What Determines an Algebraic Variety?

János Kollár, Max Lieblich, Martin Olsson, Will Sawin

• Publisher: Princeton University Press
• Year: 2023
• Print ISBN:
  9780691246819
  9780691246802
• eBook ISBN:
  9780691246833
• Topic: Matemáticas
• Subtopic: Geometría algebraica

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.

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