K3s and Fano 3-folds, Tom and Jerry -- Miles Reid

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

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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 <a
href=http://arxiv.org/abs/math/?0202092>[Singa]</a>.  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 database (see <a
href=http://anansi.maths.warwick.ac.uk/grdb/>[GRDB]</a>).

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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.

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<a href=http://arxiv.org/abs/math/?0202092>[Singa]</a> 
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
math.AG/0202092, 29 pp.

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<a href=http://anansi.maths.warwick.ac.uk/grdb/>[GRDB]</a> 
Gavin Brown, Graded ring database, see
  http://anansi.maths.warwick.ac.uk/grdb/