Quasi-smooth Fano 3-fold hypersurfaces in the 95 families

In 1979 M. Reid announced the 95 families of K3 surfaces in three dimensional weighted projective spaces. After this, A. R. Fletcher, who was a Ph.D. student of M. Ried, announced the 95 families of weighted Fano threefold hypersurfaces in his MPIM preprint in 1988. These are quasi-smooth hypersurfaces of degrees $d$ with only terminal singularities in weighted projective spaces $\mathbb P(1, a_1, a_2, a_3, a_4)$, where $d=a_1+a_2+a_3+a_4$. The 95 families are determined by quadruples of non-decreasing positive integers $(a_1, a_2, a_3, a_4)$. In late nineties, these 95 families were revived and attracted birational geometers to study their properties such as birational rigidity, groups of birational automorphisms, elliptic fibration structures, and so forth. In particular, A. Corti, A. Pukhlikov, and M. Reid proved that a general hypersurface in each of the 95 families of weighted Fano threefold hypersurfaces is birationally rigid.
In my talk, I will discuss the birational rigidity for every quasi-smooth hypersurface.