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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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