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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.

14J45 Fano varieties
14J30 \(3\)-folds
14E30 Minimal model program (Mori theory, extremal rays)
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