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Algebraic Geometry I: Complex Projective Varieties de David Mumford

Algebraic Geometry I: Complex Projective Varieties: Classics in Mathematics

Autor David Mumford
en Limba Engleză Paperback – 15 feb 1995

Ne-a atras atenția acest volum fundamental, Algebraic Geometry I, care ocupă un loc central în bibliografia de cercetare a matematicii moderne. În cadrul programului de studiu al geometriei algebrice, această lucrare servește drept punte între intuiția geometrică a școlii italiene clasice și rigoarea algebrică impusă de reformele lui Zariski și Grothendieck. Găsim în această ediție — un reprint al textului din 1976, corectat în 1981 — o expunere care refuză să sacrifice vizualizarea geometrică în favoarea formalismului abstract, rămânând fidelă spiritului în care David Mumford și-a construit întreaga carieră.

Notăm cu interes modul în care autorul structurează progresia materialului. Primele capitole definesc varietățile afine și proiective, abordând concepte precum spațiul tangent, dimensiunea și punctele singulare, înainte de a trece la structura complexă a corespondențelor și la demonstrarea Teoremei lui Chow. Această abordare oferă o claritate conceptuală rară asupra modului în care seturile analitice definite intern și extern se întrepătrund. Cartea acoperă aceeași arie tematică precum Methods of Algebraic Geometry: Volume 2 de W. V. D. Hodge, dar cu o abordare mult mai ancorată în varietățile complexe și în utilizarea tehnicilor de uniformizare analitică, spre deosebire de tratamentul pur geometric al lui Hodge.

În contextul operei autorului, Algebraic Geometry I reprezintă fundamentul teoretic pe care s-au sprijinit ulterior lucrări specializate precum Geometric Invariant Theory. Dacă faimoasa sa „Carte Roșie”, The Red Book of Varieties and Schemes, oferea o introducere accesibilă în scheme, volumul de față aprofundează studiul varietăților complexe proiective, pregătind terenul pentru ceea ce trebuia să fie Algebraic Geometry II. Este o lectură densă, dar esențială pentru înțelegerea evoluției disciplinei în secolul XX.

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Specificații

ISBN-13: 9783540586579
ISBN-10: 3540586571
Pagini: 204
Ilustrații: X, 186 p.
Dimensiuni: 155 x 235 x 11 mm
Greutate: 0.34 kg
Ediția:Reprint of the 1st ed. Berlin, Heidelberg, New York 1976. Corr. 2nd printing 1981.
Editura: Springer Berlin, Heidelberg
Colecția Springer
Seria Classics in Mathematics

Locul publicării:Berlin, Heidelberg, Germany

Public țintă

Research

De ce să citești această carte

Recomandăm această lucrare oricărui cercetător sau student avansat care dorește să înțeleagă geometria algebrică dincolo de formalismul pur algebric. Scrisă de un laureat al Medaliei Fields, cartea oferă o perspectivă istorică și tehnică asupra varietăților complexe, facilitând tranziția de la geometria clasică la metodele moderne de analiză. Cititorul câștigă o bază solidă în studiul sistematic al suprafețelor și varietăților, beneficiind de rigoarea seriei „Classics in Mathematics”.

Despre autor

David Mumford s-a născut în 1937 în Anglia și și-a dedicat cea mai mare parte a carierei academice Universității Harvard, unde deține titlul de Higgins Professor of Mathematics. Este recunoscut la nivel mondial ca unul dintre cei mai influenți matematicieni ai secolului XX, fiind distins cu Medalia Fields în 1974. Munca sa din anii '60 și '70 a revoluționat geometria algebrică, în special prin dezvoltarea teoriei spațiilor moduli și a teoriei invarianților geometrici. Contribuțiile sale au reușit să unească intuiția geometrică profundă cu rigoarea algebrei comutative, lăsând o moștenire durabilă prin manualele și monografiile sale de referință.

Descriere scurtă

Let me begin with a little history. In the 20th century, algebraic geometry has gone through at least 3 distinct phases. In the period 1900-1930, largely under the leadership of the 3 Italians, Castelnuovo, Enriques and Severi, the subject grew immensely. In particular, what the late 19th century had done for curves, this period did for surfaces: a deep and systematic theory of surfaces was created. Moreover, the links between the "synthetic" or purely "algebro-geometric" techniques for studying surfaces, and the topological and analytic techniques were thoroughly explored. However the very diversity of tools available and the richness of the intuitively appealing geometric picture that was built up, led this school into short-cutting the fine details of all proofs and ignoring at times the time­ consuming analysis of special cases (e. g. , possibly degenerate configurations in a construction). This is the traditional difficulty of geometry, from High School Euclidean geometry on up. In the period 1930-1960, under the leadership of Zariski, Weil, and (towards the end) Grothendieck, an immense program was launched to introduce systematically the tools of commutative algebra into algebraic geometry and to find a common language in which to talk, for instance, of projective varieties over characteristic p fields as well as over the complex numbers. In fact, the goal, which really goes back to Kronecker, was to create a "geometry" incorporating at least formally arithmetic as well as projective geo­ metry.

Cuprins

1. 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 BirationalGeometry 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.

Recenzii

"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, 821

Notă biografică

Biography of  David Mumford
David Mumford was born on June 11, 1937 in England and has been associated with Harvard University continuously from entering as freshman to his present position of Higgins Professor of Mathematics.
Mumford worked in the fields of Algebraic Gemetry in the 60's and 70's, concentrating especially on the theory of moduli spaces: spaces which classify all objects of some type, such as all curves of a given genus or all vector bundles on a fixed curve of given rank and degree. Mumford was awarded the Fields Medal in 1974 for his work on moduli spaces and algebraic surfaces. He is presently working on the mathematics of pattern recognition and artificial intelligence.