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We construct a deformation family for each of the 34 Hilbert series of index 2 Fano 3-folds. In 18 cases we construct two different families, distinguished by the topology of their general members.We produce X by Type I [Pap04, PR04] and Type II unprojections [Pap06] from particular Fano 3-folds Z in codimension 3 and from Fano hypersurfaces respectively. The key is to have X invariant under γ. We discuss how and when this is possible in Sections 4, 5, 7. The details are summarised in Table 1 in 1.3.

We classify Sarkisov links from index 1 Fano 3-folds anticanonically embedded in codimension 4 that start from so-called Type I Tom centres. We apply this to compute the Picard rank of many such Fano 3-folds.

We show that five of Reid's Fano 3-fold hyperurfaces containing at least one compound Du Val singularity of type cA n have pliability at least two. The two elements of the pliability set are the singular hypersurface itself, and another non-isomorphic Fano hypersurface of the same degree, embedded in the same weighted projective space, but with different compound Du Val singularities. The birational map between them is the composition of two birational links initiated by blowing up two Type I centres on a codimension 4 Fano 3-fold of P 2 × P 2 -type having Picard rank 2.

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