Abstract
Let \((X,L)\) be a complex polarized manifold of dimension \(n\ge 3\). In this paper, we give a classification of \((X,L)\) with \(n=3\) and \(h^{0}(K_{X}+2L)=2\). In order to classify these \((X,L)\), we study a classification of \((X,L)\) which satisfies \(h^{0}(K_{X}+(n-1)L)=2\) and \(\kappa (K_{X}+(n-2)L)=-\infty \). Moreover we also classify \((X,L)\) with \(n=3\), \(\kappa (K_{X}+L)\ge 0\) and \(h^{0}(K_{X}+2L)=3\).
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Notes
If \(X\) is smooth, then the sectional genus \(g(X,L)\) is defined by the following formula: \(g(X,L)=1+\frac{1}{2}(K_{X}+(n-1)L)\cdot L^{n-1}\).
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This research was partially supported by the Grant-in-Aid for Scientific Research (C) (No. 20540045), Japan Society for the Promotion of Science, Japan.
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Fukuma, Y. On classification of polarized 3-folds \((X,L)\) with \(h^{0}(K_{X}+2L)=2\) . Beitr Algebra Geom 55, 77–103 (2014). https://doi.org/10.1007/s13366-013-0161-7
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DOI: https://doi.org/10.1007/s13366-013-0161-7