Fano threefolds of large Fano index and large degree (Q2841113)

Let \(X\) be a \(\mathbb{Q}\)-\textit{Fano \(3\)-fold}, i.e. \(X\) has terminal \(\mathbb{Q}\)-factorial singularities, \(\text{Pic}(X) = \mathbb{Z}\) and the divisor \(-K_X\) is ample. The quantity \(\text{q}\mathbb{Q}(X) := \max\{q\in\mathbb{Z}~|~ -K_X\sim_{\mathbb{Q}}qA,\;A\;\text{is a Weil divisor}\}\) is called \(\mathbb{Q}\)-\textit{Fano index}. Let also \(g(X) := \dim |-K_X| - 1\) be the \textit{genus} of \(X\). The author of the paper under review had shown that \(q:=\text{q}\mathbb{Q}(X)\in\{1,\ldots,11,13,17,19\}\) (see \textit{Y. Prokhorov} [Doc. Math., J. DMV 15, 843--872 (2010; Zbl 1218.14031)], {K. Suzuki} [Manuscr. Math. 114, No. 2, 229--246 (2004; Zbl 1063.14049)]). Moreover, he had also proved that \(q\neq 10\), while for \(q\geq 9\) all \(X\) turned out to be weighted projective spaces (see the explicit list in Theorem 1.1 in the text).NEWLINENEWLINEPresently, the author studies those \(X\) which have \(3\leq q\leq 9\) and \(g(X)>4\). It turns out (see Theorem 1.2) that some (partial) classification is also possible here, so that in addition to weighted projective spaces \(X\) can be a hypersurface in these. This complements similar results on the classification of \(\mathbb{Q}\)-Fano \(3\)-folds obtained by G. Brown and K. Suzuki (we refer to the literature sited in the paper).NEWLINENEWLINEThe proof of the main result relies on a computer-based description of the basket of singularities of \(X\) (via the algorithm implemented in \textit{G. Brown} and \textit{K. Suzuki} [Japan J. Ind. Appl. Math. 24, No. 3, 241--250 (2007; Zbl 1166.14027)]) and, roughly, goes as follows. Considering the diagram (2.2) in the text and using numerical relations (2.5), the author shows that the pair \((X,A_k)\) is canonical for some \(k<q\) and generic \(A_k\in|kA|\), given the restrictions on \(q\) and \(g\) as above. This forces \(A_k\) to be a del Pezzo surface, having (non-trivial) Du Val singularities, which yields \(\text{Cl}(A_k)=\mathbb{Z}\) and \(A_k\) be a weighted complete intersection. The author then concludes by Theorem 3.8 (the \textit{hyperplane section principle}).