Algebraic geometry Books

Book SynopsisThis is a two-volume collection presenting the selected works of Herbert Busemann, one of the leading geometers of the twentieth century and one of the main founders of metric geometry, convexity theory and convexity in metric spaces. Busemann also did substantial work (probably the most important) on Hilbert’s Problem IV. These collected works include Busemann’s most important published articles on these topics. Volume I of the collection features Busemann’s papers on the foundations of geodesic spaces and on the metric geometry of Finsler spaces. Volume II includes Busemann’s papers on convexity and integral geometry, on Hilbert’s Problem IV, and other papers on miscellaneous subjects. Each volume offers biographical documents and introductory essays on Busemann’s work, documents from his correspondence and introductory essays written by leading specialists on Busemann’s work. They are a valuable resource for researchers in synthetic and metric geometry, convexity theory and the foundations of geometry. Table of ContentsPreface.- Introduction to Volume I.- List of publications of Herbert Busemann.- Acknowledgements.- Essays.- A. Papadpoulos: Herbert Busemann (1905-1994).- A. Papadopoulos and M. Troyanov: On three early papers by Herbert Busemann on the foundations of geometry.- M. Troyanov: On Pasch's Axiom and Desargues' Theorem in Busemann's work.- V. N. Berestovskiy: Busemann's results, ideas, questions on locally compact homogeneous geodesic spaces.- A. Papadopoulos and S. Yamada: Busemann's problems on G-spaces.- Busemann's metric theory of timelike spaces.- A. Papadopoulos: Chronogeometry.- W. M. Boothby: Review of Busemann's book The geometry of Geodesics.- F. A. Ficken: Review of Busemann's book Metric Methods in Finsler Spaces and in the Foundations of Geometry.- Busemann's papers on the foundations of geodesic spaces and on the metric geometry of Finsler spaces.

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Book SynopsisThis two volume work on Positivity in Algebraic Geometry contains a contemporary account of a body of work in complex algebraic geometry loosely centered around the theme of positivity. Topics in Volume I include ample line bundles and linear series on a projective variety, the classical theorems of Lefschetz and Bertini and their modern outgrowths, vanishing theorems, and local positivity. Volume II begins with a survey of positivity for vector bundles, and moves on to a systematic development of the theory of multiplier ideals and their applications. A good deal of this material has not previously appeared in book form, and substantial parts are worked out here in detail for the first time. At least a third of the book is devoted to concrete examples, applications, and pointers to further developments. Volume I is more elementary than Volume II, and, for the most part, it can be read without access to Volume II. Table of ContentsNotation and Conventions.- One: Ample Line Bundles and Linear Series.- to Part One.- 1 Ample and Nef Line Bundles.- 2 Linear Series.- 3 Geometric Manifestations of Positivity.- 4 Vanishing Theorems.- 5 Local Positivity.- Appendices.- A Projective Bundles.- B Cohomology and Complexes.- B.1 Cohomology.- B.2 Complexes.- References.- Glossary of Notation.

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Book SynopsisTwo volume work containing a contemporary account on "Positivity in Algebraic Geometry". Both volumes also available as hardcover editions as Vols. 48 and 49 in the series "Ergebnisse der Mathematik und ihrer Grenzgebiete". A good deal of the material has not previously appeared in book form. Volume II is more at the research level and somewhat more specialized than Volume I. Volume II contains a survey of positivity for vector bundles, and moves on to a systematic development of the theory of multiplier ideals and their applications. Contains many concrete examples, applications, and pointers to further developmentsTrade ReviewFrom the reviews: "The main theme of this ... monograph is a comprehensive description of the fields of complex algebraic geometry connected with the notion of positivity. ... The book is written for mathematicians interested in the modern development of algebraic geometry." (EMS Newsletter, September, 2006)Table of ContentsNotation and Conventions.- Two: Positivity for Vector Bundles.- 6 Ample and Nef Vector Bundles.- 6.1 Classical Theory.- 6.1.A Definition and First Properties.- 6.1.B Cohomological Properties.- 6.1.C Criteria for Amplitude.- 6.1.D Metric Approaches to Positivity of Vector Bundles.- 6.2 Q-Twisted and Nef Bundles.- 6.2.A Twists by Q-Divisors.- 6.2.B Nef Bundles.- 6.3 Examples and Constructions.- 6.3.A Normal and Tangent Bundles.- 6.3.B Ample Cotangent Bundles and Hyperbolicity.- 6.3.C Picard Bundles.- 6.3.D The Bundle Associated to a Branched Covering.- 6.3.E Direct Images of Canonical Bundles.- 6.3.F Some Constructions of Positive Vector Bundles.- 6.4 Ample Vector Bundles on Curves.- 6.4.A Review of Semistability.- 6.4.B Semistability and Amplitude.- Notes.- 7 Geometric Properties of Ample Bundles.- 7.1 Topology.- 7.1.A Sommese’s Theorem.- 7.1.B Theorem of Bloch and Gieseker.- 7.1.C A Barth-Type Theorem for Branched Coverings.- 7.2 Degeneracy Loci.- 7.2.A Statements and First Examples.- 7.2.B Proof of Connectedness of Degeneracy Loci.- 7.2.C Some Applications.- 7.2.D Variants and Extensions.- 7.3 Vanishing Theorems.- 7.3.A Vanishing Theorems of Griffiths and Le Potier.- 7.3.B Generalizations.- Notes.- 8 Numerical Properties of Ample Bundles.- 8.1 Preliminaries from Intersection Theory.- 8.1.A Chern Classes for Q-Twisted Bundles.- 8.1.B Cone Classes.- 8.1.C Cone Classes for Q-Twists.- 8.2 Positivity Theorems.- 8.2.A Positivity of Chern Classes.- 8.2.B Positivity of Cone Classes.- 8.3 Positive Polynomials for Ample Bundles.- 8.4 Some Applications.- 8.4.A Positivity of Intersection Products.- 8.4.B Non-Emptiness of Degeneracy Loci.- 8.4.C Singularities of Hypersurfaces Along a Curve.- Notes.- Three: Multiplier Ideals and Their Applications.- 9 Multiplier Ideal Sheaves.- 9.1 Preliminaries.- 9.1.A Q-Divisors.- 9.1.B Normal Crossing Divisors and Log Resolutions.- 9.1.C The Kawamata—Viehweg Vanishing Theorem.- 9.2 Definition and First Properties.- 9.2.A Definition of Multiplier Ideals.- 9.2.B First Properties.- 9.3 Examples and Complements.- 9.3.A Multiplier Ideals and Multiplicity.- 9.3.B Invariants Arising from Multiplier Ideals.- 9.3.C Monomial Ideals.- 9.3.D Analytic Construction of Multiplier Ideals.- 9.3.E Adjoint Ideals.- 9.3.F Multiplier and Jacobian Ideals.- 9.3.G Multiplier Ideals on Singular Varieties.- 9.4 Vanishing Theorems for Multiplier Ideals.- 9.4.A Local Vanishing for Multiplier Ideals.- 9.4.B The Nadel Vanishing Theorem.- 9.4.C Vanishing on Singular Varieties.- 9.4.D Nadel’s Theorem in the Analytic Setting.- 9.4.E Non-Vanishing and Global Generation.- 9.5 Geometric Properties of Multiplier Ideals.- 9.5.A Restrictions of Multiplier Ideals.- 9.5.B Subadditivity.- 9.5.C The Summation Theorem.- 9.5.D Multiplier Ideals in Families.- 9.5.E Coverings.- 9.6 Skoda’s Theorem.- 9.6.A Integral Closure of Ideals.- 9.6.B Skoda’s Theorem: Statements.- 9.6.C Skoda’s Theorem: Proofs.- 9.6.D Variants.- Notes.- 10 Some Applications of Multiplier Ideals.- 10.1 Singularities.- 10.1.A Singularities of Projective Hypersurfaces.- 10.1.B Singularities of Theta Divisors.- 10.1.C A Criterion for Separation of Jets of Adjoint Series.- 10.2 Matsusaka’s Theorem.- 10.3 Nakamaye’s Theorem on Base Loci.- 10.4 Global Generation of Adjoint Linear Series.- 10.4.A Fujita Conjecture and Angehrn—Siu Theorem.- 10.4.B Loci of Log-Canonical Singularities.- 10.4.C Proof of the Theorem of Angehrn and Siu.- 10.5 The Effective Nullstellensatz.- Notes.- 11 Asymptotic Constructions.- 11.1 Construction of Asymptotic Multiplier Ideals.- 11.1.A Complete Linear Series.- 11.1.B Graded Systems of Ideals and Linear Series.- 11.2 Properties of Asymptotic Multiplier Ideals.- 11.2.A Local Statements.- 11.2.B Global Results.- 11.2.C Multiplicativity of Plurigenera.- 11.3 Growth of Graded Families and Symbolic Powers.- 11.4 Fujita’s Approximation Theorem.- 11.4.A Statement and First Consequences.- 11.4.B Proof of Fujita’s Theorem.- 11.4.C The Dual of the Pseudoeffective Cone.- 11.5.- Notes.- References.- Glossary of Notation.

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Book SynopsisFrom the reviews: "Although several textbooks on modern algebraic geometry have been published in the meantime, Mumford's "Volume I" is, together with its predecessor the red book of varieties and schemes, now as before one of the most excellent and profound primers of modern algebraic geometry. Both books are just true classics!" Zentralblatt Trade Review"In the 20th century, algebraic geometry has undergone several revolutionary changes with respect to its conceptual foundations, technical framework, and intertwining with other branches of mathematics. Accordingly the way it is taught has gone through distinct phases. The theory of algebraic schemes, together with its full-blown machinery of sheaves and their cohomology, being for now the ultimate stage of this evolution process in algebraic geometry, had created -- around 1960 -- the urgent demand for new textbooks reflecting these developments and (henceforth) various facets of algebraic geometry. ... It was David Mumford, who at first started the project of writing a textbook on algebraic geometry in its new setting. His mimeographed Harvard notes ntroduction to algebraic geometry: Preliminary version of the first three chapters (bound in red) were distributed in the mid 1960's, and they were intended as the first stage of a forthcoming, more inclusive textbook. For some years, these mimeographed notes represented the almost only, however utmost convenient and abundant source for non-experts to get acquainted with the basic new concepts and ideas of modern algebraic geometry. Their timeless utility, in this regard, becomes apparent from the fact that two reprints of them have appeared, since 1988, as a proper book under the title he red book of varieties and schemes' ( Lect. Notes Math. 1358). In the process of exending his Harvard notes to a comprehensive textbook, the author's teaching experiences led him to the didactic conclusion that it would be better to split the book into two volumes, thereby starting with complex projective varieties (in volume I), and proceeding with schemes and their cohomology (in volume II). -- In 1976, the author published the first volume under the title lgebraic geometry. I: Complex projective varieties where the corrections concerned the wiping out of some misprints, inconsistent notations, and other slight inaccuracies.The book under review is an unchanged reprint of this corrected second edition from 1980. Although several textbooks on modern algebraic geometry have been published in the meantime, Mumford's "Volume I" is, together with its predecessor the red book of varieties and schemes now as before, one of the most excellent and profound primers of modern algebraic geometry. Both books are just true classics!"Zentralblatt MATH, 821Table of Contents1. Affine Varieties.- §1A. Their Definition, Tangent Space, Dimension, Smooth and Singular Points.- §1B. Analytic Uniformization at Smooth Points, Examples of Topological Knottedness at Singular Points.- §1C. Ox,X a UFD when x Smooth; Divisor of Zeroes and Poles of Functions.- 2. Projective Varieties.- §2A. Their Definition, Extension of Concepts from Affine to Projective Case.- §2B. Products, Segre Embedding, Correspondences.- §2C. Elimination Theory, Noether’s Normalization Lemma, Density of Zariski-Open Sets.- 3. Structure of Correspondences.- §3A. Local Properties—Smooth Maps, Fundamental Openness Principle, Zariski’s Main Theorem.- §3B. Global Properties—Zariski’s Connectedness Theorem, Specialization Principle.- §3C. Intersections on Smooth Varieties.- 4. Chow’s Theorem.- §4A. Internally and Externally Defined Analytic Sets and their Local Descriptions as Branched Coverings of ?n.- §4B. Applications to Uniqueness of Algebraic Structure and Connectedness.- 5. Degree of a Projective Variety.- §5A. Definition of deg X, multxX, of the Blow up Bx(X), Effect of a Projection, Examples.- §5B. Bezout’s Theorem.- §5C. Volume of a Projective Variety ; Review of Homology, DeRham’s Theorem, Varieties as Minimal Submanifolds.- 6. Linear Systems.- §6A. The Correspondence between Linear Systems and Rational Maps, Examples; Complete Linear Systems are Finite-Dimensional.- §6B. Differential Forms, Canonical Divisors and Branch Loci.- §6C. Hilbert Polynomials, Relations with Degree.- Appendix to Chapter 6. The Weil-Samuel Algebraic Theory of Multiplicity.- 7. Curves and Their Genus.- §7A. Existence and Uniqueness of the Non-Singular Model of Each Function Field of Transcendence Degree 1 (after Albanese).- §7B. Arithmetic Genus = Topological Genus; Existence of Good Projections to ?1, ?2, ?3.- §7C. Residues of Differentials on Curves, the Classical Riemann-Roch Theorem for Curves and Applications.- §7D. Curves of Genus 1 as Plane Cubics and as Complex Tori ?/L.- 8. The Birational Geometry of Surfaces.- §8A. Generalities on Blowing up Points.- §8B. Resolution of Singularities of Curves on a Smooth Surface by Blowing up the Surface; Examples.- §8C. Factorization of Birational Maps between Smooth Surfaces; the Trees of Infinitely Near Points.- §8D. The Birational Map between ?2 and the Quadric and Cubic Surfaces; the 27 Lines on a Cubic Surface.- List of Notations.

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Book SynopsisMumford's famous "Red Book" gives a simple, readable account of the basic objects of algebraic geometry, preserving as much as possible their geometric flavor and integrating this with the tools of commutative algebra. It is aimed at graduates or mathematicians in other fields wishing to quickly learn aboutalgebraic geometry. This new edition includes an appendix that gives an overview of the theory of curves, their moduli spaces and their Jacobians -- one of the most exciting fields within algebraic geometry.Trade Review"This is the second edition of a famous and well-known introduction to algebraic geometry, written to show that the language of schemes is fundamentally geometrical and clearly expressing the intuition of algebraic geometry. ... This book can strongly be recommended to anybody interested in algebraic geometry and willing to learn about varieties and schemes and their main problems."EMS Newsletter, Vol. 37, Sept. 2000Table of ContentsVarieties.- Preschemes.- Local properties of schemes.- References.- Appendix: Curves and their Jacobians.- Survey of work on the Schottky problem up to 1996 (by Enrico Arbarello).- References.- Guide to the Literature and References (Curves and Their Jacobians).- Supplementary Bibliography on the Schottky Problem.

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Book SynopsisThe section conjecture in anabelian geometry, announced by Grothendieck in 1983, is concerned with a description of the set of rational points of a hyperbolic algebraic curve over a number field in terms of the arithmetic of its fundamental group. While the conjecture is still open today in 2012, its study has revealed interesting arithmetic for curves and opened connections, for example, to the question whether the Brauer-Manin obstruction is the only one against rational points on curves. This monograph begins by laying the foundations for the space of sections of the fundamental group extension of an algebraic variety. Then, arithmetic assumptions on the base field are imposed and the local-to-global approach is studied in detail. The monograph concludes by discussing analogues of the section conjecture created by varying the base field or the type of variety, or by using a characteristic quotient or its birational analogue in lieu of the fundamental group extension.Trade ReviewFrom the book reviews:“The book under review, resulting from the author’s dissertation … is both a research monograph and a thorough presentation of the arithmetic and geometry of Grothendieck’s section conjecture from the foundations to the current state of the art. … It will be useful not only to specialists, as it is accessible to anyone familiar with the basics of modern algebraic geometry and the theory of algebraic fundamental groups.” (Marco A. Garuti, Mathematical Reviews, May, 2014)Table of ContentsPart I Foundations of Sections.- 1 Continuous Non-abelian H1 with Profinite Coefficients.-2 The Fundamental Groupoid.- 3 Basic Geometric Operations in Terms of Sections.- 4 The Space of Sections as a Topological Space.- 5 Evaluation of Units.- 6 Cycle Classes in Anabelian Geometry.- 7 Injectivity in the Section Conjecture.- Part II Basic Arithmetic of Sections.- 7 Injectivity in the Section Conjecture.- 8 Reduction of Sections.- 9 The Space of Sections in the Arithmetic Case and the Section Conjecture in Covers.- Part III On the Passage from Local to Global.- 10 Local Obstructions at a p-adic Place.- 11 Brauer-Manin and Descent Obstructions.- 12 Fragments of Non-abelian Tate–Poitou Duality.- Part IV Analogues of the Section Conjecture.- 13 On the Section Conjecture for Torsors.- 14 Nilpotent Sections.- 15 Sections over Finite Fields.- 16 On the Section Conjecture over Local Fields.- 17 Fields of Cohomological Dimension 1.- 18 Cuspidal Sections and Birational Analogues.

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Book SynopsisThe concept of moduli goes back to B. Riemann, who shows in [68] that the isomorphism class of a Riemann surface of genus 9 ~ 2 depends on 3g - 3 parameters, which he proposes to name "moduli". A precise formulation of global moduli problems in algebraic geometry, the definition of moduli schemes or of algebraic moduli spaces for curves and for certain higher dimensional manifolds have only been given recently (A. Grothendieck, D. Mumford, see [59]), as well as solutions in some cases. It is the aim of this monograph to present methods which allow over a field of characteristic zero to construct certain moduli schemes together with an ample sheaf. Our main source of inspiration is D. Mumford's "Geometric In­ variant Theory". We will recall the necessary tools from his book [59] and prove the "Hilbert-Mumford Criterion" and some modified version for the stability of points under group actions. As in [78], a careful study of positivity proper­ ties of direct image sheaves allows to use this criterion to construct moduli as quasi-projective schemes for canonically polarized manifolds and for polarized manifolds with a semi-ample canonical sheaf.Table of ContentsLeitfaden.- Classification Theory and Moduli Problems.- Notations and Conventions.- 1 Moduli Problems and Hilbert Schemes.- 1.1 Moduli Functors and Moduli Schemes.- 1.2 Moduli of Manifolds: The Main Results.- 1.3 Properties of Moduli Functors.- 1.4 Moduli Functors for ?-Gorenstein Schemes.- 1.5 A. Grothendieck’s Construction of Hilbert Schemes.- 1.6 Hilbert Schemes of Canonically Polarized Schemes.- 1.7 Hilbert Schemes of Polarized Schemes.- 2 Weakly Positive Sheaves and Vanishing Theorems.- 2.1 Coverings.- 2.2 Numerically Effective Sheaves.- 2.3 Weakly Positive Sheaves.- 2.4 Vanishing Theorems and Base Change.- 2.5 Examples of Weakly Positive Sheaves.- 3 D. Mumford’s Geometric Invariant Theory.- 3.1 Group Actions and Quotients.- 3.2 Linearizations.- 3.3 Stable Points.- 3.4 Properties of Stable Points.- 3.5 Quotients, without Stability Criteria.- 4 Stability and Ampleness Criteria.- 4.1 Compactifications and the Hilbert-Mumford Criterion.- 4.2 Weak Positivity of Line Bundles and Stability.- 4.3 Weak Positivity of Vector Bundles and Stability.- 4.4 Ampleness Criteria.- 5 Auxiliary Results on Locally Free Sheaves and Divisors.- 5.1 O. Gabber’s Extension Theorem.- 5.2 The Construction of Coverings.- 5.3 Singularities of Divisors.- 5.4 Singularities of Divisors in Flat Families.- 5.5 Vanishing Theorems and Base Change, Revisited.- 6 Weak Positivity of Direct Images of Sheaves.- 6.1 Variation of Hodge Structures.- 6.2 Weakly Semistable Reduction.- 6.3 Applications of the Extension Theorem.- 6.4 Powers of Dualizing Sheaves.- 6.5 Polarizations, Twisted by Powers of Dualizing Sheaves.- 7 Geometric Invariant Theory on Hilbert Schemes.- 7.1 Group Actions on Hilbert Schemes.- 7.2 Geometric Quotients and Moduli Schemes.- 7.3 Methods to Construct Quasi-Projective Moduli Schemes.- 7.4 Conditions for the Existence of Moduli Schemes: Case (CP).- 7.5 Conditions for the Existence of Moduli Schemes: Case (DP).- 7.6 Numerical Equivalence.- 8 Allowing Certain Singularities.- 8.1 Canonical and Log-Terminal Singularities.- 8.2 Singularities of Divisors.- 8.3 Deformations of Canonical and Log-Terminal Singularities.- 8.4 Base Change and Positivity.- 8.5 Moduli of Canonically Polarized Varieties.- 8.6 Moduli of Polarized Varieties.- 8.7 Towards Moduli of Canonically Polarized Schemes.- 9 Moduli as Algebraic Spaces.- 9.1 Algebraic Spaces.- 9.2 Quotients by Equivalence Relations.- 9.3 Quotients in the Category of Algebraic Spaces.- 9.4 Construction of Algebraic Moduli Spaces.- 9.5 Ample Line Bundles on Algebraic Moduli Spaces.- 9.6 Proper Algebraic Moduli Spaces for Curves and Surfaces.- References.- Glossary of Notations.

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Book SynopsisThis book explores the theory and application of locally nilpotent derivations, a subject motivated by questions in affine algebraic geometry and having fundamental connections to areas such as commutative algebra, representation theory, Lie algebras and differential equations. The author provides a unified treatment of the subject, beginning with 16 First Principles on which the theory is based. These are used to establish classical results, such as Rentschler's Theorem for the plane and the Cancellation Theorem for Curves.More recent results, such as Makar-Limanov's theorem for locally nilpotent derivations of polynomial rings, are also discussed. Topics of special interest include progress in classifying additive actions on three-dimensional affine space, finiteness questions (Hilbert's 14th Problem), algorithms, the Makar-Limanov invariant, and connections to the Cancellation Problem and the Embedding Problem.A lot of new material is included in this expanded second edition, such as canonical factorization of quotient morphisms, and a more extended treatment of linear actions. The reader will also find a wealth of examples and open problems and an updated resource for future investigations.Table of ContentsIntroduction.- 1 First Principles.- 2 Further Properties of LNDs.- 3 Polynomial Rings.- 4 Dimension Two.- 5 Dimension Three.- 6 Linear Actions of Unipotent Groups.- 7 Non-Finitely Generated Kernels.- 8 Algorithms.- 9 Makar-Limanov and Derksen Invariants.- 10 Slices, Embeddings and Cancellation.- 11 Epilogue.- References.- Index.

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Book SynopsisThis is a textbook meant to be used at the advanced undergraduate or graduate level. It is an introduction to the theory of Lie groups and Lie algebras. The book treats real and p-adic groups in a unified manner. The first chapter outlines preliminary material that is used in the rest of the book. The second chapter is on analytic functions and is of an elementary nature; this material is included to cater to students who may not be familiar with p-adic fields. The third chapter introduces analytic manifolds and contains standard material; the only notable feature being that it covers both real and p-adic analytic manifolds. Chapters 4 and 5 are on Lie groups. All the standard results on Lie groups are proved here. Some of the proofs are different from those in the earlier literature. The last two chapters are on Lie algebras and cover their structure theory as found in the first of the Bourbaki volumes on the subject. Some proofs here are new.

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Book SynopsisQuesto breve libro propone con uno spirito via via d’immagine storiografica e di dettaglio matematico, la nascita e l’evoluzione del concetto di curvatura: le sue origini ancestrali nella meccanica, nell’astronomia, nella geodesia, e infine, chiaramente nella geometria. Gli aspetti tecnici, a volte estremamente semplici, altre volte complessi, sono sempre accompagnati da spiegazioni che si sperano esaurienti.È ben noto che su entrambi i versanti culturali proposti nel libro, molto si è scritto e ad altissimo livello; qui, c’è un tentativo di sintesi, della storiografia e della matematica sul tema della curvatura. Il racconto del filo che intercorre tra Huygens, Gauss, Riemann, Christoffel, Ricci Curbastro, Levi-Civita e infine Einstein, è stato sicuramente già ben proposto sul versante puramente storico o in quello prettamente matematico: è una speranza che la narrazione qui presentata, con questi punti di vista intrecciati, sia infine soddisfacente. Il tentativo andava fatto. L’augurio forte è che gli argomenti narrati risultino coinvolgenti per il lettore, spingendolo ad esplorare autonomamente altri aspetti magari nascosti nelle pieghe della nozione di curvatura e del mondo che ci vive attorno. Il volume muove inizialmente dal racconto di qualche frammento di cosmologia antica e medioevale. Tutto ciò è solo apparentemente estraneo al corpo vivo della materia: ritroveremo per esempio che la concezione cosmologica di Dante, riassunta qui matematicamente, propose un universo come un’ipersfera 3-dimensionale che, quasi incidentalmente, risulterà proprio il modello cosmologico offerto da Einstein nel 1917 per il suo universo chiuso e statico. Ed è proprio la curvatura che domina quella scena, oggetto matematico protagonista della teoria della relatività generale einsteniana. I personaggi prima elencati vengono comunque narrati anche nelle loro salienti vicende umane, a volte altamente drammatiche, come accadde per esempio per Riemann e Tullio Levi-Civita. In un certo senso, la storia della curvatura accompagna la storia dell’umanità.Benché inizialmente sia stato generato da un disegno didattico, il volume è indirizzato ad un pubblico non necessariamente studentesco, con una cultura scientifica di base.Table of Contents1 Tracce di cosmologia.- 2 Prima di Gauss.- 3 Gauss.- 4 Riemann.- 5 Christoffel.- 6 Ricci Curbastro.- 7 Levi-Civita.- 8 Tracce di geometria differenziale.- 9 Einstein.

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Book SynopsisThis thesis deals with specific features of the theory of holomorphic dynamics in dimension 2 and then sets out to study analogous questions in higher dimensions, e.g. dealing with normal forms for rigid germs, and examples of Kato 3-folds.The local dynamics of holomorphic maps around critical points is still not completely understood, in dimension 2 or higher, due to the richness of the geometry of the critical set for all iterates.In dimension 2, the study of the dynamics induced on a suitable functional space (the valuative tree) allows a classification of such maps up to birational conjugacy, reducing the problem to the special class of rigid germs, where the geometry of the critical set is simple. In some cases, from such dynamical data one can construct special compact complex surfaces, called Kato surfaces, related to some conjectures in complex geometry.Table of ContentsIntroduction.-1.Background.- 2.Dynamics in 2D.- 3.Rigid germs in higher dimension.- 4 Construction of non-Kahler 3-folds.- References.- Index.

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Book SynopsisSeveral generations of students of algebraic geometry have learned the subject from David Mumford's fabled "Red Book" containing notes of his lectures at Harvard University. Their genesis and evolution are described in the preface as:Initially notes to the course were mimeographed and bound and sold by the Harvard math department with a red cover. These old notes were picked up by Springer and are now sold as the "Red book of Varieties and Schemes". However, every time I taught the course, the content changed and grew. I had aimed to eventually publish more polished notes in three volumes...This book contains what Mumford had then intended to be Volume II. It covers the material in the "Red Book" in more depth with several more topics added. The notes have been brought to the present form in collaboration with T. Oda.

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Book SynopsisThis volume contains the proceedings of the international colloquium organized by the Tata Institute of Fundamental Research in January 2016, one of a series of colloquia going back to 1956.The talks at the colloquium covered a wide spectrum of mathematics, ranging over algebraic geometry, topology, algebraic $K$-theory and number theory. Algebraic theory, $\mathbb{A}^1$-homotopy theory and topological $K$-theory formed important sub-streams in this colloquium. Several branches of $K$-theory, like algebraic cycles, triangulated categories of motives, motivic cohomology, motivic homotopy theory, Chow groups of varieties, Euler class theory, equivariant $K$-theory as well as classical $K$-theory have developed considerably in recent years, giving rise to newer directions to the subject as well as proving results of ``classical'' interest. The colloquium brought together experts in these various branches and their talks covered this wide spectrum, highlighting the interconnections and giving a better perspective of the whole subject area.This volume contains refereed articles by leading experts in these fields and includes original results as well as expository materials in these areas.

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Book SynopsisThis book deals with the classical theory of Nevanlinna on the value distribution of meromorphic functions of one complex variable, based on minimum prerequisites for complex manifolds. The theory was extended to several variables by S. Kobayashi, T. Ochiai, J. Carleson, and P. Griffiths in the early 1970s. K. Kodaira took up this subject in his course at The University of Tokyo in 1973 and gave an introductory account of this development in the context of his final paper, contained in this book. The first three chapters are devoted to holomorphic mappings from C to complex manifolds. In the fourth chapter, holomorphic mappings between higher dimensional manifolds are covered. The book is a valuable treatise on the Nevanlinna theory, of special interests to those who want to understand Kodaira's unique approach to basic questions on complex manifolds.Table of ContentsPreface1. Nevanlinna Theory of One Variable (1)1.1 metrics of compact Rimann surfaces1.2 integral formula1.3 holomorphic maps over compact Riemann surfaces whose genus are greater than 21.4 holomorphic maps over Riemann sphreres1.5 Defect relation2. Schwarz--Kobayashi's Lemma2.1 Schwarz--Kobayashi's Lemma2.2 holomorphic maps over algebraic varieties (general type)2.3 hyperbolic measures3. Nevanlinna Theory of One Variable (2)3.1 holomorphic maps over Riemann shpres3.2 the first main theorem3.3 the second main theorem4. Nevanlinna Theory of Several Variables4.1 Biebelbach's example4.2 the first main theorem4.3 the second main theorem4.4 defect relation4.5 applicationsReferences

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Book Synopsis0 Preliminaries.- 1 Enriques surfaces: generalities.- 2 Linear Systems on Enriques Surfaces.- 3 Projective Models of Enriques Surfaces.- 4 Genus One Fibrations.- 5 Moduli Spaces.- Appendix A: Automorphic Forms and Moduli Spaces by S. Kondo.

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Book SynopsisOverview of Work on Sigma Function from Historical Viewpoint.- Curves in Weierstrass Canonical Form (W-curves).- Theory of Sigma Function.- Application of the Sigma Function Theory to Integrable Systems.

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a huge range and FREE tracked UK delivery on ALL orders.

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Book SynopsisThis is the first volume of a two-volume book that offers an in-depth, and essentially self-contained, treatment of the arithmetic theory of algebraic groups. It is accessible to graduate students and researchers in number theory, algebraic geometry, and related areas.Trade Review'The original English version of the book 'Algebraic Groups and Number Theory' by Platonov and Rapinchuk was a go to reference for graduate students and senior researchers alike working in areas of arithmetic and algebraic groups, discrete subgroups of Lie groups, and connections with number theory. The second edition, which will be split into two volumes, and also co-authored with I. Rapinchuk, is a welcome and timely update to the original. The first volume of the second edition, consists of an update to chapters 1-5 of the original with an additional section 4.9 to include new material on the structure of extensions of arithmetic groups. There is no doubt in my mind that this first volume of the second edition will again take on the role of a go to text for those working in an area of huge ongoing interest and importance, and be at the forefront training new generations of mathematicians working in the areas of arithmetic and algebraic groups, discrete subgroups of Lie groups, and connections with number theory.' Alan Reid, Rice University'The arithmetic theory of algebraic groups is a beautiful area of mathematics: a crossroad of number theory, groups, geometry, representation theory, and more. Not surprisingly it attracted some of the greatest mathematicians of the last few generations. The first edition of the book 'Algebraic Groups and Number Theory' by Vladimir Platonov and Andrei Rapinchuk which came out in the early 90s has quickly become the standard reference of the field. It presents in a clear way several deep topics. The book was one of the reasons the area attracted more researchers and expanded to new directions. This made an updated version much needed. The original authors and Igor Rapinchuk should be thanked by the mathematical community for carrying out this monumental job.' Alex Lubotzky, Hebrew University of JerusalemTable of Contents1. Algebraic number theory; 2. Algebraic groups; 3. Algebraic groups over locally compact fields; 4. Arithmetic groups and reduction theory; 5. Adeles; Bibliography; Index.

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Book SynopsisAlgebraic geometry is, essentially, the study of the solution of equations and occupies a central position in pure mathematics. This short and readable introduction to algebraic geometry will be ideal for all undergraduate mathematicians coming to the subject for the first time. With the minimum of prerequisites, Dr Reid introduces the reader to the basic concepts of algebraic geometry including: plane conics, cubics and the group law, affine and projective varieties, and non-singularity and dimension. He is at pains to stress the connections the subject has with commutative algebra as well as its relation to topology, differential geometry, and number theory. The book arises from an undergraduate course given at the University of Warwick and contains numerous examples and exercises illustrating the theory.Trade Review"Before Reid's volume there was hardly anything to recommend at the undergraduate level...Reid's book is fun; it is filled with examples, applications, asides, gossip...What it does, it does well, and there is nothing comparable." Choice"...at a level advanced undergraduates will understand and appreciate." Mathematics Magazine"...the author leads the student on a lively, interesting, down-to-earth tour of the fundamental algebraic geometry...with some welcome, provocative comments..." American Mathematical MonthlyTable of Contents1. Playing with plane curves; 2. The category of affine varieties; 3. Applications; Index.

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Book SynopsisThe study of (special cases of) elliptic curves goes back to Diophantos and Fermat, and today it is still one of the liveliest centres of research in number theory. This book, which is addressed to beginning graduate students, introduces basic theory from a contemporary viewpoint but with an eye to the historical background. The central portion deals with curves over the rationals: the Mordell-Weil finite basis theorem, points of finite order (Nagell-Lutz) etc. The treatment is structured by the local-global standpoint and culminates in the description of the Tate-Shafarevich group as the obstruction to a Hasse principle. In an introductory section the Hasse principle for conics is discussed. The book closes with sections on the theory over finite fields (the 'Riemann hypothesis for function fields') and recently developed uses of elliptic curves for factoring large integers. Prerequisites are kept to a minimum; an acquaintance with the fundamentals of Galois theory is assumed, but no Trade Review'… an excellent introduction … written with humour.' Monatshefte für MathematikTable of ContentsIntroduction; 1. Curves of genus: introduction; 2. p-adic numbers; 3. The local-global principle for conics; 4. Geometry of numbers; 5. Local-global principle: conclusion of proof; 6. Cubic curves; 7. Non-singular cubics: the group law; 8. Elliptic curves: canonical form; 9. Degenerate laws; 10. Reduction; 11. The p-adic case; 12. Global torsion; 13. Finite basis theorem: strategy and comments; 14. A 2-isogeny; 15. The weak finite basis theorem; 16. Remedial mathematics: resultants; 17. Heights: finite basis theorem; 18. Local-global for genus principle; 19. Elements of Galois cohomology; 20. Construction of the jacobian; 21. Some abstract nonsense; 22. Principle homogeneous spaces and Galois cohomology; 23. The Tate-Shafarevich group; 24. The endomorphism ring; 25. Points over finite fields; 26. Factorizing using elliptic curves; Formulary; Further reading; Index.

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Book SynopsisThis work consists of two courses on the moduli spaces of vector bundles. The first is introductory, and assumes very little background; the second is more advanced and takes the reader into current areas of research. This a treatment of vector bundles that will be welcomed by experienced algebraic geometers and novices alike.Trade Review'The whole book is well written and is a valuable addition to the literature … It is essential purchase for all libraries maintaining a collection in algebraic geometry, and strongly recommended for individual researchers and graduate students with an interest in vector bundles.' Peter Newstead, Bulletin of the London Mathematical SocietyTable of ContentsPart I. Vector Bundles On Algebraic Curves: 1. Generalities; 2. The Riemann-Roch formula; 3. Topological; 4. The Hilbert scheme; 5. Semi-stability; 6. Invariant geometry; 7. The construction of M(r,d); 8. Study of M(r,d); Part II. Moduli Spaces Of Semi-Stable Sheaves On The Projective Plane; 9. Introduction; 10. Operations on semi-stable sheaves; 11. Restriction to curves; 12. Bogomolov's theorem; 13. Bounded families; 14. The construction of the moduli space; 15. Differential study of the Shatz stratification; 16. The conditions for existence; 17. The irreducibility; 18. The Picard group; Bibliography.

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Book SynopsisDeveloped over more than a century, and still an active area of research today, the classification of algebraic surfaces is an intricate and fascinating branch of mathematics. In this book Professor Beauville gives a lucid and concise account of the subject, following the strategy of F. Enriques, but expressed simply in the language of modern topology and sheaf theory, so as to be accessible to any budding geometer. This volume is self contained and the exercises succeed both in giving the flavour of the extraordinary wealth of examples in the classical subject, and in equipping the reader with most of the techniques needed for research.Trade Review‘… a lucid and concise account of the subject.’ L’Enseignement MathématiqueTable of ContentsIntroduction; Notation; Part I. The Picard Group and the Riemann-Roch Theorem: Part II. Birational Maps: Part III. Ruled Surfaces: Part IV. Rational Surfaces: Part V. Castelnuovo’s Theorem and Applications: Part VI. Surfaces With pg = 0 and q > 1: Part VII. Kodaira Dimension: Part VIII. Surfaces With k = 0: Part IX. Surfaces With k = 1 and Elliptic Surfaces: Part X. Surfaces of General Type: Appendix A. Characteristic p; Appendix B. Complex surfaces; Appendix C. Further reading; References; Index.

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Book SynopsisOne of the major discoveries of the last two decades of the twentieth century in algebraic geometry is the realization that the theory of minimal models of surfaces can be generalized to higher dimensional varieties. This generalization, called the minimal model program or Mori's program, has developed into a powerful tool with applications to diverse questions in algebraic geometry and beyond. This book provides the a comprehensive introduction to the circle of ideas developed around the program, the prerequisites being only a basic knowledge of algebraic geometry. It will be of great interest to graduate students and researchers working in algebraic geometry and related fields.Trade ReviewReview of the hardback: '… the book is very crisply written, unusually easy to read for a book covering advanced material, and is moreover very concise for the book for reference, but is also an ideal book on which to base a series of seminars for research students, or indeed for research students to read by themselves.' P. M. H. Wilson, Bulletin of the London Mathematical SocietyReview of the hardback: '… a very good survey of present research.' European Mathematical SocietyReview of the hardback: 'I can recommend it to anyone wanting to get a deeper knowledge than just getting a survey of some facts on the classification theory.' M. Coppens, Niew Archief voor WiskundeReview of the hardback: '… a very good survey of present research … a very clear presentation of the subject.' EMSTable of Contents1. Rational curves and the canonical class; 2. Introduction to minimal model program; 3. Cone theorems; 4. Surface singularities; 5. Singularities of the minimal model program; 6. Three dimensional flops; 7. Semi-stable minimal models.

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Book SynopsisMatroids provides a unified way to understand graph theory, linear algebra and combinatorics via finite geometry. This informal text provides a comprehensive introduction to matroid theory that emphasizes its connections to geometry and is suitable for undergraduates. It includes over 300 exercises, examples and projects suitable for independent study.Trade Review"The authors write in an entertaining, conversational style, and the text is often peppered with humorous footnotes. Nearly 300 exercises and scores of references will benefit motivated readers." -J. T. Saccoman, ChoiceTable of Contents1. A tour of matroids; 2. Cryptomorphisms; 3. New matroids from old; 4. Graphic matroids; 5. Finite geometry; 6. Representable matroids; 7. Other matroids; 8. Matroid minors; 9. The Tutte polynomial; Projects; Appendix: matroid axiom systems; Bibliography; Index.

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Book SynopsisThis is a completely self-contained modern introduction to Kaehlerian geometry and Hodge structure. The author proves the Kaehler identities, which leads to the hard Lefschetz theorem and the Hodge index theorem. Aimed at students, the text is complemented by exercises which provide useful results in complex algebraic geometry.Trade Review'This introductory text to Hodge theory and Kahlerian geometry is an excellent and modern introduction to the subject, shining with comprehensiveness, strictness, clarity, rigor, thematic steadfastness of purpose, and catching enthusiasm for this fascinating field of contemporary mathematical research. This book is exceedingly instructive, inspiring, challenging and user-friendly, which makes it truly outstanding and extremely valuable for students, teachers, and researchers in complex geometry.' Zentralblatt MATH'I would recommend anyone interested in learning about a topic in complex differential or algebraic geometry to read Voisin's volumes. She has done a remarkably good job.' Proceedings of the Edinburgh Mathematical Society'… this book is going to become a very common reference in this field … useful for both a student trying to learn the subject as well as the researcher that can find a wealth of results in a clear and compact format. The exposition is very precise and the introduction that precedes each chapter helps the reader to focus on the main ideas in the text.' Mathematical Reviews'The book provides a very satisfying exposition of all the methods of studying algebraic cycles that have come out of Hodge theory.' Bulletin of the London Mathematical Society'Mathematical rewards [await] those who invest their mathematical energies in this beautiful pair of volumes.' Bulletin of the AMSPrize Winner Cambridge University Press congratulates Claire Voisin, winner of the 2007 Ruth Lyttle Satter Prize in Mathematics!Table of ContentsIntroduction; Part I. Preliminaries: 1. Holomorphic functions of many variables; 2. Complex manifolds; 3. Kähler metrics; 4. Sheaves and cohomology; Part II. The Hodge Decomposition: 5. Harmonic forms and cohomology; 6. The case of Kähler manifolds; 7. Hodge structures and polarisations; 8. Holomorphic de Rham complexes and spectral sequences; Part III. Variations of Hodge Structure: 9. Families and deformations; 10. Variations of Hodge structure; Part IV. Cycles and Cycle Classes: 11. Hodge classes; 12. Deligne-Beilinson cohomology and the Abel-Jacobi map; Bibliography; Index.

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Book SynopsisThe 2003 second volume of this self-contained account of Kaehlerian geometry and Hodge theory continues Voisin's study of topology of families of algebraic varieties and the relationships between Hodge theory and algebraic cycles. Aimed at researchers, the text includes exercises providing useful results in complex algebraic geometry.Trade Review'All together, the author has maintained her masterly style also throughout this second, much more advanced volume, just as expected. The entire two-volume text is highly instructive, inspiring, reader-friendly and generally outstanding. Without any doubt, these two volumes must be seen as an indispensible standard text on transcendental algebraic geometry for advanced students, teachers, and also researchers in this contemporary field of mathematics. The author provides, simultaneously and in a unique manner, both a complete didactic exposition and an up-to-date presentation of the subject, which is still a rather exceptional feature in the textbook literature.' Zentralblatt MATH'The book provides a very satisfying exposition of all the methods of studying algebraic cycles that have come out of Hodge theory.' Bulletin of the London Mathematical Society'I would recommend anyone interested in learning about a topic in complex differential or algebraic geometry to read Voisin's volumes. She has done a remarkably good job.' Proceedings of the Edinburgh Mathematical Society'… this book is going to become a very common reference in this field … useful for both a student trying to learn the subject as well as the researcher that can find a wealth of results in a clear and compact format. The exposition is very precise and the introduction that precedes each chapter helps the reader to focus on the main ideas in the text.' Mathematical Reviews'Mathematical rewards [await] those who invest their mathematical energies in this beautiful pair of volumes.' Bulletin of the AMSPrize Winner Cambridge University Press congratulates Claire Voisin, winner of the 2007 Ruth Lyttle Satter Prize in Mathematics!Table of ContentsIntroduction. Part I. The Topology of Algebraic Varieties: 1. The Lefschetz theorem on hyperplane sections; 2. Lefschetz pencils; 3. Monodromy; 4. The Leray spectral sequence; Part II. Variations of Hodge Structure: 5. Transversality and applications; 6. Hodge filtration of hypersurfaces; 7. Normal functions and infinitesimal invariants; 8. Nori's work; Part III. Algebraic Cycles: 9. Chow groups; 10. Mumford' theorem and its generalisations; 11. The Bloch conjecture and its generalisations; References; Index.

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