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On sectional genus of quasi-polarized 3-folds

Author: Yoshiaki Fukuma
Journal: Trans. Amer. Math. Soc. 351 (1999), 363-377
MSC (1991): Primary 14C20; Secondary 14J99
MathSciNet review: 1487615
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Abstract: Let $X$ be a smooth projective variety over $\mathbb{C}$ and $L$ a nef-big (resp. ample) divisor on $X$. Then $(X,L)$ is called a quasi-polarized (resp. polarized) manifold. Then we conjecture that $g(L)\geq q(X)$, where $g(L)$ is the sectional genus of $L$ and $q(X)=\operatorname{dim}H^{1}(\mathcal{O}_{X})$ is the irregularity of $X$. In general it is unknown whether this conjecture is true or not, even in the case of $\operatorname{dim}X=2$. For example, this conjecture is true if $\operatorname{dim}X=2$ and $\operatorname{dim}H^{0}(L)>0$. But it is unknown if $\operatorname{dim}X\geq 3$ and $\operatorname{dim}H^{0}(L)>0$. In this paper, we prove $g(L)\geq q(X)$ if $\operatorname{dim}X=3$ and $\operatorname{dim}H^{0}(L)\geq 2$. Furthermore we classify polarized manifolds $(X,L)$ with $\operatorname{dim}X=3$, $\operatorname{dim}H^{0}(L)\geq 3$, and $g(L)=q(X)$.

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Additional Information

Yoshiaki Fukuma
Affiliation: Department of Mathematics, Faculty of Science, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo 152, Japan
Address at time of publication: Department of Mathematics, College of Education, Naruto University of Education, Takashima, Naruto-cho, Naruto-shi 772-8502, Japan

Received by editor(s): February 5, 1997
Article copyright: © Copyright 1999 American Mathematical Society

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