Complexes, unprojection and Fano 3-folds
(Magma Demonstration)
Gavin Brown
Warwick/Kent
The homogeneous coordinate ring of a non-hyperelliptic curve in
its canonical embedding is the epitome of the Gorenstein ring.
In 1983, Kustin and Miller described an algebraic method of making
big Gorenstein rings from small ones. Their method ties together
two simple free resolutions, and its main application is to build
rings in codimension 4 (or depth 4) where structure theorems are
not known. I will demonstrate this calculation using Magma's new
machinery for complexes of modules over affine algebras.
The Kustin--Miller method was reinterpreted in 2003 by Papadakis
and Reid in geometrical terms: it is an `unprojection', the inverse
of a Gorenstein projection, and it is computed by functions having
poles of very particular type. The geometry reveals a heirarchy
of such unprojections, and I will give examples (probably by Coughlan,
Papadakis, Reid) in various geometrical contexts where Gorenstein
rings abound.
@article {MR725084,
AUTHOR = {Kustin, Andrew R. and Miller, Matthew},
TITLE = {Constructing big {G}orenstein ideals from small ones},
JOURNAL = {J. Algebra},
FJOURNAL = {Journal of Algebra},
VOLUME = {85},
YEAR = {1983},
NUMBER = {2},
PAGES = {303--322},
ISSN = {0021-8693},
CODEN = {JALGA4},
MRCLASS = {13D25 (13H10 14M05)},
MRNUMBER = {MR725084 (85f:13014)},
MRREVIEWER = {P. Schenzel},
}
@article {MR2047681,
AUTHOR = {Papadakis, Stavros Argyrios and Reid, Miles},
TITLE = {Kustin-{M}iller unprojection without complexes},
JOURNAL = {J. Algebraic Geom.},
FJOURNAL = {Journal of Algebraic Geometry},
VOLUME = {13},
YEAR = {2004},
NUMBER = {3},
PAGES = {563--577},
ISSN = {1056-3911},
MRCLASS = {14M05},
MRNUMBER = {MR2047681},
}