The 4-intersection unprojection format

Unprojection theory is a philosophy due to Miles Reid, which becomes a useful tool in algebraic geometry for the construction and the study of new interesting geometric objects such as algebraic surfaces and 3-folds. In this present work we introduce a new format of unprojection, which we call the 4-intersection format. It is specified by a codimension 2 complete intersection ideal I which is contained in four codimension 3 complete intersection ideals $J_1,J_2,J_3,J_4$ and leads to the construction of codimension 6 Gorenstein rings. As an application, we construct three families of codimension 6 Fano 3-folds embedded in weighted projective space which correspond to the entries with identifier numbers 29376, 9176 and 24198 respectively in the Graded Ring Database.