# Quaternary Quartic forms and Gorenstein rings

Gregorz Kapustka  G. Kapustka: Department of Mathematics and Informatics,
Jagiellonian University, Kraków, Poland<EMAIL_ADDRESS>,  Michał
Kapustka  M. Kapustka: Institute of Mathematics of the Polish Academy of
Sciences, Warszawa, Poland<EMAIL_ADDRESS>,  Kristian Ranestad
Ranestad: Matematisk institutt, Universitetet i Oslo, Oslo, Norway
<EMAIL_ADDRESS>,  Hal Schenck  Schenck: Mathematics Department, Auburn
University, Auburn, USA<EMAIL_ADDRESS>,  Mike Stillman  Stillman:
Mathematics Department, Cornell University, Ithaca, USA<EMAIL_ADDRESS>and  Beihui Yuan  Yuan: Mathematics Department, Cornell University, Ithaca,
USA<EMAIL_ADDRESS>
###### Abstract.

A quaternary quartic form, a quartic form in four variables, is the dual socle
generator of an Artinian Gorenstein ring of codimension and regularity 4. We
present a classification of quartic forms in terms of rank and powersum
decompositions which corresponds to the classification by the Betti tables of
the corresponding Artinian Gorenstein rings. This gives a stratification of
the space of quaternary quartic forms which we compare with the Noether-
Lefschetz stratification. We discuss various phenomena related to this
stratification. We study the geometry of powersum varieties for a general form
in each stratum. In particular, we show that the powersum variety $VSP(F,9)$
of a general quartic with singular middle catalecticant is again a quartic
surface, thus giving a rational map between two divisors in the space of
quartics. Finally, we provide various explicit constructions of general
Artinian Gorenstein rings corresponding to each stratum and discuss their
lifting to higher dimension. These provide constructions of codimension four
varieties, which include canonical surfaces, Calabi-Yau threefolds and Fano
fourfolds. In the particular case of quaternary quartics, our results yield
answers to questions posed by Geramita in [Geramita1], Iarrobino-Kanev in
[IK], and Reid in [ReidGeneral].

###### Key words and phrases:

Apolar Ideal, Gorenstein Ring, Macaulay Inverse System, Calabi-Yau

###### 2000 Mathematics Subject Classification:

Primary 13E10, Secondary 14J32, 13H10, 13D02

G. Kapustka supported by Narodowe Centrum Nauki 2018/30/E/ST1/00530.

M. Kapustka supported by Narodowe Centrum Nauki 2018/31/B/ST1/02857

Schenck supported by NSF 2006410

Stillman and Yuan supported by NSF 1502294

###### Contents

  1. 1 Introduction
    1. 1.1 Organization of the paper
    2. 1.2 Notation.
  2. 2 Preliminaries
    1. 2.1 Apolarity and syzygies
    2. 2.2 Apolarity in codimension $3$ or $4$ and regularity $3$
    3. 2.3 Geometry: Catalecticant and Secant varieties
      1. 2.3.1 Geometry of the catalecticant variety and the Gorenstein parameter scheme
      2. 2.3.2 Motivating Questions
      3. 2.3.3 Secant varieties to the Veronese variety and power sum decompositions
    4. 2.4 Gorenstein rings of low codimension: History and Techniques
    5. 2.5 The doubling construction
    6. 2.6 Bilinkage and unprojection
      1. 2.6.1 Biliaison
      2. 2.6.2 Unprojections
      3. 2.6.3 The Kustin-Miller construction
    7. 2.7 AG varieties of codimension $3$ in quadrics
  3. 3 Apolar curves and Betti tables of point sets
  4. 4 The quadratic part of the apolar ideal
    1. 4.1 Syzygy schemes and preparatory lemmas
    2. 4.2 A case by case characterization of the quadratic component
      1. 4.2.1 Case $b_{12}=2$
      2. 4.2.2 Case $b_{12}=3$
      3. 4.2.3 Case $b_{12}=4$
      4. 4.2.4 Case $b_{12}=5$
      5. 4.2.5 Case $b_{12}=6$
  5. 5 Rank and powersum presentations of quartics
    1. 5.1 $VSP(F,9)$ for a general quartic form of rank $9$
      1. 5.1.1 Apolarity on $Q$.
  6. 6 Stratifications of the space of quaternary quartics
    1. 6.1 The Betti stratification
    2. 6.2 Relations to Noether-Lefschetz varieties
  7. 7 Codimension three varieties in quadrics
    1. 7.1 The Artinian case
    2. 7.2 AG-liftings to curves
    3. 7.3 AG-liftings to Calabi-Yau threefolds; Lagrangian degeneracy loci
    4. 7.4 Degenerations of liftings
  8. 8 Irreducible liftings
  9. 9 Reducible liftings
  10. A Betti tables for the inverse system of a quarternary quartic
  11. B Betti table subschemes of the Hilbert scheme
    1. B.1 Strongly stable ideals with a given Hilbert polynomial
    2. B.2 Generic initial ideals
    3. B.3 The Groebner family and the Groebner stratum of an initial ideal
    4. B.4 The Schreyer resolution and minimal Betti numbers
    5. B.5 The components of the Betti table loci of degree $6$

##  1\. Introduction

Quaternary quartic forms are dual socle generators of Artinian Gorenstein
rings. In this paper we investigate two aspects of this construction. First,
we study the relation between decompositions of the form and the resolution of
the Artinian ring as a quotient of ${\mathbb{C}}[x_{0},\ldots,x_{3}]$. Second,
we investigate graded quotients $S/I$ of a polynomial ring
$S={\mathbb{C}}[x_{0},\ldots,x_{n}]$ by a homogeneous ideal $I$ whose Artinian
reduction is Gorenstein with dual socle generator a quaternary quartic form.
In particular, we consider quotients $S/I$ that are arithmetically Gorenstein
(AG). These rings are defined by the property that if $I$ is of codimension
$c$, then $\operatorname{Ext}^{i}_{S}(S/I,S)=0$ if $i\neq c$, and the
canonical module $\omega_{S/I}$ of $S/I$ satisfies

(1) $\omega_{S/I}\simeq\operatorname{Ext}^{c}_{S}(S/I,S(-n-1))\simeq
S(-n-1+\operatorname{codim}(I)+\operatorname{reg}(S/I))/I.$

The Artinian reduction has dual socle generator a quaternary quartic form when
$S/I$ has codimension and (Castelnuovo-Mumford) regularity both $4$.

The motivation for studying this situation is twofold. First, when $X=V(I)$ is
a threefold, then $X$ is a Calabi-Yau (CY) variety; such varieties play a
central role in string theory. We extend constructions of Calabi-Yau
threefolds that appear as AG varieties in ${\mathbb{P}}^{7}$ in Section 8.

Second, while there is a classical structure theory for AG rings of
codimension at most $3$, in codimension $4$ there are many open questions.
Recent work in [SSY] shows that a codimension $=4=$ regularity AG ring has one
of $16$ possible Betti tables (see 1.2). These 16 possibilities are listed in
tables 6 and 7 in Appendix A. Our study extends the characterization of these
AG rings and their relation to known constructions of Gorenstein rings in
codimension $4$.

Our main object of investigation throughout the paper is the quartic form, as
a dual socle generator of an Artinian Gorenstein ring. The geometry of a
homogeneous form $F\in R={\mathbb{C}}[y_{0},\ldots,y_{n}]$ is most often
thought of as that of the associated hypersurface $V(F)$ in projective space.
However, there are also zero schemes $\Gamma$ associated to $F$, via the
apolarity construction of Macaulay. We discuss apolarity in detail in Section
2. The starting point is the role of the form as a dual socle generator:

Elements of $S$ act on elements of $R$ by differentiation:

$x_{i}(y_{j})=\frac{\partial}{\partial(y_{i})}(y_{j})=\delta_{ij}.$

We work over ${\mathbb{C}}$, but our results hold over any field of
characteristic zero; the operation above is essentially the same as
contraction. Consider the annihilator of $F$ under this action

$F^{\perp}:=\operatorname{ann}_{S}(F)=\\{G\in S|\ G(F)=0\\}.$

We denote by $A_{F}:=S/F^{\perp}$ the quotient and call it the apolar ring of
$F$. It is clearly Artinian, in fact also Gorenstein. Macaulay shows (see
Lemma 2.2 in the next section), that apolarity gives a bijection between
homogeneous polynomials $F\in R_{j}$, up to rescaling, and graded Artinian
Gorenstein rings $S/F^{\perp}$ with socle in degree $j$.

A geometric aspect of a form that is best understood via apolarity is the
powersum presentation. A representation of $F=l_{1}^{d}+\ldots+l_{s}^{d}$ as a
sum of powers of linear forms $l_{i}$ yields a zero scheme
$\Gamma=\\{[l_{1}],...,[l_{s}]\\}\subset{\mathbb{P}}(R_{1})$ of degree $s$.
Via apolarity, the ideal $I_{\Gamma}$ is naturally an ideal in $S$, such that
the homogeneous coordinate ring $S/I_{\Gamma}$ of $\Gamma$ admits a surjection
$S/I_{\Gamma}\to A_{F}$.

We use minimal power sum presentations to analyze a stratification of the set
of quartics. By [SSY], for a quaternary quartic form $F$ that is nondegenerate
(not annihilated by a linear form), there are $16$ different Betti tables for
$A_{F}$, and for $i\in\\{1,2,3\\}$, the Betti numbers

(2)
$\dim_{{\mathbb{C}}}\operatorname{Tor}^{S}_{i}(A_{F},{\mathbb{C}})_{i+1}=b_{i,i+1}(A_{F}),$

i.e. the quadratic part of the Betti table, determine the entire Betti table
of $A_{F}$. We write $B_{[b_{12}b_{23}b_{34}]}$ to denote such a Betti table.
One goal of this paper is to describe the loci of quartic forms in
${\mathbb{P}}^{3}$ with a specific Betti table:

######  Definition 1.1.

Let

${\mathcal{F}}_{[b_{12}b_{23}b_{34}]}=\\{F\in{\mathbb{P}}(\operatorname{Sym}_{4}({\mathbb{C}}^{4}))={\mathbb{P}}^{34}\mid
S/F^{\perp}\mbox{ has Betti table }B_{[b_{12}b_{23}b_{34}]}\\}.$

We say that $F$ is of type ${[b_{12}b_{23}b_{34}]}$ if
$F\in{\mathcal{F}}_{[b_{12}b_{23}b_{34}]}$ to reflect this, or the equivalent
condition $[F]\in{\mathcal{F}}_{[b_{12}b_{23}b_{34}]}$, and write
${\mathcal{F}}_{B}$ when the type is not fixed.

We identify the irreducible components of ${\mathcal{F}}_{B}$ for all $B$
arising as Betti tables of quaternary quartics. The main ingredient in our
argument is an analysis of two different subideals of $F^{\perp}$. In Section
3 we consider the ideals of finite sets of points that give minimal power sum
presentations of quaternary quartics, and in Section 4 we consider the
subideal $Q_{F}$ defined by the quadrics in $F^{\perp}$.

Since the Betti table of $A_{F}$ is determined by the quadratic part, we will
describe the locus ${\mathcal{F}}_{B}$ in terms of the schemes defined by
$Q_{F}$ for $F\in{\mathcal{F}}_{B}$. We find, somewhat surprisingly, the
quadratic ideals $Q_{F}$ are saturated, and identify these ideals by their set
of generators in the space of quadrics.

######  Definition 1.2.

Let

${\mathcal{G}}_{[b_{12}b_{23}b_{34}]}=\\{G\in\mathbf{Gr}(b_{12},S_{2})\mid\mbox{
the Betti table of }S/I_{G}=B^{Q}_{[b_{12}b_{23}b_{34}]}\\},$

where $I_{G}$ is the ideal generated by $G$, seen as a subspace of the space
of quadrics, and $B^{Q}_{[b_{12}b_{23}b_{34}]}$ is one of the possible Betti
tables of quadratic ideals of the form $Q_{F}$ for some form $F$ of type
$[b_{12}b_{23}b_{34}]$. These Betti tables are classified in Section 4.

As in Definition 1.1, we say that a quadratic ideal $Q_{F}$ is of type
${[b_{12}b_{23}b_{34}]}$ if $Q_{F}$ corresponds to a point in
${\mathcal{G}}_{[b_{12}b_{23}b_{34}]}$, and write ${\mathcal{G}}_{B}$ when the
type is not fixed.

In Section 6, we identify the ${\mathcal{F}}_{B}$ strata in
${\mathbb{P}}^{34}$. While many of the ${\mathcal{F}}_{B}$ are irreducible,
not all are, and the geometry of the ${\mathcal{G}}_{B}$ loci is key to
unraveling this. The two cases which require additional analysis are
${\mathcal{F}}_{[300]}$ and ${\mathcal{F}}_{[441]}$.

We identify three irreducible subsets of ${\mathcal{F}}_{[300]}$, which we
designate as $[300a],[300b]$ and $[300c]$. Proposition 6.3 shows that the
corresponding loci satisfy

${\mathcal{F}}_{[300c]}\subseteq\overline{{\mathcal{F}}_{[300b]}}\subseteq\overline{{\mathcal{F}}_{[300a]}}.$

The corresponding loci ${\mathcal{G}}_{[300a]}$ and ${\mathcal{G}}_{[300b]}$
in ${\mathcal{G}}_{[300]}$ are irreducible and coincide, so we denote them
both by ${\mathcal{G}}_{[300ab]}$. The locus ${\mathcal{G}}_{[300c]}$ is also
irreducible, and corresponds to a Hilbert scheme distinct from that of
${\mathcal{G}}_{[300ab]}$.

Similarly, we show that ${\mathcal{F}}_{[441]}$ has two irreducible
components, which we designate as ${\mathcal{F}}_{[441a]}$ and
${\mathcal{F}}_{[441b]}$. In this case the corresponding loci for the
quadratic components ${\mathcal{G}}_{[441a]}$ and ${\mathcal{G}}_{[441b]}$ are
distinct components of the same Hilbert scheme. Our first main result is the
following:

######  Theorem 1.3.

There are $19$ irreducible Betti strata for nondegenerate quaternary quartics.
They satisfy the following closure relations below.

  1. (1)

${\mathcal{F}}_{[562]}$ $\subset\overline{\mathcal{F}_{[551]}}$
$\subset\overline{\mathcal{F}_{[550]}}$

  2. (2)

${\mathcal{F}}_{[441a]}$ $\subset\overline{\mathcal{F}_{[420]}}$

  3. (3)

${\mathcal{F}}_{[441b]}$ $\subset\overline{\mathcal{F}_{[430]}}$
$\subset\overline{\mathcal{F}_{[420]}}$

  4. (4)

$\mathcal{F}_{[320]}$ $\subset\overline{\mathcal{F}_{[300b]}}$

  5. (5)

$\mathcal{F}_{[300c]}$ $\subset\overline{\mathcal{F}_{[300b]}}$

  6. (6)

$\mathcal{F}_{[331]}$ $\subset\overline{\mathcal{F}_{[310]}}$
$\subset\overline{\mathcal{F}_{[300b]}}$

  7. (7)

$\mathcal{F}_{[210]}$ $\subset\overline{\mathcal{F}_{[200]}}$

  8. (8)

$\mathcal{F}_{[683]}$ $\subset\overline{\mathcal{F}_{[550]}}$
$\subset\overline{\mathcal{F}_{[420]}}$
$\subset\overline{\mathcal{F}_{[300b]}}$
$\subset\overline{\mathcal{F}_{[300a]}}$
$\subset\overline{\mathcal{F}_{[200]}}$
$\subset\overline{\mathcal{F}_{[100]}}$

  9. (9)

$\mathcal{F}_{[683]}$ $\subset\overline{\mathcal{F}_{[551]}}$

  10. (10)

$\mathcal{F}_{[550]}$ $\subset\overline{\mathcal{F}_{[430]}}$

  11. (11)

$\mathcal{F}_{[683]}$ $\subset\overline{\mathcal{F}_{[562]}}$
$\subset\overline{\mathcal{F}_{[430]}}$
$\subset\overline{\mathcal{F}_{[300c]}}$

  12. (12)

$\mathcal{F}_{[562]}$ $\subset\overline{\mathcal{F}_{[441b]}}$

  13. (13)

$\mathcal{F}_{[552]}$ $\subset\overline{\mathcal{F}_{[441a]}}$
$\subset\overline{\mathcal{F}_{[310]}}$
$\subset\overline{\mathcal{F}_{[021]}}$

  14. (14)

${\mathcal{F}_{[441a]}}$ $\subset\overline{\mathcal{F}_{[331]}}$
$\subset\overline{\mathcal{F}_{[210]}}$

  15. (15)

${\mathcal{F}_{[420]}}$ $\subset\overline{\mathcal{F}_{[400]}}$
$\subset\overline{\mathcal{F}_{[300a]}}$

The proofs of these containments appear in Section 6 where Figure 1 depicts
this graphically. Non-containments appear in Propositions 6.9 and 6.11. This
answers questions of Geramita and Iarrobino-Kanev concerning Gorenstein
parameter schemes and Catalecticant varieties in the case of quartics as
explained in Section 2.3.2, as well as questions of Reid about codimension $4$
Gorenstein subschemes, as discussed in Section 2.4.

The Fermat quartic, $y_{0}^{4}+y_{1}^{4}+y_{2}^{4}+y_{3}^{4}$, is of type
$[683]$. It is a nonsingular quartic, so a corollary of Theorem 1.3 is that
the general quartic in every Betti stratum is nonsingular. We compare in
Section 6.2 the Betti stratification with the Noether-Lefschetz locus, and
show the relation between Betti strata and so called Noether-Lefschetz
varieties of quartic surfaces.

A crucial issue in the analysis of the 19 irreducible Betti strata is the
geometry of the set of power sum presentations of a general form in each
stratum. In particular, the power sum presentations of degree equal to the
rank of the form.

######  Definition 1.4.

The rank of a form $F\in R_{d}={\mathbb{C}}[y_{0},\ldots,y_{n}]_{d}$, denoted
$\operatorname{r}(F)$, is the minimal $s$ such that

$F=l_{1}^{d}+\ldots+l_{s}^{d}$

for some linear forms $l_{i}\in R_{1}$.

Apolarity defines a duality between the linear forms $S_{1}\subset S$ and
$R_{1}\subset R$. The linear forms $l_{i}\in R_{1}$ define hyperplanes
$V(l_{i})\subset{\mathbb{P}}(S_{1})$, and points
$[l_{i}]\in{\mathbb{P}}(R_{1})$. The set of points $\\{[l_{1}],...,[l_{s}]\\}$
is a point in the Hilbert scheme $Hilb_{s}({\mathbb{P}}(R_{1}))$. We have the

######  Definition 1.5.

The variety of sums of powers presenting $F$ is the closure

$VSP(F,s)=\overline{\\{\\{[l_{1}],\ldots,[l_{s}]\\}\in
Hilb_{s}({\mathbb{P}}(R_{1}))\mid\exists\lambda_{i}\in{\mathbb{C}}:F=\lambda_{1}l_{1}^{d}+\ldots+\lambda_{s}l_{s}^{d}\\}}$

of power sums presenting $F$ in the Hilbert scheme.

We will be particularly interested in the case when $s$ is the rank
$\operatorname{r}(F)$ of $F$, and $F$ is a general member of each type. In the
special cases when $VSP(F,\operatorname{r}(F))$ is a point, the form is called
identifiable. Angelini-Chiantini-Vannieuwenhoven find criteria for
identifiability of quartic forms in [ACV].

Our second main result is Proposition 5.5 of Section 6 which describes
$VSP(F,\operatorname{r}(F))$ for the general form in each stratum. Our results
appear in Tables 4 and 6 of Section 6, which we reproduce below:

| Betti table B | $r$ | VSP(F,r)
---|---|---
[683] | $4$ | one point
[550] | $5$ | one point
[400] | $8$ | ${\mathbb{P}}^{3}$
[320] | $7$ | ${\mathbb{P}}^{1}$
[300a] | $8$ | one point
[300b] | $7$ | one point
[300c] | $7$ | ${\mathbb{P}}^{1}$
[200] | $8$ | two points
[100] | $9$ | K3 Surface
[000] | $10$ | Fivefold
| Betti table B | $r$ | VSP(F,r)
---|---|---
[562] | $5$ | ${\mathbb{P}}^{1}$
[551] | $5$ | one point
[441a] | $6$ | ${\mathbb{P}}^{1}$
[441b] | $6$ | ${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$
[430] | $6$ | ${\mathbb{P}}^{1}$
[420] | $6$ | one point
[331] | $7$ | $V_{22}$
[310] | $7$ | ${\mathbb{P}}^{1}$
[210] | $8$ |  $V_{22}$ $\cup\;{\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$
|  |

Here $V_{22}$ stands for the Fano threefold of degree 22 constructed by Mukai
as $VSP(F_{3},6)$, the variety of sums of powers of a general ternary quartic
$F_{3}$, see [Mukai].

The description of $VSP(F,9)$ for a general quartic form of rank $9$ as a K3
surface was a long standing question. It is tempting to compare it to the
result by Mukai, in [Mukai], that $VSP(F,10)$ for a general ternary sextic
form is a K3 surface. We ask:

######  Question 1.6.

Is $VSP(F,\operatorname{r}(F))$ a K3 surface, when $F$ is a general form in
$H^{0}(S,-2K_{S})$ on any toric surface $S$?

The results in the table leaves another open question:

######  Question 1.7.

What kind of variety is $VSP(F,10)$, when $F$ is a general quartic form?

Finally, we study AG projective varieties: varieties $X=V(I)$ such that $S/I$
is AG, whose homogeneous coordinate rings $A(X)$ are liftings of $A_{F}$, that
is

$A_{F}\cong A(X)/(\ell_{0},...,\ell_{{\rm dim}X}),$

where the $\ell_{i}$ are a regular sequence of linear forms. Thus $A(X)$ and
$A_{F}$ have the same Betti table. In that case, $A_{F}$ will be called an
Artinian reduction of $X$, and we shall call the variety $X$ an AG-lifting (or
lifting) of $A_{F}$ or $F$. If $X$ is a cone over a variety $Y$, then $A(X)$
is trivially a lifting of $A(Y)$, so we consider only non-trivial liftings:
those liftings $A(X)$ where $X$ is not a cone.

For example, an elliptic normal curve is an AG-lifting of a quadratic form,
and a projectively normal $K3$-surface is an AG-lifting of a cubic form. A
smooth threefold $X$ that is a lifting of a quartic form is a Calabi-Yau
threefold, i.e. $K_{X}\simeq\mathcal{O}_{X}$ and
$H^{i}(\mathcal{O}_{X})=0\mbox{ if }i\neq 0,3$.

The codimension $4$ case was studied systematically in [B] and [CGKK].
Patience Ablett, cf. [Ablett], recently found liftings to curves in
${\mathbb{P}}^{5}$ of quartics $F$ whose Betti table is in Table 7 in Appendix
A. Our third main result complements the lifting results of [CGKK, SSY,
Ablett]. We prove in Corollary 8.6 that a general Artinian Gorenstein ring
$A_{F}$ with a Betti table from Table 6 lifts to a smooth curve. It remains an
open question in the case of many Betti tables considered in this paper to
understand to which dimension we can lift them to smooth varieties.

We find liftings to (possibly reducible) Calabi-Yau threefolds $X$ of $A_{F}$
for each type of nondegenerate quartic $F$. In some cases of irreducible
liftings, the constructions allow liftings to higher dimensions to Fano
varieties.

Our first approach, used to find an AG-lifting of a quaternary quartic form
$F$ with a smooth quadric in $F^{\perp}$, is to restrict our attention to such
a quadric. More precisely, in this case the restriction $F_{Q}^{\perp}$ of
$F^{\perp}$ to the quadric $Q$ is a codimension $3$ ideal, that that is
related to a Pfaffian ideal (see Section 7). We extend these constructions to
higher dimensional quadrics using the spinor bundles and Lagrangian
degeneration loci (see Section 2.7). In particular, by describing the liftings
of the generic $A_{F}$ with Betti table $[100]$, we extend and complete the
results of [SSY, CGKK].

For instance, let $V$ be a $8$-dimensional vector space with a nondegenerate
quadratic form $q:V\to{\mathbb{C}}$ and let $Q_{6}=\\{q=0\\}$ be the
$6$-dimensional quadric defined by $q$ in ${\mathbb{P}}(V)$. Consider a
Lagrangian subbundle $\mathcal{E}\subset V\otimes{\mathcal{O}}_{Q_{6}}$. Let
$W\subset V$ be a Lagrangian subspace such that $W\cap\mathcal{E}(x)$ is odd-
dimensional for all $x\in Q_{6}$. Then, by [EPW1]*Thm 2.1, for sufficiently
general $W$,

$X=\\{x\in Q_{6}|\ {\rm dim}(W\cap\mathcal{E}(x))=3\\}$

is a smooth subvariety of $X$ of codimension $3$.

######  Proposition 1.8.

Suppose $\mathcal{E}=3\mathcal{S}_{1}$ or $\mathcal{S}_{1}\oplus
2\mathcal{S}_{2}$, where $S_{1}$ and $S_{2}$ are spinor bundles on $Q_{6}$.
Then for each choice of $\mathcal{E}$ and for sufficiently general Lagrangian
subspace $W\subset V$,

$X=\\{x\in Q_{6}|\ {\rm dim}(W\cap\mathcal{E}(x))=3\\}$

is a Calabi-Yau threefold of degree $19$. Moreover, $X$ is AG in
$\mathbb{P}^{7}$ and its coordinate ring has Betti table $[100]$.

Note that the construction in Proposition 1.8 resembles the Tom and Jerry
situation for del Pezzo varieties of degree $6$. A del Pezzo surface of degree
$6$ lifts to a hyperplane section of ${\mathbb{P}}^{2}\times{\mathbb{P}}^{2}$
(Tom), but also of
${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$ (Jerry).

A similar phenomenon occurs for Artinian Gorenstein rings $A_{F}$ already at
the level of lifting to points. For instance, for the general ring $A_{F}$
with Betti table of type [562] or [320], the family of liftings to sets of
points $X\subset{\mathbb{P}}^{4}$ has at least two irreducible components. We
give a geometric description of these loci in Sections 8 and 9.

A second approach to find liftings is the doubling construction, described in
Section 2.5. We prove that in most cases (except $[300a]$) the Artinian
Gorenstein ring is obtained as a doubling of a configuration of points
computing its rank. Geometrically, in positive dimension, a doubling of an
arithmetic Cohen Macaulay variety $Y$ is a variety $X$ obtained as a divisor
in a twist of the anti-canonical system of $Y$. Note that, anticanonical
divisors in arithmetic Cohen Macaulay fourfolds are Calabi-Yau threefolds (see
Proposition 2.14). For Artinian Gorenstein rings of type $[000]$ we consider,
in Proposition 8.4, a fourfold $Y\subset{\mathbb{P}}^{7}$ defined by the
maximal minors of a $3\times 5$ matrix $A$ of linear forms. When $A$ has rank
$1$ at some point, then $Y$ has an effective anticanonical divisor. For a
general $A$ with a rank $1$ point, the anticanonical divisor $X$ is unique and
is a two nodal AG Calabi-Yau threefold of degree $20$. The same construction
in ${\mathbb{P}}^{5}$ yields a halfcanonical curve $X$ of degree $20$, that is
the lifting of a general quartic form $F$ of type $[000]$. Smooth Calabi-Yau
threefolds of degree $20$ in ${\mathbb{P}}^{7}$ can be defined by $3\times 3$
minors of a $4\times 4$ matrix of linear forms. These are liftings of special
quartic forms $F$ of type $[000]$.

Bilinkage, or biliaison, as explained in Section 2.6, provides a third method
of constructing AG-liftings. As an example we consider Calabi-Yau threefolds
with Betti tables $[300a]$, $[300b]$ and $[300c]$. The first is obtained by
elementary biliaison of height $2$ from $\mathbb{P}^{3}$ inside an irreducible
intersection of three quadrics. These threefolds cannot be constructed by a
doubling in a complete intersection as discussed in Section 2.19. The second
case occurs when the intersection of the quadrics is reducible: The union of a
$\mathbb{P}^{4}$ and a fourfold $Y$ of degree $7$. In this case, a Calabi-Yau
threefold can be constructed by elementary biliaison of height 2 from the
cubic threefold $\mathbb{P}^{4}\cap Y$ in the fourfold $Y$. However, we show
that a biliaison of height $2$ cannot be used to construct an example of type
$[300c]$. Instead we can use a doubling construction, or a biliaison of height
$1$, see Section 8.

Constructions of liftings for quaternary quartic forms $F$ of each type to
reduced, but not necessarily irreducible manifolds $X$ are summarised in
Sections 8 and 9. Liftings of dimension three are particularly interesting as
they are necessarily Calabi-Yau manifolds. Some of these constructions allow
further liftings to dimensions higher than $3$, i.e. to AG Fano varieties of
dimensions at least $4$. The following table presents the maximal dimensions
of smooth liftings of general Artinian Gorenstein rings of the given type that
we are able to construct, see in particular Proposition 8.6, Corollary 8.7.

$\stackrel{{\scriptstyle\mbox{Betti table}}}{{\mbox{B}}}$ | $\stackrel{{\scriptstyle\mbox{dimension of possible}}}{{\mbox{smooth lifting}}}$
---|---
[683] | 3
[550] | 6
[400] | $\infty$
[320] | 4
[300a] | 3
[300b] | 2
[200] | 4
[100] | 1 (3)
[000] | 1 (4)

For the types $[100]$ and $[000]$, there are special rings $A_{F}$ that lift
to higher dimensions as marked in parenthesis in the table. Also, some rings
$A_{F}$ lift to different families of higher dimensional varieties. For
details see Section 8.

An analysis of Betti tables for Artinian Gorenstein rings $A_{F}$ of
codimension $=4$ and regularity $=5$, as well as of codimension $=5$ and
regularity $=4$, is a natural and interesting extension of this study, but is
beyond the scope of this paper.

###  1.1. Organization of the paper

  1. (1)

In Section 2 we give an overview of theory and constructions of Gorenstein
rings, and comment on how the rings $A_{F}$ of regularity $4$ fit with and
inform the general theory. We then move to a discussion of the geometric
connections to catalecticant varieties and secant varieties to Veronese
varieties. The section closes with a description of apolarity and powersum
presentations of forms, and the key role played by syzygies.

  2. (2)

In Section 3 we find the Betti tables of point sets $\Gamma$ that compute the
rank for a general $F\in{\mathcal{F}}_{B}$, the set of forms with Betti table
$B$.

  3. (3)

In Section 4 we find the Betti tables of the ideals $Q_{F}\subset F^{\perp}$,
for any $F\in{\mathcal{F}}_{B}$. We show that, except for two Betti tables
$B$, the set ${\mathcal{G}}_{B}$ of these ideals form irreducible sets in the
Hilbert scheme.

  4. (4)

In Section 5 we compute for each Betti table $B$, the rank $s=r(F)$ and the
$VSP(F,s)$ for a general form $F\in{\mathcal{F}}_{B}$ whenever $s<9$. We
compute the dimension of ${\mathcal{F}}_{B}$ and show that all but one are
irreducible.

  5. (5)

In Section 6 we present the stratification (defined by the Betti tables) of
the space ${\mathbb{P}}(\operatorname{Sym}(S_{4}))\simeq{\mathbb{P}}^{34}$ of
quartic forms in four variables.

  6. (6)

In Section 7 we consider the construction of codimension $4$ AG rings
supported on a quadric. We use Pfaffian and Lagrangian constructions of
codimension $3$ varieties using vector bundles supported on this quadrics. We
construct in this way liftings of some of the Artinian rings.

  7. (7)

In Section 8 and Section 9 we complete the task started in Section 7 and
construct liftings of rings $A_{F}$ to families of AG threefolds for forms $F$
in each family ${\mathcal{F}}_{B}$. We use here the doubling and the bilinkage
constructions. In some cases we show that the Artinian reduction of the
general AG threefold in the family is a general form in ${\mathcal{F}}_{B}$,
while for other families of threefolds, all the Artinian reductions belong to
a proper subset of ${\mathcal{F}}_{B}$.

  8. (8)

In Appendix A we give the 16 Betti tables for Artinian Gorenstein rings
appearing in [SSY], and in Appendix B we describe the computational tools
underlying this paper: Generic initial ideals, the Groebner family and
Groebner strata, and the Schreyer resolution. We illustrate these concepts
with a running example: the case of six points. The scripts which implement
the relevant computations are part of a package in Macaulay2, [M2]*Package
QuaternaryQuartics.

###  1.2. Notation.

We present the numerical information of the minimal free resolution of a
graded $S={\mathbb{C}}[x_{0},\ldots,x_{n}]$-module

$0\leftarrow M\leftarrow F_{0}\leftarrow F_{1}\leftarrow\ldots\leftarrow
F_{n+1}\leftarrow 0$

with $F_{i}=\bigoplus_{j\in{\mathbb{Z}}}S(-j)^{b_{ij}}$ in the form of a Betti
table (see [GeomSyz]*§1B):

$\begin{matrix}0:&b_{00}&b_{11}&b_{22}&\ldots&b_{n+1,n+1}&\cr
1:&b_{01}&b_{12}&b_{23}&\ldots&b_{n+1,n+2}&\cr\vdots&\vdots&\vdots&\vdots&\ldots&\vdots&\cr
m:&b_{0m}&b_{1,m+1}&b_{2,m+2}&\ldots&b_{n+1,n+1+m}&\cr\end{matrix}$

The Castelnuovo-Mumford regularity $\operatorname{reg}(M)$ of $M$ is $m$, and
we denote $b_{ij}=0$ with a $-$.

######  Example 1.9.

For the ideal $I_{C}$ of the twisted cubic in ${\mathbb{P}}^{3}$, we have

$0\rightarrow
S(-3)^{2}\xrightarrow{\left[\\!\begin{array}[]{cc}-x_{2}&x_{3}\\\
x_{1}&-x_{2}\\\
-x_{0}&x_{1}\end{array}\\!\right]}S(-2)^{3}\xrightarrow{\left[\\!\begin{array}[]{ccc}x_{1}^{2}-x_{0}x_{2}&x_{1}x_{2}-x_{0}x_{3}&x_{2}^{2}-x_{1}x_{3}\end{array}\\!\right]}S\rightarrow
S/I_{C}\rightarrow 0.$

The Betti table for this resolution is:

total | 1 | 3 | 2 |
---|---|---|---|---
0 | 1 | – | – |
1 | – | 3 | 2 |

We work with arithmetically Gorenstein rings, and will call such rings AG or
Gorenstein; CI will denote a complete intersection. Several of our results
address, in the case of quaternary quartics, questions posed by Iarrobino-
Kanev in [IK]. In terms of notation, the rings we call $S$ and $R$ are denoted
$R$ and $D$ in [IK]; we use $H$ to denote an Artinian Hilbert function, while
[IK] use $T$. We use $A_{F}$ to denote $S/F^{\perp}$, where $F$ is a quartic
in ${\mathbb{C}}[x_{0},\ldots,x_{3}]$. Our focus is on the case where $F$ is
nondegenerate, which means that $F^{\perp}$ contains no linear form.

##  2\. Preliminaries

###  2.1. Apolarity and syzygies

Consider the rings $R={\mathbb{C}}[y_{0},\ldots,y_{n}]$ and
$S={\mathbb{C}}[x_{0},\ldots,x_{n}]$ appearing in the introduction, and the
action of $S$ on $R$ by differentiation:

$x^{\alpha}(y^{\beta})=\alpha!\binom{\beta}{\alpha}y^{\beta-\alpha}$

if $\beta\geq\alpha$ and 0 otherwise. Here $\alpha$ and $\beta$ are multi-
indices, $\binom{\beta}{\alpha}=\prod\binom{\beta_{i}}{\alpha_{i}}$ and so on;
this defines a natural duality between $R_{d}$ and $S_{d}$. We can interchange
the roles of $R$ and $S$ by defining

$y^{\beta}(x^{\alpha})=\beta!\binom{\alpha}{\beta}x^{\alpha-\beta}.$

We set ${\mathbb{P}}^{n}={\mathbb{P}}(R_{1})$, identifying a point
$a=(a_{0}:\ldots:a_{n})\in{\mathbb{P}}^{n}$ with a linear form $l_{a}=\sum
a_{i}y_{i}\in R_{1}$ up to scalar. Then

$l_{a}^{d}(G)=G(l_{a}^{d})=d!G(a),$

for any $G\in S_{d}$. In particular if $m\geq d$

$None$ $G(l_{a}^{m})=0\iff G(a)=0$

So the ideal of a subscheme $\Gamma\subset{\mathbb{P}}^{n}$ is a homogenous
ideal in $S$. Similarly, if $P_{a}=\sum a_{i}x_{i}\in S_{1}$, then

$P_{a}^{d}(F)=F(P_{a}^{d})=d!F(a).$

######  Definition 2.1.

The ideal of differential operators that annihilates $F\in R$, written
$F^{\bot}=\\{G\in S|G(F)=0\\}$, is called the apolar ideal of $F$.

The ring

$A_{F}=S/F^{\bot}$

is Artinian and Gorenstein, with the socle of $A_{F}$ in degree $d=\deg(F)$.
Indeed

$P_{a}(G(F))=0\hskip 3.0pt\forall P_{a}\in S_{1}\iff G(F)=0\hskip 3.0pt\mbox{
or }\hskip 3.0ptG\in S_{d}.$

In particular the socle of $A_{F}$ is 1-dimensional, and $A_{F}$ is
Gorenstein; it is called the apolar Artinian Gorenstein ring or simply the
apolar ring of $F$.

Conversely for a graded Gorenstein ring $A=S/I$ with socle degree $d$,
multiplication in $A$ induces a linear form $\phi\colon S_{d}\to{\mathbb{C}}$
which can be identified with a homogeneous polynomial $F\in R$ of degree $d$.
This is:

######  Lemma 2.2.

[Macaulay 1916]. The map $F\mapsto A_{F}$ is a bijection between homogeneous
forms $F\in R_{d}$ and graded Artinian Gorenstein quotient rings $A_{F}=S/I$
of S with socle degree d.

The polynomial $F$ is sometimes called the dual socle generator or the dual
polynomial of $A_{F}$ ([CAwvtAG]*Thm. 21.6 and Exercise 21.7).

######  Definition 2.3.

A subscheme $\Gamma\subset{\mathbb{P}}^{n}={\mathbb{P}}(R_{1})$ is said to be
apolar to $F$ if the homogeneous ideal $I_{\Gamma}\subset F^{\bot}\subset S$.

By the duality between $R_{d}$ and $S_{d}$, this apolarity may be interpreted
via the Veronese map

$v_{d}:{\mathbb{P}}(R_{1})\to{\mathbb{P}}(R_{d});\quad[l]\mapsto[l^{d}].$

For $\Gamma\subset{\mathbb{P}}^{n}$ the image
$v_{d}(\Gamma)\subset{\mathbb{P}}(R_{d})$ spans a linear subspace defined by
the space of forms $(I_{\Gamma})_{d}\subset S_{d}$ of degree $d$ in the ideal
$I_{\Gamma}$, i.e.

$\langle
v_{d}(\Gamma)\rangle=(I_{\Gamma})_{d}^{\perp}\subset{\mathbb{P}}(R_{d}).$

The fundamental lemma is:

######  Lemma 2.4 (Apolarity).

Let $\Gamma\subset{\mathbb{P}}(R_{1})$ be any subscheme, then $[F]\in\langle
v_{d}(\Gamma)\rangle\subset{\mathbb{P}}(R_{d})$ if and only if $\Gamma$ is
apolar to $F$.

In particular, the relationship between a form $F$ and its powersum
presentations is given by:

######  Lemma 2.5.

$F=l_{1}^{d}+\ldots+l_{s}^{d}$ where
$\Gamma=\\{[l_{1}],\ldots,[l_{s}]\\}\subset{\mathbb{P}}(R_{1})$ if and only if
$\Gamma$ is apolar to $F$.

So the rank of a form (Definition 1.4) is the minimal length of an apolar
scheme $\Gamma$ that is a reduced set of points. Note that there is a
generalization of the notion of rank to the setting where $\Gamma$ is not
reduced:

######  Definition 2.6.

The cactus rank $\operatorname{cr}(F)$ is the minimal length of a subscheme
apolar to $F$.

######  Remark 2.7.

Clearly $\operatorname{cr}(F)\leq\operatorname{r}(F)$, and they often coincide
when $d$ and $n$ or $\operatorname{r}(F)$ are small. The cactus rank is
computed by a scheme that is locally Gorenstein [BB]*Proposition 2.2, Lemma
2.3. In an irreducible family of forms, both the rank and the cactus rank
attains its minimum in a general member. Both of them may however increase in
special members.

To find the rank of a form is, in general, very hard. In low degrees, the
syzygies of $A_{F}$ may sometimes be used to determine $\operatorname{r}(F)$
and $VSP(F,s)$, as illustrated by our results.

###  2.2. Apolarity in codimension $3$ or $4$ and regularity $3$

For a general ternary cubic form $F\in
R_{3}={\mathbb{C}}[y_{0},y_{1},y_{2}]_{3}$ the apolar ring is a complete
intersection

$A_{F}\cong S/(q_{1},q_{2},q_{3})$

with quadrics $q_{i}$. The Betti table of $A_{F}$ is

$\begin{matrix}1&-&-&-&\cr-&3&-&-&\cr-&-&3&-&\cr-&-&-&1&\cr\end{matrix}.$

Any pencil of quadrics $\langle q,q^{\prime}\rangle\subset\langle
q_{1},q_{2},q_{3}\rangle$ defines a scheme of length $4$, in general four
points. No ideal of three points is contained in $F^{\bot}$, so the rank of
$F$ is $4$ and $VSP(F,4)$ is a projective plane.

If $F=l_{1}^{3}+l_{2}^{3}+l_{3}^{3}$ has rank $3$, i.e. $F$ is a Fermat cubic,
the Betti table of $A_{F}$ is

$\begin{matrix}1&-&-&-&\cr-&3&2&-&\cr-&2&3&-&\cr-&-&-&1&\cr\end{matrix},$

and the three quadric generators in $F^{\bot}$ are the $2\times 2$ minors of a
$2\times 3$ matrix of linear forms which generate the ideal of three points

$\\{[l_{1}],[l_{2}],[l_{3}]\\}\subset{\mathbb{P}}(R_{1})={\mathbb{P}}^{2}.$

For example, changing variables we may assume $l_{i}=x_{i}$, and so it is easy
to see that

$F^{\perp}=\langle
x_{0}x_{1},x_{0}x_{2},x_{1}x_{2},x_{0}^{3}-x_{1}^{3},x_{0}^{3}-x_{2}^{3}\rangle.$

The set of ternary cubic forms that are nondegenerate (have no linear form in
the apolar ideal) has a stratification defined by the Betti table. The forms
with a Fermat Betti table lie in the closure of the forms with a complete
intersection Betti table.

For a general quaternary cubic $F\in
R_{3}={\mathbb{C}}[y_{0},y_{1},y_{2},y_{3}]_{3}$ the apolar ring $A_{F}$ has
Betti table

$\begin{matrix}1&-&-&-&-&\cr-&6&5&-&-&\cr-&-&5&6&-&\cr-&-&-&-&1&\cr\end{matrix}.$

Five of the six quadrics in the ideal are the $4\times 4$ Pfaffians of a skew-
symmetric $5\times 5$ \- matrix. They define the ideal of five points. This
decomposition of syzygies is familiar for a general canonical curve
$C\subset{\mathbb{P}}^{5}$ of genus $6$, which is a complete intersection of a
unique Del Pezzo surface of degree $5$ and a non unique hyperquadric, cf.
[ACGH 1985], [Schreyer 1986], [Mukai 1988]. So $F$ has rank at most $5$. If it
has rank $4$, the ideal $F^{\bot}$ would contain six quadrics that vanish on
the four points. But $F^{\bot}$ is generated by six quadrics so this is
impossible. So $F$ has a unique presentation as a sum of five cubes
(Sylvesters Pentahedral Theorem). There are two more possible Betti tables of
rings $A_{F}$ of codimension $4$ and regularity $3$. In the first case the six
quadrics define a scheme of length $4$, so if this scheme is smooth, $F$ has
rank $4$ and is a Fermat cubic, with Betti table

$\begin{matrix}1&-&-&-&-&\cr-&6&8&3&-&\cr-&3&8&6&-&\cr-&-&-&-&1&\cr\end{matrix}.$

In the second case three of the six quadrics are reducible with a common
linear factor. So, assuming that these three quadrics are
$x_{0}x_{1},x_{0}x_{2},x_{0}x_{3}$, the form $F$ decomposes into
$F=y_{0}^{3}+F_{0}(y_{1},y_{2},y_{3})$. If the apolar ideal of the ternary
form $F_{0}$ is determinantal, then it is apolar to a scheme of length $3$,
and hence $F$ is apolar to a scheme of length $4$. Then the apolar ring
$A_{F}$ has the Betti table of a Fermat cubic. If the apolar ideal of the
ternary form $F_{0}$ is not determinantal, it is a complete intersection
$(2,2,2)$, in which case $F_{0}$ has rank $4$ and $F$ has rank $5$. The form
$F_{0}$ has a projective plane of different decompositions as a sum of four
cubes, so $F$ has a projective plane of different decompositions as a sum of
five cubes. The Betti table for $A_{F}$ is:

$\begin{matrix}1&-&-&-&-&\cr-&6&6&1&-&\cr-&1&6&6&-&\cr-&-&-&-&1&\cr\end{matrix}$

As in the ternary case, there is a stratification in the space of quaternary
cubics of the set of nondegenerate cubic forms defined by the Betti tables of
their apolar ring.

Let ${\mathcal{F}}_{\operatorname{fe}}\subset{\mathbb{P}}^{19}$ be the set of
cubic quaternary forms $F$ with the Betti table for $A_{F}$ of a Fermat cubic,
and similarly, let ${\mathcal{F}}_{\operatorname{pl}}$ be the set of forms
with the Betti table of a general form that is a sum of a rank $1$ form and a
ternary form. Finally, let ${\mathcal{F}}_{\operatorname{gen}}$ be the set of
forms with the Betti table of a general form. Then
${\mathcal{F}}_{\operatorname{gen}}$ is dense in ${\mathbb{P}}^{19}$, the
subset ${\mathcal{F}}_{\operatorname{pl}}$ is irreducible of dimension $16$,
and the subset ${\mathcal{F}}_{\operatorname{fe}}$ is irreducible of dimension
$15$. Furthermore, there is a stratification of the form below:

${\mathcal{F}}_{\operatorname{fe}}\subset\overline{{\mathcal{F}}_{\operatorname{pl}}}\subset\overline{{\mathcal{F}}_{\operatorname{gen}}}={\mathbb{P}}(R_{3})={\mathbb{P}}^{19}.$

Recent results on decompositions for cubics in the real setting appear in
[MM].

###  2.3. Geometry: Catalecticant and Secant varieties

We now review classical connections between power sum decompositions, secant
varieties to Veronese varieties and catalecticant matrices. In addition to the
book of Iarrobino-Kanev [IK], other good references are Geramita’s papers
[Geramita1], [Geramita2].

####  2.3.1. Geometry of the catalecticant variety and the Gorenstein
parameter scheme

######  Definition 2.8.

Following [IK], and the definitions of apolarity in 2.1, we write
$\mathcal{H}(d,n+1)$ for the set of Hilbert functions for an Artinian
Gorenstein algebra $S/F^{\perp}$ of socle degree $d={\rm deg}\;F$ when
$S={\mathbb{C}}[x_{0},...,x_{n}]$, and define $\operatorname{Gor}(H)$ as the
set of polynomials $F\in R_{d}={\mathbb{C}}[y_{0},...,y_{n}]_{d}$ such that
$S/F^{\perp}$ has Hilbert function $H$. Then

(3) $R_{d}=\bigcup\limits_{H\in\mathcal{H}(d,n+1)}\operatorname{Gor}(H)$

######  Definition 2.9.

Let $F\in R_{d}$ and $G\in S_{d-u}$, so that $G\circ F\in R_{u}$. If we write
$F$ with indeterminate coefficients $\\{a_{1},\ldots,a_{m}\\}$ with
$m=\binom{n+d}{d}$, then the matrix defining the map

$S_{d-u}\longrightarrow R_{u}\mbox{ via }G\mapsto G\circ F$

is called the catalecticant, and is written $\operatorname{Cat}(u,d-u,n+1)$.
Let $I_{r}(\operatorname{Cat}(u,d-u,n+1))\subset{\mathbb{C}}[a_{1},..,a_{m}]$
be the ideal generated by the $(r\times r)$-minors of
$\operatorname{Cat}(u,d-u,n+1)$. Next, for a Hilbert function

$H=(1=h_{0},h_{1},h_{2},\ldots,h_{d}=1)\in\mathcal{H}(d,n+1),$

define the affine scheme

$\operatorname{{\bf
Gor}}_{\leq}(H)={\mathbb{V}}(\sum\limits_{u=1}^{d-1}I_{h_{u}+1}(\operatorname{Cat}(u,d-u,n+1))).$

The scheme $\operatorname{{\bf Gor}}_{\leq}(H)$ is a sum of determinantal
ideals of catalecticants. It has an open subset $\operatorname{{\bf Gor}}(H)$
consisting of $F$ such that the Hilbert function of $S/F^{\perp}$ is $H$;
$\operatorname{Gor}(H)$ is the corresponding reduced affine scheme.

It follows from Definition 2.9 that the locus in ${\mathbb{P}}(R_{d})$ where
$\operatorname{rk}(\operatorname{Cat}(u,d-u,n+1))=k$ corresponds to those
$F\in R_{d}$ which have Hilbert function $\dim(S/F^{\perp})_{d-u}=k$.

######  Example 2.10.

For the case of quartics in four variables, we are interested in the non-
degenerate situation: that is, where no linear form annihilates $F$. Thus, the
Hilbert function of $S/F^{\perp}$ satisfies

$H(S/F^{\perp})=(1,4,h_{2},4,1),\mbox{ with }h_{2}\in\\{4,\ldots,10\\}.$

Therefore $10-h_{2}$ is the dimension of the vector space of quadrics apolar
to $F$, i.e. $10-h_{2}={\rm dim}F_{2}^{\perp}$, and the relevant catalecticant
matrix is $\operatorname{Cat}(2,2,4)$, given by

$\left[\begin{matrix}a_{0}&a_{1}&a_{2}&a_{3}&a_{4}&a_{5}&a_{6}&a_{7}&a_{8}&a_{9}\\\
a_{1}&a_{4}&a_{5}&a_{6}&a_{10}&a_{11}&a_{12}&a_{13}&a_{14}&a_{15}\\\
a_{2}&a_{5}&a_{7}&a_{8}&a_{11}&a_{13}&a_{14}&a_{16}&a_{17}&a_{18}\\\
a_{3}&a_{6}&a_{8}&a_{9}&a_{12}&a_{14}&a_{15}&a_{17}&a_{18}&a_{19}\\\
a_{4}&a_{10}&a_{11}&a_{12}&a_{20}&a_{21}&a_{22}&a_{23}&a_{24}&a_{25}\\\
a_{5}&a_{11}&a_{13}&a_{14}&a_{21}&a_{23}&a_{24}&a_{26}&a_{27}&a_{28}\\\
a_{6}&a_{12}&a_{14}&a_{15}&a_{22}&a_{24}&a_{25}&a_{27}&a_{28}&a_{29}\\\
a_{7}&a_{13}&a_{16}&a_{17}&a_{23}&a_{26}&a_{27}&a_{30}&a_{31}&a_{32}\\\
a_{8}&a_{14}&a_{17}&a_{18}&a_{24}&a_{27}&a_{28}&a_{31}&a_{32}&a_{33}\\\
a_{9}&a_{15}&a_{18}&a_{19}&a_{25}&a_{28}&a_{29}&a_{32}&a_{33}&a_{34}\\\
\end{matrix}\right]$.

The catalecticant relates to the rank of $F$ as follows. First,

$\begin{array}[]{ccc}[F]\in{\mathbb{V}}(I_{s+1}(\operatorname{Cat}(2,2,4)))&\leftrightarrow&h_{2}(F)=\dim(S/F^{\perp})_{2}=\operatorname{rk}(\operatorname{Cat}(2,2,4)(F)\leq\operatorname{r}(F)\\\
&\leftrightarrow&\exists\geq 10-\operatorname{r}(F)\mbox{ independent quadrics
apolar to }F.\end{array}$

Secondly, if $h_{2}(F)=s$, then the space of quadrics in $S$ annihilating $F$
has dimension $10-s$. But then any $0$-dimensional scheme $\Gamma$ that is
apolar to $F$ contains at most $10-s$ independent quadrics. The length of
$\Gamma$ is therefore at least $s$. In particular, by Lemma 2.5 and Remark
2.7,

$h_{2}(F)\leq\operatorname{cr}(F)\leq\operatorname{r}(F).$

Assume, for a moment, that the quadrics in $F^{\perp}$ define a
$0$-dimensional scheme $\Gamma$. If the length of $\Gamma$ equals $h_{2}(F)$,
then $\Gamma$ computes the cactus rank, i.e. $\operatorname{cr}(F)=h_{2}(F)$.
If the length of $\Gamma$ is strictly larger than $h_{2}(F)$, then, in some
cases $\Gamma$ computes the cactus rank, in other cases it does not.
Quaternary quartic forms $F$ of type [300a] are of the first kind, while forms
$F$ of type [300b] are of the second kind, see Section 5.

In any case, the scheme defined by the quadrics in $F^{\perp}$ plays a crucial
role in our analysis of schemes apolar to $F$, see Section 4.

####  2.3.2. Motivating Questions

We highlight some of the questions posed in [Geramita1], [Geramita2], and
[IK]:

  1. (1)

Problem A of the introduction to [IK] asks how to determine the length of a
power sum decomposition of a general polynomial, when there is a unique
decomposition of minimal length, and how the closure of the power sum locus
relates to catalecticant matrices. We obtain results for quaternary quartics
in Section 6.

  2. (2)

Problem B of the introduction to [IK] asks when $\operatorname{Gor}(H)$ is
irreducible, and for a description of the Zariski closure and dimension of
$\operatorname{Gor}(H)$. We obtain results for quaternary quartics in Section
6.1.

  3. (3)

In [Geramita1], Geramita asks under what conditions the ideal of minors of the
catalecticant is prime. In Proposition 6.1 we use geometric reasoning to show
that $I_{7}(\operatorname{Cat}(2,2,4))$ and $I_{8}(\operatorname{Cat}(2,2,4))$
are not prime.

  4. (4)

Conjecture 3.20 of [IK] gives a formula for the dimension of the tangent space
to $\operatorname{{\bf Gor}}(H)$ at a generic point $[F]$, when $\deg(F)$ is
even. In the case where $n+1=4$ and $\deg(F)=4$, the conjectured affine
dimensions are

$h_{2}=$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$
---|---|---|---|---|---|---|---
$\dim T_{[F]}=$ | $16$ | $20$ | $25$ | $29$ | $32$ | $34$ | $35$

We prove that the conjecture holds for $n+1=4=\deg(F)$ in Proposition 5.5.

####  2.3.3. Secant varieties to the Veronese variety and power sum
decompositions

Let $n=\binom{m+d}{d}$. The Veronese map

(4) ${\mathbb{P}}^{m}={\mathbb{P}}(V^{*})\stackrel{{\scriptstyle
v_{d}}}{{\longrightarrow}}{\mathbb{P}}^{n-1}={\mathbb{P}}(\operatorname{Sym}_{d}(V^{*}))$

is defined by $v_{d}=[f_{1}:\cdots:f_{n}]$ with the $f_{i}$ a basis for
$\operatorname{Sym}_{d}(V^{*})$. The map can be thought of intrinsically as

$[l]\in{\mathbb{P}}(V^{*})\mapsto[l^{d}]\in{\mathbb{P}}(\operatorname{Sym}_{d}(V^{*})).$

Hence, a point on the image of $v_{d}$ corresponds to a power of a linear
form, a point on the secant line $\alpha[l_{1}^{d}]+\beta[l_{2}^{d}]$
corresponds to $F=\alpha l_{1}^{d}+\beta
l_{2}^{d}\in\operatorname{Sym}_{d}(V^{*})$ which can be written as a sum
$F=(al_{1})^{d}+(bl_{2})^{d}$ of two powers of linear forms, where
$a^{d}=\alpha,b^{d}=\beta$, and so on. Therefore, we may interpret the
$s^{th}$ secant variety to the Veronese variety as

(5)
$\operatorname{Sec}_{s}(v_{d}({\mathbb{P}}(V^{*})))=\overline{\\{[F]\in{\mathbb{P}}(\operatorname{Sym}_{d}(V^{*}))\mid
F=l_{1}^{d}+\cdots l_{s}^{d},\;{\rm for\;some}\;l_{1},...,l_{s}\in V^{*}\\}}$

######  Definition 2.11.

The border rank $\operatorname{br}(F)$ of $F\in\operatorname{Sym}_{d}(V^{*})$
is the smallest $s$ such that $F$ lies in the closure of
$\operatorname{Sec}_{s}(v_{d}({\mathbb{P}}(V^{*})))$.

The relationship between the catalecticant and the secant variety to the
Veronese variety is classical:

######  Lemma 2.12.

[Raicu]*Lemma 2.1 For $1\leq i\leq d$ and $s\geq 1$,

$I_{s+1}(\operatorname{Cat}(i,d-i,n))\subseteq
I(\operatorname{Sec}_{s}(v_{d}({\mathbb{P}}^{n-1}))).$

So in the situation of interest to us,
$I_{s+1}(\operatorname{Cat}(2,2,4))\subseteq
I(\operatorname{Sec}_{s}(v_{4}({\mathbb{P}}^{3}))).$

######  Remark 2.13.

The border rank is clearly a lower bound for rank. If the cactus rank of $F$
is computed by a scheme that is a flat limit of smooth schemes, then the
border rank is also at most the cactus rank. In ${\mathbb{P}}^{3}$ any finite
locally Gorenstein scheme is smoothable in this sense. Finally, by Lemma 2.12,
the border rank is at least the catalecticant rank $h_{2}(F)$. Therefore, if
$F$ is a quaternary quartic, then

$h_{2}(F)\leq\operatorname{br}(F)\leq\operatorname{cr}(F)\leq\operatorname{r}(F).$

###  2.4. Gorenstein rings of low codimension: History and Techniques

For a Gorenstein ring $A=S/I$ of codimension $1$ or $2$, $I$ must be a
complete intersection, and in codimension $3$ the Buchsbaum-Eisenbud theorem
[be1975generic] shows that $I$ is given by the Pfaffians of a skew-symmetric
matrix. In particular, knowing $I$ is equivalent to knowing the matrix of
first syzygies. This is obviously also true in codimension $4$: if $F_{i}$ are
the modules in a minimal free resolution, since

$\sum_{i=0}^{4}(-1)^{i}\operatorname{rk}(F_{i})=0,$

the minimal free resolution of a codimension $4$ Gorenstein ring $A=S/I$ will
have the form

$0\longrightarrow S^{1}\stackrel{{\scriptstyle
d_{4}}}{{\longrightarrow}}F_{3}\stackrel{{\scriptstyle
d_{3}}}{{\longrightarrow}}F_{2}\stackrel{{\scriptstyle
d_{2}}}{{\longrightarrow}}F_{1}\stackrel{{\scriptstyle
d_{1}}}{{\longrightarrow}}S^{1}\longrightarrow S/I\longrightarrow 0,$

with $F_{1}\simeq S^{k+1}\simeq F_{3}$ and $F_{2}\simeq S^{2k}$. In [KMmzeit],
Kustin-Miller show that there is a symmetric map

$F_{2}\stackrel{{\scriptstyle s}}{{\longrightarrow}}F_{2}^{*}$

of the form

$\left[\\!\begin{array}[]{cc}0&I_{k}\\\ I_{k}&0\end{array}\\!\right].$

The map $s$ induces a non-degenerate symmetric quadratic form on $F_{2}$,
giving it the structure of an even orthogonal module. This structure is
further explored by Reid in [ReidGeneral], who introduces Spin-Hom varieties,
and also by Celikbas-Laxmi-Weyman in [celikbas2020family]. Reid makes several
remarks at the conclusion of [ReidGeneral] on the structure of codimension $4$
Gorenstein ideals for various values of $k$:

  1. (1)

$k=3$: Since $I$ is codimension $4$, $I$ must be a complete intersection.

  2. (2)

$k=4$: Kunz shows in [Ku] that there are no Gorenstein almost complete
intersections.

  3. (3)

$k=5$: Reid asks if any such ideal is a hypersurface section of Pfaffians.

  4. (4)

$k=6$: Reid asks if any such ideal is a Kustin-Miller unprojection.

As shown by Vasconcelos-Villereal in [VV], for $k=5$ the answer to Reid’s
question is yes when $I$ is generically a complete intersection. Reid also
asks if any Gorenstein codimension $4$ ideal with an even number of generators
(so $k$ odd) is always a hypersurface section of Pfaffians. Proposition 9.2
gives a negative answer to this question. The ideals which provide the answer
appear as type $[310]$ in the appendix, and cannot be lifted to irreducible
threefolds; we show that such ideals can be lifted to a reducible Gorenstein
threefold with components of degrees 6 and 11. An irreducible threefold giving
a negative answer is provided by ideals with Betti table of type $[200]$.
Celikbas-Laxmi-Weyman give a negative answer to $(4)$ in [laxmi2021spinor].

A standard way to build Gorenstein rings is via Nagata’s technique of
idealization, which takes as input a ring $T$ and the canonical module
$\omega_{T}$. Foxby, Gulliksen and Reiten show the idealization of a level
algebra $T$ yields $T\ltimes\omega_{T}$ which is standard graded and
Gorenstein. The idealization satisfies

$\operatorname{codim}(T\ltimes\omega_{T})=\operatorname{codim}(T)+\operatorname{type}(\omega_{T}),$

so does not produce interesting objects in codimension $4$. However, there is
a similar construction known as doubling, and in [celikbas2020family]
Celikbas-Laxmi-Weyman use doubling to construct many examples of codimension
$4$ Gorenstein ideals with a small number of generators. We next review the
doubling construction, as well as another way of constructing codimension $4$
Gorenstein rings: the bilinkage construction.

###  2.5. The doubling construction

Assume $S={\mathbb{C}}[x_{0},\ldots,x_{n}]$ and $I\subseteq S$ is codimension
$c$ with $S/I$ Cohen-Macaulay and a minimal free resolution
$(G_{\bullet},d_{\bullet})$. By [CAwvtAG]*§21,

(6) $\omega_{S/I}=\operatorname{Ext}^{c}_{S}(S/I,S(-n-1))$

is the _canonical module_ of $S/I$. Assume, furthermore, that $S/I$ is
Gorenstein at all minimal primes (i.e. satisfies the condition ${\rm G}_{0}$
cf.[Mar], [Har]) and consider maps

$\psi:\omega_{S/I}(-\gamma)\to S/I.$

If ${\rm im}(\psi)$ is an ideal $J$ of codimension $c+1$, then, by
[CMrings]*Proposition 3.3.18, $J$ is a Gorenstein ideal. The ideal $J$ is then
called a _doubling_ of $I$ with $\psi$. By abuse of notation we also say in
this case that the divisor ${\mathbb{V}}(J)$ is a doubling in
${\mathbb{V}}(I)$. It is not true that every Artinian Gorenstein ideal of
codimension $c+1$ is a doubling of some codimension $c$ ideal.

Letting $M^{*}$ denote $\operatorname{Hom}_{S}(M,S(-n-1))$, $G_{\bullet}^{*}$
is a minimal free resolution of $\omega_{S/I}$. A map
$\psi:\omega_{S/I}(-\gamma)\to S/I$ yields an induced map
$\psi_{\bullet}:G_{\bullet}^{*}(-\gamma)\to G_{\bullet}$:

(7)
$\textstyle{G_{\bullet}^{*}(-\gamma):}$$\textstyle{\omega_{S/I}(-\gamma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{G_{c}^{*}(-\gamma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{0}}$$\textstyle{G_{c-1}^{*}(-\gamma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{1}}$$\scriptstyle{d_{c}^{*}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{c-1}^{*}}$$\textstyle{G_{1}^{*}(-\gamma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{c-1}}$$\scriptstyle{d_{2}^{*}}$$\textstyle{G_{0}^{*}(-\gamma)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{c}}$$\scriptstyle{d_{1}^{*}}$$\textstyle{\hskip
18.06749ptG_{\bullet}:}$$\textstyle{S/I}$$\textstyle{G_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{1}}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{2}}$$\textstyle{G_{c-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{c-1}}$$\textstyle{G_{c}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{c}}$

If $\psi$ is injective, the mapping cone of $\psi$ gives a resolution of
$S/(I+J)$. If it is minimal, then one can read off the Betti table of $S/J$
from the Betti table of $S/I$.

######  Proposition 2.14.

Let $Y\subset{\mathbb{P}}^{n}$ be arithmetically Cohen-Macaulay subscheme of
codimension $c$ and regularity $r$, and assume that $Y$ is generically
Gorenstein. Assume a map

$\psi\in\operatorname{Hom}_{Y}(\omega_{Y}(-\gamma),{\mathcal{O}}_{Y}),\;{\rm
with}\;\gamma\geq 2r+c-n$

is an inclusion. Then the mapping cone of $\psi$ gives a minimal resolution of
the divisor $X=\\{\psi=0\\}$ as a doubling in $Y$.

###### Proof.

By [CMrings]*Proposition 3.3.18 a section
$\psi\in\operatorname{Hom}_{Y}(\omega_{Y}(-\gamma),{\mathcal{O}}_{Y})$, where
$\omega_{Y}(-\gamma)$ is invertible and not isomorphic to ${\mathcal{O}}_{Y}$,
is equivalent to the identification of $\omega_{Y}(-\gamma)$ with a
codimension $1$ ideal of ${\mathcal{O}}_{Y}$. Denote by $X$ the divisor
defined by this ideal. Hence we may identify the doubling construction of
Equation 7 using $\omega_{Y}$ with the standard mapping cone construction
associated to the exact sequence

$0\to\omega_{Y}(-\gamma)\to\mathcal{O}_{Y}\to\mathcal{O}_{X}\to 0,$

which gives a minimal resolution of the structure sheaf of $X$, assuming there
is no cancellation in the mapping cone resolution. To see that there is no
cancellation, let $G_{\bullet}$ and $G_{\bullet}^{*}(-\gamma)$ be the minimal
free resolutions appearing in Equation 7. Let $-a_{i}$ be the maximal negative
degree shift in $G_{i}$, and let $b_{i}^{\vee}$ be the maximal degree shift in
$G_{i}^{*}(-\gamma)$. Hence a sufficient condition for there to be no
cancellation in the mapping cone is that

$\begin{array}[]{ccc}b_{0}^{\vee}&<&-a_{c}\\\ b_{1}^{\vee}&<&-a_{c-1}\\\
\vdots&<&\vdots\\\ b_{c-1}^{\vee}&<&-a_{1}\\\
b_{c}^{\vee}&<&-a_{0}\end{array}$

Since $b_{i}^{\vee}=a_{i}-n-1-\gamma$, this means it is sufficient to show

(8) $\gamma>a_{i}+a_{c-i}-n-1\mbox{ for all }i.$

Since the regularity of $\mathcal{O}_{Y}$ is $r$, the maximum shift in $G_{i}$
is $r+i$. Thus, $a_{i}\leq r+i$, and the maximum value for the right hand side
in Equation 8 is $(r+i)+(r+c-i)-n-1=2r+c-n-1$. Therefore the condition that

(9) $\gamma\geq 2r+c+1-n-1=2r+c-n$

is sufficient to force exactness of the mapping cone. ∎

We formulate this geometrically as

######  Corollary 2.15.

Let $Y\subset{\mathbb{P}}^{n}$ be arithmetically Cohen-Macaulay subscheme of
regularity $r$, that is generically Gorenstein. Assume that a map
$\psi\in\operatorname{Hom}_{Y}(\omega_{Y}(\dim(Y)-2r),\mathcal{O}_{Y})$ is an
inclusion, then it induces a minimal resolution of the divisor
$X=\\{\psi=0\\}$ as a doubling of $Y$.

######  Example 2.16.

Consider the case of an ideal $I_{\Gamma}$ of six points. If $\Gamma$ does not
contain four collinear points, then Proposition 3.2 shows that $S/I_{\Gamma}$
has regularity $2$ and is of type (5), (6), (7a) or (7b) in Table 1. A
computation shows that a general element of
$\operatorname{Hom}(\omega_{S/I}(-4),S/I)$ is injective. Therefore Corollary
2.15 gives the corresponding doublings, which appear in Table 7 of Appendix A
as, respectively, Type 2.4, Type 2.3, Type 2.6.

Now suppose the six points span ${\mathbb{P}}^{3}$ and four are collinear; to
cut out the four points $I_{\Gamma}$ must contain a quartic and $S/I_{\Gamma}$
has Betti table

$\begin{matrix}&&&&\cr
1&-&-&-&\cr-&5&6&2&\cr-&-&-&-&\cr-&1&2&1&\cr\cr\end{matrix}.$

A computation shows that a general element of
$\operatorname{Hom}(\omega_{S/I}(-2),S/I)$ is not injective.

When $\gamma\geq 3$ a general element is injective. The four Betti tables
below are the results of the doubling construction with a general element of
$\operatorname{Hom}(\omega_{S/I}(-\gamma),S/I)$ for $\gamma\in\\{3,4,5,6\\}$.

$\begin{matrix}&&&&&\cr
1&1&-&-&-&\cr-&3&5&2&-&\cr-&-&-&-&-&\cr-&2&5&3&-&\cr-&-&-&1&1&\cr\cr\end{matrix}\;\;\;\;\;\;\begin{matrix}&&&&\cr
1&-&-&-&-\cr-&6&8&3&-\cr-&-&-&-&-&\cr-&3&8&6&-\cr-&-&-&-&1\cr\cr\end{matrix}\;\;\;\;\;\;\begin{matrix}&&&&\cr
1&-&-&-&-\cr-&5&6&2&-\cr-&1&2&1&-\cr-&1&2&1&-\cr-&2&6&5&-\cr-&-&-&-&1\cr\cr\end{matrix}\;\;\;\;\;\;\begin{matrix}&&&&\cr
1&-&-&-&-\cr-&5&6&2&-\cr-&-&-&-&-\cr-&2&4&2&-\cr-&-&-&-&-\cr-&2&6&5&-\cr-&-&-&-&1\cr\cr\end{matrix}$

For $\gamma=3$, the leftmost Betti table shows that cancellation occurs in the
mapping cone. For the two middle Betti tables, $\gamma\in\\{4,5\\}$ and no
cancellation occurs in the mapping cone. However, the potential for
cancellation exists. In the rightmost Betti table, $\gamma=6=2r$ and there is
no potential for cancellation. The mapping cone resolution is minimal, as
guaranteed by Corollary 2.15.

######  Construction 2.17.

We can construct a doubling of a finite scheme $\Gamma$ or more generally an
an arithmetic Cohen-Macaulay scheme with ideal $I_{\Gamma}$ and codimension
$c$. If

$\psi\in\operatorname{Hom}_{S/I_{\Gamma}}(\omega_{S/{I_{\Gamma}}}(-\gamma),S/I_{\Gamma})$

is nonzero and $\gamma$ satisfies condition (9), then $\psi$ induces a
doubling of $I_{\Gamma}$, so we suggest a way to find such maps $\psi$.

Let $J\subset I_{\Gamma}$ be a complete intersection of codimension $c$ and
type $(d_{1},...,d_{c})$ , then

(10)
$\omega_{S/I_{\Gamma}}=\operatorname{Ext}^{c}_{S}(S/I_{\Gamma},S(-n-1))\simeq\operatorname{Hom}_{S}(S/I_{\Gamma},S/J(d_{1}+...+d_{c}-n-1)).$

On the other hand, let $I_{P}=J\colon I_{\Gamma}$. Then

(11) $I_{P}\simeq\operatorname{Hom}_{S}(I_{\Gamma},J).$

By [CAwvtAG]*Theorem 21.23,

(12) $I_{P}\otimes S/J\simeq\operatorname{Hom}_{S/J}(S/I_{\Gamma},S/J).$

Hence

(13) $I_{P}\otimes
S/I_{\Gamma}\simeq\omega_{S/I_{\Gamma}}(-d_{1}-...-d_{c}+n+1).$

Now consider maps

$\psi^{\prime}\in\operatorname{Hom}_{S}(I_{P}(d_{1}+\dots+d_{c}-n-1-\gamma),S),$

and their restrictions

$\psi=\psi^{\prime}|_{S/I_{\Gamma}}\in\operatorname{Hom}_{S/I_{\Gamma}}(I_{P}/I_{\Gamma}(d_{1}+\dots+d_{c}-n-1-\gamma),S/I_{\Gamma}).$

If $\psi$ is nonzero, then $I_{\Gamma}+\mathrm{im}(\psi^{\prime})$ is a
doubling of $I_{\Gamma}$.

If $\dim\Gamma\geq 1$, then it makes sense to talk about divisors on $\Gamma$.
We proceed as above and denote by $P$ the scheme defined by $J:I_{\Gamma}$.
Note that in particular, this means $P$ is determined by the choice of $J$. In
this formulation a doubling $X$ of $\Gamma$ is a divisor bilinked to
$P_{\Gamma}:=P\cap\Gamma$ in $\Gamma$,

$X\in|P_{\Gamma}-(d_{1}+\dots+d_{c}-n-1-\gamma)H|$

as discussed below in 2.6.

######  Proposition 2.18.

For every Betti table $B$ for a codimension = 4 = regularity Gorenstein ideal,
there is an $F\in{\mathcal{F}}_{B}$, such that $F^{\perp}$ can be obtained by
a doubling.

###### Proof.

Proposition 3.2 classifies reduced zero-schemes $\Gamma$ in ${\mathbb{P}}^{3}$
with $4\leq\deg(\Gamma)\leq 9$ satisfying certain geometric constraints, and
proves that in these situations $\Gamma$ has regularity $2$, except for the
$(2,2,2)$ complete intersection. In the case of regularity $2$, we are in the
setting of Corollary 2.15, and a computation with an element of each class
shows there is a $\psi$ yielding an inclusion, cf. Macaulay2, [M2]*Package
QuaternaryQuartics. The complete intersection $(2,2,2,2)$ is trivially a
doubling of the complete intersection $(2,2,2)$. ∎

For a geometric perspective on this, in Corollary 8.6 we prove that a general
Artinian Gorenstein ideal from a family of Betti table type in Table 6, with
the exception of $[300a]$, can be obtained by doubling. For other recent work
on doubling of ACI’s, see [laxmi].

Here, let us illustrate the interesting case of Betti table of type [300]. We
will see that ideals with the same Betti table may admit a different behavior
with respect to the doubling construction.

######  Example 2.19 (Type [300a] is not a doubling).

A quartic $F$ of type [300a] has rank $8$ and $A_{F}$ has Betti table

$\begin{matrix}1&-&-&-&-&\cr-&3&-&-&-&\cr-&4&12&4&-&\cr-&-&-&3&-&\cr-&-&-&-&1&\cr\end{matrix}.$

The quadric part $J$ of $F^{\perp}$ is a complete intersection that is a
scheme of length $8$, which is a minimal apolar scheme to $F$. In that case
the doubling of $J$ is always a complete intersection. Hence, $F^{\perp}$ is
not a doubling of $J$. There is however a family of rings $A_{F}$ with Betti
table of type $[300]$ which is the doubling of an almost complete intersection
ideal $I_{\Gamma}$ of codimension $3$.

######  Example 2.20 (Type [300b] is a doubling).

This example shows that if $F$ is of type [300b] (see Definition 1.1), then
$F^{\perp}$ can be constructed as a doubling from its apolar ideal. Assume
$I_{\Gamma}$ defines seven points in a complete intersection of three
quadrics. Then $I_{\Gamma}$ is an almost complete intersection, and
$S/I_{\Gamma}$ has Betti diagram

$\displaystyle\begin{matrix}1&-&-&-&\cr-&3&-&-&\cr-&1&6&3&\cr\end{matrix}.$

Let $I_{P}=J:I_{\Gamma}$, where $J\subset I_{\Gamma}$ is a complete
intersection $(2,2,2)$. Then $I_{P}$ defines one point $P$ in
${\mathbb{P}}^{3}$. Note that $\operatorname{reg}S/I_{\Gamma}=2$, so by
condition (9), $\gamma\geq 4$ guarantees that the mapping cone has no
cancellation. If we want to obtain a doubling having regularity $4$, we choose
$\gamma=4$. In particular, a nonzero map
$\psi^{\prime}\in\operatorname{Hom}_{S}(I_{P}(-2),S)$ defines a doubling of
$I_{\Gamma}$.

Now assume $S/F^{\perp}$ has Betti table $[300b]$ and that $F^{\perp}$
contains the ideal $I_{\Gamma}$. If
$\psi^{\prime}\in\operatorname{Hom}_{S}(I_{P}(-2),F^{\perp})$ is nonzero, then
$I_{\Gamma}+\mathrm{im}(\psi^{\prime})\subseteq F^{\perp}$ is a doubling of
$I_{\Gamma}$. Because $A_{F}$ and $S/(I_{\Gamma}+\mathrm{im}(\psi^{\prime}))$
have the same Betti table, they are actually equal. Hence, $F^{\perp}$ is a
doubling of $I_{\Gamma}$.

######  Example 2.21 (Type [300c] is a doubling).

Let $I_{\Gamma}$ be an ideal defining seven points with three on a line. Then
$S/I_{\Gamma}$ has Betti table

$\displaystyle\begin{matrix}1&-&-&-&\cr-&3&-&-&\cr-&1&6&3&\cr\end{matrix},$

and $S/Q_{\Gamma}$, where $Q_{\Gamma}\subseteq I_{\Gamma}$ is the ideal
generated by the quadrics in $I_{\Gamma}$, has Betti table

(14) $\begin{matrix}&&&&\cr 1&-&-&-&\cr-&3&-&-&\cr-&-&4&2&\cr\cr\end{matrix}.$

In this case $Q_{\Gamma}$ is the ideal of a line and four points.

We may take $J\subseteq I_{\Gamma}$ to be a complete intersection $(2,2,3)$
and proceed as in the construction 2.17. Then $S/I_{P}$ where
$I_{P}=J:I_{\Gamma}$, has Betti table

(15) $\begin{matrix}&&&&\cr 1&-&-&-&\cr-&5&5&-&\cr-&-&-&1&\cr\cr\end{matrix}.$

As in the previous example, $\gamma=4$ guarantees that we get a doubling of
$I_{\Gamma}$ which has regularity $4$. Therefore, a nonzero map
$\psi^{\prime}\in\operatorname{Hom}_{S}(I_{P}(-1),S)$, defines an ideal
$\mathrm{im}(\psi^{\prime})+I_{\Gamma}$ that is a doubling of $I_{\Gamma}$ of
type [300c].

###  2.6. Bilinkage and unprojection

Another classical construction of Gorenstein rings of codimension $4$ is by
means of bilinkage.

######  Definition 2.22.

Let $R$ be a Gorenstein ring, and let $A$ and $B$ be ideals of R. Then $A$ and
$B$ are said to be linked by an ideal $I$ if $I\subset A,B$ and if $A=(I:B)$
and $B=(I:A)$. We will similarly say that schemes defined by $A$ and $B$ are
linked via the scheme defined by $I$

Linkage is a very general construction that is usually narrowed down by adding
assumptions on $I$. In this way one distinguishes different types of linkage.
When $I$ is a complete intersection it is the classical linkage described in
[PS85]. The most general case is when $I$ is Cohen-Macaulay and generically
Gorenstein [CMlinkage]. A composition of two linkages of the same type is
called a bilinkage. This notion has been introduced because varieties related
by an even number of linkages share more common properties with each other.

Linkages are helpful in the construction of varieties as they provide a way to
relate two varieties (given by $A$ and $B$) of which one can be easier to
construct than the other. We will consider only special types of bilinkages
and use them to construct AG rings in Section 9.

####  2.6.1. Biliaison

Building on the notion of bilinkage Hartshorne in [Har2] introduces the
notions of Gorenstein biliaison and generalised biliaison.

######  Definition 2.23.

Let $X$ and $Y$ be equidimensional closed subschemes of dimension $r$ of
$\mathbb{P}^{n}$ We say that $X$ is obtained by a biliaison of height $h$ from
$Y$ if there exists an ACM scheme $Z$ in $\mathbb{P}^{n}$, of dimension $r+1$,
containing $X$ and $Y$, and so that $X\in Y+hH$ as (generalized cf. [Har])
divisors on $Z$, where $H$ denotes the hyperplane class. When $Z$ is
arithmetically Gorenstein we call it a Gorenstein biliason (cf. [Har2]).

######  Remark 2.24.

A biliaison between $X$ and $Y$ in $Z$ can be seen as a composition of two
linkages in the following way. Let $I_{X},I_{Y},I_{Z}$ be the corresponding
ideals. Choose a hypersurface $L_{d}$ of degree $d$ containing $X$. Then
$I_{X}$ is linked to some $I_{T}$ in $I_{Z}+<L_{d}>$. Then consider a linkage
of $I_{T}$ by $I_{Z}+L_{d+h}$ for $L_{d+h}$ a hypersurface of degree $d+h$
containing $T$. The latter yields an ideal $I_{Y}^{\prime}$ with $Y^{\prime}$
a divisor in $Z$ such that $Y^{\prime}\simeq Y$. Choosing $d$ large enough we
will be able to construct $Y^{\prime}=Y$.

######  Remark 2.25.

Note that the construction of doubling in Example 2.17 is in fact done by
means of biliaison in $\Gamma$ of $P\cap\Gamma$, where $P$ is linked to
$\Gamma$ via a complete intersection. The biliaison for some choices of
complete intersections might however be of height $0$, meaning that
$P\cap\Gamma$ is already a doubling.

####  2.6.2. Unprojections

A special case of biliaison is provided by the unprojection technique of
Papadakis and Reid [PR]. The unprojection in projective space is originally
the inverse of the central projection from a point on it. So having the data
of the image of a projection and the proper image of the indeterminacy point,
we can reconstruct the original variety.

Let $A$ be a local Gorenstein ring, and $I$ an ideal defining
$X={\mathbb{V}}(I)\subset\operatorname{Spec}(A)$ a Gorenstein subscheme of
codimension $1$. As in [PR]*Lem. 1.1 we have a natural injective map
$s:I\to\omega_{A}$ to the canonical module. Choose a set of generators
$f_{1},\dots,f_{k}$ of $I$ and write $s(f_{i})=g_{i}\in\omega_{A}=A$ for the
corresponding generators of $\omega_{A}$. Then the unprojection ring (which is
Gorenstein by [PR]*Thm. 1.5) of $I$ in $A$ is

$A[s]=A[x]/(xf_{i}-g_{i}),$

where $x$ is an indeterminate; $A[s]$ is usually no longer local. This
construction of unprojection can be globalised to the projective case
[PR]*§2.4. Indeed, when $A=S/J$ with $J$ homogeneous (as in our case) we can
consider the projective scheme $\operatorname{Proj}(A)$ and the subscheme
defined by $I$. This allows us to lift local Gorenstein schemes to subvarietes
in projective space.

The unprojection of $D\subset X\subset\mathbb{P}^{N}$ can be obtained as a
Gorenstein biliaison [PS74] of the cone $D_{1}\subset\mathbb{P}^{N+1}$ (cf.
[KK1]*§5).

####  2.6.3. The Kustin-Miller construction

Another special case is the biliason from a complete intersection of
codimension 4 in a complete intersection of codimension 3. This construction
has been performed in [kustin1982structure], were Kustin and Miller use
biliaison to provide a generic family of codimension $4$ Gorenstein ideals.
This construction has also later become the first model of unprojection
studied by Papadakis and Reid (cf. [PR]*§2.8). A nice summary of Kustin-
Miller’s construction appears in [laxmi2021spinor]*§6, so our treatment here
is terse. Let $\tilde{S}={\mathbb{K}}[x_{i},a_{jk},v]$ be a polynomial ring,
where $1\leq i,j\leq 4$ and $1\leq j\leq 3$. The Kustin-Miller family is
associated to a $3\times 4$ matrix $M=\begin{pmatrix}a_{jk}\end{pmatrix}$, a
$4$-vector $\mathbf{x}=\begin{pmatrix}x_{i}\end{pmatrix}$ and a variable $v$.
Let

$\displaystyle\begin{pmatrix}q_{1}\\\ q_{2}\\\
q_{3}\end{pmatrix}=M\mathbf{x},~{}\begin{pmatrix}h_{1}\\\ h_{2}\\\ h_{3}\\\
h_{4}\end{pmatrix}=\mathbf{x}v+\wedge^{3}M.$

Then minimal free resolution for $I=\langle
q_{1},q_{2},q_{3},h_{1},h_{2},h_{3},h_{4}\rangle$ is given by

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{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{141.654pt}{-6.92305pt}\pgfsys@lineto{118.454pt}{-6.92305pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},$ $\mbox{ where
}d_{1}=\begin{bmatrix}q_{1},q_{2},q_{3},h_{1},h_{2},h_{3},h_{4}\end{bmatrix},s=\begin{bmatrix}0&I_{6}\\\
I_{6}&0\end{bmatrix}\mbox{ and }d_{2}=\begin{bmatrix}Q&\wedge^{2}M&-vI_{3}\\\
0&\mathcal{K}_{2}&M^{t}\end{bmatrix}.$

The blocks of $d_{2}$ are

$\displaystyle Q$ $\displaystyle=\begin{bmatrix}-q_{2}&-q_{3}&0\\\
q_{1}&0&-q_{3}\\\ 0&q_{1}&q_{2}\end{bmatrix},$ $\displaystyle\wedge^{2}M$
$\displaystyle=\begin{bmatrix}M_{23;34}&M_{23;24}&M_{23;23}&M_{23;14}&M_{23;13}&M_{23;12}\\\
-M_{13;34}&-M_{13;24}&-M_{13;23}&-M_{13;14}&-M_{13;13}&-M_{13;12}\\\
M_{12;34}&M_{12;24}&M_{12;23}&M_{12;14}&M_{12;13}&M_{12;12}\end{bmatrix},$
$\displaystyle\mathcal{K}_{2}$
$\displaystyle=\begin{bmatrix}-x_{2}&x_{3}&-x_{4}&0&0&0\\\
x_{1}&0&0&-x_{3}&x_{4}&0\\\ 0&-x_{1}&0&x_{2}&0&-x_{4}\\\
0&0&x_{1}&0&-x_{2}&x_{3}\end{bmatrix}.$

######  Remark 2.26.

Example 2.19 i.e. example of type [300a] is a specialization of a Kustin-
Miller construction. In this case,

$\displaystyle\mathbf{x}=\begin{pmatrix}x_{0}\\\ x_{1}\\\ x_{2}\\\
x_{3}\end{pmatrix},~{}M=\begin{bmatrix}x_{0}&0&0&-x_{3}\\\
0&x_{1}&0&-4x_{3}\\\ 0&0&x_{2}&-9x_{3}\end{bmatrix},~{}v=0.$

In particular, for these examples there is a biliaison in a complete
intersection of three quadrics relating these ideals with ideals of a
codimension $4$ complete intersection of linear forms.

To deal both with doubling and bilinkage we use the adjunction relation
between the canonical divisor on a union of varieties of the same dimension
and the canonical divisor on each component. Algebraically, this is the
following linkage relation in Gorenstein rings that we have already seen:

######  Theorem 2.27.

[CAwvtAG]*Thm. 21.23 Let $A$ be a local Gorenstein ring and $I\subset A$ an
ideal of codimension $0$. Let $B=A/I$ and $J=(0:I)$. Then $J=Hom_{A}(B,A)$. If
$B$ is Cohen-Macaulay, then $J$ is a canonical module for B. In particular,
$B$ is Gorenstein and $J$ is a canonical module for $B$ if and only if $J$ is
a principal ideal in $A$.

If $X\cup X^{\prime}$ is an arithmetic Gorenstein scheme, e.g. a complete
intersection, and the component $X$ is arithmetic Cohen-Macaulay. Then one may
apply this theorem to the homogeneous coordinate rings $A$ and $B$ of $X\cup
X^{\prime}$ and $X$, respectively.

Geometrically we formulate this in terms of sheaves:

$\omega_{X}\otimes\mathcal{O}_{X}(X\cap X^{\prime})\equiv\omega_{X\cup
X^{\prime}}\otimes\mathcal{O}_{X},$

or in terms of Cartier divisors on $X$, when $X$ is arithmetic Gorenstein

$K_{X}+X^{\prime}\cap X=K_{X\cup X^{\prime}}|_{X}.$

For generalizations to arithmetic Cohen-Macaulay varieties $Y$ and $X$ that
are Gorenstein in codimension $0$, and generalized divisors see [Har, Mar]).
We do not use Theorem 2.27 in any proofs, but both in algebraic and in
geometric formulation it is an important guide in constructions of reducible
varieties with trivial canonical bundle: Each component intersects the union
of the others in an anticanonical divisor.

###  2.7. AG varieties of codimension $3$ in quadrics

In Section 7 we construct codimension $4$ AG varieties as varieties of
codimension $3$ in a smooth quadric. On the quadric these varieties are
Pfaffian or Lagrangian degeneracy loci. This allows us to lift Artinian
Gorenstein rings of codimension and regularity $4$ to Calabi-Yau varieties in
quadrics.

For example recall that Reid constructed codimension $4$ subvarieties by “Tom
and Jerry” unprojection introduced [Reid]. The unprojection data are a
codimension $3$ scheme $Y$ defined by a $5\times 5$ Pfaffian ideal, containing
a codimension $4$ complete intersection. This data is strongly related to the
lifting of a del Pezzo surface of degree $6$ to the Fano threefold
$\mathbb{P}^{2}\times\mathbb{P}^{2}$ or
$\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$.

In Section 7, we interpret these constructions by describing embeddings of
$\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$ and
$\mathbb{P}^{2}\times\mathbb{P}^{2}$ in a $6$-dimensional quadric in
$\mathbb{P}^{7}$. We use the Pfaffian and Lagrangian degeneration
constructions [EPW] involving spinor bundles on quadrics.

Recall from [Ot] that a spinor bundle on a smooth quadric $Q_{n}$ is obtained
as a pullback of the universal bundle by the natural maps

$Q_{2n+1}\to G(2^{k},2^{k+1}),$

and two maps

$Q_{2n}\to G(2^{k-1},2^{k})$

defined by maximal isotropic linear subspaces of the quadrics in [Ot]*Cor.
1.2. On a $(2n-1)$-dimensional quadric there is one spinor bundle, of rank
$(2n-1)$, while for a $2n$-dimensional quadric there are two spinor bundles of
rank $(2n-1)$. On $Q_{1}=\mathbb{P}^{1}$ it is $\mathcal{O}(1)$. On
$Q_{2}=\mathbb{P}^{1}\times\mathbb{P}^{1}$ they are $\mathcal{O}(1,0)$ and
$\mathcal{O}(0,1)$. On $Q_{3}$, which is the Lagrangian Grassmannian
$LG(2,4)$, it is the tautological quotient bundle. On $Q_{4}=G(2,4)$ they are
the tautological quotient bundle and the dual of the tautological subbundle.

##  3\. Apolar curves and Betti tables of point sets

To determine the rank and the variety of minimal power sum presentations of a
quaternary quartic $F$, we will compare the Betti numbers of $A_{F}$ and the
Betti numbers of possible apolar sets of points. If a form $F$ is apolar to a
curve of small degree, then the rank of $F$ is often computed by a set of
points on this curve.

######  Lemma 3.1.

Assume a quaternary quartic form $F$ is apolar to a curve $C$.

  1. (1)

If $C$ is a line and $F$ is not apolar to any scheme of length less than $3$,
then $F$ has rank $3$ and $VSP(F,3)={\mathbb{P}}^{1}$.

  2. (2)

Let $C$ be a conic or two intersecting lines and assume $F$ is not apolar to
any scheme of length less than $5$. Then $F$ has cactus rank $5$, and if $C$
is smooth, $F$ has rank $5$ and $VSP(F,5)={\mathbb{P}}^{1}$.

  3. (3)

Let $C$ be a twisted cubic or a reduced curve defined by a determintal net of
quadrics and assume $F$ is not apolar to any scheme of length less than $7$.
Then $F$ has cactus rank $7$, and if $C$ is smooth, then $F$ has rank $7$ and
$VSP(F,7)={\mathbb{P}}^{1}$.

  4. (4)

Assume that $C$ is an elliptic quartic curve or any reduced complete
intersection of two quadrics, and that $F$ is not apolar to any scheme of
length less than $8$. Then $F$ is apolar to a scheme $Z$ of length $8$ on $C$.

    1. (a)

If $C$ is smooth and $2Z\in C$ is a quartic section of $C$, then $Z$ is the
unique apolar scheme of length $8$ to $C$.

    2. (b)

If $C$ is smooth and $2Z\in C$ is not a quartic section of $C$, then $F$ is
apolar to exactly two schemes of length $8$.

###### Proof.

When the quartic form $F$ is apolar to a curve $C$, the rank of $F$ is the
minimal cardinality of point sets on $C$ apolar to $F$. When $C$ is rational
of degree $d$, then the restriction $F_{C}$ is a binary form of degree $4d$.
Any binary form of degree $4d$ is apolar to a scheme of length $\leq 2d+1$,
and if it is not apolar to a scheme of length strictly less, then it is apolar
to a pencil of schemes of rank $2d+1$ of which the general is smooth, cf.
[Sylvester, RS]. Furthermore, since the $(2d-1)$-secant variety of $C$ is a
hypersurface, in a pencil of forms $F$ there is at least one form $F_{0}$ of
cactus rank $\leq 2d$.

Let $C$ be a reducible curve $C=C_{1}\cup C_{2}$, whose components intersect
in one point $p$ and assume $F$ is apolar to $C$, then $F$ has a pencil of
decompositions $F=F_{1}+F_{2}$, where $F_{1}$ is apolar to $C_{1}$ and $F_{2}$
is apolar to $C_{2}$. So if $C$ is a reducible conic or twisted cubic, we may
assume $C_{1}$ is a line, and choose $F_{1}$ to have cactus rank $2$. In the
conic case, we conclude that $F_{2}$ has rank $3$, and therefore $F$ has
cactus rank $5$. In the twisted cubic case, we conclude similarly that $F_{2}$
has rank $5$ if $C_{2}$ is irreducible. If $C_{2}$ is reducible, then $F_{2}$
is apolar to a reducible conic and therefore, as above, of cactus rank $7$.
Summing up, we see that $F$ has cactus rank $9$ also in this reducible case.

In the fourth case of the lemma, the curve $C$ is a complete intersection of
two quadric surfaces. When $C$ is nonsingular, it is an elliptic quartic
curve, and the restriction $F_{C}$ is a form on the sections of a line bundle
${\mathcal{O}}_{C}(D)$ of degree $16$ on $C$. The linear system $|D|$ embeds
$C\subset{\mathbb{P}}(H^{0}({\mathcal{O}}_{C}(D))^{*})$, and $F_{C}$, up to
scalar, is a point in this ambient space.

The line bundle ${\mathcal{O}}_{C}(D)$ decomposes into line bundles
${\mathcal{O}}_{C}(B)\otimes{\mathcal{O}}_{C}(D-B)$ where $B$ has degree $8$.
Each decomposition defines a determinantal hypersurface in
${\mathbb{P}}^{15}={\mathbb{P}}(H^{0}({\mathcal{O}}_{C}(D))^{*})$ defined by
an $8\times 8$-determinant with linear entries, such that each row vanishes on
a divisor in $|B|$ and each column vanishes on a divisor in $|D-B|$, cf.
[Fisher]*Thm. 1.3, Lemma 2.9 and [Room]. The pencil of decompositions
generates a pencil of hypersurfaces, one through each point in
${\mathbb{P}}^{15}$ outside the intersection of the hypersurfaces. The latter
is the variety of $7$-secant ${\mathbb{P}}^{6}$’s to $C$. A general point on a
determinantal hypersurface lies in the spans of a unique divisor in $|B|$ and
in $|D-B|$ respectively. So $2B=D$ and the two divisors coincide.

If $C$ is an irreducible but singular complete intersection of degree $4$,
then $C$ is the projection of a rational normal quartic curve. In the
$4$-tuple embedding $v_{4}(C)$, it is a projection of a rational normal curve
of degree $16$, from a point on a secant. The $7$-secant variety to this
rational normal curve is a hypersurface that maps onto the linear span of
$v_{4}(C)$, so $F$ must lie in a $7$-secant, and therefore have cactus rank at
most $8$. When $C$ is a reducible complete intersection of two quadrics, then
the components are rational and any component intersect the unions of residual
components in a scheme of length $2$. So we may assume the $C=C_{1}\cup
C_{2}$, where $C_{1}$, and $C_{2}$ are possibly reducible rational curves of
degree $d_{1}$ and $d_{2}$ with $d_{1}+d_{2}=4$. The form $F$ now has a net of
decompositions $F=F_{1}+F_{2}$ such that the $F_{i}$ are apolar to $C_{i}$.
Now there is, as above, a finite set of decompostions such that $F_{1}$ has
cactus rank $2d_{1}$ and $F_{2}$ has cactus rank $2d_{2}$. We conclude that
$F$ has cactus rank $2(d_{1}+d_{2})=8$. ∎

In Section 5 we will compute the rank of a quaternary quartic form $F$ under
the assumption that it is general among the forms apolar to the space of
quadrics in $F^{\perp}$. The rank will be the cardinality of a set of points
$\Gamma$ contained in the locus defined by the quadrics in $F^{\perp}$.

When the quadrics in $F^{\perp}$ generate the ideal of a connected curve, this
curve is one of the curves covered by Lemma 3.1 and $\Gamma$ is a set of
points as described in that lemma. Otherwise, the set of points $\Gamma$ will
satisfy certain generality conditions as listed in the next proposition.

| $\begin{matrix}&&&&\cr 1&-&-&-&\cr-&6&8&3&\cr\cr\end{matrix}$
---
1) four points that span ${\mathbb{P}}^{3}$
| $\begin{matrix}&&&&\cr 1&-&-&-&\cr-&5&5&-&\cr-&-&-&1&\cr\cr\end{matrix}$
---
2) five points, no
four in a plane
| $\begin{matrix}&&&&\cr 1&-&-&-&\cr-&5&5&1&\cr-&-&1&1&\cr\cr\end{matrix}$
---
3) five points, four in a
plane, no three collinear
| $\begin{matrix}&&&&\cr 1&-&-&-&\cr-&5&6&2&\cr-&1&2&1&\cr\cr\end{matrix}$
---
4) five points,
three in a line
| $\begin{matrix}&&&&\cr 1&-&-&-&\cr-&4&2&-&\cr-&-&3&2&\cr\cr\end{matrix}$
---
5) six points,
no five in a plane
| $\begin{matrix}&&&&\cr 1&-&-&-&\cr-&4&3&-&\cr-&1&3&2&\cr\cr\end{matrix}$
---
6) six points,
three in a line
| $\begin{matrix}&&&&\cr 1&-&-&-&\cr-&4&4&1&\cr-&2&4&2&\cr\cr\end{matrix}$
---
7a) six points, five in a plane, or
7b) three on each of two skew lines
| $\begin{matrix}&&&&\cr 1&-&-&-&\cr-&3&-&-&\cr-&1&6&3&\cr\cr\end{matrix}$
---
8a) seven points in a CI, or
8b) seven points, three collinear
| $\begin{matrix}&&&&\cr 1&-&-&-&\cr-&3&1&-&\cr-&2&6&3&\cr\cr\end{matrix}$
---
9) seven points,
five in a plane
| $\begin{matrix}&&&&\cr 1&-&-&-&\cr-&3&2&-&\cr-&3&6&3&\cr\cr\end{matrix}$
---
10) seven points on a
determinantal cubic curve
| $\begin{matrix}&&&&\cr 1&-&-&-&\cr-&3&3&1&\cr-&4&7&3&\cr\cr\end{matrix}$
---
11) seven points,
six in a plane
| $\begin{matrix}&&&&\cr
1&-&-&-&\cr-&3&-&-&\cr-&-&3&-&\cr-&-&-&1&\cr\end{matrix}$
---
12) CI of eight points
| $\begin{matrix}&&&&\cr 1&-&-&-&\cr-&2&-&-&\cr-&4&9&4&\cr\cr\end{matrix}$
---
13) eight points, no six
in a plane, not a CI
| $\begin{matrix}&&&&\cr 1&-&-&-&\cr-&2&1&-&\cr-&5&9&4&\cr\cr\end{matrix}$
---
14) eight points, six in a plane, or
five in a plane and three in a line
| $\begin{matrix}&&&&\cr 1&-&-&-&\cr&1&-&-&\cr-&7&12&5&\cr\cr\end{matrix}$
---
15) nine points on
a unique quadric

Table 1. Betti tables $B_{\Gamma}$ of certain point sets
$\Gamma\subset{\mathbb{P}}^{3}$
CI denotes complete intersection

######  Proposition 3.2.

Let $\Gamma\subset{\mathbb{P}}^{3}$ be a set of $n$ distinct points, $4\leq
n\leq 9$, that span ${\mathbb{P}}^{3}$. Assume furthermore that no subset of
four of the points lie on a line, no subset of six lie in a smooth or singular
conic, no subset of seven lie in a plane, no subset of eight lie in a
determinantal net of quadrics, and no subset of nine lie on two quadrics. Then
$S/I_{\Gamma}$ has one of the Betti tables in Table 1. For each table, we
describe a set of points which realizes the table.

###### Proof.

The proof has two parts. We first show that a configuration of reduced points
satisfying the hypotheses has regularity $2$, with the obvious exception of
#12. In the second part, we show that only the Betti tables in Table 1
actually appear.

First part: since $\Gamma$ is Cohen-Macaulay, the Betti table is preserved for
the Artinian reduction, the quotient of the coordinate ring of $\Gamma$ by a
general linear form. The degree of $\Gamma$ is the degree of the Artininan
reduction $A_{\Gamma}$ of $S/I_{\Gamma}$. Since $Ext^{i}(A_{\Gamma},S)\neq 0$
only at the last position, the regularity is attained at the last step of the
free resolution; if the largest shift at the last position is $S(-a)$, then
the regularity is $a-3$. On the other hand, the largest shift contributes a
$t^{a}$ in the numerator of the Hilbert series, which is divisible by
$(1-t)^{3}$. Indeed, for any finite length quotient of a polynomial ring, the
regularity is attained at the last step of the resolution, so is equal to the
socle degree.

We write $\Delta H$ for the Hilbert function of the Artinian reduction
$A_{\Gamma}$. As $\Gamma$ is Cohen-Macaulay and spans ${\mathbb{P}}^{3}$,
$\Delta H$ begins $(1,3)$. With our hypotheses, the possible values for
$\Delta H$ are listed in table 2.

$\deg=4$ | $\deg=5$ | $\deg=6$ | $\deg=7$ | $\deg=8$ | $\deg=9$
---|---|---|---|---|---
$(1,3)$ | $(1,3,1)$ | $(1,3,2)$ | $(1,3,3)$ | $(1,3,4)$ | $(1,3,5)$
|  | $(1,3,1,1)$ | $(1,3,2,1)$ | $(1,3,3,1)$ | $(1,3,4,1)$
|  |  | $(1,3,1,1,1)$ | $(1,3,2,2)$ | $(1,3,3,2)$
|  |  |  | $(1,3,2,1,1)$ | $(1,3,3,1,1)$
|  |  |  | $(1,3,1,1,1,1)$ | $(1,3,2,2,1)$
|  |  |  |  | $(1,3,2,1,1,1)$
|  |  |  |  | $(1,3,1,1,1,1,1)$
Table 2. Hilbert function $\Delta H$ of Artinian reductions

To see that these are the possible values for $\Delta H$, note that if
$h_{2}\in\\{1,2\\}$, Macaulay growth says all higher $h_{i}$ have to be
(respectively) $\leq 1$ or $2$. If $h_{2}\in\\{3,4\\}$, there is no room due
to degree constraints for any increase: If $h_{2}=4$, then
$\deg(A_{\Gamma})\geq 8$, so the only possibilities if $h_{2}=4$ are degree
$8$ and $\Delta H=(1,3,4)$, or degree $9$ and $\Delta H=(1,3,4,1)$. If
$h_{2}=3$, then $\deg(A_{\Gamma})\geq 7$. If $\deg(A_{\Gamma})=8$ then $\Delta
H=(1,3,3,1)$, and if $\deg(A_{\Gamma})=9$ then $\Delta H=(1,3,3,2)$ or
$(1,3,3,1,1)$. We now bring Theorem 3.3 and 3.6 of Bigatti-Geramita-Migliore
[BGM] into play. Theorem 3.3 of [BGM]: Suppose $Y$ is a closed, nonempty
reduced subscheme of ${\mathbb{P}}^{r+1}$, and for some $m$ and two
consecutive values $d,d+1\in{\mathbb{Z}}$ the Artinian reduction $\Delta H$
satisfies

$\Delta H(Y,d)=\binom{d+m-1}{d}\mbox{ and }\Delta H(Y,d+1)=\binom{d+m}{d+1}.$

Then $Y$ is the disjoint union of a scheme $Y_{1}$ and a scheme $Y_{2}$, where
$Y_{1}$ lies in a ${\mathbb{P}}^{m}$ and $Y_{2}$ is a finite set of points.
Furthermore

$\Delta H(Y_{1},t)=\Delta H({\mathbb{P}}^{m},t)\mbox{ if }t\leq d+1,\mbox{ and
}\Delta H(Y,t)\mbox{ if }t\geq d.$

Theorem 3.6 of [BGM]: Suppose $Y$ is a reduced scheme in ${\mathbb{P}}^{r+1}$,
and for some $d$,

$\Delta H(Y,d)=\Delta H(Y,d+1)=s,\mbox{ where }d\geq s.$

Then it must be the case that $\dim(Y)\leq 1$, and $(I_{Y})_{\leq d}$ is the
saturated ideal of a reduced curve of degree $s$ (not necessarily unmixed)

Applying Theorem 3.3 to any of the $\Delta H$ above which have two consecutive
values $(1,1)$ means that there is a subscheme lying on a line containing $k$
points, where $k$ is the position of the last value $1$. So for example,
$(1,3,1,1)$ forces $\Gamma$ to contain four collinear points, and
$(1,3,1,1,1,1,1)$ forces $\Gamma$ to contain seven collinear points. Hence the
hypothesis that $\Gamma$ does not contain four collinear points means that we
have reduced the possible Hilbert functions $\Delta H$ for an Artinian
reduction of nine or fewer reduced points which span ${\mathbb{P}}^{3}$ to the
set

$\begin{array}[]{cccccc}(1,3)&(1,3,1)&(1,3,2)&(1,3,3)&(1,3,4)&(1,3,5)\\\
(1,3,2,1)&(1,3,3,1)&(1,3,2,2)&(1,3,4,1)&(1,3,3,2)&(1,3,2,2,1).\end{array}$

By Theorem 3.6 of [BGM], if $\Delta H$ has consecutive values $(2,2)$, then
the quadratic component of $I_{\Gamma}$ lies on a curve of degree $2$. So the
hypothesis that $\Gamma$ has no subset of six points in a plane conic (smooth
or singular) rules out $(1,3,2,2)$ and $(1,3,2,2,1)$. It remains to deal with
the cases $(1,3,2,1)$, which has degree $7$, and $(1,3,4,1),(1,3,3,2)$, which
are of degree $9$.

$\Delta H=(1,3,3,2)$:

The Hilbert function of $\Gamma$ is $(1,4,7,9,9,\cdots)$, so
$\dim(I_{\Gamma})_{2}=3$. The hypothesis that no nine points lie on an
intersection of two quadrics precludes this case.

$\Delta H=(1,3,4,1)$:

The Hilbert function of $\Gamma$ is $(1,4,8,9,9,\cdots)$, so
$\dim(I_{\Gamma})_{2}=2$. The hypothesis that no nine points lie on an
intersection of two quadrics precludes this case.

$\Delta H=(1,3,2,1)$:

The seven points have Hilbert function $(1,4,6,7,7,\ldots)$, so
$\dim(I_{\Gamma})_{2}=4$. In particular, the points do not impose independent
conditions on quadrics. Checking the value of $h_{3}$ shows there must be at
least three linear syzygies on the quadrics. The codimension of $I_{2}$ cannot
be $1$, for then $I_{2}=L\cdot(x,y,z,w)$ and $I_{\Gamma}$ is not saturated. So
$\operatorname{codim}(I_{2})\geq 2$, and two of the four quadrics yield a CI.
If the codimension is exactly $2$, then the remaining quadrics drop the degree
from $4$ down to at most $2$, yielding a plane conic or a line, which are
ruled out by our hypotheses. If the $\operatorname{codim}(I_{2})=3$, then
three of the quadrics are a CI. Since $h_{3}=7$, there are at least three
linear syzygies on $I_{2}$. Let $Q$ denote the $4^{th}$ quadric and $J$ the
three quadrics yielding a CI. Then $J:Q$ contains three linear forms, so the
Betti table for the mapping cone of $I_{2}$ is

$\begin{matrix}&&&&&\cr
1&-&-&-&-&\cr-&4&3&3&1\cr-&-&3&-&-\cr-&-&-&1&-\cr\cr\end{matrix}$.

This ideal is not saturated, so this case cannot arise with our hypotheses.

We have shown that a set of points $\Gamma\subseteq{\mathbb{P}}^{3}$
satisfying the hypotheses of the lemma either has $\Delta H=(1,3,3,1)$, or has
Artinian reduction with Hilbert function in the set

(16) $\\{(1,3),(1,3,1),(1,3,2),(1,3,3),(1,3,4),(1,3,5)\\}.$

We now tackle the second part of the proof: showing that the Betti tables in
Table 1 are the only ones that occur. In what follows, we restrict our
attention to the regularity $2$ cases, and deal with $\Delta H=(1,3,3,1)$ at
the end. By Equation 16, the Hilbert series of the Artinian reductions are all
of the form

$1+3t+h_{2}t^{2},\mbox{ with }h_{2}\in\\{0,1,2,3,4,5\\}.$

As our reductions are all of regularity $2$, we can also read off the Hilbert
series from the free resolutions, yielding

$\frac{1-b_{12}t^{2}-b_{13}t^{3}+b_{23}t^{3}+b_{24}t^{4}-b_{34}t^{4}-b_{35}t^{5}}{(1-t)^{3}}.$

Comparing, we obtain the equalities

$\begin{array}[]{ccc}6-b_{12}&=&h_{2}\\\ 8-b_{23}+b_{13}&=&3h_{2}\\\
3-b_{34}+b_{24}&=&3h_{2}\\\ b_{35}&=&h_{2}\;.\end{array}$

Therefore the Betti tables of the Artinian reduction $A_{\Gamma}$ are of the
form

$\begin{matrix}&&&&\cr
1&-&-&-\cr-&6-h_{2}&b_{23}&b_{34}\cr-&3h_{2}-8+b_{23}&3h_{2}-3+b_{34}&h_{2}\cr\cr\end{matrix}$.

We now examine the various possibilities for $h_{2}$. If
$h_{2}\in\\{0,3,4,5\\}$ the analysis is straightforward. For
$h_{2}\in\\{1,2\\}$, we will apply Macaulay’s theorem to the degree $2$ piece
$I_{2}$ of the ideal of the Artinian reduction.

Case $h_{2}\in\\{0,3,4,5\\}$:

For $h_{2}=0$ the regularity of $A_{\Gamma}$ is $1$ and so the Hilbert
function determines the Betti table. For $h_{2}\in\\{4,5\\}$ the analysis is
also straightforward: since $\dim(I_{\Gamma})_{2}\leq 2$,
$b_{23}\in\\{0,1\\}$. For $h_{2}=3$ we find $b_{23}\leq 3$, with $b_{23}=3$
only if the ideal of the Artinian reduction is of the form $L\cdot(x,y,z)$
with $L$ a linear form.

Case $h_{2}=2$:

We have $h^{\langle 2\rangle}_{2}=2$, and therefore

$h_{3}(I_{2})=10-4\cdot 3+b_{23}\leq 2,\mbox{ so }b_{23}\leq 4.$

From the Betti table, $b_{13}\geq 0$ forces $b_{23}\geq 2$, so for $h_{2}=2$,

$b_{23}\in\\{2,3,4\\}$

When $b_{23}=2$, there cannot be a second linear syzygy (it would have rank 2,
which is impossible), so the top row of the Betti table must be $(4,2,0)$.
When $b_{23}\in\\{3,4\\}$, we consider the generic initial ideal (GIN). The
GIN must be of the form (see Appendix B).

$\langle x^{2},xy,y^{2},xz,yz^{2},z^{3}\rangle,$

which has Betti table

$\begin{matrix}1&-&-&-&\cr-&4&4&1&\cr-&2&4&2&\cr\cr\end{matrix}$ .

The Betti table of $I$ must be a subtable of this. We know
$b_{23}\in\\{2,3,4\\}$, and we have dealt with the $b_{23}=2$ case above. To
rule out top row $(4,3,1)$ note that the linear second syzygy would have to
have rank $3$, so would force $I_{2}$ to contain $L\cdot(x,y,z)$. Hence the
three linear first syzygies are on the ideal $J=L\cdot(x,y,z)$, forcing the
fourth quadric $q$ to be a nonzero divisor on $J$, yielding a mapping cone
resolution with Betti table

$\begin{matrix}&&&&&\cr 1&-&-&-&-\cr-&4&3&1&-\cr-&-&3&3&1\cr\cr\end{matrix}$,

which is clearly impossible.

To rule out at top row of $(4,4,0)$, we continue to consider the rank of the
linear syzygies; as we are in three variables the rank must be either $2$ or
$3$. First, suppose we have a rank $3$ first syzygy, which after a change of
variables is $(x,y,z)$. Considering the quadrics as a syzygy on this, we see
that three quadrics are combinations of the three Koszul syzygies on
$(x,y,z)$, which makes them the $2\times 2$ of a $2\times 3$ matrix; in
particular one first syzygy of rank $3$ forces a second. Let $J$ be the ideal
of the $2\times 2$ minors, and $Q$ the $4^{th}$ quadric. Since
$\operatorname{codim}(J)=2$, this means $Q$ is a nonzero divisor on $J$, so
the resolution is a mapping cone, which means there are only two linear first
syzygies, a contradiction. Thus, all first syzygies have rank $2$, so each
linear first syzygy involves only two generators of $I_{2}$, of the form
$L\cdot(l_{1},l_{2})$. WLOG choose $L_{1}\cdot(l_{1},l_{2})$ for the first two
generators, and $L_{2}\cdot(l_{3},l_{4})$ as the second two generators,
yielding the syzygies

$[l_{2},-l_{1},0,0]^{t},[0,0,l_{4},-l_{3}]^{t}.$

But now there are two additional rank $2$ syzygies among the four generators,
forcing the ideal to have the form $\langle ac,ad,bc,bd\rangle$. This has a
linear second syzygy, so a top row $(4,4,0)$ is impossible. This completes the
analysis for $h_{2}=2$.

Case $h_{2}=1$:

We apply Macaulay’s theorem to $I_{2}$, and find that $b_{23}\in\\{5,6\\}$. In
this case, the GIN must be $\langle x^{2},xy,y^{2},xz,yz,z^{3}\rangle$, which
has Betti table

$\begin{matrix}1&-&-&-&\cr-&5&6&2\cr-&1&2&1\cr\cr\end{matrix}$.

Now, we know $b_{23}\in\\{5,6\\}$. If $b_{23}=6$, we need to rule out
$b_{34}\in\\{0,1\\}.$ If $b_{34}=0$, then $b_{24}=0$, which means the cubic
generator is not in any syzygy, hence a nonzero divisor on $I_{2}$. Since
$\operatorname{codim}(I_{2})=3$, this is impossible. Next, if $b_{34}=1$, this
means there is a single linear second syzygy on $I_{2}$, necessarily (see
earlier remarks) of rank $3$. Let $C$ be the unique cubic, and consider the
mapping cone resolution of $I_{2}+C$. The linear syzygy represented by
$b_{24}=1$ must involve $C$ (otherwise we are in the situation above), so in
$I_{2}:C$ we have one linear form. However, $I_{2}$ is also contained in
$I_{2}:C$, so in the mapping cone, the resolution of $S(-3)/I_{2}:C$ begins

$S(-4)\oplus S(-5)^{a}\rightarrow S(-3)\rightarrow S(-3)/I_{2}:C\rightarrow
0,\mbox{ with }a\geq 1.$

There can be no cancellation in the top row, and so this yields a resolution
for $S/I_{2}+C$ which is inconsistent with the Betti table. Our final case is
when $b_{23}=5$. We know there is a cancellation of $b_{13}$, so the Betti
table is a subset of

$\begin{matrix}&&&&\cr 1&-&-&-&\cr-&5&5&2\cr-&0&2&1\cr\cr\end{matrix}$.

The cases $b_{34}=b_{24}\in\\{0,1\\}$ occur, so we just need to rule out the
case $b_{34}=2$. As before, the rank of the linear second syzygies must be
$3$. Therefore, $L\cdot(x,y,z)\in I_{2}$, so let $I=\langle
Lx,Ly,Lz,Q_{1},Q_{2}\rangle$. As $\operatorname{codim}(L,Q_{1},Q_{2})=3$,
$\Gamma$ contains a $(2,2)$ CI on the plane ${\mathbb{V}}(L)$.

Therefore the fifth point of $\Gamma$ must be $(0,0,0,1)$, and
$Q_{1},Q_{2}\in\langle x,y,z\rangle$. Writing $Q_{1}=ax+by+cz$, we see there
is a linear syzygy $aLx+bLy+cLz-LQ_{1}=0$, and similarly for $Q_{2}=dx+ey+fz$.
Hence the matrix of first syzygies on $I$ is

$\displaystyle\begin{bmatrix}y&z&0&a&d\\\ -x&0&z&b&e\\\ 0&-x&-y&c&f\\\
0&0&0&-L&0\\\ 0&0&0&0&-L\\\ \end{bmatrix}.$

A linear second syzygy involves only the first three columns, so $b_{34}=2$ is
impossible.

We close by analyzing the situation when $\Delta H=(1,3,3,1)$. The Hilbert
function of $\Gamma$ is $(1,4,7,8,8,\ldots)$, so $\dim(I_{\Gamma})_{2}=3$. The
arguments above show $b_{23}\leq 3$; if $b_{23}=3$ then
$(I_{\Gamma})_{2}=L\cdot(x,y,z)$ and there are seven points on a
${\mathbb{P}}^{2}$, which is precluded by our hypotheses.

When $b_{23}=2$, if there is a rank $3$ first syzygy, then as in the analysis
above, $(I_{\Gamma})_{2}$ is the $2\times 2$ minors of a $2\times 3$ matrix.
If both first syzygies are of rank $2$, then $(I_{\Gamma})_{2}=\langle
l_{1}l_{2},l_{1}l_{3},l_{2}l_{3}\rangle$ and again $(I_{\Gamma})_{2}$ is the
$2\times 2$ minors of a $2\times 3$ matrix. Both these cases are precluded by
the hypothesis that there are no eight points on a determinantal net. So if
$\Delta H=(1,3,3,1)$, our hypotheses imply that $I_{\Gamma}$ is a $(2,2,2)$
CI, which concludes the proof. ∎

##  4\. The quadratic part of the apolar ideal

Consider a quartic form $F$ (in four variables, as usual), such that
$F^{\perp}$ contains no linear forms. We write $Q_{F}$ for the ideal generated
by the quadrics in $F^{\perp}$; recall that $A_{F}$ denotes $S/F^{\perp}$. In
this section, we describe the possible Betti tables of $Q_{F}$ over all such
$F$, and show that they are the ideal of a scheme $Z$ apolar to $F$. In
particular, the ideals $Q_{F}$ are all saturated.

###  4.1. Syzygy schemes and preparatory lemmas

######  Lemma 4.1.

Let $F$ be a quartic form and assume $J\subseteq F^{\perp}$ is an ideal
generated by quadrics. If $\operatorname{reg}S/J<4$, then the saturation
$J^{\operatorname{sat}}=(J:\mathfrak{m}^{\infty})$ is also in $F^{\perp}$.

###### Proof.

The $0$-th local cohomology $H^{0}_{\mathfrak{m}}(S/J)$ is isomorphic to
$J^{\operatorname{sat}}/J$. By [GeomSyz]*Theorem 4.3,
$\operatorname{reg}S/J<4$ implies $H^{0}_{\mathfrak{m}}(S/J)$ vanishes at
degree $4$, which means $J^{\operatorname{sat}}_{4}=J_{4}$. Suppose that $g\in
J^{\operatorname{sat}}$, but not in $J$; since
$J^{\operatorname{sat}}_{4}=J_{4}$, $\deg(g)\in\\{1,2,3\\}$. First consider
$\deg(g)=3$. Since $J^{\operatorname{sat}}_{4}=J_{4}$, $\mathfrak{m}\cdot g\in
J_{4}$ so $(\mathfrak{m}\cdot g)\circ F=0$. If $g\circ F$ is nonzero, then it
is a linear form, so $\mathfrak{m}$ cannot annihilate it. Hence we must have
$g\circ F=0$, so $g\in F^{\perp}$. The cases for $\deg(g)\in\\{1,2\\}$ work
the same way, hence for $g\in J^{\operatorname{sat}},g\circ F=0.$ ∎

We say $J$ is saturated at degree $4$, if $J_{4}^{\operatorname{sat}}=J_{4}$,
as in the lemma. Now we assume $J=Q_{F}$ is the quadratic part of $F^{\perp}$,
hence $b_{i,i+1}(S/Q_{F})=b_{i,i+1}(A_{F})$ for all $i$. For the $16$
Gorenstein Betti tables of $A_{F}$, we now conduct a case-by-case analysis of
the possible Betti tables for $S/Q_{F}$. From [SSY] we know that the numbers
$b_{i,i+1}(A_{F})$ determine the entire Betti table of $A_{F}$, and that
$0\leq b_{12}\leq 6$; hence if $b_{12}\in\\{0,1\\}$ the Betti table is fixed.
In the upcoming analysis, Lemma 4.1 will allow us to rule out certain
possibilities.

######  Lemma 4.2.

The Betti tables below cannot occur for $S/Q_{F}$ when $F$ is a quaternary
quartic:

$\begin{matrix}1&-&-&-&-\cr-&3&-&-&-\cr-&-&5&4&1\cr\end{matrix}$

and

$\begin{matrix}1&-&-&-&-&\cr-&4&2&-&-&\cr-&-&4&4&1&\cr\end{matrix}.$

###### Proof.

The Hilbert polynomial of $S/Q_{F}$ for the first Betti table is $2t+2$, and
there are two possibilities in the Hilbert scheme: the ideal of two skew
lines/double line in a smooth quadric, or the union of a plane conic and a
(possibly embedded) point. In the second case, any net of quadrics in the
ideal would contain a pencil of reducible ones, and hence a linear syzygy. In
the first case, after a change of variables, the ideal is either of the form
$\langle x_{0}x_{2},x_{0}x_{3},x_{1}x_{2},x_{1}x_{3}\rangle$ or $\langle
x_{0}^{2},x_{1}x_{1},x_{1}^{2},x_{0}l_{0}(x_{2},x_{3})+x_{1}l_{1}(x_{2},x_{3})\rangle$
with $l_{i}$ independent. In both cases only quadrics are needed to saturate
the ideal; by Lemma 4.1 this cannot occur.

For the second Betti table the Hilbert polynomial of $S/Q_{F}$ is $t+3$. The
only possibility is a line and two (possibly embedded) points, and in this
case the saturation is generated by five quadrics, so again by Lemma 4.1 the
corresponding Betti table cannot occur for $S/Q_{F}$. ∎

In our investigation of $Q_{F}$ we will make extensive use of the theory of
ideals generated by quadrics and how they are determined by their syzygies.
Let us recall some definitions and basic results from [vB07a]. Let $f$ be a
p-th linear syzygy of an ideal $I\subset\operatorname{Sym}(V)$ generated by
quadrics, so that $f\in\operatorname{Tor}_{p+1}^{S}(S/I,{\mathbb{C}})_{p+2}$.
We define $I_{f}$ to be the subideal of $I$ generated by those elements of $I$
which are involved in $f$ (see [vB07a]*Def.2.2 for the formal definition).
Following [vB07a], we define $G$ to be the vector space of $(p-1)$-st syzygies
involved in $f$; note that these syzygies are all linear, and the dimension of
$G$ is equal to the rank of $f$. Finally, for a vector space $G$ we define the
$k$-th generic syzygy ideal ${\operatorname{Gensyzid}_{k}(G)}$ of $G$ to be
the quadratic ideal generated by the image of the natural inclusion

$\wedge^{k+1}G^{*}\to
G^{*}\otimes\wedge^{k}G^{*}\subset\operatorname{Sym}_{2}(G^{*}\oplus\wedge^{k}G^{*})$

in $\operatorname{Sym}(G^{*}\oplus\wedge^{k}G^{*})$.

In this notation, if $f$ is a p-th syzygy of rank $p+k+1$ in $I$ and $G$ the
space of $(p-1)$-st syzygies involved in $f$, then (cf. the proof of
[vB07a]*Thm 3.4), the ideal $I_{f}$ is obtained as the image of the generic
k-th syzygy ideal of $G$ under a natural map
$\varphi:G^{*}\oplus\wedge^{k}G^{*}\to V$, which in particular maps the
elements of the $G^{*}$ component to their corresponding linear forms given by
the syzygy $f$. More precisely, we have the following diagram:

${{\operatorname{Gensyzid}_{k}(G)}}$${{I_{f}}}$${\operatorname{Sym}(G^{*}\oplus\wedge^{k}G^{*})}$${\operatorname{Sym}(V)}$$\scriptstyle{\operatorname{Sym}(\varphi)}$

In this way, the scheme defined by $I_{f}$ appears as a cone over a linear
section of the scheme defined by the generic syzygy ideal. From this point of
view, to describe $I_{f}$, it is good to know what are the generic k-th syzygy
ideals for a vector space $G$. For the following, see [vB07a] and references
therein:

  1. (1)

the 0-th generic syzygy ideal of $G$ is the ideal of a union of a hyperplane
and a point in $\mathbb{P}(G^{*}\oplus\mathbb{C})$.

  2. (2)

the 1-st generic syzygy ideal of $G$ is the ideal of
$\mathbb{P}^{1}\times\mathbb{P}(G^{*})\subset\mathbb{P}(G^{*}\oplus\mathbb{G}^{*})$
i.e. it is given by $2\times 2$ minors of a $2\times\dim G$ matrix of
coordinates.

  3. (3)

the 2-nd generic syzygy ideal of $G$ is the ideal of the union of a
Grassmannian $G(G^{*}\oplus\mathbb{C},2)$ and a linear space
$\mathbb{P}(\wedge^{2}G^{*})$ in $\mathbb{P}(G^{*}\oplus\wedge^{2}G^{*})$.

From the above we deduce the following.

######  Proposition 4.3.

Let $I\subset\operatorname{Sym}(V)=S={\mathbb{C}}[x_{0},\ldots,x_{3}]$ be an
ideal generated by quadrics. Assume that $I$ admits a $p$-th linear syzygy $f$
of rank $p+k+1$ with $1\leq p\leq 2$. Then $0\leq p\leq p+k\leq 3$ and we have
the following possibilities for the syzygy ideal $I_{f}$:

  1. (1)

$p=1$, $k=0$ and $S/I_{f}$ has Betti table

$\begin{matrix}1&-&-&\cr-&2&1&\cr\end{matrix}$

and $I_{f}$ is the ideal of a plane and a possibly embedded line.

  2. (2)

$p=1$, $k=1$ and $S/I_{f}$ has Betti table

$\begin{matrix}1&-&-&\cr-&3&2&\cr\end{matrix}$

and $I_{f}$ is generated by $2\times 2$ minors of $2\times 3$ linear forms and
is the ideal of a curve of degree 3 with Hilbert polynomial $3t+1$.

  3. (3)

$p=1$, $k=2$ and $I_{f}$ is generated by four $4\times 4$ Pfaffians of a skew-
symmetric $5\times 5$ matrix of linear forms and it is contained in a unique
ideal generated by Pfaffians of a skew-symmetric $5\times 5$ matrix of linear
forms.

  4. (4)

$p=2$, $k=0$ and $S/I_{f}$ has Betti table

$\begin{matrix}1&-&-&-&\cr-&3&3&1&\cr\end{matrix}$

and $I_{f}$ is the ideal of a plane and a possibly embedded point.

  5. (5)

$p=2$, $k=1$ and we have one of three possibilities

    1. (a)

$S/I_{f}$ has Betti table

$\begin{matrix}1&-&-&-&\cr-&4&4&1&\cr\end{matrix}$

and $I_{f}$ is the ideal of two skew lines (or a double line),

    2. (b)

$S/I_{f}$ has Betti table

$\begin{matrix}1&-&-&-&\cr-&5&6&2&\cr\end{matrix}$

and $I_{f}$ is the ideal of a line and a (possibly embedded) scheme of length
2,

    3. (c)

$S/I_{f}$ has Betti table

$\begin{matrix}1&-&-&-&\cr-&6&8&3&\cr\end{matrix}$

and $I_{f}$ is the ideal of a scheme of length 4.

###### Proof.

We begin by noting that the rank of a linear syzygy of an ideal in a ring with
four variables cannot exceed 4. Furthermore it is well known (see [vB07a])
that we must have $k\geq 0$. Now, with the notation and results above, to
derive the syzygy ideal from the generic syzygy ideal we need a map

(17) $\varphi:G^{*}\oplus\wedge^{k}G^{*}\to V,$

that on the $G^{*}$ component corresponds to associating to an element of
$G^{*}$ the corresponding linear form corresponding by $f$ to that syzygy. In
particular, we may assume by choosing a basis of $G$ that $\varphi$ on the
component $G^{*}$ is given by a vector of coordinates.

If $k=0$, we choose a basis $(g_{1},..,g_{p+1},c)$ of $G^{*}\oplus\mathbb{C}$
such that for $i\in\\{1,\dots,p+1\\}$ we have $\varphi(g_{i})=x_{i}$ and
denote $l:=\varphi(c)$. In this case, the generic syzygy ideal
$\operatorname{Gensyzid}_{0}(G)$ is the ideal $\langle
g_{1}c,\dots,g_{p+1}c\rangle$ hence $I_{f}=\langle
x_{1}l,\dots,x_{p+1}l\rangle$ and is the ideal of a hyperplane defined by
$\\{l=0\\}$ and a codimension ${p+1}$ linear space defined by
$\\{x_{1}=\dots=x_{p+1}=0\\}$.

For $p=1$, the ring $S/I_{f}$ has Betti table

$\begin{matrix}1&-&-&\cr-&2&1&\cr\end{matrix},$

whereas for $p=2$, the ring $S/I_{f}$ has Betti table

$\begin{matrix}1&-&-&-&\cr-&3&3&1&\cr\end{matrix}.$

If $k=1$, we choose a basis
$(g_{1},\dots,g_{p+2},g^{\prime}_{1},\dots,g^{\prime}_{p+2})$ of $G^{*}\oplus
G^{*}$ such that for $i\in\\{1,\dots,p+2\\}$ we have $\varphi(g_{i})=x_{i}$
and denote $l_{i}:=\varphi(g^{\prime}_{i})$. In this case, the generic syzygy
ideal $\operatorname{Gensyzid}_{1}(G)$ is the ideal generated by $2\times 2$
minors of the matrix:

(18) $N=\begin{pmatrix}g_{1}&\dots&g_{p+2}\\\
g^{\prime}_{1}&\dots&g^{\prime}_{p+2}\end{pmatrix}.$

Hence $I_{f}$ is generated by $2\times 2$ minors of the matrix:

(19) $\varphi(N)=\begin{pmatrix}x_{1}&\dots&x_{p+2}\\\
l_{1}&\dots&l_{p+2}\end{pmatrix}.$

Now, if $p=1$, the three minors of $\varphi_{N}$ generating $I_{f}$ must be
linearly independent and $S/I_{f}$ has Betti table

$\begin{matrix}1&-&-&\cr-&3&2&\cr\end{matrix}.$

$I_{f}$ is the ideal of a possibly non-reduced curve of degree 3 with Hilbert
polynomial $3t+1$.

If $p=2$, we define $s$ to be the least number that $\varphi(N)$ can be
written in the form such that $l_{i}=0$ for $s+1\leq i\leq 4$ and $l_{s}\neq
0$. A matrix is said to be $1-$generic [GeomSyz] if it cannot be modified by
elementary row/column operations to have a zero entry. Note that $\varphi(N)$
cannot be $1$-generic by [GeomSyz]*Theorem 6.4, so $s\leq 3$.

If $s=1$, then $I_{f}=\langle ly,lz,lw\rangle$, so $\operatorname{rk}f=3$.
However, our hypothesis is $\operatorname{rk}(f)=4$, contradiction.

If $s=2$, then

(20) $\varphi(N)=\begin{pmatrix}x&y&z&w\\\ l_{1}&l_{2}&0&0\end{pmatrix},$

where $l_{1}$ and $l_{2}$ are not proportional.
If $\det\begin{pmatrix}x&y\\\ l_{1}&l_{2}\end{pmatrix}\in\langle z,w\rangle$,
then $S/I_{f}$ has Betti table

$\begin{matrix}1&-&-&-&\cr-&4&4&1&\cr\end{matrix}.$

and is the ideal of two skew lines: $\\{z=w=0\\}$ and $\\{l_{1}=l_{2}=0\\}$
(or a double line if $l_{1},l_{2}\in\langle z,w\rangle$). If
$\det\begin{pmatrix}x&y\\\ l_{1}&l_{2}\end{pmatrix}\not\in\langle z,w\rangle$,
then $S/I_{f}$ has Betti table

$\begin{matrix}1&-&-&-&\cr-&5&6&2&\cr\end{matrix}.$

and is the ideal of a line $\\{l_{1}=l_{2}=0\\}$ and a scheme of length $2$
defined by $\\{z=t=xl_{2}-yl_{1}=0\\}$.

Conversely, if $S/I_{f}$ has Betti table

$\begin{matrix}1&-&-&-&\cr-&5&6&2&\cr\end{matrix}$

or

$\begin{matrix}1&-&-&-&\cr-&4&4&1&\cr\end{matrix},$

then $s=2$. Reason: If $s=3$ and $b_{12}\leq 5$, then there exists numbers
$a_{1},\dots,a_{6}$ such that

(21)
$(a_{1}l_{1}+a_{2}l_{2}+a_{3}l_{3})w+a_{4}(xl_{2}-yl_{1})+a_{5}(xl_{3}-zl_{1})+a_{6}(yl_{3}-zl_{2})=0.$

This is equivalent to

(22)
$\begin{pmatrix}x&y&z&w\end{pmatrix}\begin{pmatrix}0&-a_{4}&-a_{5}&-a_{1}\\\
a_{4}&0&-a_{6}&-a_{2}\\\ a_{5}&a_{6}&0&-a_{3}\\\
a_{1}&a_{2}&a_{3}&0\end{pmatrix}\begin{pmatrix}l_{1}\\\ l_{2}\\\ l_{3}\\\
0\end{pmatrix}=0.$

If $w\not\in\langle l_{1},l_{2},l_{3}\rangle$, then after changing
coordinates, we may assume $\langle l_{1},l_{2},l_{3}\rangle\subseteq\langle
x,y,z\rangle$. Then $(a_{1}l_{1}+a_{2}l_{2}+a_{3}l_{3})w=0$ and
$a_{4}(xl_{2}-yl_{1})+a_{5}(xl_{3}-zl_{1})+a_{6}(yl_{3}-zl_{2})=0$, which
implies $a_{1}l_{1}+a_{2}l_{2}+a_{3}l_{3}=0$. Therefore $s\leq 2$.
If $w\in\langle l_{1},l_{2},l_{3}\rangle$, then after changing coordinates, we
may assume $l_{3}=w$ and $\langle l_{1},l_{2}\rangle\subseteq\langle
x,y,z\rangle$. Making use of Equation 21, we have $a_{3}=0$,
$a_{1}l_{2}+a_{2}l_{2}+a_{5}x+a_{6}y=0$ and
$a_{4}(xl_{2}-yl_{1})-a_{5}zl_{1}+a_{6}zl_{2}=0$. Therefore,
$\det\begin{pmatrix}a_{4}x+a_{6}z&a_{4}y+a_{5}z\\\
l_{1}&l_{2}\end{pmatrix}=0$. This means
$(l_{1},l_{2})=c\cdot(a_{4}x+a_{6}z,a_{4}y+a_{5}z)$ and we conclude $s\leq 2$.

If $s=3$, then $b_{12}(I_{f})=6$, the Betti table of $S/I_{f}$ is

$\begin{matrix}1&-&-&-&\cr-&6&8&3&\cr\end{matrix}$

and $I_{f}$ is the ideal of a scheme of length 4.

Finally for $k=2$, $p=1$, we choose a basis
$(g_{1},\dots,g_{4},g^{\prime}_{12},\dots,g^{\prime}_{34})$ of
$G^{*}\oplus\wedge^{2}G^{*}$ such that for $i\in\\{1,\dots,4\\}$ we have
$\varphi(g_{i})=x_{i}$ and for $i,j\in\\{1,\dots,4\\}$ with $i<j$ denote
$l_{ij}:=\varphi(g^{\prime}_{ij})$. In this case, the generic syzygy ideal
$\operatorname{Gensyzid}_{2}(G)$ is the ideal generated by four $4\times 4$
Pfaffians involving the first row of the skew-symmetric matrix

(23) $N=\begin{pmatrix}0&g_{1}&g_{2}&g_{3}&g_{4}\\\
-g_{1}&0&g^{\prime}_{12}&g^{\prime}_{13}&g^{\prime}_{14}\\\
-g_{2}&-g^{\prime}_{12}&0&g^{\prime}_{23}&g^{\prime}_{24}\\\
-g_{3}&-g^{\prime}_{13}&-g^{\prime}_{23}&0&g^{\prime}_{34}\\\
-g_{4}&-g^{\prime}_{14}&-g^{\prime}_{24}&-g^{\prime}_{34}&0\end{pmatrix}.$

Hence $I_{f}$ is generated by four Pfaffians involving the first row of the
matrix:

(24) $\varphi(N)=\begin{pmatrix}0&x_{1}&x_{2}&x_{3}&x_{4}\\\
-x_{1}&0&l_{12}&l_{13}&l_{14}\\\ -x_{2}&-l_{12}&0&l_{23}&l_{24}\\\
-x_{3}&-l_{13}&-l_{23}&0&l_{34}\\\
-x_{4}&-l_{14}&-l_{24}&-l_{34}&0\end{pmatrix}.$

Furthermore the four Pfaffians are linearly independent and since $\langle
x_{1}\dots x_{4}\rangle$ define an Artinian ideal, the saturation of $I_{f}$
contains the ideal $\operatorname{Pf}(4,\varphi(N))$. Finally, the first
linear syzygies of $I_{f}$ must be combinations of rows of $\varphi(N)$ with
first entry 0, which implies that $f$ is the unique (up to scaling) first
linear syzygy of $I_{f}$.

To prove the last statement, it is enough to note that the ideal defined by
all $4\times 4$ Pfaffians of a $5\times 5$ skew-symmetric matrix of linear
forms is saturated, so whenever it contains $I_{f}$ it must also contain, and
hence be equal to, $\operatorname{Pf}(4,\varphi(N))$. ∎

######  Remark 4.4.

Note that if in Proposition 4.3 we omit the assumption $p\leq 2$, we have one
more case: $k=0$, $p=3$. In that case, with the same arguments we deduce that
$I_{f}$ is the ideal generated by all quadrics vanishing in a plane. These
kinds of ideals cannot appear in our investigation of $Q_{F}$, so we omit that
case in Proposition 4.3.

We now list some additional lemmas concerning higher dimensional spaces of
syzygies that will be useful in our investigation of $Q_{F}$.

######  Lemma 4.5.

If $I$ is an ideal generated by four quadrics in
${\mathbb{C}}[x_{0},\ldots,x_{3}]$, such that all the first linear syzygies
are of rank 4, then it admits at most one linear syzygy. So the first two rows
of the Betti table of $S/I$ are:

$\begin{matrix}1&-&-&\cr-&4&1&\cr\end{matrix}$

###### Proof.

The corank $1$ (i.e. rank at most $3$) locus is in this case a hypersurface
(defined by the determinant of a matrix with four rows corresponding to
variables and four columns corresponding to the number of generators of the
ideal) in the space of all first linear syzygies. Hence it can be empty only
if the space of first linear syzygies is $1$-dimensional, and in that case we
can have no more linear syzygies. ∎

######  Lemma 4.6.

If $J$ is an ideal generated by three quadrics with a pencil of syzygies of
rank $2$, then $S/J$ has Betti table

$\begin{matrix}1&-&-&-&\cr-&3&3&1&\cr\end{matrix}$

In this case $J$ is the ideal of the union of a plane and a (possibly
embedded) point.

###### Proof.

A rank $2$ syzygy involves two generators of the ideal that have a common
linear factor, so we may assume the two generators are
$x_{0}x_{1},x_{0}x_{2}$. Any other rank 2 syzygy would involve a third
generator and a quadric in the span of the two first. Now if these form a
pencil of rank $2$ syzygies the third generator needs to have a common linear
factor with every quadric in the span of the two first. Therefore, the three
generators of $J$ have a common linear factor, three linear syzygies of rank
$2$ and a Betti table as stated. ∎

######  Lemma 4.7.

If $S/Q_{F}$ has $b_{12}=4$ and at least three independent rank $2$ first
linear syzygies, then $Q_{F}$ has a subideal $J$ with Betti table for $S/J$

$\begin{matrix}1&-&-&\cr-&3&2&\cr\end{matrix},$

or a subideal $J^{\prime}$ with Betti table for $S/J^{\prime}$

$\begin{matrix}1&-&-&-\cr-&3&3&1\cr\end{matrix}.$

###### Proof.

Assume $s_{1},s_{2},s_{3}$ are three rank $2$ first linear syzygies. If
$s_{1}$ and $s_{2}$ involve all four generators of $Q_{F}$, then we may assume
$s_{1}$ involves $(q_{1},q_{2})=(x_{0}l_{1},x_{0}l_{2})$ and $s_{2}$ involves
$(q_{3},q_{4})=(l_{0}l_{3},l_{0}l_{4})$. If $l_{0}=x_{0}$, then $J=\langle
q_{1},q_{2},q_{3}\rangle$ and $S/J$ has Betti table

$\begin{matrix}1&-&-&-\cr-&3&3&1\cr\end{matrix}.$

If $l_{0}\neq x_{0}$, then $(q_{3},q_{4})=(x_{1}l_{3},x_{1}l_{4})$. Then the
third linear syzygy $s_{3}$ must involve some quadric $q_{5}\in\langle