.e., X is covered by rational curves. -
Convergence of Quotients of Af Algebras in Quantum Propinquity by Convergence of Ideals
CONVERGENCE OF QUOTIENTS OF AF ALGEBRAS IN QUANTUM PROPINQUITY BY CONVERGENCE OF IDEALS KONRAD AGUILAR ABSTRACT. We provide conditions for when quotients of AF algebras are quasi- Leibniz quantum compact metric spaces building from our previous work with F. Latrémolière. Given a C*-algebra, the ideal space may be equipped with natural topologies. Next, we impart criteria for when convergence of ideals of an AF al- gebra can provide convergence of quotients in quantum propinquity, while intro- ducing a metric on the ideal space of a C*-algebra. We then apply these findings to a certain class of ideals of the Boca-Mundici AF algebra by providing a continuous map from this class of ideals equipped with various topologies including the Ja- cobson and Fell topologies to the space of quotients with the quantum propinquity topology. Lastly, we introduce new Leibniz Lip-norms on any unital AF algebra motivated by Rieffel’s work on Leibniz seminorms and best approximations. CONTENTS 1. Introduction1 2. Quantum Metric Geometry and AF algebras4 3. Convergence of AF algebras with Quantum Propinquity 10 4. Metric on Ideal Space of C*-Inductive Limits 18 5. Metric on Ideal Space of C*-Inductive Limits: AF case 27 6. Convergence of Quotients of AF algebras with Quantum Propinquity 39 6.1. The Boca-Mundici AF algebra 43 7. Leibniz Lip-norms for Unital AF algebras 56 Acknowledgements 59 References 59 1. INTRODUCTION The Gromov-Hausdorff propinquity [29, 26, 24, 28, 27], a family of noncommu- tative analogues of the Gromov-Hausdorff distance, provides a new framework to study the geometry of classes of C*-algebras, opening new avenues of research in noncommutative geometry by work of Latrémolière building off notions intro- duced by Rieffel [37, 45]. -
Ample Families, Multihomogeneous Spectra, and Algebraization of Formal Schemes
AMPLE FAMILIES, MULTIHOMOGENEOUS SPECTRA, AND ALGEBRAIZATION OF FORMAL SCHEMES HOLGER BRENNER AND STEFAN SCHROER¨ Abstract. Generalizing homogeneous spectra for rings graded by natural numbers, we introduce multihomogeneous spectra for rings graded by abelian groups. Such homogeneous spectra have the same completeness properties as their classical counterparts, but are possibly nonseparated. We relate them to ample families of invertible sheaves and simplicial toric varieties. As an application, we generalize Grothendieck’s Algebraization Theorem and show that formal schemes with certain ample families are algebraizable. Introduction A powerful method to study algebraic varieties is to embed them, if possible, into some projective space Pn. Such an embedding X ⊂ Pn allows you to view points x ∈ X as homogeneous prime ideals in some N-graded ring. The purpose of this paper is to extend this to divisorial varieties, which are not necessarily quasiprojective. The notion of divisorial varieties is due to Borelli [2]. The class of divisorial vari- eties contains all quasiprojective schemes, smooth varieties, and locally Q-factorial varieties. Roughly speaking, divisoriality means that there is a finite collection of invertible sheaves L1,..., Lr so that the whole collection behaves like an ample invertible sheaf. Such collections are called ample families. Our main idea is to define homogeneous spectra Proj(S) for multigraded rings, that is, for rings graded by an abelian group of finite type. We obtain Proj(S) by patching affine pieces D+(f) = Spec(S(f)), where f ∈ S are certain homogeneous elements. Roughly speaking, we demand that f has many homogeneous divisors. Multihomogeneous spectra share many properties of classical homogeneous spectra.