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

Finding the 16 betti tables possible for quartic forms in 4 variables, and examples -- Material from Table 6 and 7 of Appendix 1

Section 2: Basic constructions

Doubling Examples -- Doubling of each type of set of points
Doubling Examples for ideals of 6 points -- For an ideal $I_{\Gamma}$ of six points we compute possible doublings of $I_{\Gamma}$. See Example 2.16 in [QQ] for details
Example Type [300a] -- An example of an apolar ideal of a quartic that cannot be obtained as a doubling of it's apolar set
Example Type [300b] -- An example of doubling construction
Example Type [300c] -- The third family of type [300]

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

Finding the possible betti tables for points in P^3 with given geometry -- Material from Section 3 of [QQ]

Section 4: the quadratic part of the apolar ideal

Finding all possible betti tables for quadratic component of inverse system for quartics in 4 variables -- Material from Section 4 of [QQ]

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

Finding the Betti stratum of a given quartic -- the 19 Betti strata
Noether-Lefschetz examples -- examples from Section 6.2 in [QQ]

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

Type [000], CY of degree 20 -- lifting to a 3-fold with two singular points
Singularities of lifting of type [300b] -- The lifting of [300b] defined by biliaison acquires singularities in dimension 3
Half canonical degree 20 -- Computation which supports the proof of Proposition 8.4

Section 9: Construction and lifting of AG varieties

Type [210], CY of degree 18 via linkage -- lifting to a 3-fold with components of degrees 11, 6, 1
Type [310], CY of degree 17 via linkage -- lifting to a 3-fold with components of degrees 11, 6
Type [331], CY of degree 17 via linkage -- lifting to a 3-fold with components of degrees 13 and 4
Type [420], CY of degree 16 via linkage -- lifting to an irreducible 3-fold
Type [430], CY of degree 16 via linkage -- lifting to a 3-fold with components of degrees 10, 6
Type [441a], CY of degree 16 -- lifting to a 3-fold with components of degrees 12, 4
Type [441b], CY of degree 16 -- lifting to a 3-fold with components of degrees 8, 8
Type [551], CY of degree 15 via linkage -- lifting to a 3-fold with components of degrees 11 and 4
Type [562] with lifting of type I, a CY of degree 15 via linkage -- lifting to a 3-fold with components of degree 8, 7
Type [562] with a lifting of type II, a CY of degree 15 via linkage -- lifting to a 3-fold with components of degrees 7, 4, 4

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

Hilbert scheme of 6 points in projective 3-space -- Betti table loci

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

Gregorz Kapustka
Michal Kapustka
Kristian Ranestad
Hal Schenck
Mike Stillman <[email protected]>
Beihui Yuan

Version

This documentation describes version 0.99 of QuaternaryQuartics.

Source code

The source code from which this documentation is derived is in the file QuaternaryQuartics.m2. The auxiliary files accompanying it are in the directory QuaternaryQuartics/.

Exports

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

For the programmer

The object QuaternaryQuartics is a package.