In the book of Iskovskikh and Prokhorov it seems not known wether the $V_1$, an hypersurface of degree $6$ in the weighted projective space $\mathbb{P}(3,2,1,1,1)$, is rational or not. Is there any progress since then ?
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You mean degree 6 in $\mathbb{P}(3,2,1,1,1)$. It is not rational, and its birational automorphisms are biregular. This has been proved by M. Grinenko, Mori structures on a Fano threefold of index 2 and degree 1, Proc. Steklov Inst. Math. 246 (2004), 103-128. The russian version is available here.
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$\begingroup$ Thanks a lot! Yes $\mathbb{P}(3,2,1,1,1)$, it is changed now... $\endgroup$ Commented Oct 11, 2016 at 12:58