A remark on the Kawamata rationality theorem
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).
scientific article; zbMATH DE number 469078