This package implements the construction of the Kustin-Miller complex [1]. This is the fundamental construction of resolutions in unprojection theory [2].
The main goal of unprojection theory is to provide a substitute for structure theorems not available for Gorenstein rings of codimension >3.
It has been applied in various cases to construct new varieties, e.g., in [3], [4] for Campedelli surfaces and Calabi-Yau varieties.
We provide a general command kustinMillerComplex for the Kustin-Miller construction and demonstrate it at several examples connecting unprojection theory and combinatorics like stellar subdivisions of simplicial complexes [5], minimal resolutions of Stanley-Reisner rings of boundary complexes Δ(d,m) of cyclic polytopes of dimension d on m vertices [6], and the classical (non-monomial) Tom example of unprojection [2].
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] Papadakis, Kustin-Miller unprojection with complexes, J. Algebraic Geometry 13 (2004) 249-268, http://arxiv.org/abs/math/0111195
For constructing new varieties see for example:
[3] J. Neves and S. Papadakis, A construction of numerical Campedelli surfaces with ZZ/6 torsion, Trans. Amer. Math. Soc. 361 (2009), 4999-5021.
[4] 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:
[5] 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:
[6] J. Boehm, S. Papadakis: On the structure of Stanley-Reisner rings associated to cyclic polytopes, http://arxiv.org/abs/0912.2152
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
Key functions and data types:
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
Types and functions used to compare with the combinatorics:
Complex -- The class of all simplicial complexes
Face -- The class of all faces of simplicial complexes
complexToIdeal -- Compute the Stanley-Reisner ideal associated to a simplicial complex
idealToComplex -- Compute the Stanley-Reisner complex associated to a monomial square free ideal
delta -- The boundary complex of a cyclic polytope
stellarSubdivision -- Compute the stellar subdivision of a simplicial complex
Installation:
Put the file KustinMiller.m2 somewhere into the path of Macaulay2 (usually into the directory .Macaulay2/code inside your home directory, type path in M2 to see the path) and do inside M2
installPackage "KustinMiller"