What Determines an Algebraic Variety? : (ams-216) / János Kollár [and three others].

Format:

Book

Author/Creator:

Kollár, János, author.

Series:

Annals of mathematics studies ; Volume 216.

Annals of Mathematics Studies ; Volume 216

Language:

English

Subjects (All):

Algebraic varieties.

Mathematics.

Physical Description:

1 online resource (241 pages)

Edition:

First edition.

Place of Publication:

Princeton, New Jersey : Princeton University Press, [2023]

Summary:

"A pioneering new nonlinear approach to a fundamental question in algebraic geometry. One of the crowning achievements of nineteenth-century mathematics was the proof that the geometry of lines in space uniquely determines the Cartesian coordinates, up to a linear ambiguity. What Determines an Algebraic Variety? develops a nonlinear version of this theory, offering the first nonlinear generalization of the seminal work of Veblen and Young in a century. While the book uses cutting-edge techniques, the statements of its theorems would have been understandable a century ago; despite this, the results are totally unexpected. Putting geometry first in algebraic geometry, the book provides a new perspective on a classical theorem of fundamental importance to a wide range of fields in mathematics.Starting with basic observations, the book shows how to read off various properties of a variety from its geometry. The results get stronger as the dimension increases. The main result then says that a normal projective variety of dimension at least 4 over a field of characteristic 0 is completely determined by its Zariski topological space. There are many open questions in dimensions 2 and 3, and in positive characteristic"-- Provided by publisher.

"In this monograph, the authors approach a rarely considered question in the field of algebraic geometry: to what extent is an algebraic variety X determined by the underlying Zariski topological space |X|? Before this work, it was believed that the Zariski topology could give only coarse information about X. Using three reconstruction theorems, the authors prove -- astoundingly -- that the variety X is entirely determined by the Zariski topology when the dimension is at least two. It offers both new techniques, as this question had not been previously studied in depth, and future paths for application and inquiry"-- Provided by publisher.

Contents:

Cover

Contents

Preface

Acknowledgments

1. Introduction

1.1 From lines and planes to the Zariski topology of Pn

1.2 The main theorem

1.3 Organization of the book

2. Preliminaries

2.1 Algebraic varieties

2.2 Examples and speculations

2.3 Scheme-theoretic formulation

2.4 Survey of related results

2.5 Terminology and notation

3. The fundamental theorem of projective geometry

3.1 The fundamental theorem of definable projective geometry

3.2 A variant fundamental theorem

3.3 The probabilistic fundamental theorem of projective geometry

4. Divisorial structures and definable linear systems

4.1 Divisorial structures

4.2 Remarks on divisors

4.3 Definable subspaces in linear systems

5. Reconstruction from divisorial structures: infinite fields

5.1 Reduction to the quasi-projective case

5.2 The quasi-projective case

5.3 Counterexamples in dimension 1

6. Reconstruction from divisorial structures: finite fields

6.1 The Bertini-Poonen theorem

6.2 Preparatory lemmas

6.3 Reconstruction over finite fields

7. Topological geometry

7.1 Pencils

7.2 Fibers of finite morphisms

7.3 Topological pencils

7.4 Degree functions and algebraic pencils

7.5 Degree functions and linear equivalence

7.6 Uncountable fields

8. The set-theoretic complete intersection property

8.1 Summary of results

8.2 Set-theoretic complete intersection property

8.3 Mordell-Weil fields

8.4 Reducible scip subsets

8.5 Projective spaces

8.6 Appendix: Special fields

9. Linkage

9.1 Linkage of divisors

9.2 Preparations: Sections and their zero sets

9.3 Néron's theorem and consequences

9.4 Linear similarity

9.5 Bertini-Hilbert dimension

9.6 Linkage of divisors and residue fields

9.7 Minimally restrictive linking and transversality.

9.8 Recovering linear equivalence

9.9 Appendix: Weakly Hilbertian fields

10. Complements, counterexamples, and conjectures

10.1 A topological Gabriel theorem

10.2 Examples over finite fields

10.3 Surfaces over locally finite fields

10.4 Real Zariski topology

10.5 Countable noetherian topologies

10.6 Conjectures

11. Appendix

11.1 Bertini-type theorems

11.2 Complete intersections

11.3 Picard group, class group, and Albanese variety

Bibliography

Index of Notation

Index of Terminology.

Notes:

Includes bibliographical references.

Description based on publisher supplied metadata and other sources.

Description based on print version record.

ISBN:

9780691246833

0691246831

OCLC:

1378391192