Explicit Birational Geometry of Fano threefold complete intersections

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. This allows, in particular, the construction of new examples of Fano 3-folds of codimension 6 which are realised as complete intersections in fake weighted projective spaces. 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 $A_2$ singularities.