Towards a general theory of unprojection
We propose a general definition of unprojection, and prove that it indeed generalizes previous efforts.
math.AG↗
arXiv subjects
Stavros Papadakis
Publications and source records attributed to Stavros Papadakis.
Kustin--Miller unprojection without complexes
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 (A. Kustin and M. Miller, Constructing big Gorenstein ideals from small ones, J. Algebra 85 (1983) 303--322), 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. The final section contains examples, and discusses briefly the applications to graded rings and birational geometry that motivate this study.
math.AG↗
Kustin-Miller unprojection with complexes
A main ingredient for Kustin-Miller unprojection, as developed in (S. Papadakis and M. Reid, Kustin-Miller unprojection without complexes, math.AG/0011094), is the module Hom_R(I, \om_R), where R is a local Gorenstein ring and I a codimension one ideal with R/I Gorenstein. We prove a method of calculating it in a relative setting using resolutions. We give three applications. In the first we generalise a result of (F. Catanese et al., Embeddings of curves and surfaces, Nagoya Math. J. 154 (1999), 185-220). The second and the third are about Tom and Jerry, two families of Gorenstein codimension four rings with 9x16 resolutions.
math.AG↗