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  3. The Geometry of Cubic Hypersurfaces
  • Cited by 24
Publisher:
Cambridge University Press
Online publication date:
June 2023
Print publication year:
2023
Online ISBN:
9781009280020
DOI:
https://doi.org/10.1017/9781009280020
Subjects:
Mathematics (general), Mathematics, Mathematical Physics, Geometry and Topology
Series:
Cambridge Studies in Advanced Mathematics (206)
Information

Book description

Cubic hypersurfaces are described by almost the simplest possible polynomial equations, yet their behaviour is rich enough to demonstrate many of the central challenges in algebraic geometry. With exercises and detailed references to the wider literature, this thorough text introduces cubic hypersurfaces and all the techniques needed to study them. The book starts by laying the foundations for the study of cubic hypersurfaces and of many other algebraic varieties, covering cohomology and Hodge theory of hypersurfaces, moduli spaces of those and Fano varieties of linear subspaces contained in hypersurfaces. The next three chapters examine the general machinery applied to cubic hypersurfaces of dimension two, three, and four. Finally, the author looks at cubic hypersurfaces from a categorical point of view and describes motivic features. Based on the author's lecture courses, this is an ideal text for graduate students as well as an invaluable reference for researchers in algebraic geometry.

Reviews

‘What a beautiful book. Several turning points in algebraic geometry have in their background a cubic hypersurface. This superb exposition, by one of the masters in the field, takes the reader, in a friendly manner, through the fascinating, and occasionally mysterious, properties of these geometrical objects, and through them offers a glimpse of the underlying Hodge theory, of the theory of periods, the theory of motives, and the theory of derived categories. Graduate students, researchers, and colleagues will all find this unified treatment of cubic hypersurfaces profoundly inspiring.’

Enrico Arbarello - Accademia Nazionale dei Lincei

‘This is just a fantastic book for students and experts alike. The geometry of cubics is a wonderful mix of the classical and the modern; Huybrechts consolidates the diverse results into a coherent account for the first time. His famously lucid writing clearly conveys the beauty of the geometry of these varieties. The book describes a plethora of techniques culminating in new (and really surprising) viewpoints on the subject.’

Richard Thomas - Imperial College London

‘This exceedingly well written monograph covers material ranging from the very beginning of algebraic geometry, the 27 lines on a cubic surface, to highly relevant topical issues. The book will be a most valuable companion to algebraic geometers from graduate students to active researchers.’

Klaus Hulek - Leibniz University Hannover

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