The generic finiteness of the $m$-canonical map for 3-folds of general type

Abstract

Let $X$ be a projective minimal threefold of general type with only $\mathbb{Q}$-factorial terminal singularities. We study the generic finiteness of the $m$-canonial map for such 3-folds. Suppose $P_g(X)\ge 2$ and $q(X)\ge 3$. We prove that the $m$-canonical map is generically finite for $m\ge 3$, which is a supplement to Kollár's result. Suppose $P_g(X)\ge 5$. We prove that the 3-canonical map is generically finite, which improves Meng Chen's result.