On the birational geometry of sextic threefold hypersurface in $\mathbf{P}(1,1,2,2,3)$

We investigate birational properties of hypersurfaces of degree $6$ in the weighted projective space $\mathbf{P}(1,1,2,2,3)$. In particular, we prove that any such quasi-smooth hypersurface is not rational.
Comment: 14 pages