Linear system on the fano threefold

Abstract

Let X be a smooth projective threefold whose anticanonical division $-K_X$ is ample, i.e., Fano threefold. In this paper, we studied the linear system $$\mid$-nK_X$\mid$$ for a positive integer n. In Theorem 4, we studied the cases that $\-nK_X$\mid$$ has no base-points and the cases that $$\mid$-nK_X$\mid$$ generate the birational map. In Proposition 5, we studied the possible exceptional cases given in Theorem 4. Some results in this paper are already known, but we have gave brief proofs for those results.

Keywords

• linear system;
• birational map;
• base point;
• Fano threefold

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