DOCSLIB.ORG

Distribution of Rational Points on Algebraic Varieties

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Distribution of Rational Points on Algebraic Varieties Distribution of rational points on algebraic varieties by Sho Tanimoto A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Mathematics New York University May 2012 Professor Yuri Tschinkel To my wife, Mei. iii Acknowledgements First and foremost it is my pleasure to thank my advisor Yuri Tschinkel for his pa- tience and guidance throughout my Ph.D.. He has influenced my development from an ordinary math student to a professional researcher in mathematics. I learned from him, his patience, work ethic, creativity, and broad outlook in mathematics, and these have shaped me as a professional mathematician. I would also like to thank my second advisor Fedor Bogomolov. His passion, intuition, creativity, and the view of mathematics have inspired me in my research. His generosity has supported my study and life, and without it, I would not have completed this dissertation. I would like to thank Sylvain Cappell for his warmest support which he has given me since I was a first year student at Courant. Brendan Hassett is a great inspiration for my research, and I also benefited from conversations and discus- sions with Benjamin Bakker, David Harvey, Sonal Jain, Ilya Karzhemanov, Brian Lehmann, and Tony Varilly. Many thanks to my fellow students in our algebraic geometry and number theory group, Michael Burr, Edgar Costa, Lukas Koehler, Fedor Soloviev, Jordan Thomas, and Pankaj Vishe, for their insights, questions, help, and friendship. In particular, Michael and Jordan helped me to improve the exposition of this thesis, and Lukas did a fantastic job to activate our algebraic geometry seminar. I would also like to thank friends with whom I shared time as a graduate student at Courant, especially Andrea, Chaitu, Dingyu, Felix, Hantaek, Ivan, Jong Ho, Jungwoon, Kela, Nengli, Shoshana, and Sukbin. I owe a special debt of gratitude to Tamar Arnon for her warmest support throughout my Ph.D.. Going back to Japan, I would like to thank my teachers at my college, Takao iv Fujita, Shihoko Ishii, and Nobushige Kurokawa for their support and encourage- ment. I learned a great deal of algebraic geometry from my advisor Takao Fujita when I was an undergraduate student at Tokyo Institute of Technology. Ryoichi Aoki, my physics teacher at my high school, is one who let me consider a career path in academia. He always kept his door open for me, and he shared time with me to discuss my questions in physics. Finally, I thank my family: my parents, Shin and Mariko, my brother Ryo, and our lovely cats, Chao and Mie have supported me via skype for the last five years and provided me an ideal atmosphere when I was in Japan. And my wife Mei, for me, she is everything. She has supported and encouraged me all the time with her love via skype, phone, and shipments, and without her support, I could not have achieved any accomplishment which today I have. I thank her with all my love, and I wish the best luck for our continuing journey. v Abstract We study the asymptotic formulas for the number of rational points of bounded height on equivariant compactifications of solvable linear algebraic groups over a number field, and we introduce the notion of balanced line bundles. In the first part, we discuss Height zeta functions of equivariant compactifica- tions of solvable linear algebraic groups and show that the main term of a Height zeta function coincides with the prediction of Manin's conjecture by using geomet- ric integration techniques developed by Chambert-Loir and Tschinkel. Then we study Height zeta functions of equivariant compactifications of the ax + b-group and prove Manin's conjecture under some mild geometric conditions. In the second part, we introduce the notion of balanced line bundles and give a systematic study of this notion in terms of birational geometry. We identify balanced big line bundles for many cases, e.g., del Pezzo surfaces, Fano 3-folds of Picard rank one, flag varieties, toric varieties, and equivariant compactifications of homogeneous spaces. The last result has important arithmetic applications in [CLT02] and [GTBT11]. vi Contents Dedication . iii Acknowledgements . iv Abstract . vi Introduction . 1 I Height zeta functions 8 1 Height functions 9 1.1 Metrics and Heights . 9 1.2 Measures . 16 2 Manin's conjecture and Height zeta functions 22 2.1 Manin's conjecture . 23 2.2 Height zeta functions . 30 3 Height zeta functions of equivariant compactifications of semi- direct products of algebraic groups 37 3.1 Geometry of equivariant compactifications . 38 3.2 The main terms of Height zeta functions of equivariant compactifi- cations of solvable algebraic groups . 46 vii 3.3 Height zeta functions of equivariant compactifications of the ax + b- group . 64 II Balanced line bundles 99 4 Balanced line bundles 100 4.1 Generalities . 100 4.2 Balanced line bundles . 112 5 Fano varieties 115 5.1 Del Pezzo surfaces . 115 5.2 Del Pezzo surface fibrations . 120 5.3 Fano threefolds . 122 6 Equivariant geometry 125 6.1 Generalized flag varieties . 125 6.2 Equivariant compactifications of homogeneous spaces . 129 6.3 Toric varieties . 137 Bibliography 146 viii Introduction One of the classical problems in mathematics is to study integral or rational solutions to Diophantine equations, i.e., integral or rational solutions to polynomial equations with integer coefficients. The most famous Diophantine equations are the Fermat equations : xn + yn = zn; where n is a positive integer. There are many traditional questions concerning Diophantine equations, which include: 1. (Existence) Do integral or rational solutions exist? 2. (Finiteness) If solutions exist, are there finitely or infinitely many? 3. (Distribution) If there are infinitely many solutions, how are they distributed? For example, one could determine their density in various topologies or asymptotic formulae for the counting functions of solutions with respect to appropriate size functions. 4. (Construction) Construct some or all solutions effectively. The solutions to each of these problems have lead to intensive and long studies, and they are still active areas in modern mathematics. Although there are many elementary and ad hoc approaches to these questions, modern research in this field focuses on intrinsic properties of Diophantine equations, e.g., their geometry. 1 Diophantine geometry Note that answers to the above questions are stable under changes of coordi- nates. Perhaps this is the first observation to support the philosophy that \Geom- etry determines Arithmetic", appearing in [HS00]. The subject which studies the phenomena realizing this philosophy is called Diophantine geometry, and is one of main topics in current mathematics. One of the most important invariants in Diophantine geometry is the canonical line bundle KX of a smooth projective variety X, and we believe that this geometric invariant roughly determines the behavior of the distribution of rational points, at least after taking a finite extension. To illustrate this idea, consider the case of curves: Let F be a number field and C a smooth projective curve defined over F . The behavior of the distribution of rational points is determined by one geometric invariant, the genus g of C: • When g = 0, −KC is ample and C(F ) is either empty or infinite; • When g = 1, KC is trivial and C(F ) is either empty or a finitely generated abelian group. C(F ) is infinite if and only if the Mordell-Weil rank, the rank of C(F ), is positive. • When g ≥ 2, KC is ample and C(F ) is finite. (Mordell-Faltings theorem) In particular, after taking a finite extension of F , C(F ) is infinite when −KX is ample or trivial. Based on this evidence, we believe in the following principle which is the most important guiding principle in Diophantine geometry: The positivity of the canonical line bundle (or the anticanonical line bundle) determines the distribution of rational points on X. 2 We hope that the principle applies to higher dimensional cases as well, and thus we consider the following classification of smooth projective varieties: n • (Fano-type): −KX is ample, e.g., P , del Pezzo surfaces, and Fano manifolds; • (Intermediate type): KX is trivial, e.g., abelian varieties and K3 surfaces; • (General type): KX is ample, e.g., projective manifolds of general type. My research interest lies in Fano and intermediate type varieties which are expected to have infinitely many rational points, at least after taking a finite extension. More precisely, I am interested in the distribution of rational points on these varieties. Distribution of rational points There are many problems involving the distribution of rational points on al- gebraic varieties, but here we would like to mention two problems: Let X be a smooth projective variety defined over a number field F . One can ask the following two questions: • (Potential density): Is X(F ) dense in the Zariski topology, possibly after taking a finite extension of F ? • (Asymptotic formulae): Let L = (L; k · k) be a big, adelically metrized line bundle on X and HL the associated height (see Definitions 4 and 6 in Subsection 1.1.2). What is the asymptotic formula of the counting function of rational points of bounded height? In other words, determine ◦ ◦ N(X ; L; B) := #fx 2 X (F ) j HL(x) 6 Bg ∼ ??? as B ! 1, for a suitable Zariski open subset X◦. 3 The answer to the potential density problem is believed to be affirmative for Fano and Intermediate type varieties. It is possible that X(F ) is empty due to the arithmetic obstructions, e.g., the Hasse principle and the Brauer-Manin obstruc- tions (see, for example, [Man86]). To focus on the effect of geometry, it is essential to allow one to take a finite extension.
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]
  • 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]
  • 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]
  • 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]
  • Fano 3-Folds
    Fano 3-folds Shigeru MUKAI Abstract: In the beginning of this century, G. Fano initiated the study of 3-dimensional projective varieties g+1 1 X2g¡2 ½ P with canonical curve sections in connection with the LÄurothproblem. After a quick review of a modern treatment of Fano's approach (x1), we discuss a new approach to Fano 3-folds via vector bundles, which has revealed their relation to certain homogeneous spaces (xx2 and 3) and varieties of sums of powers (xx5 and 6). We also give a new proof of the gunus bound of prime Fano 3-folds (x4). In the maximum genus 13 3 (g = 12) case, Fano 3-folds X22 ½ P yield a 4-dimensional family of compacti¯cations of C (x8). A compact complex manifold X is Fano if its ¯rst Chern class c1(X) is positive, or equivalently, its anticanonical line bundle OX (¡KX ) is ample. If OX (¡KX ) is generated by global sections and ©j¡KX j is birational, then its image is called the anticanonical model of g¡1 X. In the case dim X = 3, every smooth curve section C = X \ H1 \ H2 ½ P of the anticanonical model X ½ Pg+1 is canonical, that is, embedded by the canonical linear system g+1 jKC j. Conversely, every projective 3-fold X2g¡2 ½ P with a canonical curve section is 1 ¡ 3 obtained in this way. The integer 2 ( KX ) + 1 is called the genus of a Fano 3-fold X since it is equal to the genus g of a curve section of the anticanonical model.
    [Show full text]
  • Algebraic Geometry of Curves
    Algebraic Geometry of Curves Andrew Kobin Fall 2016 Contents Contents Contents 0 Introduction 1 0.1 Geometry and Number Theory . .2 0.2 Rational Curves . .4 0.3 The p-adic Numbers . .6 1 Algebraic Geometry 9 1.1 Affine and Projective Space . .9 1.2 Morphisms of Affine Varieties . 15 1.3 Morphisms of Projective Varieties . 19 1.4 Products of Varieties . 21 1.5 Blowing Up . 22 1.6 Dimension of Varieties . 24 1.7 Complete Varieties . 26 1.8 Tangent Space . 27 1.9 Local Parameters . 32 2 Curves 33 2.1 Divisors . 34 2.2 Morphisms Between Curves . 36 2.3 Linear Equivalence . 38 2.4 Differentials . 40 2.5 The Riemann-Hurwitz Formula . 42 2.6 The Riemann-Roch Theorem . 43 2.7 The Canonical Map . 45 2.8 B´ezout'sTheorem . 45 2.9 Rational Points of Conics . 47 3 Elliptic Curves 52 3.1 Weierstrass Equations . 53 3.2 Moduli Spaces . 55 3.3 The Group Law . 56 3.4 The Jacobian . 58 4 Rational Points on Elliptic Curves 61 4.1 Isogenies . 62 4.2 The Dual Isogeny . 65 4.3 The Weil Conjectures . 67 4.4 Elliptic Curves over Local Fields . 69 4.5 Jacobians of Hyperelliptic Curves . 75 5 The Mordell-Weil Theorem 77 5.1 Some Galois Cohomology . 77 5.2 Selmer and Tate-Shafarevich Groups . 80 5.3 Twists, Covers and Principal Homogeneous Spaces . 84 i Contents Contents 5.4 Descent . 89 5.5 Heights . 97 6 Elliptic Curves and Complex Analysis 99 6.1 Elliptic Functions . 99 6.2 Elliptic Curves .
    [Show full text]
  • Fano 3-Folds. Ii Udc 513.6 V
    H3B.AKcm.HayK CCCP Math. USSR Izvestija Cep.Ma-r.ToM 42(1978), Bun. 3 Vol. 12 (1978), No. 3 FANO 3-FOLDS. II UDC 513.6 V. A. ISKOVSKIH Abstract. In this paper Fano 3-folds of the principal series Vjg—2 'n P^ are studied. A classification is given of trivial (i.e. containing a trigonal canonical curve) 3-folds of this kind. Among all Fano 3-folds of the principal series these are distinguished by the property that they are not the intersection of the quadrics containing them. It turns out that the ge- nus g of such 3-folds does not exceed 10. Fano 3-folds of genus one (i.e. with Pic V ~ Z) containing a line are described. It is proved that they exist for g < 10 and g = 12. Their rationality for g = 1 and g > 9 is established by direct construction. Bibliography: 21 titles. Introduction This is a continuation of our previous article, Part I [8]. In this part we will study +1 Fano 3-folds of the principal series (that is, smooth 3-folds V2g_2 C P^ of degree 2g - 2) such that the anticanonical class ~KV is the class of the hyperplane section. The ground field k is algebraically closed, and although this hypothesis is not essential in this article, we will assume that k has characteristic char k = 0; we mainly require this to be able to refer to Part I. In § 1 we give a new treatment of certain classical results relating to the elementary properties of Fano 3-folds of the principal series.
    [Show full text]
  • LECTURE 1. DEFINITIONS and FIRST EXAMPLES in This Lecture We Introduce the Varieties That Are Rational Or Close to Being Rationa
    LECTURE 1. DEFINITIONS AND FIRST EXAMPLES In this lecture we introduce the varieties that are rational or close to being rational and discuss some first easy examples. 1. Rational and unirational varieties: definitions Let k be a fixed field. We will be mostly interested in the case when k is algebraically closed. However, for certain key arguments, it is important to have the definitions and some basic facts developed over arbitrary fields. We will also be mostly interested in the case when char(k) = 0. Whenever this simplifies the arguments, we will restrict to this case. A variety over k is a scheme of finite type over k, which is integral and separated. n Definition 1.1. A variety X over k is rational (over k) if it is birational to Pk (of course, in this case n = dim(X)); equivalently, the function field k(X) is isomorphic over k to the purely transcendental extension k(t1; : : : ; tn). It is clear from definition that in order to determine whether X is rational, we may replace X by any variety that is birational to X. For example, we may assume that X is affine. However, this is rarely useful, as most known techniques are of global nature. It is more useful to replace X by a projective model (after taking an affine open subset and taking the closure in a suitable projective case). When char(k) = 0, it follows from Hironaka's theorem on resolution of singularities that we may replace X by a smooth, projective variety in the same birational equivalence class.
    [Show full text]
  • Rational and Non-Rational Algebraic Varieties: Lectures of J\'Anos Koll\'Ar
    RATIONAL AND NON-RATIONAL ALGEBRAIC VARIETIES: LECTURES OF JANOS´ KOLLAR´ By Karen E. Smith with an Appendix by Joel Rosenberg July 14, 1997 Introduction Rational varieties are among the simplest possible algebraic varieties. Their study is as old as algebraic geometry itself, yet it remains a remarkably difficult area of research today. These notes offer an introduction to the study of rational varieties. We begin with the beautiful classical geometric approach to finding ex- amples of rational varieties, and end with some subtle algebraic arguments that have recently established non-rationality of varieties that otherwise share many of their traits. In lecture one, rationality and unirationality are defined, and illustrated with a series of examples. We also introduce some easily computable invariants, the pluri- genera, that vanish for all rational varieties. Using the plurigenera, one develops a sense how rare rational varieties are. For example, we will immediately see that no smooth projective hypersurface whose degree exceeds its embedding dimension can be rational: no plane cubic curve is rational, no space quartic surface is rational, and so on. arXiv:alg-geom/9707013v1 15 Jul 1997 The second and third lectures focus on the rationality question for smooth cubic surfaces over arbitrary fields, an issue thoroughly explored by B. Segre in the forties. This is a pretty story, depending subtly on the field of definition. It was already understood one hundred years ago that every cubic surface is rational over the complex numbers; however, the situation is quite complicated over the rational numbers. In the second lecture, we construct examples of non-rational smooth cubics over Q by considering the orbits of the Galois group of Q¯ /Q on the twenty seven lines on the cubic surface.
    [Show full text]