Annals of Mathematics Studies
(AMS-216)
- 240 pages
- English
- PDF
- Available on iOS & Android
About This Book
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.
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Table of contents
- Cover
- Contents
- Preface
- Acknowledgments
- 1. Introduction
- 2. Preliminaries
- 3. The fundamental theorem of projective geometry
- 4. Divisorial structures and definable linear systems
- 5. Reconstruction from divisorial structures: infinite fields
- 6. Reconstruction from divisorial structures: finite fields
- 7. Topological geometry
- 8. The set-theoretic complete intersection property
- 9. Linkage
- 10. Complements, counterexamples, and conjectures
- 11. Appendix
- Bibliography
- Index of Notation
- Index of Terminology