Cubic forms. Algebra, geometry, arithmetic. Transl. from the Russian by M. Hazewinkel. 2nd ed. (English) Zbl 0582.14010

Manin, Yu. I.

North-Holland Mathematical Library, Vol. 4. Amsterdam-New York-Oxford: North-Holland. X, 326 p. $ 64.75; Dfl. 175.00 (1986).

In the ten years since this book was published in English (1974; for a review see the Russian original 1972; Zbl 0255.14002) there has been important progress in a number of topics related to its subject. Were this book to be written anew, its title could be ”Algebraic varieties close to the rational ones. Algebra, geometry, arithmetic”. In fact, this class of varieties has crystallized as a natural domain for the methods developed and expounded in ”Cubic forms”.
In this edition the original text is left intact, except for a few corrections, but an appendix is added together with a list of references to original papers, mainly of the last decade. This appendix sketches some of the most essential new results, constructions and ideas, including the solutions of the Lüroth and Zariski problems, the theory of the descent and obstructions to the Hasse principle on rational varieties, and recent applications of K-theory to arithmetic. Proofs are omitted since their complete presentation would demand a new book.

MSC:

14Jxx	Surfaces and higher-dimensional varieties
14-02	Research exposition (monographs, survey articles) pertaining to algebraic geometry
14M20	Rational and unirational varieties
14Gxx	Arithmetic problems in algebraic geometry; Diophantine geometry
14E30	Minimal model program (Mori theory, extremal rays)
11E76	Forms of degree higher than two

Keywords:

Lüroth problem; Zariski problems; descent; Hasse principle on rational varieties; K-theory

Citations:

Zbl 0277.14014; Zbl 0255.14002