(Based on joint work with Gavin Brown and Michael Kerber)

I first recall briefly the graded ring approach to polarised varieties, the orbifold Riemann–Roch formula giving their Hilbert series. The general program for classifying ℚ-K3 surfaces and ℚ-Fano 3-folds was outlined in [Singa]. There has been considerable progress on this in the last 2 or 3 years due to Gavin Brown and others. In particular, the combinatorics of the Hilbert series, and the large number of cases that happen, are handled by a convenient online Magma database (see [GRDB]).

A main aim is to treat the 142 numerical families of codim 4 Fano 3-folds by Gorenstein projections of the simplest type. The database tells us that 116 of the 142 numerical families have one or more numerical Type I projection to codim 3. We show that in each of these 116 x (one or more) cases, there is at least one Tom and one Jerry construction, leading to quasismooth Fano 3-folds that are not isomorphic; in particular, in each of these numerical cases the Hilbert scheme of codim 4 Fanos has at least two irreducible components containing quasismooth Fanos.

[Singa] S. Altnok, G. Brown and M. Reid, Fano 3-folds, K3 surfaces and graded rings, in Topology and geometry: commemorating SISTAG (National Univ. of Singapore, 2001), Ed. A. J. Berrick and others, Contemp. Math. 314, AMS, 2002, pp. 25–53, preprint arXiv:math/0202092v1, 29 pp.

[GRDB] Gavin Brown, Graded ring database