An application of Noether-Fano inequalities (Q1127649)

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).