  This work is part of the Graded Ring Database project [GRDB], and is a sequel
to [Altinok's 1998 PhD thesis] and [Altinok, Brown and Reid, Fano 3-folds, K3
surfaces and graded rings, in SISTAG (Singapore, 2001), Contemp. Math. 314,
2002, pp. 25-53]. We introduce a strategy based on Kustin-Miller unprojection
that constructs many hundreds of Gorenstein codimension 4 ideals with 9x16
resolutions (that is, 9 equations and 16 first syzygies). Our two basic games
are called Tom and Jerry; the main application is the biregular construction of
most of the anticanonically polarised Mori Fano 3-folds of Altinok's thesis.
There are 115 cases whose numerical data (in effect, the Hilbert series) allow
a Type I projection. In every case, at least one Tom and one Jerry construction
works, providing at least two deformation families of quasismooth Fano 3-folds
having the same numerics but different topology.

Algebraic Geometry;Commutative Algebra