# 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

## Certification

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 .