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 ?



Rationality of $V_1$ fano threefold
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](http://www.mathnet.ru/links/0f52fb02ec3b740b5ed48033fd5717be/tm149.pdf).


