This package implements the construction of the Kustin-Miller complex [1]. This is the fundamental construction of resolutions in unprojection theory [2]. For details on the computation of the Kustin-Miller complex see [3].
Gorenstein rings with an embedding codimension at most 2 are known to be complete intersections, and those with embedding codimension 3 are described by the theorem of Buchsbaum and Eisenbud as Pfaffians of a skew-symmetric matrix; general structure theorems in higher codimension are lacking and the main goal of unprojection theory is to provide a substitute for a structure theorem.
Unprojection theorey has been applied in various cases to construct new varieties, for example, in [4] in the case of Campedelli surfaces and [5] in the case of Calabi-Yau varieties.
We provide a general command kustinMillerComplex for the Kustin-Miller complex construction and demonstrate it on several examples connecting unprojection theory and combinatorics such as stellar subdivisions of simplicial complexes [6], minimal resolutions of Stanley-Reisner rings of boundary complexes Δ(d,m) of cyclic polytopes of dimension d on m vertices [7], and the classical (non-monomial) Tom example of unprojection [2].
This package requires the package SimplicialComplexes.m2 version 1.2 or higher, so install this first.
References:
For the Kustin-Miller complex see:
[1] A. Kustin and M. Miller, Constructing big Gorenstein ideals from small ones, J. Algebra 85 (1983), 303-322.
[2] S. Papadakis, Kustin-Miller unprojection with complexes, J. Algebraic Geometry 13 (2004) 249-268, http://arxiv.org/abs/math/0111195
[3] J. Boehm, S. Papadakis: Implementing the Kustin-Miller complex construction, http://arxiv.org/abs/1103.2314
For constructing new varieties see for example:
[4] J. Neves and S. Papadakis, A construction of numerical Campedelli surfaces with ZZ/6 torsion, Trans. Amer. Math. Soc. 361 (2009), 4999-5021.
[5] J. Neves and S. Papadakis, Parallel Kustin-Miller unprojection with an application to Calabi-Yau geometry, preprint, 2009, 23 pp, http://arxiv.org/abs/0903.1335
For the stellar subdivision case see:
[6] J. Boehm, S. Papadakis: Stellar subdivisions and Stanley-Reisner rings of Gorenstein complexes, http://arxiv.org/abs/0912.2151
For the case of cyclic polytopes see:
[7] J. Boehm, S. Papadakis: On the structure of Stanley-Reisner rings associated to cyclic polytopes, http://arxiv.org/abs/0912.2152, to appear in Osaka J. Math.
Examples:
Cyclic Polytopes -- Minimal resolutions of Stanley-Reisner rings of boundary complexes of cyclic polytopes
Stellar Subdivisions -- Stellar subdivisions and unprojection
Tom -- The Tom example of unprojection
Jerry -- The Jerry example of unprojection
Key user functions:
The central function of the package is:
kustinMillerComplex -- The Kustin-Miller complex construction
Also important is the function to represent the unprojection data as a homomorphism:
unprojectionHomomorphism -- Compute the homomorphism associated to an unprojection pair
Functions used in the examples to compare with the combinatorics:
delta -- The boundary complex of a cyclic polytope
stellarSubdivision -- Compute the stellar subdivision of a simplicial complex
Version 1.4 of this package was accepted for publication in volume 4 of the journal The Journal of Software for Algebra and Geometry: Macaulay2 on 2012-05-07, in the article Implementing the Kustin-Miller complex construction. That version can be obtained from the journal or from the Macaulay2 source code repository, svn://svn.macaulay2.com/Macaulay2/trunk/M2/Macaulay2/packages/KustinMiller.m2, release number 14712.
This documentation describes version 1.4 of KustinMiller.
The source code from which this documentation is derived is in the file KustinMiller.m2.