What determines an algebraic variety? / János Kollár, Max Lieblich, Martin Olsson, Will Sawin.

"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 Algeb...

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Bibliographic Details
Main Authors: Kollár, János (Author)
Lieblich, Max, 1978- (Author)
Olsson, Martin C. (Author)
Sawin, Will, 1993- (Author)
Language:English
Published: Princeton, New Jersey : Princeton University Press, 2023.
Series:Annals of mathematics studies ; no.216.
Subjects:
Algebraic varieties.
Physical Description:viii ; 226 pages ; 24 cm.
Format: Book
Contents:
  • Introduction
  • From lines and planes to the Zariski topology of {P} ^{n}
  • The main theorum
  • Organization of this book
  • Preliminaries
  • Algebraic varieties
  • Examples and speculations
  • Scheme-theoretic formulation
  • Survey of related results
  • Terminology and notation
  • The fundamental theorem of projective geometry
  • A varient fundamental theorem
  • The probabilistic fundamental theorem of projective geometry
  • Divisorial structures and definable linear systems
  • Divisorial structures
  • Remarks on divisors
  • Definable subspaces in linear systems
  • Reconstruction from divisorial structures: infinite fields
  • Reduction to the quasi-projective case
  • The quasi-projective case
  • Counterexamples in dimension 1
  • Reconstruction from divisorial structures: finite fields
  • The Bertini-Poonen theorem
  • Preparatory lemmas
  • Reconstruction over finite fields
  • Topological geometry
  • Pencils
  • Fibers of finite morphisms
  • Topological pencils
  • Degree functions and algebraic pencils
  • Degree functions and linear equivalents
  • Uncountable fields
  • The set-theoretic complete intersection property
  • Summary of results
  • Set-theoretic complete intersection property
  • Mordell-Weil fields
  • Reducible scip subsets
  • Projective spaces
  • Appendix: special fields
  • Linkage
  • Linkage of divisors
  • Preparations: sections and their zero sets
  • Néron's theorem and consequences
  • Linear similarity
  • Bertini-Hilbert dimension
  • Linkage of divisors and residue fields
  • Minimally restrictive linking and transversality
  • Recovering linear equivalence
  • Appendix: weakly Hilbertian fields
  • Complements, counterexamples, and conjectures
  • A topological Gabriel theorem
  • Examples over finite fields
  • Surfaces over locally finite fields
  • Real Zariski topology
  • Countable noetherian topologies
  • Conjectures
  • Appendix
  • Bertini-type theorems
  • Complete itnersections
  • Picard group, class group, and Albanese variety.