Kustin–Miller unprojection without complexes

By Stavros A. Papadakis, M. Reid. 2000

100 citations recorded by the source corpus.

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; see also Papadakis [P1] and Reid [R3]–[R4].

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/da46e251ab8fb8ed5d62f5c904c25f9cea153a36

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. Gavin Brown
2. Michael Kerber
3. M. Reid
4. Lucia Caporaso
5. Nathan Pflueger

In Geometry and Topology

1. Gavin Brown
2. Michael Kerber
3. M. Reid
4. Lucia Caporaso
5. M. Kawakita
6. Nathan Pflueger

Citations in the corpus

• Fano 3-folds, K3 surfaces and graded rings (2002, unknown): “There are two different solutions to this problem, called Tom and Jerry, treated in detail in Papadakis [P]–[ P1 ] (see also [Ki], Examples 6.4 and 6.8, and Section 8). In these two cases, we specialise the skew 5 × 5 matrix defining V ′ to”
• Fano 3-folds in codimension 4, Tom and Jerry. Part I (2010, methodology): “dimension 4 remains one of the frontiers of science: there is no automatic structure theory, and deformations are almost always obstructed. Type I projection and Kustin–Miller unprojection (see [KM], [PR], [Ki]) is a substitute that is sometimes adequate. This paper addresses codimension 4 Fano 3-fo”