A theorem on pluricanonical maps of nonsingular minimal threefolds of general type (Q1273102)

Let \(X\) be a complex nonsingular minimal projective threefold of general type. The nature of pluricanonical maps of \(X\) is very important to the classification theory. It is well-known that \(\Phi_{|mK_X |}\) is a birational map for \(m\geq 7\). \textit{M. Chen} [J. Math. Soc. Japan 50, No. 3, 615-621 (1998; Zbl 0916.14020)] proved that a 6-canonical map of \(X\) is a birational map onto its image. In this paper, we mainly study the following problem: Problem. What is the greatest positive integer \(m_0\) such that \(|m_0K_X|\) is composed of a pencil of surfaces for some \(X\), i.e., \(\dim\Phi_{|m_0K_X|}(X)=1\)? \textit{X. Benveniste} [Am. J. Math. 108, 433-449 (1986; Zbl 0601.14035)] proved that \(m_0\leq 3\). We can easily see that \(m_0\geq 1\) [see \textit{M. Chen} and \textit{Z. J. Chen}, Math. Proc. Camb. Philos. Soc. 125, No. 1, 83-87 (1999)]. Our result in this paper is that \(m_0\leq 2\). Main theorem. Let \(X\) be a nonsingular minimal projective threefold of general type. Then \(\dim\Phi_{|mK_X |}(X)\geq 2\) for \(m\geq 3\).