An application of Noether-Fano inequalities
type: Item
instance of: scholarly article
title: An application of Noether-Fano inequalities
zbMATH Open document ID: 0939.14024
DOI: 10.1007/PL00004406
author: Nobuyoshi Takahashi
published in: Mathematische Zeitschrift
publication date: 1998-11-08T00:00:00Z
review text: From the introducton: When a birational map between two varieties is given, one problem is to describe it concretely. Restricting to birational maps between Mori fiber spaces of dimension three (e.g. Cremona transformations of a projective space), an answer is given, at least theoretically, by the so-called Sarkisov program which says that they are decomposed into ``birational links''. The starting point of the Sarkisov program is to prove the ``Noether-Fano inequalities'', an analogue of the classical Noether inequality in the context of the minimal model theory. -- \textit{A. Bruno} and \textit{K. Matsuki} [Int. J. Math. 8, No. 4, 451-494 (1997; Zbl 0903.14001)] and the author in his master thesis (``Sarkisov program for log surfaces'', Tokyo 1995) developed the log Sarkisov program, i.e. an extension of the Sarkisov program for log varieties. In this paper, the author proves the following result (theorem 2.3) as an application of log Noether-Fano inequalities for the birational geometry of birational pairs of Kodaira dimension 0: Let \(X\) be a Fano manifold (over \(\mathbb{C})\) with \(\rho(X)=1\), \(S\in|-K_X |\) a smooth hypersurface with \(\text{Pic}(X)\to \text{Pic}(S)\) surjective, \(X'\) a \(\mathbb{Q}\)-factorial terminal variety with \(\rho(X')=1\) and \(\Phi:X\to X'\) a birational map. Then \(K_{X'}+S'\) is nef, where \(S':= \Phi_*S\), and if \(K_{X'}+ S'\equiv 0\), then \(\Phi\) is an isomorphism. In remark 2.4, we give an example of a smooth quartic surface which is mapped to itself by a nonlinear Cremona transformation of \(\mathbb{P}^3\). We also give a brief remark on automorphisms of open subvarieties of projective spaces (remark 2.5).
Mathematics Subject Classification ID: 14J45
Mathematics Subject Classification ID: 14E05
Mathematics Subject Classification ID: 14E30
zbMATH DE Number: 1185889
zbMATH Keywords: numerical effective divisor
zbMATH Keywords: minimal model theory
zbMATH Keywords: log Noether-Fano inequalities
zbMATH Keywords: birational geometry
zbMATH Keywords: Fano manifold
MaRDI profile type: Publication
full work available at URL: pl00004406
OpenAlex ID: W2060532114
description: scientific article; zbMATH DE number 1185889