Constructing prime Q-Fano threefolds of codimension four via key varieties related with P²× P²-fibrations

In the previous papers [Tak8,Tak6], we construct the affine varieties ΣA¹³ and ΠA¹⁴ whose partial projectivizations have P²²-fibrations. In this paper, we produce prime Q-Fano 3-folds of anticanonical codimension 4 belonging to 23 (resp. 8) classes of Graded Ring Database [GRDB] as weighted complete intersections of weighted projectivizations of ΣA¹³ (resp. ΠA¹⁴ or the cone over it). Together with the results of [CD] and [Tak5], prime Q-Fano 3-folds of anticanonical codimension 4 are constructed for 141 classes among the 143 classes in [GRDB].