The Geometry of Cubic Hypersurfaces

A detailed introduction to cubic hypersurfaces, applying diverse techniques to a central class of algebraic varieties.

Daniel Huybrechts (Author)

9781009280006, Cambridge University Press

Hardback, published 29 June 2023

458 pages
23.5 x 15.9 x 3.2 cm, 0.83 kg

'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

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.

1. Basic facts
2. Fano varieties of lines
3. Moduli spaces
4. Cubic surfaces
5. Cubic threefolds
6. Cubic fourfolds
7. Derived categories of cubic hypersurfaces
References
Subject index.

Subject Areas: Geometry [PBM]