Towards a general theory of unprojection
Let \(R\) be a Gorenstein ring. \textit{Unprojection} is a method to construct a new Gorenstein ring \(S\) of codimension bigger than the codimension of \(R\). From the algebraic point of view it is a substitute for a structure theory of Gorenstein rings in low codimension. From the geometric point of view it gives a graded ring method to construct terminal \(\mathbb{Q}\)-factorial Fano 3-folds and Sarkisov birational links between them. There are currently four types of unprojection, namely unprojection of types \textit{I, II, III} and \textit{IV}. The simplest case is unprojection of type \textit{I} due \textit{A. Kustin} and \textit{M. Miller} [J. Algebra 85, 303--322 (1983; Zbl 0522.13011)] and \textit{S. A. Papadakis} and \textit{M. Reid} [J. Algebr. Geom. 13, No. 3, 563--577 (2004; Zbl 1071.14047)]. In previous works the author has also studied cases of unprojection of types \textit{II} and \textit{III}. In this paper the author introduces a general definition of unprojection. In particular, given a commutative ring with unit \(\mathcal{O}_X\), and \(I_D \subset \mathcal{O}_X\) an ideal containing an \(\mathcal{O}_X\)-regular element \(q \in I_D\), he defines the unprojection algebra \(\mathrm{unpr}_{\mathcal{O}_X}I_D\) and he shows that this definition agrees with the definitions given in the special cases treated in previous works by himself and M. Reid.
scientific article