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KustinMiller :: KustinMiller

KustinMiller -- Unprojection and the Kustin-Miller complex construction

Description

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 $\Delta(d,m)$ 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

Authors

Certification a gold star

Version 1.4 of this package was accepted for publication in volume 4 of 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.

Version

This documentation describes version 1.4 of KustinMiller.

Source code

The source code from which this documentation is derived is in the file KustinMiller.m2.

Exports

  • Functions and commands
    • delta -- Boundary complex of cyclic polytope.
    • isExactRes -- Test whether a chain complex is an exact resolution.
    • kustinMillerComplex -- Compute Kustin-Miller resolution of the unprojection of I in J
    • resBE -- Buchsbaum-Eisenbud resolution
    • stellarSubdivision -- Compute the stellar subdivision of a simplicial complex.
    • unprojectionHomomorphism -- Compute the homomorphism associated to an unprojection pair
  • Methods
    • "delta(ZZ,PolynomialRing)" -- see delta -- Boundary complex of cyclic polytope.
    • "isExactRes(ChainComplex)" -- see isExactRes -- Test whether a chain complex is an exact resolution.
    • "kustinMillerComplex(ChainComplex,ChainComplex,PolynomialRing)" -- see kustinMillerComplex -- Compute Kustin-Miller resolution of the unprojection of I in J
    • "kustinMillerComplex(Ideal,Ideal,PolynomialRing)" -- see kustinMillerComplex -- Compute Kustin-Miller resolution of the unprojection of I in J
    • "resBE(Matrix)" -- see resBE -- Buchsbaum-Eisenbud resolution
    • "stellarSubdivision(SimplicialComplex,Face,PolynomialRing)" -- see stellarSubdivision -- Compute the stellar subdivision of a simplicial complex.
    • "unprojectionHomomorphism(Ideal,Ideal)" -- see unprojectionHomomorphism -- Compute the homomorphism associated to an unprojection pair
  • Symbols
    • Jerry -- The Kustin-Miller complex for Jerry
    • Tom -- The Kustin-Miller complex for Tom

For the programmer

The object KustinMiller is a package.