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].

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"

Authors

Version

This documentation describes version 1.0 of KustinMiller.

Source code

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

A more recent version (however dependent on the Macaulay2 package SimplicialComplexes.m2 to be installed first) can be obtained at KustinMiller.tar.

Exports

Types
- Complex -- The class of all simplicial complexes.
- Face -- The class of faces of simplicial complexes.
Functions and commands
- complex -- Create a complex.
- complexToIdeal -- Compute the Stanley-Reisner ideal.
- delta -- Boundary complex of cyclic polytope.
- differentials -- Generate the differentials of the Kustin-Miller resolution
- dimension -- The dimension of a simplicial complex or a face of a simplicial complex.
- dualComplex -- Dualize a chain complex
- face -- Generate a face.
- faceIdeal -- The ideal of a face of a simplicial complex.
- faces -- Returns the faces of a complex
- facets -- The facets of a simplicial complex.
- fvector -- Returns the F-vector of a complex
- idealToComplex -- Compute the Stanley-Reisner complex.
- isExactRes -- Test whether a chain complex is exact.
- isFace -- Test whether a face is a face of a given complex.
- isSubface -- Test whether a face is a subface of another face.
- kustinMillerComplex -- Compute Kustin-Miller resolution of the unprojection of I in J
- link -- The link of a face of a complex.
- resBE -- Buchsbaum-Eisenbud resolution
- shiftComplex -- Shift the indexing of a chain complex
- stellarSubdivision -- Compute the stellar subdivision of a simplicial complex.
- substituteComplex -- Substitute a complex to a different ring.
- substituteFace -- Substitute a face to a different ring.
- unprojectionHomomorphism -- Compute the homomorphism associated to an unprojection pair
- vertices -- The vertices of a face of a simplicial complex.
Methods
- Complex == Complex -- Compare two complexes.
- complexToIdeal(Complex), see complexToIdeal -- Compute the Stanley-Reisner ideal.
- dimension(Complex), see dimension -- The dimension of a simplicial complex or a face of a simplicial complex.
- dimension(Face), see dimension -- The dimension of a simplicial complex or a face of a simplicial complex.
- Face == Face -- Compare two faces.
- faceIdeal(Face), see faceIdeal -- The ideal of a face of a simplicial complex.
- faces(Complex), see faces -- Returns the faces of a complex
- faces(Complex,ZZ), see faces -- Returns the faces of a complex
- facets(Complex), see facets -- The facets of a simplicial complex.
- fvector(Complex), see fvector -- Returns the F-vector of a complex
- isFace(Face,Complex), see isFace -- Test whether a face is a face of a given complex.
- isSubface(Face,Face), see isSubface -- Test whether a face is a subface of another face.
- link(Face,Complex), see link -- The link of a face of a complex.
- net(Face) -- Printing faces or cones.
- ring(Complex) -- The underlying polynomial ring of a simplicial complex.
- ring(Face) -- The underlying polynomial ring of a face.
- stellarSubdivision(Complex,Face,PolynomialRing), see stellarSubdivision -- Compute the stellar subdivision of a simplicial complex.
- substituteComplex(Complex,PolynomialRing), see substituteComplex -- Substitute a complex to a different ring.
- substituteFace(Face,PolynomialRing), see substituteFace -- Substitute a face to a different ring.
- vertices(Face), see vertices -- The vertices of a face of a simplicial complex.
Symbols
- subRing -- Option to consider the link as a complex on its natural set of vertices
- verbose -- Option to print intermediate data