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

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

[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