On pluricanonical maps for 3-folds of general type (Q5903092)

Let \(K_ X\) be the canonical divisor of a non singular projective 3- dimensional variety X/\({\mathbb{C}}\) and let \(\Phi_{| nK_ X|}\) be the rational map associated to the linear system \(| nK_ X|\). \(K_ X\) is said to be ''nef'' if \(K_ X\cdot C\geq 0\) for any curve \(C\subset X\) and it is said to be ''big'' if X is of general type. Main result: if \(K_ X\) is nef and big then \((i)\quad \Phi_{| nK_ X|}\) is birational with the possible exceptions of \((a)\quad \chi ({\mathcal O}_ X)=0\) and \(K^ 3_ X=2\) or \((b)\quad | 3K_ X|\) is composed of pencils dim \(\Phi\) \({}_{| 3K_ X|}(X)=1\); \((ii)\quad \Phi_{| nK_ X|}\) is birational for \(n\geq 8\).