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

##### By: Kollár, János [author.].

##### Contributor(s): Lieblich, Max [author.] | Olsson, Martin C [author.] | Sawin, Will [author.].

Material type: TextSeries: Annals of mathematics studies ; Number 216.Publisher: Princeton : Princeton University Press, 2023Description: viii, 226 pages; USD 165.00 24 cms.Content type: text Media type: unmediated Carrier type: volumeISBN: 9780691246802; 9780691246819.Subject(s): Algebraic varieties | MATHEMATICS / Geometry / AlgebraicAdditional physical formats: Online version:: What determines an algebraic variety?DDC classification: 516.3/53 Other classification: MAT012010Item type | Current location | Call number | Status | Date due | Barcode | Item holds |
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Book | Chennai Mathematical Institute General Stacks | 516.353 KOL (Browse shelf) | Available | 11140 |

Includes bibliographical references and index.

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.

"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.