On the pluricanonical map of threefolds of general type

Abstract:

Let $X$ be a smooth minimal threefold of general type and let $n$ be an integer $>1$. Assume that the image of the pluricanonical map $\Phi _{n}$ of $X$ is a curve. Then a simple computation shows that $n$ is necessarily $2$ or $3$. When $n=2$ with a numerical condition or when $n=3$, we obtain two inequalities $\chi (\mathcal {O}_{X})\leq \text {min}\{-1,2-2q_{1}\}$ and $q_{1}\leq \dfrac {3}{14}{K_{X}}^{3}+1$, where $q_{1}$ is the irregularity of $X$ and $\chi (\mathcal {O}_{X})$ is the Euler characteristic of $X$.