### School of Mathematics and Statistics

### Miles Reid

University of Warwick

### Birational geometry and graded rings.

**Friday 5th January, 12-1pm,
Carslaw 275. **

In algebraic geometry, an affine variety is studied in terms of its
coordinate ring, and a projective variety in terms of its homogeneous
coordinate ring. The lecture will run through some basic and more
advanced examples of how rings are used to study varieties.

The canonical ring of a regular algebraic surface of general type or
the anticanonical ring of a Fano variety is a Gorenstein ring; in
simple cases a Gorenstein ring is a hypersurface, a codimension 2
complete intersection, or a codimension 3 Pfaffian. We now have
additional techniques based on the idea of projection in birational
geometry that produce results in codimension 4, even though there is
at present no useable structure theory for the ring.

For more information, see the e-print
Stavros Papadakis and Miles Reid, Kustin-Miller unprojection without
complexes, math.AG/0011094, 15 pp. submitted to J. Algebraic Geometry

Gorenstein projections play a key role in birational geometry; the
typical example is the linear projection of a del Pezzo surface of degree
d to one of degree d-1, but variations on the same idea provide many of
the classical and modern birational links between Fano 3-folds. The
inverse operation is the Kustin-Miller unprojection theorem (A. Kustin
and M. Miller, Constructing big Gorenstein ideals from small ones, J.
Algebra 85 (1983) 303--322), which constructs "more complicated"
Gorenstein rings starting from "less complicated" ones (increasing the
codimension by 1). We give a clean statement and proof of their
theorem, using the adjunction formula for the dualising sheaf in place
of their complexes and Buchsbaum-Eisenbud exactness criterion. Our
methods are scheme theoretic and work without any mention of the
ambient space. They are thus not restricted to the local situation, and
are well adapted to generalisations.