Intersection Theory
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Bézout's Theorem
Bézout's theorem Toni Annala Contents 1 Introduction 2 2 Bézout's theorem 4 2.1 Ane plane curves . .4 2.2 Projective plane curves . .6 2.3 An elementary proof for the upper bound . 10 2.4 Intersection numbers . 12 2.5 Proof of Bézout's theorem . 16 3 Intersections in Pn 20 3.1 The Hilbert series and the Hilbert polynomial . 20 3.2 A brief overview of the modern theory . 24 3.3 Intersections of hypersurfaces in Pn ...................... 26 3.4 Intersections of subvarieties in Pn ....................... 32 3.5 Serre's Tor-formula . 36 4 Appendix 42 4.1 Homogeneous Noether normalization . 42 4.2 Primes associated to a graded module . 43 1 Chapter 1 Introduction The topic of this thesis is Bézout's theorem. The classical version considers the number of points in the intersection of two algebraic plane curves. It states that if we have two algebraic plane curves, dened over an algebraically closed eld and cut out by multi- variate polynomials of degrees n and m, then the number of points where these curves intersect is exactly nm if we count multiple intersections and intersections at innity. In Chapter 2 we dene the necessary terminology to state this theorem rigorously, give an elementary proof for the upper bound using resultants, and later prove the full Bézout's theorem. A very modest background should suce for understanding this chapter, for example the course Algebra II given at the University of Helsinki should cover most, if not all, of the prerequisites. In Chapter 3, we generalize Bézout's theorem to higher dimensional projective spaces. -
Automorphisms of a Symmetric Product of a Curve (With an Appendix by Najmuddin Fakhruddin)
Documenta Math. 1181 Automorphisms of a Symmetric Product of a Curve (with an Appendix by Najmuddin Fakhruddin) Indranil Biswas and Tomas´ L. Gomez´ Received: February 27, 2016 Revised: April 26, 2017 Communicated by Ulf Rehmann Abstract. Let X be an irreducible smooth projective curve of genus g > 2 defined over an algebraically closed field of characteristic dif- ferent from two. We prove that the natural homomorphism from the automorphisms of X to the automorphisms of the symmetric product Symd(X) is an isomorphism if d > 2g − 2. In an appendix, Fakhrud- din proves that the isomorphism class of the symmetric product of a curve determines the isomorphism class of the curve. 2000 Mathematics Subject Classification: 14H40, 14J50 Keywords and Phrases: Symmetric product; automorphism; Torelli theorem. 1. Introduction Automorphisms of varieties is currently a very active topic in algebraic geom- etry; see [Og], [HT], [Zh] and references therein. Hurwitz’s automorphisms theorem, [Hu], says that the order of the automorphism group Aut(X) of a compact Riemann surface X of genus g ≥ 2 is bounded by 84(g − 1). The group of automorphisms of the Jacobian J(X) preserving the theta polariza- tion is generated by Aut(X), translations and inversion [We], [La]. There is a universal constant c such that the order of the group of all automorphisms of any smooth minimal complex projective surface S of general type is bounded 2 above by c · KS [Xi]. Let X be a smooth projective curve of genus g, with g > 2, over an al- gebraically closed field of characteristic different from two. -
Fundamental Algebraic Geometry
http://dx.doi.org/10.1090/surv/123 hematical Surveys and onographs olume 123 Fundamental Algebraic Geometry Grothendieck's FGA Explained Barbara Fantechi Lothar Gottsche Luc lllusie Steven L. Kleiman Nitin Nitsure AngeloVistoli American Mathematical Society U^VDED^ EDITORIAL COMMITTEE Jerry L. Bona Peter S. Landweber Michael G. Eastwood Michael P. Loss J. T. Stafford, Chair 2000 Mathematics Subject Classification. Primary 14-01, 14C20, 13D10, 14D15, 14K30, 18F10, 18D30. For additional information and updates on this book, visit www.ams.org/bookpages/surv-123 Library of Congress Cataloging-in-Publication Data Fundamental algebraic geometry : Grothendieck's FGA explained / Barbara Fantechi p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 123) Includes bibliographical references and index. ISBN 0-8218-3541-6 (pbk. : acid-free paper) ISBN 0-8218-4245-5 (soft cover : acid-free paper) 1. Geometry, Algebraic. 2. Grothendieck groups. 3. Grothendieck categories. I Barbara, 1966- II. Mathematical surveys and monographs ; no. 123. QA564.F86 2005 516.3'5—dc22 2005053614 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. -
Associating Abelian Varieties to Weight-2 Modular Forms: the Eichler-Shimura Construction
Ecole´ Polytechnique Fed´ erale´ de Lausanne Master’s Thesis in Mathematics Associating abelian varieties to weight-2 modular forms: the Eichler-Shimura construction Author: Corentin Perret-Gentil Supervisors: Prof. Akshay Venkatesh Stanford University Prof. Philippe Michel EPF Lausanne Spring 2014 Abstract This document is the final report for the author’s Master’s project, whose goal was to study the Eichler-Shimura construction associating abelian va- rieties to weight-2 modular forms for Γ0(N). The starting points and main resources were the survey article by Fred Diamond and John Im [DI95], the book by Goro Shimura [Shi71], and the book by Fred Diamond and Jerry Shurman [DS06]. The latter is a very good first reference about this sub- ject, but interesting points are sometimes eluded. In particular, although most statements are given in the general setting, the book mainly deals with the particular case of elliptic curves (i.e. with forms having rational Fourier coefficients), with little details about abelian varieties. On the other hand, Chapter 7 of Shimura’s book is difficult, according to the author himself, and the article by Diamond and Im skims rapidly through the subject, be- ing a survey. The goal of this document is therefore to give an account of the theory with intermediate difficulty, accessible to someone having read a first text on modular forms – such as [Zag08] – and with basic knowledge in the theory of compact Riemann surfaces (see e.g. [Mir95]) and algebraic geometry (see e.g. [Har77]). This report begins with an account of the theory of abelian varieties needed for what follows. -
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 . -
AN EXPLORATION of COMPLEX JACOBIAN VARIETIES Contents 1
AN EXPLORATION OF COMPLEX JACOBIAN VARIETIES MATTHEW WOOLF Abstract. In this paper, I will describe my thought process as I read in [1] about Abelian varieties in general, and the Jacobian variety associated to any compact Riemann surface in particular. I will also describe the way I currently think about the material, and any additional questions I have. I will not include material I personally knew before the beginning of the summer, which included the basics of algebraic and differential topology, real analysis, one complex variable, and some elementary material about complex algebraic curves. Contents 1. Abelian Sums (I) 1 2. Complex Manifolds 2 3. Hodge Theory and the Hodge Decomposition 3 4. Abelian Sums (II) 6 5. Jacobian Varieties 6 6. Line Bundles 8 7. Abelian Varieties 11 8. Intermediate Jacobians 15 9. Curves and their Jacobians 16 References 17 1. Abelian Sums (I) The first thing I read about were Abelian sums, which are sums of the form 3 X Z pi (1.1) (L) = !; i=1 p0 where ! is the meromorphic one-form dx=y, L is a line in P2 (the complex projective 2 3 2 plane), p0 is a fixed point of the cubic curve C = (y = x + ax + bx + c), and p1, p2, and p3 are the three intersections of the line L with C. The fact that C and L intersect in three points counting multiplicity is just Bezout's theorem. Since C is not simply connected, the integrals in the Abelian sum depend on the path, but there's no natural choice of path from p0 to pi, so this function depends on an arbitrary choice of path, and will not necessarily be holomorphic, or even Date: August 22, 2008. -
Intersection Theory on Regular Schemes Via Alterations and Deformation to the Normal Cone Dissertation
Intersection Theory on Regular Schemes via Alterations and Deformation to the Normal Cone Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften (Dr. rer. nat.) der Fakultat¨ fur¨ Mathematik der Universitat¨ Regensburg vorgelegt von Andreas Weber aus Regensburg im Jahr 2015 Promotionsgesuch eingereicht am 13. April 2015. Die Arbeit wurde angeleitet von Prof. Dr. Klaus K¨unnemann. Pr¨ufungsausschuss: Vorsitzender: Prof. Dr. Harald Garcke 1. Gutachter: Prof. Dr. Klaus K¨unnemann 2. Gutachter: Prof. Dr. Walter Gubler weiterer Pr¨ufer: Prof. Dr. Uwe Jannsen Contents Contents 3 1 Introduction 5 2 Chow Groups of S-schemes 11 2.1 The S-Dimension . 11 2.2 Chow Groups . 14 3 Resolution of Singularities and Alterations 17 3.1 Assumption on Alterations . 17 3.2 State of the Art . 18 4 Intersection Theory with Supports on Regular Schemes 21 4.1 Bivariant Classes and Orientations . 21 4.2 Alterations and BQ-Orientations . 37 4.3 Intersection Theory with Supports on BQ-orienting Schemes . 39 5 Comparison to other Approaches to Intersection Theory 47 5.1 Intersection with Divisors . 47 5.2 Smooth Schemes over a Dedekind scheme . 49 A Fulton's Theory for S-schemes 53 A.1 Proper push-forward and flat pull-back . 53 A.2 Intersection with Divisors . 56 A.3 Cones, Chern and Segre classes . 56 A.4 Deformation to the Normal bundle . 63 A.5 Refined Gysin homomorphisms . 65 A.6 Intersection theory for smooth schemes over a one-dimensional base . 69 Bibliography 71 Chapter 1 Introduction k For a Noetherian separated regular scheme X, the Chow group CHY (X) of algebraic cycles of codimension k with supports in a closed subset Y of X is given as k k k CHY (X) := ZY (X) = RatY (X); k i.e. -
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 . -
Waring-Type Problems for Polynomials Algebra Meets Geometry Alessandro Oneto
Waring-type problems for polynomials Algebra meets Geometry Alessandro Oneto Waring-type problems for polynomials Algebra meets Geometry Alessandro Oneto ©Alessandro Oneto, Stockholm University 2016 e-mail: [email protected] ISBN: 978-91-7649-424-0 Printed by Holmbergs, Malmö 2016 Distributor: Department of Mathematics, Stockholm University CONTENTS 1 INTRODUCTION 3 1.1 ADDITIVE DECOMPOSITIONS OF INTEGERS ............ 3 1.2 ADDITIVE DECOMPOSITIONS OF POLYNOMIALS .......... 4 1.2.1 CLASSICAL WARING DECOMPOSITIONS .......... 5 1.2.2 d-TH WARING DECOMPOSITIONS .............. 7 1.2.3 WARING-LIKE DECOMPOSITIONS .............. 8 1.2.4 REAL WARING DECOMPOSITIONS ............. 9 1.3 GEOMETRIC INTERPRETATION ................... 10 1.3.1 SECANT VARIETIES ..................... 10 1.3.2 CLASSICAL WARING PROBLEM:VERONESE VARIETIES . 12 1.3.3 d-TH WARING PROBLEM: VARIETIES OF POWERS . 14 1.3.4 WARING-LIKE PROBLEMS: VARIETIES OF µ-POWERS . 17 1.3.5 TERRACINI’S LEMMA .................... 18 2 APOLARITY THEORY AND POINTS CONFIGURATIONS 23 2.1 APOLARITY THEORY ......................... 23 2.2 HILBERT FUNCTIONS OF CONFIGURATIONS OF REDUCED POINTS 27 2.3 WARING LOCI OF HOMOGENEOUS POLYNOMIALS . 29 2.3.1 QUADRICS .......................... 31 2.3.2 MONOMIALS ......................... 32 2.3.3 BINARY FORMS ....................... 34 2.3.4 PLANE CUBICS ....................... 37 2.4 WARING LOCI AND THE STRASSEN CONJECTURE . 43 2.5 APOLARITY LEMMA: POWER IDEALS AND FAT POINTS . 49 2.5.1 IDEALS OF FAT POINTS ................... 49 2.5.2 INVERSE SYSTEMS OF IDEALS FAT POINTS. 51 2.6 HILBERT FUNCTIONS OF CONFIGURATIONS OF FAT POINTS . 53 2.6.1 DOUBLE POINTS: THE ALEXANDER–HIRSCHOWITZ THEO- REM ............................. 55 2.7 SPECIAL CONFIGURATIONS OF FAT POINTS . -
The Historical Development of Algebraic Geometry∗
The Historical Development of Algebraic Geometry∗ Jean Dieudonn´e March 3, 1972y 1 The lecture This is an enriched transcription of footage posted by the University of Wis- consin{Milwaukee Department of Mathematical Sciences [1]. The images are composites of screenshots from the footage manipulated with Python and the Python OpenCV library [2]. These materials were prepared by Ryan C. Schwiebert with the goal of capturing and restoring some of the material. The lecture presents the content of Dieudonn´e'sarticle of the same title [3] as well as the content of earlier lecture notes [4], but the live lecture is natu- rally embellished with different details. The text presented here is, for the most part, written as it was spoken. This is why there are some ungrammatical con- structions of spoken word, and some linguistic quirks of a French speaker. The transcriber has taken some liberties by abbreviating places where Dieudonn´e self-corrected. That is, short phrases which turned out to be false starts have been elided, and now only the subsequent word-choices that replaced them ap- pear. Unfortunately, time has taken its toll on the footage, and it appears that a brief portion at the beginning, as well as the last enumerated period (VII. Sheaves and schemes), have been lost. (Fortunately, we can still read about them in [3]!) The audio and video quality has contributed to several transcription errors. Finally, a lack of thorough knowledge of the history of algebraic geometry has also contributed some errors. Contact with suggestions for improvement of arXiv:1803.00163v1 [math.HO] 1 Mar 2018 the materials is welcome. -
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 -
ULRICH BUNDLES on CUBIC FOURFOLDS Daniele Faenzi, Yeongrak Kim
ULRICH BUNDLES ON CUBIC FOURFOLDS Daniele Faenzi, Yeongrak Kim To cite this version: Daniele Faenzi, Yeongrak Kim. ULRICH BUNDLES ON CUBIC FOURFOLDS. 2020. hal- 03023101v2 HAL Id: hal-03023101 https://hal.archives-ouvertes.fr/hal-03023101v2 Preprint submitted on 25 Nov 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ULRICH BUNDLES ON CUBIC FOURFOLDS DANIELE FAENZI AND YEONGRAK KIM Abstract. We show the existence of rank 6 Ulrich bundles on a smooth cubic fourfold. First, we construct a simple sheaf E of rank 6 as an elementary modification of an ACM bundle of rank 6 on a smooth cubic fourfold. Such an E appears as an extension of two Lehn-Lehn-Sorger-van Straten sheaves. Then we prove that a general deformation of E(1) becomes Ulrich. In particular, this says that general cubic fourfolds have Ulrich complexity 6. Introduction An Ulrich sheaf on a closed subscheme X of PN of dimension n and degree d is a non-zero coherent sheaf F on X satisfying H∗(X, F(−j)) = 0 for 1 ≤ j ≤ n. In particular, the cohomology table {hi(X, F(j))} of F is a multiple of the cohomology table of Pn.