Rings and varieties

I leave the title and abstract as vague as possible, so that I can talk about whatever I feel like on the day. Many varieties of interest in the classification of varieties are obtained as Spec or Proj of a Gorenstein ring. In codimension $\le 3$, the well known structure theory provides explicit methods of calculating with Gorenstein rings. In contrast, there is no useable structure theory for rings of codimension $\ge 4$. Nevertheless, in many cases, Gorenstein projection (and its inverse, Kustin-Miller unprojection) provide methods of attacking these rings. These methods apply to sporadic classes of canonical rings of regular algebraic surfaces, and to more systematic constructions of $Q$-Fano 3-folds, Sarkisov links between these, and the 3-folds flips of Type A of Mori theory.
For introductory tutorial material, see my website + surfaces + Graded rings and the associated homework.
For applications of Gorenstein unprojection, see “Graded rings and birational geometry” on my website + 3-folds, or the more recent paper.
Gavin Brown, Michael Kerber and Miles Reid, Fano 3-folds in codimension 4, Tom and Jerry (unprojection constructions of $Q$-Fano 3-folds), Composition to appear, arXiv:1009.4313.