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}(6,2,1,1,1)$$\mathbb{P}(3,2,1,1,1)$, is rational or not. Is there any progress since then ?
IMeasy
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