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Unprojection is a theory due to Reid which constructs more complicated rings starting from simpler data. In the present work, we develop a new method of unprojection. Starting from a codimension 3 ideal defined by the Pfaffians of a 5 × 5 skewsymmetric matrix, we describe a number of unprojection formats that construct Gorenstein rings of codimension 6. We give two applications that construct families of Fano 3-folds embedded anticanonically in codimension 6 that existing methods do not realize.

Tom and Jerry triples with an application to Fano 3-folds