Explicity birational geometry of Fano 3-folds of high index

We complete the analysis on the birational rigidity of quasismooth Fano 3-fold deformation families appearing in the Graded Ring Database as a complete intersection. When such a deformation family X has Fano index at least 2 and is minimally embedded in a weighted projective space in codimension 2, we determine which cyclic quotient singularity is a maximal centre. If a cyclic quotient singularity is a maximal centre, we construct a Sarkisov link to a non-isomorphic Mori fibre space or a birational involution. We define linear cyclic quotient singularities on X and prove that these are maximal centres by explicitly computing Sarkisov links centred at them. It turns out that each X has a linear cyclic quotient singularity leading to a new birational model. As a consequence, we show that if X is birationally rigid then its Fano index is 1. If the new birational model is a strict Mori fibre space, we determine its fibration type explicitly. In this case, a general member of X is birational to a del Pezzo fibration of degrees 1, 2 or 3 or to a conic bundle Y/S where S is a weighted projective plane with at most A₂ singularities.