Graded rings and birational geometry

By M. Reid. 2000

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Birational Geometry of 3-fold Mori Fibre Spaces (2003, methodology): “Starting from Y 2,3, we construct Y as an unprojection, that is, the contraction of the Weil divisor � ⊂ Y ; see [ Rei ], [ABR02] for details.”
Fano 3-folds, K3 surfaces and graded rings (2002, unknown): “This second stage is the hard part, and we pass over it for the present (see [PR], [P1] and [ Ki ]), although in time the computer will have something useful to add here too.”
Birational geometry of quartic 3-folds II: The importance of being -factorial (2004, methodology): “This can be easily seen with the unprojection method developed by Reid, [ Un ].”
Kustin–Miller unprojection without complexes (2000, methodology): “The Tom and Jerry families of unprojection (Section 5), which were defined and named by Reid, play a more important role in algebraic geometry because they appear in hundreds of known examples; it is an open problem ([ Ki ], Problem 8.3) whether all the currently known Gorenstein codimension 4 rings”