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QuaternaryQuartics -- code to support the paper 'Quaternary Quartic Forms and Gorenstein Rings'

Description

This package contains code and examples for the paper [QQ] Quaternary Quartic Forms and Gorenstein Rings, by Grzegorz Kapustka, Michal Kapustka, Kristian Ranestad, Hal Schenck, Mike Stillman and Beihui Yuan, referenced below.

We study the space of quartic forms in four variables, interleaving the notions of: rank, border rank, annihilator of the quartic form, Betti tables, and Calabi-Yau varieties of codimension 4.

Section 1: Generating the Betti tables

Section 2: Basic constructions

Section 3: betti tables for points in P^3 with given geometry

Section 4: the quadratic part of the apolar ideal

Section 5: VSP(F,9) for a general quadric form of rank 9

  • VSP(F_Q,9) -- Computation appearing in the proof of Theorem 5.16 in [QQ]

Section 6: Stratification of the space of quaternary quartics

Section 7: Codimension three varieties in quadrics

  • Pfaffians on quadrics -- compute the quartic and betti table corresponding to a pfaffian ideal in a quadric

Section 8: Irreducible liftings

Section 9: Construction and lifting of AG varieties

Appendix 2: Components of the Betti table loci in Hilbert schemes of points

References

[QQ] Quaternary Quartic Forms and Gorenstein Rings, by Grzegorz Kapustka, Michal Kapustka, Kristian Ranestad, Hal Schenck, Mike Stillman and Beihui Yuan. (arxiv:2111.05817) 2021.

Authors

Version

This documentation describes version 0.99 of QuaternaryQuartics.

Citation

If you have used this package in your research, please cite it as follows:

@misc{QuaternaryQuarticsSource,
  title = {{QuaternaryQuartics: A \emph{Macaulay2} package. Version~0.99}},
  author = {Gregorz Kapustka and Michal Kapustka and Kristian Ranestad and Hal Schenck and Mike Stillman and Beihui Yuan},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages}}
}

Exports

  • Functions and commands
  • Methods
    • bettiStrataExamples(Ring) -- see bettiStrataExamples -- a hash table consisting of examples for each of the 19 Betti strata
    • doubling(ZZ,Ideal) -- see doubling -- implement the doubling construction
    • nondegenerateBorels(ZZ,Ring) -- see nondegenerateBorels -- construct all nondegenerate strongly stable ideals of given length
    • pointsIdeal(Matrix) -- see pointsIdeal -- create an ideal of points
    • pointsIdeal(Ring,Matrix) -- see pointsIdeal -- create an ideal of points
    • quartic(Matrix) -- see quartic -- a quartic given by power sums of linear forms
    • quartic(Matrix,Ring) -- see quartic -- a quartic given by power sums of linear forms
    • quarticType(RingElement) -- the Betti stratum a specific quartic lies on
    • randomBlockMatrix(List,List,List) -- see randomBlockMatrix -- create a block matrix with zero, identity and random blocks
    • randomHomomorphism(List,Module,Module) -- see randomHomomorphism -- create a random homomorphism between graded modules
    • randomHomomorphism(ZZ,Module,Module) -- see randomHomomorphism -- create a random homomorphism between graded modules
    • randomPoints(Ring,ZZ) -- see randomPoints -- create a matrix whose columns are random points
    • randomPoints(Ring,ZZ,ZZ) -- see randomPoints -- create a matrix whose columns are random points
    • smallerBettiTables(BettiTally) -- see smallerBettiTables -- Find all (potentially) smaller Betti tables that could degenerate to given table
  • Symbols

For the programmer

The object QuaternaryQuartics is a package, defined in QuaternaryQuartics.m2, with auxiliary files in QuaternaryQuartics/.


The source of this document is in QuaternaryQuartics.m2:379:0.