Algebra Seminar

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.