Geodesic
$\mathbb Q$-Fano threefolds of index~$7$
Informatics and Automation, Modern problems of mathematics, mechanics, and mathematical physics. II, Tome 294 (2016), pp. 152-166

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We show that if the inequality $\dim\mathopen|-K_X|\ge15$ holds for a $\mathbb Q$-Fano threefold $X$ of Fano index $7$, then $X$ is isomorphic to one of the following varieties: $\mathbb P(1^2,2,3)$, $X_6\subset\mathbb P(1,2^2,3,5)$, or $X_6\subset\mathbb P(1,2,3^2,4)$. 
@article{TRSPY_2016_294_a8,
     author = {Yuri G. Prokhorov},
     title = {$\mathbb Q${-Fano} threefolds of index~$7$},
     journal = {Informatics and Automation},
     pages = {152--166},
     publisher = {mathdoc},
     volume = {294},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2016_294_a8/}
}
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Yuri G. Prokhorov. $\mathbb Q$-Fano threefolds of index~$7$. Informatics and Automation, Modern problems of mathematics, mechanics, and mathematical physics. II, Tome 294 (2016), pp. 152-166. http://geodesic.mathdoc.fr/item/TRSPY_2016_294_a8/