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Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Kustin–Miller unprojection without complexes

Authors: Stavros Argyrios Papadakis and Miles Reid
Journal: J. Algebraic Geom. 13 (2004), 563-577
Published electronically: January 27, 2004
MathSciNet review: 2047681
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Abstract | References | Additional Information


Gorenstein projection plays a key role in birational geometry; the typical example is the linear projection of a del Pezzo surface of degree $d$ to one of degree $d-1$, but variations on the same idea provide many of the classical and modern birational links between Fano 3-folds. The inverse operation is the Kustin–Miller unprojection theorem, which constructs “more complicated” Gorenstein rings starting from “less complicated” ones (increasing the codimension by 1). We give a clean statement and proof of their theorem, using the adjunction formula for the dualising sheaf in place of their complexes and Buchsbaum–Eisenbud exactness criterion. Our methods are scheme theoretic and work without any mention of the ambient space. They are thus not restricted to the local situation, and are well adapted to generalisations.

Section 2 contains examples, and discusses briefly the applications to graded rings and birational geometry that motivate this study.

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Additional Information

Stavros Argyrios Papadakis
Affiliation: Mathematics Institute, University of Warwick, Coventry CV4 7AL, England
Address at time of publication: Fakultaet fuer Mathematik und Informatik, Geb. 27, Universitaet des Saarlandes, D-66123, Saarbruecken, Germany

Miles Reid
Affiliation: Mathematics Institute, University of Warwick, Coventry CV4 7AL, England

Received by editor(s): February 25, 2002
Published electronically: January 27, 2004
Additional Notes: The first author thanks the Greek State Scholarships Foundation for support. We both thank Kyoto University, RIMS for generous support and hospitality