Birational automorphisms of multidimensional algebraic manifolds
type: Item
instance of: scholarly article
title: Birational automorphisms of multidimensional algebraic manifolds
zbMATH Open document ID: 0917.14007
DOI: 10.1007/BF02363913
author: V. A. Iskovskikh
author: A. V. Pukhlikov
published in: Journal of Mathematical Sciences (New York)
publication date: 1999-08-04T00:00:00Z
review text: This survey summarizes the results of investigations during the two last decades in the field of birational automorphism groups of Fano manifolds. This subject, which is traditional for the Moscow school of algebraic geometry (and has been under active study only here in the second half of this century), has its origin in the works of the eminent Italian geometrician G. Fano. He proposed several very fruitful concepts but did not present them in a complete form: He did not prove any of his correctly guessed theorems. His reasoning has significant gaps and outright mistakes. Thus, the modern stage in multidimensional birational geometry begins with the correction of the latter. In 1971, \textit{V. A. Iskovskikh} and \textit{Yu. I. Manin} published their work: Math. USSR, Sb. 15(1971), 141-166 (1972); translation from Mat. Sb., Nov. Ser. 86 (128), 140-166 (1971; Zbl 0222.14002), where the Fano theorem about the rigidity of smooth 3-dimensional quartics was proved. The maximal singularities method, the theory proposed in this work, appeared to be very effective for the study of the birational correspondences of multidimensional algebraic manifolds with negative canonical sheaf (of Fano manifolds). In the subsequent works by \textit{V. A. Iskovskikh} [Sov. Math., Dokl. 18, 748-750 (1977); translation from Dokl. Akad. Nauk SSSR 234, 743-745 (1977; Zbl 0406.14004), Sov. Math., Dokl. 18(1977), 948-951 (1978); translation from Dokl. Akad. Nauk SSSR 235, 509-511 (1977; Zbl 0414.14025)], birational automorphism groups were described and a birational type of several series of Fano 3-folds was found: double spaces and double quadrics of index 1, the double Veronese cone of index 2, the complete intersection of a quadric and a cubic in \(\mathbb{P}^5\). The success of the maximal singularities method in 3-dimensional birational geometry posed the question on its generalization to the multidimensional case and to the case of singular Fano manifolds. Such a generalization was foun…
Mathematics Subject Classification ID: 14E07
Mathematics Subject Classification ID: 14J45
Mathematics Subject Classification ID: 14E30
Mathematics Subject Classification ID: 14J30
zbMATH DE Number: 1017574
zbMATH Keywords: birational automorphism groups
zbMATH Keywords: 3-dimensional birational geometry
zbMATH Keywords: singular Fano manifolds
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02363913
OpenAlex ID: W1965582960
description: scientific article; zbMATH DE number 1017574