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On the pluricanonical map of threefolds
of general type

Author: Dong-Kwan Shin
Journal: Proc. Amer. Math. Soc. 124 (1996), 3641-3646
MSC (1991): Primary 14E05, 14J30
MathSciNet review: 1389536
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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 $X$.

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Additional Information

Dong-Kwan Shin
Affiliation: Department of Mathematics, Konkuk University, Seoul, 143–701, Korea

Keywords: Threefold of general type, pluricanonical map, fiber space, Euler characteristic
Received by editor(s): June 12, 1995
Additional Notes: This paper is supported by KOSEF and Dae-Yang Foundation
Communicated by: Eric M. Friedlander
Article copyright: © Copyright 1996 American Mathematical Society