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VIDEO LIBRARY |
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Rings and varieties M. Reid University of Warwick, Mathematics Institute |
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Abstract: 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 <= 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 >= 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 |