A remark on the Kawamata rationality theorem
type: Item
instance of: scholarly article
title: A remark on the Kawamata rationality theorem
zbMATH Open document ID: 0803.14003
author: Mauro C. Beltrametti
author: Andrew J. Sommese
published in: Journal of the Mathematical Society of Japan
publication date: 1994-12-15T00:00:00Z
review text: Let \(\varphi : X \to Y\) be a surjective morphism with connected fibers, from the projective variety \(X\) (with at most Gorenstein rational singularities) to the normal projective variety \(Y\). Let moreover \(K_ X\) be not \(\varphi\)-nef, and let \(L\) be a \(\varphi\)-ample line bundle on \(X\). It follows from the Kawamata rationality theorem that the positive number \(t\) for which \(K_ X + t.L\) is \(\varphi\)-nef but not \(\varphi\)- ample is rational; moreover, if \(t = u/v\), where \(u\) and \(v\) are coprime, then \(u \leq m (\varphi) = \max_{y \in Y} \{\dim \varphi^{ - 1} (y)\} + 1\). If moreover \(L\) is ample and \(K_ X\) not nef, the Kawamata- Shokurov theorem implies that \(\varphi : X \to Y\) can be chosen such that \(v.K_ X + u.L\) is a preimage of an ample bundle on \(Y\). The corresponding pair \((u,v)\) is called the nef value of \((X,L)\). The main result (theorem 1.4) of the paper states that the extremal values \(u = m (\varphi)\), \(m (\varphi) - 1\) of \(u\) in a nef value \((u,v)\) of \((X,L)\) appear only in very special situations: \(X\) has to be either a scroll over \(Y\), or a quadratic fibration over \(Y\), or \(\varphi\) has to be birational (= a simultaneous contraction of a finite number of disjoint extremal rays). In addition, theorem 2.2 formulates conditions under which such a morphism \(\varphi\) with \(u = m(\varphi)\) is a projective bundle; in fact, it is enough to assume \(\text{Sing} (X) > \dim (Y)\), and \(u = m(\varphi)\) (see theorem 2.2).
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14J30
Mathematics Subject Classification ID: 14J40
zbMATH DE Number: 469078
zbMATH Keywords: Kawamata-Shokurov theorem
zbMATH Keywords: nef value
zbMATH Keywords: extremal rays
reviewed by: Atanas Iliev
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04530557
OpenAlex ID: W2106391029
DOI: 10.2969/JMSJ/04530557
description: scientific article; zbMATH DE number 469078