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 ₀^13 and ₀^14 whose partial projectivizations have P^2^2-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 ₀^13 (resp. ₀^14 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.