Кольца и многообразия

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 $\leqslant3$, 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 $\geqslant4$. 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