Pluricanonical systems on irregular 3-folds of general type
Let \(X\) be a smooth projective variety of dimension \(n\) of general type defined over the field of complex numbers. Then for all sufficiently large \(m\), the pluricanonical system \(|mK_{X}|\) defines a birational map \(\varphi_{m}:X\to \mathbb{P}^{N}\) to a projective space \(\mathbb{P}^{N}\). Then it is natural to find an effective bound for such integers \(m\). When \(X\) is a curve, \(\varphi_{m}\) is an embedding for any \(m\geq 3\). For the case of \(\dim X=2\), the map \(\varphi_{m}\) is birational for any \(m\geq 5\). In this paper, the authors studied the case of irregular \(3\)-folds and they obtained the following result. Let \(X\) be an irregular smooth projective variety of general type with \(\dim X=3\) and \(\chi(\omega_{X})>0\). Then the rational map defined by \(|mK_{X}+P|\) is birational for any \(m\geq 5\) and any \(P\in \text{Pic}^{0}(X)\).
scientific article; zbMATH DE number 5706069
scientific article