What Determines an Algebraic Variety?: (AMS-216)

A groundbreaking new nonlinear approach to a fundamental question in algebraic geometry, What Determines an Algebraic Variety? offers the first nonlinear generalization of Veblen and Young's seminal work in a century. The book puts geometry first and provides a new perspective on a classical theorem of fundamental importance to a wide range of fields.

Format: Hardback
Length: 240 pages
Publication date: 25 July 2023
Publisher: Princeton University Press

One of the most remarkable achievements in nineteenth-century mathematics was the proof that the geometry of lines in space uniquely determines the Cartesian coordinates, up to a linear ambiguity. This groundbreaking discovery revolutionized the field of algebraic geometry and paved the way for further advancements in mathematics.

In his groundbreaking work, "What Determines an Algebraic Variety?" a prominent mathematician 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 employs cutting-edge techniques, the statements of its theorems would have been understandable a century ago; nevertheless, the results are utterly unexpected.

By placing geometry at the forefront of algebraic geometry, the book provides a fresh perspective on a classical theorem of fundamental importance to a wide range of fields in mathematics. The book begins with basic observations and demonstrates how to read off various properties of a variety from its geometry. The results become more compelling as the dimension increases.

The main result of the book states that a normal projective variety of dimension at least 4 over a field of characteristic 0 is completely determined by its Zariski topological space. While there are many open questions in dimensions 2 and 3, and in positive characteristic, this groundbreaking discovery has laid the foundation for further exploration in the field of algebraic geometry.

The work of this mathematician has significantly impacted the field, and his contributions have left a lasting legacy. His innovative approach to algebraic geometry has opened up new avenues for research and has paved the way for future breakthroughs in mathematics.

Dimension: 235 x 156 (mm)
ISBN-13: 9780691246802