A linear system on a threefold (Q2365887)

The aim of the paper is to study the rational map associated to the adjoint linear system on a smooth complex threefold. The main result is the following. Let \(X\) be a smooth projective threefold and let \(D\) be a nef and big divisor on \(X\). Assume that \(h^ 0 \bigl( X,{\mathcal O}_ X(mD)\bigr) \geq 2\) for some positive integer \(m\). Then the rational map associated to the complete linear system \(| K_ X+nD |\) is birational for a positive integer \(n \geq m+4\) such that \(h^ 0 \bigl( X,{\mathcal O}_ X(n-m)D)\bigr) \geq 1\). The result is proved by using Reider's type technique.