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

Format:

Book

Author/Creator:

Kollár, János, Author.

Lieblich, Max, 1978- Author.

Olsson, Martin C., Author.

Sawin, Will, 1993- Author.

Series:

Annals of mathematics studies ; no.216.

Annals of mathematics studies ; Number 216

Language:

English

Subjects (All):

Algebraic varieties.

Physical Description:

viii ; 226 pages : illustrations (black and white) ; 24 cm.

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:

The fundamental theorem of projective geometry

Divisorial structures and definable linear systems

Reconstruction from divisorial structures: infinite fields

Reconstruction from divisorial structures: finite fields

Topological geometry

The set-theoretic complete intersection property

Linkage

Complements, counterexamples, and conjectures.

Notes:

Includes bibliographical references (pages [213]-221) and index.

Other Format:

Online version: Kollár, János. What determines an algebraic variety?

ISBN:

9780691246802

0691246807

9780691246819

0691246815

OCLC:

1346294487