Kustin-Miller complex

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Implementing the Kustin-Miller complex construction

Overview

The Kustin-Miller complex construction, due to A. Kustin and M. Miller, can be applied to a tuple of resolutions of Gorenstein rings with certain properties to obtain a new Gorenstein ring and a resolution of it. It gives a tool to construct and analyze Gorenstein rings of high codimension. We describe the Kustin-Miller complex and its implementation in the Macaulay2 package

Goal

Implementation of the Kustin-Miller complex associated to two resolutions of Gorenstein ring R/I and R/J (R a polynomial ring, I,J homogeneous ideals) with I contained in J and dim(R/J) = dim(R/I)-1.

To Do List

  • Implement the differentials of the Kustin-Miller complex
  • Implement the correct grading
  • Add Tom example
  • Add Jerry example
  • Add Stellar subdivision example
  • Add cyclic polytope example
  • Use to SimplicialComplexes package for the examples
  • Add to the simplicial complexes package a class Face
  • Minimalize the Kustin-Miller complex

References

General

A. Kustin and M. Miller, Constructing big Gorenstein ideals from small ones, J. Algebra 85 (1983)}, 303-322.

Papadakis, Kustin-Miller unprojection with complexes, J. Algebraic Geometry 13 (2004) 249-268}

Constructing new varieties

J. Neves and S. Papadakis, A construction of numerical Campedelli surfaces with ZZ/6 torsion, Trans. Amer. Math. Soc. 361 (2009), 4999-5021.

J. Neves and S. Papadakis, Parallel Kustin-Miller unprojection with an application to Calabi-Yau geometry, preprint, 2009, 23 pp

G. Brown, M. Kerber, M. Reid, Fano 3-folds in codimension 4, Tom and Jerry, Part I

Stellar subdivision case

J. Boehm, S. Papadakis: Stellar subdivisions and Stanley-Reisner rings of Gorenstein complexes

Cyclic polytopes case

J. Boehm, S. Papadakis: On the structure of Stanley-Reisner rings associated to cyclic polytopes


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