Contractions of Gorenstein polarized varieties with high nef value (Q1345948)

Let \(X\) be a normal complex projective variety of dimension \(n\) with Gorenstein terminal singularities; let \(L\) be an ample line bundle over \(X\) and let \(K_ X\) denote the canonical sheaf of \(X\). Assuming that \(K_ X\) is not nef we study the contractions of extremal faces which are supported by divisors of the form \(K_ X + \tau L\) with \(\tau \geq (n- 2)\). In other words we classify the pairs \((X,L)\) which have ``nef value'' \(=\tau (X,L) \geq (n-2)\) as well as the structure of their associate ``nef value morphisms''. In the case \(\tau = (n-2)\) we assume also that \(X\) is factorial. We study moreover the general case in which \((K_ X + \tau L)\) is nef and big but not ample and the dimension of the fibers of the nef value morphism is \(\leq r\).