DOCSLIB.ORG

Introduction to Algebraic Geometry

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

GRADUATE STUDIES IN MATHEMATICS 188 Introduction to Algebraic Geometry Steven Dale Cutkosky 10.1090/gsm/188 GRADUATE STUDIES IN MATHEMATICS 188 Introduction to Algebraic Geometry Steven Dale Cutkosky EDITORIAL COMMITTEE Dan Abramovich Daniel S. Freed (Chair) Gigliola Staffilani Jeff A. Viaclovsky 2010 Mathematics Subject Classification. Primary 14-01. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-188 Library of Congress Cataloging-in-Publication Data Names: Cutkosky, Steven Dale, author. Title: Introduction to algebraic geometry / Steven Dale Cutkosky. Other titles: Algebraic geometry Description: Providence, Rhode Island : American Mathematical Society, [2018] | Series: Gradu- ate studies in mathematics ; volume 188 | Includes bibliographical references and index. Identifiers: LCCN 2017045552 | ISBN 9781470435189 (alk. paper) Subjects: LCSH: Geometry, Algebraic. | AMS: Algebraic geometry – Instructional exposition (textbooks, tutorial papers, etc.). msc Classification: LCC QA564 .C8794 2018 | DDC 516.3/5–dc23 LC record available at https://lccn.loc.gov/2017045552 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 select pages 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 permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/publications/pubpermissions. Send requests for translation rights and licensed reprints to [email protected]. c 2018 by the author. All rights reserved. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 232221201918 To Hema, Ashok, and Maya Contents Preface xi Chapter 1. A Crash Course in Commutative Algebra 1 §1.1. Basic algebra 1 §1.2. Field extensions 6 §1.3. Modules 8 §1.4. Localization 9 §1.5. Noetherian rings and factorization 10 §1.6. Primary decomposition 13 §1.7. Integral extensions 16 §1.8. Dimension 19 §1.9. Depth 20 §1.10. Normal rings and regular rings 22 Chapter 2. Affine Varieties 27 §2.1. Affine space and algebraic sets 27 §2.2. Regular functions and regular maps of affine algebraic sets 33 §2.3. Finite maps 40 §2.4. Dimension of algebraic sets 42 §2.5. Regular functions and regular maps of quasi-affine varieties 48 §2.6. Rational maps of affine varieties 58 Chapter 3. Projective Varieties 63 §3.1. Standard graded algebras 63 v vi Contents §3.2. Projective varieties 67 §3.3. Grassmann varieties 73 §3.4. Regular functions and regular maps of quasi-projective varieties 74 Chapter 4. Regular and Rational Maps of Quasi-projective Varieties 87 §4.1. Criteria for regular maps 87 §4.2. Linear isomorphisms of projective space 90 §4.3. The Veronese embedding 91 §4.4. Rational maps of quasi-projective varieties 93 §4.5. Projection from a linear subspace 95 Chapter 5. Products 99 §5.1. Tensor products 99 §5.2. Products of varieties 101 §5.3. The Segre embedding 105 §5.4. Graphs of regular and rational maps 106 Chapter 6. The Blow-up of an Ideal 111 §6.1. The blow-up of an ideal in an affine variety 111 §6.2. The blow-up of an ideal in a projective variety 120 Chapter 7. Finite Maps of Quasi-projective Varieties 127 §7.1. Affine and finite maps 127 §7.2. Finite maps 131 §7.3. Construction of the normalization 135 Chapter 8. Dimension of Quasi-projective Algebraic Sets 139 §8.1. Properties of dimension 139 §8.2. The theorem on dimension of fibers 141 Chapter 9. Zariski’s Main Theorem 147 Chapter 10. Nonsingularity 153 §10.1. Regular parameters 153 §10.2. Local equations 155 §10.3. The tangent space 156 §10.4. Nonsingularity and the singular locus 159 §10.5. Applications to rational maps 165 Contents vii §10.6. Factorization of birational regular maps of nonsingular surfaces 168 §10.7. Projective embedding of nonsingular varieties 170 §10.8. Complex manifolds 175 Chapter 11. Sheaves 181 §11.1. Limits 181 §11.2. Presheaves and sheaves 185 §11.3. Some sheaves associated to modules 196 §11.4. Quasi-coherent and coherent sheaves 200 §11.5. Constructions of sheaves from sheaves of modules 204 §11.6. Some theorems about coherent sheaves 209 Chapter 12. Applications to Regular and Rational Maps 221 §12.1. Blow-ups of ideal sheaves 221 §12.2. Resolution of singularities 225 §12.3. Valuations in algebraic geometry 228 §12.4. Factorization of birational maps 232 §12.5. Monomialization of maps 236 Chapter 13. Divisors 239 §13.1. Divisors and the class group 240 §13.2. The sheaf associated to a divisor 242 §13.3. Divisors associated to forms 249 §13.4. Calculation of some class groups 249 §13.5. The class group of a curve 254 §13.6. Divisors, rational maps, and linear systems 259 §13.7. Criteria for closed embeddings 264 §13.8. Invertible sheaves 269 §13.9. Transition functions 271 Chapter 14. Differential Forms and the Canonical Divisor 279 §14.1. Derivations and K¨ahler differentials 279 §14.2. Differentials on varieties 283 §14.3. n-forms and canonical divisors 286 Chapter 15. Schemes 289 §15.1. Subschemes of varieties, schemes, and Cartier divisors 289 §15.2. Blow-ups of ideals and associated graded rings of ideals 293 viii Contents §15.3. Abstract algebraic varieties 295 §15.4. Varieties over nonclosed fields 296 §15.5. General schemes 296 Chapter 16. The Degree of a Projective Variety 299 Chapter 17. Cohomology 307 §17.1. Complexes 307 §17.2. Sheaf cohomology 308 §17.3. Cechˇ cohomology 310 §17.4. Applications 312 §17.5. Higher direct images of sheaves 320 §17.6. Local cohomology and regularity 325 Chapter 18. Curves 333 §18.1. The Riemann-Roch inequality 334 §18.2. Serre duality 335 §18.3. The Riemann-Roch theorem 340 §18.4. The Riemann-Roch problem on varieties 343 §18.5. The Hurwitz theorem 345 §18.6. Inseparable maps of curves 348 §18.7. Elliptic curves 351 §18.8. Complex curves 358 §18.9. Abelian varieties and Jacobians of curves 360 Chapter 19. An Introduction to Intersection Theory 365 §19.1. Definition, properties, and some examples of intersection numbers 366 §19.2. Applications to degree and multiplicity 375 Chapter 20. Surfaces 379 §20.1. The Riemann-Roch theorem and the Hodge index theorem on a surface 379 §20.2. Contractions and linear systems 383 Chapter 21. Ramification and Etale´ Maps 391 §21.1. Norms and Traces 392 §21.2. Integral extensions 393 §21.3. Discriminants and ramification 398 Contents ix §21.4. Ramification of regular maps of varieties 406 §21.5. Completion 408 §21.6. Zariski’s main theorem and Zariski’s subspace theorem 413 §21.7. Galois theory of varieties 421 §21.8. Derivations and K¨ahler differentials redux 424 §21.9. Etale´ maps and uniformizing parameters 426 §21.10. Purity of the branch locus and the Abhyankar-Jung theorem 433 §21.11. Galois theory of local rings 438 §21.12. A proof of the Abhyankar-Jung theorem 441 Chapter 22. Bertini’s Theorems and General Fibers of Maps 451 §22.1. Geometric integrality 452 §22.2. Nonsingularity of the general fiber 454 §22.3. Bertini’s second theorem 457 §22.4. Bertini’s first theorem 458 Bibliography 469 Index 477 Preface This book is an introductory course in algebraic geometry, proving most of the fundamental classical results of algebraic geometry. Algebraic geometry combines the intuition of geometry with the preci- sion of algebra. Starting with geometric concepts, we introduce machinery as necessary to model important ideas from algebraic geometry and to prove fundamental results. Emphasis is put on developing facility with connect- ing geometric and algebraic concepts. Examples are constructed or cited illustrating the scope of these results. The theory in this book is developed in increasing sophistication, giving (and refining) definitions as required to accommodate new geometric ideas. We work as much as possible with quasi-projective varieties over an algebraically closed field of arbitrary characteristic. This allows us to inter- pret varieties through their function fields. This approach and the use of methods of algebraic number theory in algebraic geometry have been cen- tral to algebraic geometry at least since the time of Dedekind and Weber (Theorie der algebraischen Functionen einer Ver¨anderlichen [46], translated in [47]). By interpreting the geometric concept of varieties through their regular functions, we are able to use the techniques of commutative algebra. Differences between the theory in characteristic 0 and positive charac- teristic are emphasized in this book. We extend our view to schemes, al- lowing rings with nilpotents, to study fibers of regular maps and to develop intersection theory. We discuss the cases of nonclosed ground fields and nonseparated schemes and some of the extra considerations which appear in these situations. A list of exercises is given at the end of many sections and chapters. xi xii Preface The classic textbooks Basic Algebraic Geometry [136] by Shafarevich, Introduction to Algebraic Geometry [116] by Mumford, and Algebraic Geometry [73] by Hartshorne, as well as the works of Zariski, Abhyankar, Serre, and Grothendieck, have been major influences on this book. The necessary commutative algebra is introduced and reviewed, begin- ning with Chapter 1, “A Crash Course in Commutative Algebra”. We state definitions and theorems, explain concepts, and give examples from com- mutative algebra for everything that we will need, proving some results and giving a few examples, but mostly giving references to books on commuta- tive algebra for proofs.
Recommended publications
  • Fundamental Algebraic Geometry

    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.
  • Algebraic Curves and Surfaces

    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 .
  • Torification and Factorization of Birational Maps 3

    Torification and Factorization of Birational Maps 3

    TORIFICATION AND FACTORIZATION OF BIRATIONAL MAPS DAN ABRAMOVICH, KALLE KARU, KENJI MATSUKI, AND JAROSLAW WLODARCZYK Abstract. Building on work of the fourth author in [69], we prove the weak factorization conjecture for birational maps in characteristic zero: a birational map between complete nonsingular varieties over an algebraically closed field K of characteristic zero is a composite of blowings up and blowings down with smooth centers. Contents 0. Introduction 1 1. Preliminaries 5 2. Birational Cobordisms 10 3. Torification 16 4. A proof of the weak factorization theorem 22 5. Generalizations 24 6. Problems related to weak factorization 27 References 28 0. Introduction We work over an algebraically closed field K of characteristic 0. We denote the multiplicative group of K by K∗. 0.1. Statement of the main result. The purpose of this paper is to give a proof for the following weak factorization conjecture of birational maps. We note that another proof of this theorem was given by the fourth author in [70]. See section 0.12 for a brief comparison of the two approaches. Theorem 0.1.1 (Weak Factorization). Let φ : X1 99K X2 be a birational map between complete nonsingular algebraic varieties X1 and X2 over an algebraically closed field K of characteristic zero, and let U ⊂ X1 be an open set where φ is an isomorphism. Then φ can be factored into a sequence of blowings up and blowings arXiv:math/9904135v4 [math.AG] 31 May 2000 down with smooth irreducible centers disjoint from U, namely, there exists a sequence of birational maps between complete nonsingular algebraic varieties ϕ1 ϕ2 ϕi ϕi+1 ϕi+2 ϕl−1 ϕl X1 = V0 99K V1 99K · · · 99K Vi 99K Vi+1 99K · · · 99K Vl−1 99K Vl = X2 where 1.
  • New York Journal of Mathematics Lang's Conjectures, Fibered

    New York Journal of Mathematics Lang's Conjectures, Fibered

    New York Journal of Mathematics New York J. Math. 2 (1996) 20–34. Lang’s Conjectures, Fibered Powers, and Uniformity Dan Abramovich and Jos´e Felipe Voloch Abstract. We prove that the fibered power conjecture of Caporaso et al. (Con- jecture H, [CHM], §6) together with Lang’s conjecture implies the uniformity of rational points on varieties of general type, as predicted in [CHM]; a few applications on the arithmetic and geometry of curves are stated. In an opposite direction, we give counterexamples to some analogous results in positive characteristic. We show that curves that change genus can have arbitrarily many rational points; and that curves over Fp(t) can have arbitrarily many Frobenius orbits of non-constant points. Contents 1. Introduction 21 1.1. A Few Conjectures of Lang 21 1.2. The Fibered Power Conjecture 22 1.3. Summary of Results on the Implication Side 22 1.4. Summary of Results: Examples in Positive Characteristic 24 1.5. Acknowledgments 25 2. Proof of Theorem 1.5 25 2.1. Preliminaries 25 2.2. Prolongable Points 25 2.3. Proof of Theorem 1.5 26 3. A Few Refinements and Applications in Arithmetic and Geometry 26 3.1. Proof of Theorem 1.6 26 3.2. Uniformity in Terms of the Degree of an Extension 27 3.3. The Geometric Case 28 4. Examples in Positive Characteristic 30 4.1. Curves that Change Genus 30 Received December 20, 1995. Mathematics Subject Classification. 14G; 11G. Key words and phrases. arithmetic geometry, Lang’s conjecture, rational points. Abramovich partially supported by NSF grant DMS-9503276.
  • Moduli of K3 Surfaces and Irreducible Symplectic Manifolds

    Moduli of K3 Surfaces and Irreducible Symplectic Manifolds

    Moduli of K3 Surfaces and Irreducible Symplectic Manifolds V. Gritsenko, K. Hulek and G.K. Sankaran Abstract The name “K3 surfaces” was coined by A. Weil in 1957 when he formu- lated a research programme for these surfaces and their moduli. Since then, irreducible holomorphic symplectic manifolds have been intro- duced as a higher dimensional analogue of K3 surfaces. In this paper we present a review of this theory starting from the definition of K3 surfaces and going as far as the global Torelli theorem for irreducible holomorphic symplectic manifolds as recently proved by M. Verbitsky. For many years the last open question of Weil’s programme was that of the geometric type of the moduli spaces of polarised K3 surfaces. We explain how this problem has been solved. Our method uses algebraic geometry, modular forms and Borcherds automorphic products. We collect and discuss the relevant facts from the theory of modular forms with respect to the orthogonal group O(2,n). We also give a detailed description of quasi pull-back of automorphic Borcherds products. This part contains previously unpublished results. Contents 1 Introduction 2 2 Periods of K3 surfaces and the Torelli theorem 4 3 Irreducible symplectic manifolds 15 4 Projective models 24 arXiv:1012.4155v2 [math.AG] 26 May 2011 5 Compactifications 26 6 Modular forms and Kodaira dimension 39 7 Orthogonal groups and reflections 46 8 The quasi pull-back of modular forms 51 9 Arithmetic of root systems 59 1 1 Introduction The history of K3 surfaces goes back to classical algebraic geometry. In modern complex algebraic geometry a K3 surface is a compact complex surface S whose canonical bundle is trivial, i.e.
  • Intersection Theory

    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.
  • 2020-2021 Annual Report

    2020-2021 Annual Report

    Institute for Computational and Experimental Research in Mathematics Annual Report May 1, 2020 – April 30, 2021 Brendan Hassett, Director, PI Mathew Borton, IT Director Ruth Crane, Assistant Director and Chief of Staff Juliet Duyster, Assistant Director Finance and Administration Sigal Gottlieb, Deputy Director Jeffrey Hoffstein, Consulting Associate Director Caroline Klivans, Deputy Director Benoit Pausader, co-PI Jill Pipher, Consulting Director Emerita, co-PI Kavita Ramanan, Associate Director, co-PI Bjorn Sandstede, Associate Director, co-PI Ulrica Wilson, Associate Director for Diversity and Outreach Table of Contents Mission ........................................................................................................................................... 5 Annual Report for 2020-2021 ........................................................................................................ 5 Core Programs and Events ............................................................................................................ 5 Participant Summaries by Program Type ..................................................................................... 7 ICERM Funded Participants ................................................................................................................. 7 ICERM Funded Speakers ...................................................................................................................... 9 All Speakers (ICERM funded and Non-ICERM funded) ................................................................
  • Arxiv:Math/9809023V1 [Math.AG] 5 Sep 1998 .W a That Say We 0

    Arxiv:Math/9809023V1 [Math.AG] 5 Sep 1998 .W a That Say We 0

    UNIFORMITY OF STABLY INTEGRAL POINTS ON PRINCIPALLY POLARIZED ABELIAN VARIETIES OF DIMENSION ≤ 2 DAN ABRAMOVICH AND KENJI MATSUKI Abstract. The purpose of this paper is to prove, assuming that the conjecture of Lang and Vojta holds true, that there is a uniform bound on the number of stably integral points in the complement of the theta divisor on a principally polarized abelian surface defined over a number field. Most of our argument works in arbitrary dimension and the restriction on the dimension ≤ 2 is used only at the last step, where we apply Pacelli’s stronger uniformity results for elliptic curves. Preliminary version, July 13, 2021. 0. Introduction 0.1. The conjecture of Lang and Vojta. Let X be a variety over a field of characteristic 0. We say that X is a variety of logarithmic general type, if there exists a desingularization X˜ → X, and a projective embedding X˜ ⊂ Y where D = Y X˜ is a divisor of normal crossings, such that the invertible sheaf ωY (D) is big. We note first that this peoperty is independent of the choices of X˜ and Y , and that it is a proper birational invariant, namely, if X′ → X is a proper birational morphism (or an inverse of such) then X is of logarithmic general type if and only if X′ is. Now let X be a variety of logarithmic general type defined over a number field K. Let S be a finite set of places in K and let OK,S ⊂ K be the ring of S-integers. Fix a model X of X over OK,S.
  • Advanced Lectures in Mathematics (ALM)

    Advanced Lectures in Mathematics (ALM)

    Advanced Lectures in Mathematics (ALM) ALM 1: Superstring Theory ALM 2: Asymptotic Theory in Probability and Statistics with Applications ALM 3: Computational Conformal Geometry ALM 4: Variational Principles for Discrete Surfaces ALM 6: Geometry, Analysis and Topology of Discrete Groups ALM 7: Handbook of Geometric Analysis, No. 1 ALM 8: Recent Developments in Algebra and Related Areas ALM 9: Automorphic Forms and the Langlands Program ALM 10: Trends in Partial Differential Equations ALM 11: Recent Advances in Geometric Analysis ALM 12: Cohomology of Groups and Algebraic K-theory ALM 13: Handbook of Geometric Analysis, No. 2 ALM 14: Handbook of Geometric Analysis, No. 3 ALM 15: An Introduction to Groups and Lattices: Finite Groups and Positive Definite Rational Lattices ALM 16: Transformation Groups and Moduli Spaces of Curves ALM 17: Geometry and Analysis, No. 1 ALM 18: Geometry and Analysis, No. 2 ALM 19: Arithmetic Geometry and Automorphic Forms ALM 20: Surveys in Geometric Analysis and Relativity ALM 21: Advances in Geometric Analysis ALM 22: Differential Geometry: Under the Influence of S.-S. Chern ALM 23: Recent Developments in Geometry and Analysis ALM 24: Handbook of Moduli, Volume I ALM 25: Handbook of Moduli, Volume II ALM 26: Handbook of Moduli, Volume III Advanced Lectures in Mathematics Volume XXV Handbook of Moduli Volume II edited by Gavril Farkas · Ian Morrison International Press 浧䷘㟨十⒉䓗䯍 www.intlpress.com HIGHER EDUCATION PRESS Advanced Lectures in Mathematics, Volume XXV Handbook of Moduli, Volume II Volume Editors: Gavril Farkas (Humboldt-Universität, Berlin) Ian Morrison (Fordham University, New York) 2010 Mathematics Subject Classification. Primary: 14D20.
  • 1875–2012 Dr. Jan E. Wynn

    1875–2012 Dr. Jan E. Wynn

    HISTORY OF THE DEPARTMENT OF MATHEMATICS BRIGHAM YOUNG UNIVERSITY 1875–2012 DR. LYNN E. GARNER DR. GURCHARAN S. GILL DR. JAN E. WYNN Copyright © 2013, Department of Mathematics, Brigham Young University All rights reserved 2 Foreword In August 2012, the leadership of the Department of Mathematics of Brigham Young University requested the authors to compose a history of the department. The history that we had all heard was that the department had come into being in 1954, formed from the Physics Department, and with a physicist as the first chairman. This turned out to be partially true, in that the Department of Mathematics had been chaired by physicists until 1958, but it was referred to in the University Catalog as a department as early as 1904 and the first chairman was appointed in 1906. The authors were also part of the history of the department as professors of mathematics: Gurcharan S. Gill 1960–1999 Lynn E. Garner 1963–2007 Jan E. Wynn 1966–2000 Dr. Gill (1956–1958) and Dr. Garner (1960–1962) were also students in the department and hold B. S. degrees in Mathematics from BYU. We decided to address the history of the department by dividing it into three eras of quite different characteristics. The first era (1875–1978): Early development of the department as an entity, focusing on rapid growth during the administration of Kenneth L. Hillam as chairman. The second era (1978–1990): Efforts to bring the department in line with national standards in the mathematics community and to establish research capabilities, during the administration of Peter L.
  • Curriculum Vitae: Brendan Hassett

    Curriculum Vitae: Brendan Hassett

    Curriculum Vitae: Brendan Hassett Brown University brendan [email protected] Institute for Computational and Experimental Research in Mathematics 121 South Main Street, 11th Floor, Box E Providence, RI 02903 USA 401 863 7010 Department of Mathematics Box 1917 151 Thayer Street Providence, Rhode Island 02912 USA 401 863 7961 Positions Director, Institute for Computational and Experimental Research in Mathemat- ics, Brown University, from July 2016 Professor of Mathematics, from July 2015 Milton Brockett Porter Professor, Rice University, July 2013 to June 2015 Chair, Department of Mathematics, July 2009 to June 2014 Professor of Mathematics, July 2006 to June 2015 Associate Professor of Mathematics, July 2003 to June 2006 Assistant Professor of Mathematics, July 2000 to June 2003 Professeur Invit´e,Universit´eParis-Sud, Orsay, March to April 2005 Visiting Scholar, Institute of Mathematical Sciences, Chinese University of Hong Kong, August 2000 to July 2001 Dickson Instructor of Mathematics, University of Chicago, October 1996 to September 2000 Visitor at the Institut Mittag-Leffler, Stockholm, January to March 1997 Education Harvard University, M.A., 1994, and Ph.D., 1996 (supervised by Joe Harris) Yale College, B.A. in mathematics, summa cum laude, 1992 Awards Fellow of the American Mathematical Society, 2014 Charles W. Duncan Jr. Achievement Award for Outstanding Faculty, Rice Uni- versity, 2009 Grants and Fellowships Simons Foundation Award 815891: September 1, 2021{August 31, 2023; Simons Bridge for Postdoctoral Fellowships at ICERM
  • Arxiv:1506.07513V2 [Math.AG] 2 Aug 2016 Flggnrltp,Adamorphism a and Type, General Log of Olo Hspprt Etepofo H Olwn Theorem: Following the of Pa Proof of 1.1

    Arxiv:1506.07513V2 [Math.AG] 2 Aug 2016 Flggnrltp,Adamorphism a and Type, General Log of Olo Hspprt Etepofo H Olwn Theorem: Following the of Pa Proof of 1.1

    A FIBERED POWER THEOREM FOR PAIRS OF LOG GENERAL TYPE KENNETH ASCHER AND AMOS TURCHET Abstract. Let f : (X,D) → B be a stable family with log canonical general fiber. We prove that, after a birational modification of the base B → B, there is a morphism from a high fibered power of the family to a pair of log general type. If in addition the general fiber is openly canonical, then there is a morphisme from a high fibered power of the original family to a pair openly of log general type. 1. Introduction We work over an algebraically closed field of characteristic 0. Analyzing the geometry of fibered powers of families of varieties has been pivotal in showing that well known conjectures in Diophantine geometry imply uniform bounded- ness of rational points on algebraic varieties of general type. In 1997, Caporaso, Harris, and Mazur, show in their celebrated paper [CHM97], that various versions of Lang’s conjecture imply uniform boundedness of rational points on curves of general type. More precisely, they show that, assuming Lang’s conjecture, for every number field K and integer g ≥ 2, there exists an integer B(K,g) such that no smooth curve of genus g ≥ 2 defined over K has more than B(K,g) rational points. Similar statements were proven for the case of surfaces of general type by Hassett [Has96], and eventually all positive dimensional varieties of general type in a series of two papers by Abramovich and Abramovich-Voloch ([Abr97a] and [AV96]). The essence of these papers, is a purely algebro-geometric statement: the proof of a ”fibered power theorem”, which analyzes the behavior of families of varieties of general type.