Fano 3-folds in codimension 4, Tom and Jerry, Part I

TL;DR

This paper develops a new unprojection strategy called Tom and Jerry to construct numerous Gorenstein codimension 4 ideals, enabling the creation of many Fano 3-folds with diverse topologies and the same numerical invariants.

Contribution

It introduces the Tom and Jerry unprojection methods for constructing Gorenstein ideals and Fano 3-folds, expanding the known families and topological variations.

Findings

Constructed hundreds of Gorenstein codimension 4 ideals.

Produced multiple deformation families of Fano 3-folds with identical numerics.

Demonstrated the effectiveness of Tom and Jerry methods in Fano 3-fold classification.

Abstract

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.