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On anti-pluricanonical systems of $$\mathbb Q$$-Fano 3-folds. (English) Zbl 1258.14016
Let $$X$$ be a terminal weak $$\mathbb Q$$-Fano threefold, i.e, $$X$$ is $$\mathbb Q$$-factorial, has at worst terminal singularities, and $$-K_X$$ is $$\mathbb Q$$-Cartier, nef and big.
In the paper under review the author studies the behaviour of the anti-pluricanonical maps $$\varphi_{-m}$$, i.e. the rational maps defined by the linear systems $$|-mK_X|$$, with $$m$$ a positive integer.
It is known, by the boundedness theorem, that there exists $$m$$ such that $$\varphi_{-m}$$ is birational onto its image for any weak $$\mathbb Q$$-Fano threefold. The main concern of this paper is to find an optimal constant $$c$$ such that, for $$m \geq c$$, $$\varphi_{-m}$$ is birational onto its image for any weak $$\mathbb Q$$-Fano threefold. In Theorem 1.1 the author finds a lower bound for this constant.
He then defines a special class of $$\mathbb Q$$-Fano threefolds of Picard number one, called “good”, for which he can prove that $$c \leq 6$$. Finally, in the last section, some applications of the birationality of $$\varphi_{-m}$$ are considered.

##### MSC:
 14E05 Rational and birational maps 14J30 $$3$$-folds 14J45 Fano varieties
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