DOCSLIB.ORG

Topics in Algebraic Geometry II. Rationality of Algebraic Varieties

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Topics in Algebraic Geometry II. Rationality of Algebraic Varieties Math 732: Topics in Algebraic Geometry II Rationality of Algebraic Varieties Mircea Mustat¸˘a∗ Winter 2017 Course Description A fundamental problem in algebraic geometry is to determine which varieties are rational, that is, birational to the projective space. Several important developments in the field have been motivated by this question. The main goal of the course is to describe two recent directions of study in this area. One approach goes back to Iskovskikh and Manin, who proved that smooth, 3-dimensional quartic hypersurfaces are not rational. This relies on ideas and methods from higher-dimensional birational geometry. The second approach, based on recent work of Claire Voisin and many other people, relies on a systematic use of decomposition of the diagonal and invariants such as Chow groups, Brauer groups, etc. to prove irrationality. Both directions will give us motivation to introduce and discuss some important concepts and results in algebraic geometry. The following is a rough outline of the course: Introduction 1. Rationality in dimension 2 (Castelnuovo's criterion). Part I 2. Introduction to Chow groups and intersection theory. 3. Decomposition of the diagonal and non-stable-rationality. 4. Examples of classes on non-stably-rational varieties. Part II 5. Introduction to singularities of pairs and vanishing theorems. 6. Birational rigidity of Fano hypersurfaces of index 1. Mircea Mustat¸˘a'sown lecture notes are available at: http://www-personal.umich.edu/~mmustata/lectures_rationality.html Contents 1 January 5 5 1.1 The L¨urothproblem and its solution for curves..........................5 1.2 Birational invariance of plurigenera.................................7 2 January 10 9 2.1 Birational invariance of plurigenera (continued)..........................9 2.2 Other properties close to rationality................................ 11 2.2.1 Stably rational varieties................................... 11 2.2.2 Uniruled varieties....................................... 11 2.3 Cubic surfaces............................................ 12 ∗Notes were taken by Takumi Murayama, who is responsible for any and all errors. Please e-mail [email protected] with any corrections. Compiled on April 29, 2017. 1 3 January 12 14 3.1 Cubic surfaces (continued)...................................... 14 3.2 Some rational cubic hypersurfaces................................. 17 4 January 17 18 4.1 Some rational cubic hypersurfaces (continued)........................... 18 4.2 Unirationality of cubic hypersurfaces................................ 21 4.2.1 Review of projections.................................... 22 5 January 19 23 5.1 Unirationality of cubic hypersurfaces (continued)......................... 23 5.1.1 Projections.......................................... 23 5.1.2 Quadric bundles....................................... 26 5.1.3 Proof of Theorem 5.1.................................... 27 6 January 24 28 6.1 Unirationality of cubic hypersurfaces (continued)......................... 28 6.1.1 Proof of Theorem 5.1 (continued).............................. 28 6.2 More on incidence correspondences................................. 29 7 January 26 32 7.1 Comments on Theorem 5.1..................................... 32 7.2 Varieties containing linear subspaces................................ 33 7.3 Unirationality of hypersurfaces of higher degree.......................... 36 8 January 31 36 8.1 General statements about (uni)rationality............................. 36 8.2 Unirationality of hypersurfaces of higher degree (continued)................... 37 9 February 2 40 9.1 Overview of topology of algebraic varieties............................. 40 9.1.1 Singular (co)homology.................................... 40 9.1.2 Complex algebraic varieties................................. 43 10 February 14 45 10.1 Overview of topology of algebraic varieties (continued)...................... 45 10.2 Rationality and unirationality for surfaces............................. 45 11 February 16 49 11.1 Rationality and unirationality for surfaces (continued)...................... 49 12 February 21 53 12.1 Shioda's examples of unirational but not rational surfaces.................... 53 12.2 More topology............................................ 55 13 February 23 58 13.1 Some cohomology computations................................... 58 13.2 An invariant that detects non-stable-rationality.......................... 59 3 13.3 How to produce examples with H (X; Z)tors 6= 0......................... 60 14 March 7 61 14.1 The Artin{Mumford example.................................... 61 14.1.1 The complete linear system of quadric hypersurfaces in P3 ............... 61 14.1.2 Geometry of the map p in (14:1).............................. 63 14.1.3 Introduction to cyclic covers................................. 64 2 15 March 9 65 15.1 The Artin{Mumford example (continued)............................. 65 16 March 21 68 16.1 The Artin{Mumford example (continued)............................. 69 16.2 Introduction to ´etalecohomology.................................. 71 16.2.1 Presheaves and sheaves in the ´etaletopology....................... 71 17 March 23 73 17.1 Introduction to ´etalecohomology (continued)........................... 73 17.1.1 Some statements from descent theory........................... 73 17.1.2 Presheaves and sheaves in the ´etalecohomology (continued)............... 74 17.2 Brauer group............................................. 75 17.2.1 Connection with singular cohomology........................... 75 17.2.2 Brauer groups and Pn-fibrations.............................. 76 18 March 28 77 18.1 Brauer groups (continued)...................................... 77 18.1.1 Brauer groups and Pn-fibrations (continued)....................... 77 18.2 Introduction to first cohomology of non-abelian sheaves..................... 78 19 March 30 81 19.1 Introduction to Chow groups.................................... 81 19.1.1 Operations on Chow groups................................. 82 19.1.2 Chern classes......................................... 85 20 April 4 86 20.1 Introduction to Chow groups (continued)............................. 86 20.1.1 Refined Gysin maps..................................... 86 20.1.2 Pullback by lci morphisms.................................. 88 20.1.3 Connection with singular cohomology........................... 89 20.2 Correspondences........................................... 90 21 April 6 91 21.1 Correspondences (continued).................................... 91 21.1.1 Actions of correspondences on Chow groups........................ 91 21.1.2 CH0 is a birational invariant................................ 92 21.1.3 Actions of correspondences on singular cohomology.................... 93 21.1.4 The specialization map................................... 94 22 April 11 95 22.1 Decomposition of the diagonal................................... 95 22.1.1 The Bloch{Srinivas theorem and stable rationality.................... 96 23 April 13 98 23.1 Decomposition of the diagonal (continued)............................. 98 23.1.1 CH 0-triviality for morphisms................................ 101 24 April 18 102 24.1 Decomposition of the diagonal (continued)............................. 102 24.1.1 A criterion for universal CH 0-triviality for morphisms.................. 102 24.1.2 Behavior of decomposition of the diagonal in families................... 103 3 25 April 25 106 25.1 Non-stable-rationality of quartic threefolds............................. 106 25.1.1 Rationality and base change................................. 106 25.1.2 Proof of Theorem 25.1.................................... 107 25.1.3 The geometry of quartic symmetroids........................... 108 25.1.4 The construction for Theorem 25.2............................. 110 References 111 4 1 January 5 The theory of rationality centers around the question: Which varieties are rational, i.e., birational to the projective space Pn? This theory is very interesting because it uses vastly different methods. On one hand, showing something is rational usually requires a lot of explicit geometric constructions that are classical in flavor. On the other hand, showing something is not rational requires more complicated invariants like Chow groups, etc., many of them arising from birational geometry. We first give a brief outline of the course: Introduction: Some definitions and easy properties; examples In particular, we will study surfaces (Castelnuovo's rationality criterion), cubic three-folds and quartic three-folds, and the unirationality of hypersurfaces of small degree. Note that for these last examples, it is already hard to show they are not rational. These examples will build intuition and help you develop tools to study more complicated cases. Part I: Stable rationality We will review some things about Chow groups and intersection theory, and then study the theory of \decomposition of the diagonal" and how it relates to stable rationality, following ideas of Voisin and others. Part II: Birational rigidity of certain Fano varieties A key example here is that of hypersurfaces of degree n in Pn, which are \extremal" Fano varieties. Birationally rigid varieties are those for which birational self-maps can all be extended to automorphisms; for the aforementioned hypersurfaces, the group of birational self-maps will be finite, and so they cannot be rational. The main ideas in this theory come from Mori theory, and so we will need to talk about invariants of singularities and vanishing
Load more

Recommended publications
  • THE CUBIC THREEFOLD and FRIENDS 1. Background On
    THE CUBIC THREEFOLD AND FRIENDS 1. Background on Threefolds Fano, around the 1910s, proved that any smooth quartic threefold • is not rational. Later on, in the 1950s, Roth criticised the proof as incomplete. In 1971, Iskovskikh and Manin provide a complete proof. In 1972, Artin and Mumford gave an example of another unirational, • but not rational Fano threefold. This is a certain double cover X of P3 ramified over a singular quartic surface. They showed that H3(X; Z) is a birational invariant and obtained that H3(X; Z) = Z2, from which they concluded that X cannot be rational. At the same time, Clemens and Griffiths showed that any smooth • cubic threefold V over a field of characteristic zero is unirational, but not rational. In the same year, Murre proved the result in char- acteristic p. The idea of Clemens and Griffiths was to consider two auxiliary varieties of a smooth cubic threefold V: The intermediate Jacobian J(V) - a principally polarised abelian variety • playing a role similar to that of the Jacobian for studying divisors on curves. the Fano variety of lines F(V) - a smooth projective surface parametris- • ing lines on V. In order to arrive at the result, they represented J(V) as the Albanese variety of F(V), and studied its theta divisors. 2. The Cohomology of the Cubic Threefold As before, let V be a cubic threefold, i.e. a hypersurface of degree three in P4. A first idea is to compare the cohomology of V with that of P3 and hope that we shall thus find a birational invariant that has different values for V and P3, respectively.
    [Show full text]
  • Moduli Spaces of Quadratic Rational Maps with a Marked Periodic Point
    J. Blanc et al. (2015) “Quadratic Maps with a Marked Periodic Point of Small Order,” International Mathematics Research Notices, Vol. 2015, No. 23, pp. 12459–12489 Advance Access Publication March 10, 2015 doi:10.1093/imrn/rnv063 Moduli Spaces of Quadratic Rational Maps with a Marked Periodic Point of Small Order Downloaded from https://academic.oup.com/imrn/article/2015/23/12459/672405 by guest on 27 September 2021 Jer´ emy´ Blanc1, Jung Kyu Canci1, and Noam D. Elkies2 1Mathematisches Institut, Universitat¨ Basel, Spiegelgasse 1, CH-4051 Basel, Switzerland and 2Department of Mathematics, Harvard University, Cambridge, MA 02138, USA Correspondence to be sent to: [email protected] The surface corresponding to the moduli space of quadratic endomorphisms of P1 with a marked periodic point of order nis studied. It is shown that the surface is rational over Q when n≤ 5 and is of general type for n= 6. An explicit description of the n= 6 surface lets us find several infinite families of quadratic endomorphisms f : P1 → P1 defined over Q with a rational periodic point of order 6. In one of these families, f also has a rational fixed point, for a total of at least 7 periodic and 7 preperiodic points. This is in contrast with the polynomial case, where it is conjectured that no polynomial endomorphism defined over Q admits rational periodic points of order n> 3. 1 Introduction A classical question in arithmetic dynamics concerns periodic, and more generally preperiodic, points of a rational map (endomorphism) f : P1(Q) → P1(Q).Apointp is said to be periodic of order n if f n(p) = p and if f i(p) = p for 0 < i < n.Apointp is said to be preperiodic if there exists a non-negative integer m such that the point f m(p) is periodic.
    [Show full text]
  • 'Models of Brauer-Severi Surface Bundles'
    MODELS OF BRAUER-SEVERI SURFACE BUNDLES ANDREW KRESCH AND YURI TSCHINKEL 1. Introduction This paper is motivated by the study of rationality properties of Mori fiber spaces. These are algebraic varieties, naturally occurring in the min- imal model program, typically via contractions along extremal rays; they are fibrations with geometrically rational generic fiber. Widely studied are conic bundles π : X S, when the generic fiber is a conic. According to the Sarkisov program→ [60], there exists a birational modification X ❴❴❴ / X π˜ π e S / S such that: (i) the general fiber ofπ ˜ is a smooth conic, (ii) the discriminant divisor ofπ ˜ is a simple normal crossinge divisor, (iii) generally along the discriminant divisor, the fiber is a union of two lines, and (iv) over the singular locus of the discriminant divisor, the fiber is a double line in the plane. Rationality of conic bundles over surfaces has been classically stud- ied by Artin and Mumford [11], who produced examples of nonrational unirational threefolds of this type, computing their Brauer groups and using its nontriviality as an obstruction to rationality. This was gen- eralized to higher-dimensional quadric bundles in [20], bringing higher unramified cohomology into the subject and providing further examples arXiv:1708.06277v1 [math.AG] 21 Aug 2017 of nonrational varieties based on these new obstructions. The specialization method, introduced by Voisin [64] and developed further by Colliot-Th´el`ene–Pirutka [21], emerged as a powerful tool in the study of stable rationality. It allows to relate the failure of stable rationality of a very general member of a flat family to the existence of special fibers with nontrivial unramified cohomology and mild singular- ities; see also [13], [63].
    [Show full text]
  • RATIONAL HOMOGENEOUS VARIETIES Giorgio Ottaviani
    RATIONAL HOMOGENEOUS VARIETIES Giorgio Ottaviani Dipartimento di Matematica, Universit`adell'Aquila Current address: Dipartimento di Matematica, Universit`adi Firenze viale Morgagni 67/A 50134 FIRENZE [email protected]fi.it x1. Introduction page 1 x2. Grassmannians and flag varieties 5 x3. Lie algebras and Lie groups 9 x4. The Borel fixed point theorem 18 x5. SL(2) 22 x6. The Cartan decomposition 26 x7. Borel and parabolic subgroups 37 x8. ABC about bundles 40 x9. Homogeneous bundles 47 x10. The theorem of Borel-Weil 51 x11. The theorem of Bott 61 x12. Stability of homogeneous bundles 65 References 69 These notes have been written for distribution to the participants to the summer school in Algebraic Geometry organized by the Scuola Matematica Universitaria in Cortona in the period 13-26 August 1995. The aim was to describe the classification of rational homogeneous varieties and to provide the theorems of Borel-Weil and Bott. Lie algebras are introduced starting from the definition and they are studied as far as it is necessary for the aim. At the other side, the knowledge of basic techniques of algebraic geometry (as sheaf cohomology) and algebraic topology was assumed. I learned most of the material of these notes from many lectures and discussions with Vincenzo Ancona and Alan Huckleberry. I am sincerely grateful to both of them and also to all the participants to the course in Cortona, especially to Raffaella Paoletti and Anke Simon for careful proofreading and for their suggestions. x1. Introduction An algebraic variety is a quasiprojective variety over the field C of complex numbers.
    [Show full text]
  • 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.
    [Show full text]
  • On the Density of Rational Points on Elliptic Fibrations
    On the density of rational points on elliptic fibrations F. A. Bogomolov Courant Institute of Mathematical Sciences, N.Y.U. 251 Mercer str. New York, (NY) 10012, U.S.A. e-mail: [email protected] and Yu. Tschinkel Dept. of Mathematics, U.I.C. 851 South Morgan str. Chicago, (IL) 60607-7045, U.S.A. e-mail: [email protected] arXiv:math/9811043v1 [math.AG] 7 Nov 1998 1 Introduction Let X be an algebraic variety defined over a number field F . We will say that rational points are potentially dense if there exists a finite extension K/F such that the set of K-rational points X(K) is Zariski dense in X. The main problem is to relate this property to geometric invariants of X. Hypothetically, on varieties of general type rational points are not potentially dense. In this paper we are interested in smooth projective varieties such that neither they nor their unramified coverings admit a dominant map onto varieties of general type. For these varieties it seems plausible to expect that rational points are potentially dense (see [2]). Varieties which are not of general type can be thought of as triple fibra- tions X → Y → Z, where the generic fiber of X → Y is rationally connected, Y → Z is a Kodaira fibration with generic fiber of Kodaira dimension 0 and the base has Kodaira dimension ≤ 0 (cf. [3]). In this paper we study mostly varieties of dimension 2 and 3. In dimension 2 the picture is as follows: Rationally connected surfaces are rational over some finite extension of F and therefore the problem has an easy solution in this case.
    [Show full text]
  • Contemporary Mathematics 334
    CONTEMPORARY MATHEMATICS 334 Topics in Algebraic Geometry and Geometric Modeling Workshop on Algebraic Geometry and Geometric Modeling July 29- August 2, 2002 Vilnius University Vilnius, Lithuania Ron Goldman Rimvydas Krasauskas Editors http://dx.doi.org/10.1090/conm/334 CoNTEMPORARY MATHEMATICS 334 Topics in Algebraic Geometry and Geometric Modeling Workshop on Algebraic Geometry and Geometric Modeling July 29- August 2, 2002 Vilnius University Vilnius, Lithuania Ron Goldman Rimvydas Krasauskas Editors American Mathematical Society Providence, Rhode Island Editorial Board Dennis DeThrck, managing editor Andreas Blass Andy R. Magid Michael Vogelius 2000 Mathematics Subject Classification. Primary 14M25, 14Qxx, 52B20, 65Dxx, 68U07; Secondary 13D02, 13P10, 14E20, 41A25, 58D99. Library of Congress Cataloging-in-Publication Data Workshop on Algebraic Geometry and Geometric Modeling (2002: Vilnius University) Topics in algebraic geometry and geometric modeling : Workshop on Algebraic Geometry and Geometric Modeling, July 29-August 2, 2002, Vilnius University, Lithuania/ Ron Goldman, Rimvydas Krasauskas, editors. p. em. -(Contemporary mathematics, ISSN 0271-4132; 334) Includes bibliographical references and index. ISBN 0-8218-3420-7 (alk. paper) 1. Curves on surfaces-Mathematical models-Congresses. 2. Algebraic geometry-Congresses. I. Goldman, Ron, 1947- II. Krasauskas, Rimvydas, 1958- III. Title. IV. Series. QA565.W76 2002 516.3'5-dc22 2003055938 Copying and reprinting. Material in this book may be reproduced by any means for edu- cational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledg- ment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale.
    [Show full text]
  • Unramified Brauer Groups of Conic Bundle Threefolds in Characteristic Two
    UNRAMIFIED BRAUER GROUPS OF CONIC BUNDLE THREEFOLDS IN CHARACTERISTIC TWO ASHER AUEL, ALESSANDRO BIGAZZI, CHRISTIAN BOHNING,¨ AND HANS-CHRISTIAN GRAF VON BOTHMER Abstract. We establish a formula for computing the unramified Brauer group of tame conic bundle threefolds in characteristic 2. The formula depends on the arrangement and residue double covers of the discriminant components, the latter being governed by Artin{Schreier theory (instead of Kummer theory in characteristic not 2). We use this to give new examples of threefold conic bundles defined over Z that are not stably rational over C. 1. Introduction Motivated by the rationality problem in algebraic geometry, we compute new obstructions to the universal triviality of the Chow group of 0-cycles of smooth projective varieties in characteristic p > 0, related to ideas coming from crystalline cohomology (see [CL98] for a survey): the p-torsion in the unramified Brauer group. In [ABBB18], we proved that p-torsion Brauer classes do obstruct the universal triv- iality of the Chow group of 0-cycles; here we focus on computing these obstructions in the case of conic bundles. In particular, we provide a formula to compute the two torsion in the unramified Brauer group of conic bundle threefolds in character- istic 2. We provide some applications showing that one can obtain results with this type of obstruction that one cannot by other means: there exist conic bundles over P2, defined over Z, that are smooth over Q and whose reduction modulo p has (1) nontrivial two torsion in the unramified Brauer group and a universally CH0-trivial resolution for p = 2, and (2) irreducible discriminant, hence trivial Brauer group, for all p > 2.
    [Show full text]
  • The Grothendieck Ring of Varieties and Piecewise Isomorphisms
    Math. Z. DOI 10.1007/s00209-009-0518-7 Mathematische Zeitschrift The Grothendieck ring of varieties and piecewise isomorphisms Qing Liu · Julien Sebag Received: 15 February 2008 / Accepted: 29 January 2009 © Springer-Verlag 2009 Abstract Let K0(Var k) be the Grothendieck ring of algebraic varieties over a field k.Let X, Y be two algebraic varieties over k which are piecewise isomorphic (i.e. X and Y admit finite partitions X1,...,Xn, Y1,...,Yn into locally closed subvarieties such that Xi is iso- morphic to Yi for all i ≤ n), then [X]=[Y ] in K0(Var k ). Larsen and Lunts ask whether the converse is true. For characteristic zero and algebraically closed field k, we answer positively this question when dim X ≤ 1orX is a smooth connected projective surface or if X contains only finitely many rational curves. 1 Introduction The main topic of this article is to study the Grothendieck ring of algebraic varieties K0(Var k) over a field k. Appeared in a letter of Grothendieck to Serre ([5], letter of 16 Aug. 1964), this ring has been deeply used for developing the theorie(s) of motivic integration. But we know very little about this ring. For example, Poonen ([33], Theorem 1) and Kollár ([21], Example 6) show that this ring is not a domain when k has characteristic zero (see also Corollary 9), and Niko Naumann ([29], Theorem 22) provides zero divisors for K0(Var k) over finite fields k. On the opposite, one can construct infinite family of classes in K0(Var k) which are algebraically independent over Z (see references in Remark 8).
    [Show full text]
  • 19 Aug 2018 Non-Rational Varieties with the Hilbert Property
    Non-rational varieties with the Hilbert Property Julian Lawrence Demeio August 21, 2018 Abstract A variety X over a field k is said to have the Hilbert Property if X(k) is not thin. We shall exhibit some examples of varieties, for which the Hilbert Property is a new result. We give a sufficient condition for descending the Hilbert Property to the quotient of a variety by the action of a finite group. Applying this result to linear actions of groups, we exhibit some examples of non-rational unirational varieties with the Hilbert Property, providing positive instances of a conjecture posed by Colliot-Th´el`ene and Sansuc. We also give a sufficient condition for a surface with two elliptic fibrations to have the Hilbert Property, and use it to prove that a certain class of K3 surfaces have the Hilbert Property, generalizing a result of Corvaja e Zannier. 1 Introduction In 1917, Emily Noether tried to approach the Inverse Galois Problem by using the Hilbert Irreducibility Theorem [Ser08, Thm. 3.4.1]. Her strategy relied on her G rationality conjecture that every field of the form k(x1,...,xk) , G being a finite group acting by permutation on the variables xi, is rational. It is now well known that this conjecture is false [Swa69]. The Hilbert Property was introduced by Colliot-Th´el`ene and Sansuc in [CTS87], in an effort to try to recover the original strategy of Noether, by generalizing the arXiv:1801.10413v3 [math.AG] 19 Aug 2018 classical Hilbert Irreducibility Theorem. A geometrically irreducible variety X over a field k is said to have the Hilbert Property if, for any finite morphism π : E X, such that X(k) π(E(k)) is not Zariski-dense in X, there exists a rational→ section of π (see [Ser08\, Ch.
    [Show full text]
  • Download (1570Kb)
    A Thesis Submitted for the Degree of PhD at the University of Warwick Permanent WRAP URL: http://wrap.warwick.ac.uk/148579 Copyright and reuse: This thesis is made available online and is protected by original copyright. Please scroll down to view the document itself. Please refer to the repository record for this item for information to help you to cite it. Our policy information is available from the repository home page. For more information, please contact the WRAP Team at: [email protected] warwick.ac.uk/lib-publications Stable rationality and degenerations of conic bundles Author: Supervisor: Alessandro Bigazzi Christian Böhning Thesis submitted to the University of Warwick for the degree of Doctor of Philosophy Department of Mathematics March 2020 Contents Acknowledgments. iii Declaration. v Abstract. vii Introduction. ix Chapter 1. Nearly rational varieties and the stable Lüroth problem. 1 1.1. Rationality and unirationality. 1 1.2. The stable Lüroth problem. 3 1.3. Currently known results. 5 Chapter 2. Chow-theoretic invariants and the degeneration method. 11 2.1. Decomposition of the diagonal and zero-cycles. 11 2.2. Mild desingularisation 13 2.3. The degeneration method. 14 2.4. Further generalisation: semi-stable degeneration method. 16 Chapter 3. Brauer groups, unramified invariants and applications to conic bundles. 25 3.1. Brauer group of fields. 25 3.2. Brauer group of schemes. 33 3.3. Residue maps and unramified cohomology. 38 3.4. Conic bundles. 44 3.5. The case of cubic threefold hypersurfaces. 56 Chapter 4. Purity and universal triviality for the p-torsion of the Brauer group in characteristic p.
    [Show full text]
  • Igor R. Shafarevich Varieties in Projective Space Third Edition
    Igor R. Shafarevich Basic Algebraic Geometry 1 Varieties in Projective Space Third Edition Basic Algebraic Geometry 1 Igor R. Shafarevich Basic Algebraic Geometry 1 Varieties in Projective Space Third Edition Igor R. Shafarevich Translator Algebra Section Miles Reid Steklov Mathematical Institute Mathematics Institute of the Russian Academy of Sciences University of Warwick Moscow, Russia Coventry, UK ISBN 978-3-642-37955-0 ISBN 978-3-642-37956-7 (eBook) DOI 10.1007/978-3-642-37956-7 Springer Heidelberg New York Dordrecht London Library of Congress Control Number: 2013945284 Mathematics Subject Classification (2010): 14-01 Translation of the 3rd Russian edition entitled “Osnovy algebraicheskoj geometrii”. MCCME, Moscow 2007, originally published in Russian in one volume © Springer-Verlag Berlin Heidelberg 1977, 1994, 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer.
    [Show full text]