The Geometry of Cubic Hypersurfaces

Cubic hypersurfaces are described by simple polynomial equations but exhibit complex behavior, making them a rich area of study in algebraic geometry. This text provides an introduction to cubic hypersurfaces, covering cohomology, Hodge theory, moduli spaces, and Fano varieties. It also examines cubic hypersurfaces from a categorical perspective and discusses motivic features. Ideal for graduate students and researchers.

Format: Hardback
Length: 458 pages
Publication date: 29 June 2023
Publisher: Cambridge University Press

Cubic hypersurfaces, despite their simplicity in mathematical description, exhibit a remarkable complexity that poses significant challenges in algebraic geometry. These surfaces are defined by polynomial equations, making them accessible to study, yet their behavior encompasses a wide range of phenomena. This comprehensive text aims to introduce cubic hypersurfaces and provide the necessary tools for their exploration.

The book begins by establishing the foundational principles for studying cubic hypersurfaces and various other algebraic varieties. It delves into cohomology and Hodge theory of hypersurfaces, exploring moduli spaces and Fano varieties of linear subspaces within these surfaces. The subsequent three chapters focus on the general machinery applied to cubic hypersurfaces of dimension two, three, and four. Finally, the author takes a categorical approach, describing motivic features and their significance in understanding cubic hypersurfaces.

Written based on the author's lecture courses, this text serves as an ideal resource for graduate students seeking to delve into the field of algebraic geometry. It also serves as a valuable reference for researchers seeking to advance their understanding of cubic hypersurfaces and their applications.

ISBN-13: 9781009280006