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$$\mathbb Q$$-Fano threefolds of large Fano index. I. (English) Zbl 1218.14031
A $$\mathbb{Q}$$-Fano variety is a normal projective variety $$X$$ with terminal $$\mathbb{Q}$$-factorial singularities such that $$-K_X$$ is ample and the Picard group $$\text{Pic}(X)$$ has rank 1. The index of a smooth Fano variety $$X$$ is the largest integer $$q(X)$$ such that $$-K_X \sim q(x) A$$, where $$A$$ is a Cartier divisor on $$X$$. If $$X$$ is singular then $$qW(X)$$ and $$q\mathbb{Q}(X)$$ are defined to be the largest integers $$q_1, q_2$$ such that $$-K_X \sim q_1A_1$$ and $$-K_X \sim_{\mathbb{Q}} q_2A_2$$, respectively, where $$A_1$$, $$A_2$$ are $$\mathbb{Q}$$-Cartier divisors on $$X$$. If $$X$$ is smooth then it is well known that $$q(X)=qW(X)=q\mathbb{Q}(X)$$ and that $$1 \leq q(X) \leq \dim X +1$$. Smooth Fano threefolds of Picard number 1 are completely classified [V. A. Iskovskikh and Yu. G. Prokhorov, Encycl. Math. Sci. 47, 1–245 (1999; Zbl 0912.14013)]. In particular, $$q(X)=\dim X+1$$ if and only if $$X \cong \mathbb{P}^n$$.
The paper under review gives a partial classification of singular $$\mathbb{Q}$$-Fano threefolds. Initially, using the results of K. Suzuki on $$qW(X)$$ [Manuscr. Math. 114, No. 2, 229–246 (2004; Zbl 1063.14049)], the author shows that $$q\mathbb{Q}(X) \in \{1,\dots,11, 13,17,19\}$$. The main result of the paper is that if $$q\mathbb{Q}(X) \geq 5$$ and under certain conditions on $$\dim |-K_X|$$, then $$X$$ is a weighted projective space. In particular, if $$q\mathbb{Q}(X)=19$$ then $$X \cong \mathbb{P}(3,4,5,7)$$, if $$q\mathbb{Q}(X)=17$$ then $$X \cong \mathbb{P}(2,3,5,7)$$. Moreover, the case $$q\mathbb{Q}(X)=10$$ does not happen.

##### MSC:
 14J45 Fano varieties 14J30 $$3$$-folds 14E30 Minimal model program (Mori theory, extremal rays)
PARI/GP
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