  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 the
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 which is contained in four codimension 3 complete intersection ideals 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.
