fully apolar to $F_{Q}$.

To determine the type of the corresponding quartic form $F$ on $R$ in most
(all) cases (except $[300]$ which is trivial) it is enough to identify a
suitable configuration $\Gamma$ of points of on $Q$ defined by elements of $J$
such that $F$ is apolar to $\Gamma$. We can then conclude if we prove that a
general quartic $F_{Q}$ apolar to $\Gamma$ can be obtained by means of our
construction. For a $(4,4)$-form $F_{J}$, we say it is of type
$[b_{12},b_{23},b_{34}]$ if there is an $F$ of this type with restriction
$F_{Q}=F_{J}$.

Let us be more precise in each case.

Type $[683]$:

In this case $J$ is a complete intersection ideal of type $(2,1),(2,1),(2,4)$
generated by $p_{1},p_{2},p_{3}$ of the corresponding bidegrees. Since it is a
complete intersection we easily check that $\langle
p_{1},p_{2},p_{3}\rangle(4,4)$ is of dimension 24 and hence of codimension 1
in $T(4,4)$. Let $F_{J}$ be the $(4,4)$-form apolar to $J$, unique up to
constant multiple. We claim that $F_{J}$ is of type $[683]$. Indeed, $\langle
p_{1},p_{2}\rangle$ defines a set $\Gamma$ of 4 general points on $Q$.
Furthermore $\langle p_{1},p_{2}\rangle(4,4)$ is a subspace of codimension 4
in $T(4,4)$ and hence is the space of $(4,4)$-forms vanishing on $\Gamma$. It
follows that $F_{J}$ is apolar to $\Gamma$ and hence is of type $[683]$.

For the generality part, consider a general quartic $F$ of rank 4 and let
$\Gamma$ be its apolar fourtuple of points, which are in general position.
Clearly $\Gamma$ lies on a smooth quadric surface and appears there as a
complete intersection of two forms $p_{1}$, $p_{2}$ of degree $(2,1)$.
Furthermore, by dimension count there must exist a form $p_{3}$ of type
$(2,4)$ annihilating $F_{Q}$. Moreover if $F_{Q}$ is general among $(4,4)$
forms apolar to $\Gamma$ then $\langle p_{1},p_{2},p_{3}\rangle$ is a complete
intersection ideal in T.

Type $[550]$:

A set of five general points $\Gamma$ lies on a smooth quadric $Q$ as a
complete intersection of two forms $p_{1}$,$p_{2}$ of respective bidegrees
$(1,2),(2,1)$. Then, $\dim_{{\mathbb{C}}}\langle p_{1},p_{2}\rangle(2,2)=4$,
$\dim_{{\mathbb{C}}}\langle p_{1},p_{2}\rangle(3,3)=11$ and $\langle
p_{1},p_{2}\rangle(4,4)=20$. We deduce that any form apolar to $\langle
p_{1},p_{2}\rangle$ is apolar to $\Gamma$ the set of five points (i.e. to the
whole ideal of $\Gamma$) defined by $p_{1},p_{2}$. Furthermore if $p_{3}$ is a
general element of $T(3,3)$ then $\dim_{{\mathbb{C}}}\langle
p_{1},p_{2},p_{3}\rangle(4,4)=24$, hence there is a form $F_{Q}$ apolar to
$\Gamma$ to which $\langle p_{1},p_{2},p_{3}\rangle$ is fully apolar in $T$.

For generality, we observe that for a general $(4,4)$ form $F_{Q}$ apolar to a
set of five general points $\Gamma$ defined by $p_{1}$,$p_{2}$ of bidegrees
$(2,1)$ and $(1,2)$ we have $\dim_{{\mathbb{C}}}F_{Q}^{\perp}(3,3)=12$ and
$\dim_{{\mathbb{C}}}\langle p_{1},p_{2}\rangle(3,3)=11$. It follows that
$F_{Q}$ admit an additional apolar $(3,3)$ form $p_{3}$. Now since $F_{Q}$ was
general $\langle p_{1},p_{2},p_{3}\rangle$ is a complete intersection.

Type $[400]$:

This case is trivial, since if $J$ is a complete intersection of three
quadrics on a quadric $Q$ and F is the unique up to constant $(4,4)$ form
apolar to all the quadrics then the stronger condition $r^{-1}(J)=F^{\bot}$ is
clearly fulfilled.

Type $[300a]$:

If $J$ is a the Pfaffian ideal of a general $\varphi$ for this type in Table
5, then one computes $\dim_{{\mathbb{C}}}J(4,4)=24$ and hence there is a
$(4,4)$-form $F_{Q}$ to which $J$ is apolar. Now, $F_{Q}$ is apolar to the set
$\Gamma$ of eight points defined by the pencil of elements in $J(2,2)$. Indeed
$\dim_{{\mathbb{C}}}\langle J(2,2)\rangle(4,4)=25-8$ which is the dimension of
the space of $(4,4)$ forms passing through 8 points. To prove that $F_{Q}$
corresponds to the family $[300a]$ it is enough to check that it is a general
$(4,4)$ form apolar to $\Gamma$.

Let $\Gamma$ be a complete intersection of three quadrics, consisting of eight
points, and assume $F$ is a general quartic apolar to $\Gamma$. Let $Q$ be a
smooth quadric that contains $\Gamma$ and let $q_{1},q_{2}\in T(2,2)$ define
$\Gamma$ on $Q$. Consider the $(4,4)$-form $F_{Q}$, the restriction of $F$ to
$Q$. Then $\dim_{{\mathbb{C}}}F_{Q}^{\bot}(2,2)=2$,
$\dim_{{\mathbb{C}}}F_{Q}^{\bot}(3,2)=\dim_{{\mathbb{C}}}F_{Q}^{\bot}(2,3)=6$
and $\dim_{{\mathbb{C}}}F_{Q}^{\bot}(3,3)=12$. Let $p_{1}$ be a general form
in $F_{Q}^{\bot}(3,2)$. Then $\langle q_{1},q_{2},p_{1}\rangle$ is a complete
intersection ideal such that $\dim_{{\mathbb{C}}}\langle
q_{1},q_{2},p_{1}\rangle(4,4)=23$. We furthermore choose $p_{2}\in
F_{Q}^{\bot}(2,3)$ such that $\langle q_{1},q_{2},p_{1},p_{2}\rangle$ admits
one syzygy in $T(3,3)$. This is possible since $\dim_{{\mathbb{C}}}\langle
q_{1},q_{2},p_{1}\rangle(3,3)=10$, $\dim_{{\mathbb{C}}}F_{Q}^{\bot}(3,3)=12$
and every form $p_{2}\in F_{Q}^{\bot}(2,3)$ generate a pencil of forms
$F_{Q}^{\bot}(3,3)$, so the set of forms $p_{2}$ such that this pencil
intersects $\langle q_{1},q_{2},p_{1}\rangle(3,3)$ in $F_{Q}^{\bot}(3,3)$ has
codimension at most $1$. To see that there is a $p_{2}$ such that $\langle
q_{1},q_{2},p_{1},p_{2}\rangle$ admits only one syzygy in $T(3,3)$ we note
that $\langle p_{2}\rangle(3,3)\subset\langle q_{1},q_{2},p_{1}\rangle(3,3)$
implies $p_{2}\in\langle q_{1},q_{2}\rangle$. Note moreover, that by possibly
replacing $q_{1},q_{2}$ by suitable linear combinations and replacing
$p_{1},p_{2}$ by adding multiples of $q_{1}$ and $q_{2}$ we can assume that
our syzygy in $T(3,3)$ is of the form $q_{2}l+p_{1}s_{1}+p_{2}s_{2}=0$ with
$l,s_{1},s_{2}$ of respective bidegrees $(1,1)$,$(0,1)$,$(1,0)$. Furthermore,
by generality, we may assume $l,s_{1},s_{2}$ to be a complete intersection. In
that case $q_{2},p_{1},p_{2}$ are the $2\times 2$ minors of a matrix

$N=\begin{pmatrix}l&s_{1}&s_{2}\\\ m_{1}&m_{2}&m_{3}\\\ \end{pmatrix}$

representing a map $T(0,0)\oplus T(-1,-1)\to T(1,0)\oplus T(0,1)\oplus
T(1,1)$. The bidegrees of the entries $m_{1},m_{2},m_{3}$ are
$(2,2),(1,2),(2,1)$ respectively. Then the only syzygy of $\langle
q_{1},q_{2},p_{1},p_{2}\rangle$ in bidegree $(4,4)$ that involves $q_{1}$ is
the Koszul syzygy with $q_{2}$, so one computes $\dim_{{\mathbb{C}}}\langle
q_{1},q_{2},p_{1},p_{2}\rangle(4,4)=24$. Thus $\langle
q_{1},q_{2},p_{1},p_{2}\rangle$ is fully apolar to $F_{Q}$. We may express
$q_{1}$ as a quadric in the ideal generated by the first row of $N$, i.e.
$q_{1}=a_{1}l+a_{2}s_{1}+a_{3}s_{2}$ where the $a_{i}$ have bidegrees
$(1,1),(2,1),(1,2)$ respectively. Then the skew-symmetric matrix

$M=\begin{pmatrix}0&0&l&s_{1}&s_{2}\\\ 0&0&m_{1}&m_{2}&m_{3}\\\
-l&-m_{1}&0&-a_{3}&a_{2}\\\ -s_{1}&-m_{2}&a_{3}&0&a_{1}\\\
-s_{2}&-m_{3}&-a_{2}&-a_{1}&0\\\ \end{pmatrix}$

representing a map from $E^{\vee}\to E$, have $q_{1},q_{2},p_{1},p_{2}$ as
four of its pfaffians. The remaining pfaffian
$p_{3}:=a_{1}m_{1}+a_{2}m_{2}+a_{3}m_{3}$ has bidegree $(3,3)$.

There are three syzygies describing $p_{3}l,p_{3}s_{1},p_{3}s_{2}$ in terms of
$q_{1},q_{2},p_{1},p_{2}$. Since $\langle l,s_{1},s_{2}\rangle(1,1)=T(1,1)$,
it follows that $\langle q_{1},q_{2},p_{1},p_{2},p_{3}\rangle(4,4)=\langle
q_{1},q_{2},p_{1},p_{2}\rangle(4,4)$ i.e. $\langle
q_{1},q_{2},p_{1},p_{2},p_{3}\rangle$ is a pfaffian ideal of the desired shape
that is fully apolar to $F_{Q}$.

Type $[300b]$:

Let $\phi$ be a general skew map for this type as described in Table 5. Among
the pfaffians generating the Pfaffian ideal $J$ we recover the $2\times 2$
minors of a general map $T\oplus T(-1,0)\to 2T(1,1)\oplus T(0,1)$. The latter
defines a set of seven points $\Gamma$ such that $I_{\Gamma}(4,4)$ is
generated by these minors. Notice that a general set of seven points in a
complete intersection $((2,2),(2,2))$ on $Q$ are defined this way.
Furthermore, we compute $\dim_{{\mathbb{C}}}J(4,4)=24$, hence $J$ is fully
apolar to some $(4,4)$ form $F_{Q}$. This form $F_{Q}$ and hence also a
corresponding $F$ is apolar to $\Gamma$.

Assume $F$ is a general quartic apolar to a general set of seven general
points $\Gamma$ and consider a smooth quadric $Q$ passing through $\Gamma$. On
$Q$ the set $\Gamma$ is contained in two $(2,2)$ forms $q_{1}$, $q_{2}$ and a
$(3,2)$ form $p_{1}$. Moreover $\Gamma$, by syzygy computation, can be
described as a degeneracy locus of a map $T\oplus T(-1,0)\to 2T(1,1)\oplus
T(0,1)$ represented by a matrix $N$. Note that $F^{\bot}$ contains a $(3,3)$
form $p$ that does not pass through any point of $\Gamma$. Since $N$ is
general we can write $p_{2}$ in terms of the three forms $l_{1}$, $l_{2}$,
$l_{3}$ of bidegrees $(2,1)$, $(2,1)$, $(1,1)$ forming the second row (i.e.
$T(-1,0)\to 2T(1,1)\oplus T(0,1)$) of $M$. Therefore, as in the previous case,
we can construct a $5\times 5$ skew symmetric matrix $M$. The first two rows
are given by $N$ completed by a zero diagonal $2\times 2$ block, and
$q_{1},q_{2},p_{1},p_{2}$ appear as four of its pfaffians. The remaining
pfaffian $p_{3}$ of $M$ has syzygies describing
$l_{1}p_{3}$,$l_{2}p_{3}$,$l_{3}p_{3}$ in terms of $q_{1},q_{2},p_{1},p_{2}$.
In particular, $\langle p_{3}\rangle(4,4)\subset\langle
q_{1},q_{2},p_{1},p_{2}\rangle(4,4)$. Hence $\langle
q_{1},q_{2},p_{1},p_{2},p_{3}\rangle$ is apolar and in consequence fully
apolar to $F$.

Other cases of size $5\times 5$:

The remaining arguments are similar, we just need to describe the set of
points $\Gamma$ computing the rank of $F$ in terms of $2\times 2$ minors of a
$2\times 3$ matrix and complete the matrix to a skew symmetric matrix using
one appropriate additional generator in $F_{Q}^{\perp}$ and its syzgies as
done above.

Type $[100]$:

The only case when the corresponding matrix is of size $7\times 7$ is type
$[100]$. In this case, let $J$ be the ideal generated by the $6\times 6$
Pfaffians of the matrix. From the resolution we compute
$\dim_{{\mathbb{C}}}J(4,4)=24$, hence $J$ is fully apolar to some $(4,4)$-form
$F_{Q}$.

Then to prove that a general quartic is obtained in this way we recall from
section 5 that for a general quartic $F$ from this family and $Q$ its apolar
quadric, we have
$\dim_{{\mathbb{C}}}F_{Q}^{\bot}(3,2)=\dim_{{\mathbb{C}}}F_{Q}^{\bot}(2,3)=6$.
Furthermore, we can choose in $F_{Q}^{\bot}(3,2)$ three elements $p_{1}$,
$p_{2}$, $p_{3}$ defining a set $\Gamma$ of 9 general points on $Q$ which are
apolar to $F$. These must have a syzygy in $T(4,3)$ and are given as three of
the minors of a $3\times 4$ matrix $N$ corresponding to a general map:
$3T(0,-1)\to T\oplus 3T(1,0)$.

Consider furthermore a general triple $(q_{1},q_{1},q_{3})$ of elements in
$F_{Q}^{\bot}(2,3)$ which admit a syzygy in $T(3,4)$, such that
$p_{1},p_{2},p_{3},q_{1},q_{1},q_{3}$ admit a syzygy in $T(3,3)$. This can be
done for dimension reasons: Given a pair of subschemes
$\Gamma,\Gamma^{\prime}\in VSP(F_{Q},9)$ apolar to $F_{Q}$. The subspace of
$I_{\Gamma}(3,3)$ generated by $I_{\Gamma}(3,2)$ and the subspace of
$I_{\Gamma^{\prime}}(3,3)$ generated by $I_{\Gamma^{\prime}}(2,3)$ are both
$6$-dimensional subspaces of the $12$-dimensional space $F_{Q}^{\perp}(3,3)$.
Since $VSP(F,9)$ is $2$-dimensional, there are pairs $\Gamma,\Gamma^{\prime}$
such that these two subspaces intersect, which means that there is a syzygy in
$T(3,3)$ among the generators in bidegree $(3,2)$ and $(2,3)$ respectively for
the ideals of $\Gamma$ and $\Gamma^{\prime}$. By possibly replacing
$p_{1},p_{2},p_{3}$ $q_{1},q_{1},q_{3}$ and making corresponding row and
column operations on the matrix $N$ we can assume:

  1. (1):

$q_{1}x+q_{2}y=p_{1}z+p_{2}t$, where $x,y$ are $(1,0)$-forms and $z,t$ are
$(0,1)$-forms.

  2. (2):

$N=\left[\begin{array}[]{cccc}x&s_{1}&s_{2}&s_{3}\\\ y&u_{1}&u_{2}&u_{3}\\\
0&l_{1}&l_{2}&l_{3}\\\ \end{array}\right]$ for some $s_{i},u_{i},l_{i}\in
T(1,1)$.

  3. (3):

The $p_{i}$ are the corresponding $3\times 3$ signed minors of $N$ of bidegree
$(3,2)$.

  4. (4):

$q_{1}s_{3}+q_{2}u_{3}+q_{3}l_{3}=0$ for $s_{3},u_{3},l_{3}$ from $N$ \-
indeed by possibly changing $q_{1},q_{2},q_{3}$ to their linear combinations
we can assume that for the syzygy in $T(3,4)$ that they admit the coefficient
at $q_{3}$ is $l_{3}$, then by column operations on $N$ involving the first
column and row operations on the first two rows we can arrange $s_{3}$,
$u_{3}$ to be the remaining coefficients in the $T(3,4)$ syzygy between
$q_{1},q_{2},q_{3}$.

  5. (5):

$l_{1},l_{2},l_{3}$ are irreducible and form a complete intersection and
similarly $s_{3},u_{3},l_{3}$ – this follows from genericity of $F$ and
$\Gamma$

Let $W=\left[\begin{array}[]{ccc}s_{1}&s_{2}&s_{3}\\\ u_{1}&u_{2}&u_{3}\\\
l_{1}&l_{2}&l_{3}\\\ z&t&0\end{array}\right]$ and $w_{i}$ be the minors of $W$
obtained by deleting the $i$-th row. Under the above assumptions we infer from
(1) and descriptions of $p_{i}$ in terms of minors of $N$ that:
$(q_{1}-w_{1})x+(q_{2}-w_{2})y=0.$ It follows that there exists $k\in T(1,3)$
such that $q_{1}-w_{1}=yk$ and $q_{2}-w_{2}=-xk$. Now from (4) we have
$(s_{3}y+u_{3}x)k+(q_{3}-w_{3})l_{3}=0,$ but since $l_{3}$ is irreducible and
has no syzygy in $T(2,1)$ with $s_{3}$, $u_{3}$ we infer $k=l_{3}h$ for $h\in
T(0,2)$. Finally note that by adding combination of the first column of $N$ to
the second and third column we can modify $s_{1},s_{2},u_{1},u_{2}$ without
changing $p_{i}$ but replacing $w_{1}$, $w_{2}$ with $w_{1}+yl_{3}(za+tb)$ and
$w_{2}-yl_{3}(za+tb)$ for any chosen $a,b\in T(0,1)$. We conclude that after
suitable choice of $a,b$ for the corresponding modification of $N$ we can
assume $h=0$ and hence $q_{1}=w_{1}$, $q_{2}=w_{2}$, $q_{3}=w_{3}$. If we now
consider the skew symmetric matrix

$\left[\begin{array}[]{cccccc}0&0&x&s_{1}&s_{2}&s_{3}\\\
&0&y&u_{1}&u_{2}&u_{3}\\\ &&0&l_{1}&l_{2}&l_{3}\\\ &&&z&t&0\\\ &&&&0&0\\\
&&&&&0\\\ \end{array}\right],$

its pfaffian ideal $J$ has the property $\langle
p_{1},p_{2},p_{3},q_{1},q_{2},q_{3}\rangle\subset J$ and

$J(4,4)=\langle
p_{1},p_{2},p_{3},q_{1},q_{2},q_{3}\rangle(4,4)=F_{Q}^{\bot}(4,4)$

i.e. $J$ is fully apolar to $F$.

Here $x,y,z,t$ are coordinates in $T$ of respective bidegrees $(1,0)$,
$(1,0)$, $(0,1)$, $(0,1)$.

∎

######  Remark 7.6.

Note that if we take $E=T(2,1)\oplus T(2,1)\oplus T(-1,1)$, then we get a
complete intersection ideal $J$ of grade 3 in $T$ with a quartic dual socle
generator $F_{Q}$ which is a restriction of $F$ from the family $[683]$.
Moreover, a general $F$ in $[683]$ can be obtained in this way. However, in
this case $r^{-1}(J)\neq F^{\bot}$ as $\dim_{{\mathbb{C}}}J(2,2)=4$

###  7.2. AG-liftings to curves

We describe liftings to curves in smooth quadric fourfolds of $A_{F}$ for
quaternary quartics $F$ of the types $[100],[200],[300],[400]$. Let $C\subset
Q\subset{\mathbb{P}}^{5}$ be an Arithmetic Gorenstein scheme of dimension 1
contained in a smooth quadric fourfold $Q$. Recall that $Q\simeq G(2,4)$. Let
furthermore $H$ be the class of a hyperplane section on $Q$.

Suppose the curve $C\subset Q$ is a Pfaffian subvariety on $Q$ defined by the
submaximal minors of a skew symmetric map between vector bundles
$E_{C}^{\vee}\to E_{C}$ of odd rank and $det(E_{C})=eH$. From [Okonek]*p. 427
the ideal sheaf of $C$ has a resolution

$0\to\mathcal{O}(-2eH)\to E_{C}^{\vee}(-eH)\to
E_{C}(-eH)\to\mathcal{I}_{C|Q}\to 0,$

and the canonical line bundle on $C$ is $K_{C}=(2e-4)H|_{C}$.

Consider the exact sequence of ideal sheaves

$0\to\mathcal{O}(-2)\to\mathcal{I}_{C}\to\mathcal{I}_{C|Q}\to 0.$

From the associated long exct sequence in cohomology, we deduce that
$C\subset\mathbb{P}^{5}$ is ACM iff $C\subset Q$ is ACM. For the Pfaffian
varieties constructed from a bundle $E$ we infer the following:

######  Lemma 7.7.

If $H^{1}(E_{C}(k))=H^{2}(E_{C}^{\vee}(k))=0$ for $k\geq 0$ then the Pfaffian
varieties on the quadric $Q$ which are associated to $E$ are arithmetically
Gorenstein.

###### Proof.

By splitting the Pfaffian sequence twisted by $k$ we deduce that the
vanishings assumed in the lemma imply $H^{1}(I_{C|Q}(k))=0$ for $k\geq 0$. By
the relative ideal sequence this implies that $H^{1}(I_{C}(k))=0$ for $k\geq
0$ and $X$ is ACM. ∎

We are now ready to provide constructions of ACM Gorenstein curves with given
Betti tables. See section 2.7 for the notation on spinor bundles.

######  Proposition 7.8.

Let be $S_{1},S_{2}$ be spinor bundles on $Q_{4}$. Pfaffian varieties
associated to general skew symmetric maps $E\to E^{\vee}$ have Betti table:

  1. (1)

Type $[100]$ for $E=3\mathcal{S}_{1}\oplus\mathcal{O}$,
$E=2\mathcal{S}_{1}\oplus\mathcal{S}_{2}\oplus\mathcal{O}$

  2. (2)

Type $[200]$ for $E=2\mathcal{S}_{1}\oplus\mathcal{O}(1)$,
$E=\mathcal{S}_{1}\oplus\mathcal{S}_{2}\oplus\mathcal{O}(1)$

  3. (3)

Type $[300]$ for $E=\mathcal{S}_{1}\oplus 2\mathcal{O}(1)\oplus\mathcal{O}$

  4. (4)

Type $[400]$ for $E=3\mathcal{O}(1)$.

###### Proof.

The following formulas relating the degree of $C$ and the Chern classes of
$E_{C}$ hold:

$c_{1}(E_{C})=3H\text{ and }\deg C=\deg(3Hc_{2}(E_{C})-c_{3}(E_{C}))$

where $H\in H^{2}(Q,\mathbb{Z})$ is the class of the hyperplane section of $Q$
and $c_{i}(E_{C})\in H^{2i}(Q,\mathbb{Z})$ are the Chern classes of $E$.

We deduce that the associated Pfaffian subvarieties of $Q_{4}$ are smooth ACM
half canonical curves of the correct degree. Furthermore, observe that for a
general codimension 2 linear section $L$ of $Q_{4}$ in each case the map
$H^{0}(Q_{4},\wedge^{2}E)\to H^{0}(L,\wedge^{2}E|_{L})$ is surjective and the
Betti table of the section of the half canonical curves by $L$ can be checked
via Macaulay2, cf. [M2]*Package QuaternaryQuartics. ∎

######  Remark 7.9.

Note that the bundle $3\mathcal{O}(1)$ defines a family of complete
intersections of four quadrics. Moreover, the varieties in the family $[320]$
are contained in a cubic scroll which is generated by quadrics of rank $4$
independent of their dimension. Hence $[320]$ does not admit Pfaffian
descriptions in $Q_{4}$.

######  Remark 7.10.

In the list of ACM bundles above we can exchange $S_{1}$ with $S_{2}$
obtaining a complete list of possible ACM bundles on $Q_{4}$ giving rise to
ACM Gorenstein half-canonical curves.

Some of these constructions admit further liftings to a smooth 6-dimensional
quadric $Q_{6}$. However, we shall need to supplement our constructions to
include Lagrangian degeneracy loci.

###  7.3. AG-liftings to Calabi-Yau threefolds; Lagrangian degeneracy loci

We construct Calabi-Yau threefolds in a smooth quadric sixfold
$Q_{6}\subset{\mathbb{P}}^{7}$ of degrees $17,18$ and $19$ that are liftings
of $A_{F}$ for quaternary quartics $F$ of type $[300],[200],[100]$
respectively.

For the degree $18$ case, we consider the Pfaffian variety of a general skew
symmetric map $E\to E^{\vee}$ where $E=S_{1}\oplus\mathcal{O}(1)$ and $S_{1}$
is a spinor bundle (see Section 2.7 for the notation). Indeed, to lift the
Pfaffians associated to
$\mathcal{S}_{1}\oplus\mathcal{S}_{2}\oplus\mathcal{O}(1)$ on $Q_{4}$ we
simply note that the bundle $S_{1}\oplus S_{2}$ on $Q_{4}$ is the restriction
of the spinor bundle on $Q_{6}$.

In order to construct Calabi-Yau threefolds of degrees $17$ and $19$ the
situation is more complicated. We discuss a Lagrangian construction and treat
in some detail the degree $19$ case. We comment briefly on the degree $17$
case in Remark 7.15.

Recall the Lagrangian construction studied in [EPW, EPW1]: 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
$\mathcal{E}\subset V\otimes{\mathcal{O}}_{Q_{6}}$ with Pfaffian line bundle
$L$, so that $L^{\otimes 2}={\rm det}\mathcal{E}$. 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 $Q_{6}$ of codimension $3$ whose sheaf of ideals has
symmetric quasi-isomorphic locally free resolutions

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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
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278.75476pt\raise-42.13776pt\hbox{\hbox{\kern 0.0pt\raise
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This is a commutative diagram such that $\alpha\beta$ is a skew symmetric map.
Moreover, $L^{\otimes 2}=det(\mathcal{E})$ such that
$\omega_{X}=(\omega_{Q_{6}}\otimes det(\mathcal{E}^{\ast}))|_{X}$.

The class of $X$ in ${Q_{6}}$ is computed by the Pragacz-Fulton formula [FP]
as
$\frac{1}{4}(c_{1}(\mathcal{E}^{\ast})c_{2}(\mathcal{E}^{\ast})-2c_{3}(\mathcal{E}^{\ast}))$.
In particular, if $\mathcal{E}$ is a spinor bundle, $\mathcal{S}_{1}$ or
$\mathcal{S}_{2}$, on $Q_{6}$ the subvariety $X$ will be a
$\mathbb{P}^{3}\subset Q_{6}$ that do not have Pfaffian resolution (see
[EPW1]*§4).

We have the following interpretation of the Tom and Jerry construction of
[Reid].

######  Proposition 7.11.

The del Pezzo Fano threefolds

$\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}\quad(Jerry)\quad{\rm
and}\quad\mathbb{P}^{2}\times\mathbb{P}^{2}\cap\mathbb{P}^{7}\quad(Tom)\quad\subset
Q_{6}$

are subcanonical subvarieties of codimension $3$ defined by the Lagrangian
construction with the bundle $\mathcal{E}=2\mathcal{S}_{1}$ for Jerry and the
bundle $\mathcal{E}=\mathcal{S}_{1}\oplus\mathcal{S}_{2}$ for Tom, where
$\mathcal{S}_{1},\mathcal{S}_{2}$ are the spinor bundles on $Q_{6}$.

###### Proof.

Observe that $\mathcal{S}_{1}\oplus\mathcal{S}_{2}$ is the restriction of the
spinor bundle on $Q_{7}$. The Lagrangian construction with that bundle defines
on $Q_{7}$ a del Pezzo fourfold of degree $6$ so the Segre embedding
$\mathbb{P}^{2}\times\mathbb{P}^{2}\subset Q_{7}$ from the classification of
del Pezzo fourfolds.

It is enough to show the other example is different. We compute the
restrictions of those degeneracy loci to the two families of isotropic
$\mathbb{P}^{3}\subset Q_{6}$. From [Ot]*Thm. 2.6 we infer the restriction of
$\mathcal{S}_{1}$ to one of those $\mathbb{P}^{3}$ is
$\Omega^{2}(2)\oplus\mathcal{O}$ and the other
$\Omega^{2}(2)\oplus\Omega^{3}(3)$ and $\mathcal{S}_{2}$ in the opposite way.
Using again the Pragacz-Fulton formula we infer
$\mathcal{S}_{1}\oplus\mathcal{S}_{2}$ gives a threefold of bidegree $(3,3)$
and $2\mathcal{S}_{1}$ of bidegree $(2,4)$.

In order to conclude we observe that a hyperplane section of
$\mathbb{P}^{2}\times\mathbb{P}^{2}$ contained in a smooth quadric $Q_{6}$
lifts to $\mathbb{P}^{2}\times\mathbb{P}^{2}$ contained in a smooth quadric
$Q_{7}$. Since in $Q_{7}$ there is one family of $\mathbb{P}^{3}$ we deduce
that Tom have always bidegree $(3,3)$ in a smooth quadric. Therefore
$2\mathcal{S}_{1}$ defines Jerry. ∎

By a spinor analogue of bilinkage we construct Calabi-Yau threefolds in the
smooth $6$-dimensional quadric $Q_{6}$.

######  Proposition 7.12.

Let $X\subset Q_{6}$ be the subvariety of codimension $3$ defined by the
Lagrangian construction with

$\mathcal{E}=3\mathcal{S}_{1}\quad{\rm
or}\quad\mathcal{E}=\mathcal{S}_{1}\oplus 2\mathcal{S}_{2}$

in diagram (28), where $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ are spinor
bundles on $Q_{6}$, then $X$ is a Calabi-Yau threefold of degree $19$ that is
Arithmetically Gorenstein (AG) in $\mathbb{P}^{7}$. Furthermore, the two
choices of bundle $\mathcal{E}$ give distinct families of polarized Calabi-Yau
threefolds.

###### Proof.

The canonical divisor is computed from the fact that
$det(3\mathcal{S}_{i})=6h$. In order to find the degree we use the Fulton-
Pragacz formula knowing that $c_{1}(\mathcal{S}_{i})=2h$,
$c_{2}(\mathcal{S}_{i})=2h^{2}$ and $c_{3}(\mathcal{S}_{i})=(2,0)$.

To prove it is AG it is enough to use the long cohomology exact sequence
corresponding to the first sequence in diagram 28. The cohomologies of the
twists of $\mathcal{S}_{i}$ are described in [Ot]*Thm. 4.3.

In order to show that the two bundles above gives different polarised Calabi-
Yau threefolds we compute as in the proof of Lemma 7.11 that they have
bidegrees $(9,10)$ and $(7,12)$ in the class of cycle of codimension $3$. ∎

######  Remark 7.13.

As in the Tom and Jerry case we see that a general AG surface of degree $19$
in ${\mathbb{P}}^{6}$ with Betti table of type $[100]$ lifts to distinct
families of AG Calabi-Yau threefolds in ${\mathbb{P}}^{7}$ explicitly
described on smooth quadrics.

Notice that Proposition 1.8 follows immediately from Proposition 7.12.

We expect that the Pfaffian Calabi-Yau threefolds of degree $19$ defined above
specialize to families constructed by unprojection in [CGKK]. Recall that in
[CGKK] such examples were constructed in quadrics of rank $6$.

######  Remark 7.14.

Let $X$ be a Calabi-Yau threefold in $Q_{6}$ constructed in Proposition 7.12.
Let $Y$ be a general codimension 2 linear section of $X$ contained in a
quadric $Q_{4}$. Then $Y$ is described by a Pfaffian resolution related to any
of the bundles in item (1) of Proposition 7.8. Since both $S_{1}$ and $S_{2}$
on $Q_{6}$ restricts to $\mathcal{S}_{1}\oplus\mathcal{S}_{2}$ on $Q_{4}$, we
consider the restriction of the diagram of sheaves on $Q_{6}$ to $Q_{4}$,
after inserting an additional component $\mathcal{O}$ :

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The added component allows for a nonminimal Lagrangian resolution, so that the
composition

$3\mathcal{S}_{1}\oplus 3\mathcal{S}_{2}\oplus\mathcal{O}\to
3{\mathcal{S}_{1}}^{\ast}\oplus 3{\mathcal{S}_{2}}^{\ast}\oplus\mathcal{O}$

becomes a skew symmetric map between odd-dimensional vector bundles.

Note that under the generality assumption the map $(3\mathcal{S}_{1}\oplus
3\mathcal{S}_{2}\oplus\mathcal{O})(-3)\to 13\mathcal{O}(-3)$ induces an
embedding $3\mathcal{S}_{1}(-3)\to 13\mathcal{O}(-3)$. Indeed, note that since
$\mathcal{S}_{1}^{*}$ has 4 sections the image of $3\mathcal{S}_{1}(-3)$ in
$26\mathcal{O}(-3)$ is contained in some non-isotropic subbundle
$12\mathcal{O}(-3)$. By Kleiman transversality, a chosen general isotropic
subspace of dimension 3 will hence avoid the image of $3\mathcal{S}_{1}(-3)$
and then the composition $3\mathcal{S}_{1}(-3)\to 13\mathcal{O}(-3)$ will be
an embedding.

But then, the resolutions of $I_{Y}$ from the diagram are not minimal. Indeed,
whenever we have a resolution

$0\to A\to B\oplus C\to D\to I\to 0,$

such that the map $B\to D$ is an embedding with cokernel $E$, i.e. inducing an
exact sequence

$0\to B\to D\to E\to 0,$

then by snake lemma we also have a resolution

$0\to A\to C\to E\to I\to 0,$

We apply this argument, using the universal sequence

$0\to S_{2}\to 4\mathcal{O}\to S_{1}^{*}\to 0$

on $Q_{4}=G(2,4)$. The diagram 29 after simplification becomes:

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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
259.60846pt\raise-41.63608pt\hbox{\hbox{\kern 0.0pt\raise
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303.69778pt\raise-41.63608pt\hbox{\hbox{\kern 0.0pt\raise
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303.69778pt\raise-41.63608pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0}$}}}}}}}\ignorespaces}}}}\ignorespaces,$

which means that $\mathcal{I}_{Y|Q}$ admits a Pfaffian resolution.

######  Remark 7.15.

We can also construct the AG Calabi–Yau threefolds of Type $[300a]$ of degree
$17$ in a smooth quadric. Indeed, a Calabi-Yau threefold of Type [300a] can be
constructed by bilinkage of $\mathbb{P}^{3}$ in a complete intersection of
three quadrics (cf. [CGKK] and Section 8). The resolution of this bilinkage on
a fixed quadric can be found tracing the resolution through the bilinkage
construction by means of the mapping cone (see [PS74]) starting from diagram
(28) with $\mathcal{E}=\mathcal{S}_{1}$. We obtain a resolution of the shape

(31) $\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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where $\mathcal{S}_{1}\oplus 2\mathcal{O}(-1)\subset 8\mathcal{O}\oplus
2\mathcal{O}(-1)\oplus 2\mathcal{O}(1)$ is embedded as a Lagrangian subbundle
of a orthogonal bundle (with natural orthogonal structure).

###  7.4. Degenerations of liftings

We considered in the previous sections stratification of the space of quartics
and the associated Artinian rings. It is natural to expect we obtain
degenerations of liftings of such Artinian rings. Let us discuss some
examples.

######  Proposition 7.16.

The complete intersection of four quadrics in $\mathbb{P}^{7}$ degenerates to
the example with resolution of type $[420]$.

###### Proof.

First $[400]$ is defined by the Pfaffians of a skewsymmetric map with the
bundle $3\mathcal{O}(1)$ on $Q_{6}$. But we can consider also the bundle
$3\mathcal{O}(1)\oplus 2\mathcal{O}$ on $Q_{6}$. It is enough to observe that
(2.4) is defined by a special skew map for $3\mathcal{O}(-1)\oplus
2\mathcal{O}\to 3\mathcal{O}(1)\oplus 2\mathcal{O}$ such that the map
$2\mathcal{O}\to 2\mathcal{O}$ is the zero map. ∎

One could expect that degenerations of Artinian rings could lead to natural
degenerations of Calabi-Yau threefolds. However the situation is more
complicated. Recall that Calabi-Yau threefolds with different Hodge number
cannot be deformation equivalent [Voisinhodge]*Prop. 9.20. We can however
prove the following:

######  Proposition 7.17.

Artinian rings with Betti table $[300a]$ degenerate to general rings with
Betti table $[320]$. The corresponding Calabi-Yau threefolds do not
degenerate, but have a common degeneration to a singular Calabi-Yau threefold
$[320]$.

###### Proof.

Recall that there can be is no smooth family containing elements of both
families since these have different Hodge numbers. However, the common
degenerations to singular Calabi-Yau threefolds can be described as follows.
The intersection of three quadrics degenerates to a degree $8$ fourfold being
a divisor in the cubic scroll (being a cone over
$\mathbb{P}^{1}\times\mathbb{P}^{2}$). Consider the Calabi-Yau fourfold is
bilinked to a $\mathbb{P}^{3}$ contained in the fourfold of degree $8$. In
fact we can choose the $\mathbb{P}^{3}$ above in two ways in the rulings of
the cubic scroll obtaining two families of degenerated Calabi-Yau threefolds
of degree $17$, being degenerations of elements in both families. ∎

##  8\. Irreducible liftings

In this section and the next we investigate liftings of the Artinian
Gorenstein rings $A_{F}$ of quaternary quartics. In the previous section, some
such liftings were found via Pfaffian and Lagrangian constructions on smooth
quadrics. In this section we treat systematically liftings of rings $A_{F}$
with Betti table in A, Table 6, leaving liftings of rings $A_{F}$ with Betti
table in A, Table 7 to the next section.

In Corollary 8.6 we shall see that a generic Artinian Gorenstein ring with a
given Betti table considered in this section lifts to a curve; this lifting is
obtained by the doubling construction. Next we attempt to give an explicit
geometric description of a general element of each family with given Betti
table. We are especially interested in a description of the lifting as
divisors in some lift of the configuration of points computing the rank. Some
of the cases follow from the classification contained in [CGKK], but we repeat
the argument here for completeness. In many cases the construction involves
bilinkage, cf.[PS74, Har], as used in [CGKK]: Let the variety
$Z\subset{\mathbb{P}}^{n}$ be arithmetic Cohen-Macaulay, let $X\subset Z$ be a
codimension $1$ arithmetic Gorenstein variety and $H$ a hyperplane section of
$Z$. Then any generalised [Har] divisor $Y\in|X+aH|,\;a\in{\mathbb{Z}}$ is
arithmetic Gorenstein and bilinked to $X$ in $Z$.

######  Remark 8.1.

Bilinkage in complete intersections give uniqueness of lifting as follows.
Assume $X,Y,Z$ are as above and $Z$ is a complete intersection. Assume that
there exists a smooth variety $\mathcal{X}$ such that
$X=\mathcal{X}\cap\mathcal{H}$, where $\mathcal{H}$ is a hyperplane. Then
there exists a complete intersection $\mathcal{Z}$ such that
$Z=\mathcal{Z}\cap\mathcal{H}$, and a variety $\mathcal{Y}$ bilinked to
$\mathcal{X}$ such that $Y=\mathcal{Y}\cap H$ and
$\mathcal{X}=\mathcal{Y}-aH_{\mathcal{Z}}$. Indeed, the existence and shape of
bilinkage in complete intersections follows from the Betti tables which do not
change via liftings. This combined with the fact that the restriction of a
bilinkage is a bilinkage proves our claim.

We now describe the liftings of the quartics $F$ with Betti tables from Table
6 in Appendix A.

Type $[683]$:

From the Betti table we deduce that the space of quadrics in $F^{\perp}$
defines a finite scheme that is a proper linear section of a variety $Z$ of
minimal degree in codimension $3$, i.e. a rational scroll of degree $4$, if
dim$(Z)>2$. A codimension $1$ AG variety $X$ on $Z$ that lifts $A_{F}$ is a
divisor of degree $14$ whose canonical divisor is a multiple of the hyperplane
section. If dim$(Z)=c$, then the canonical divisor on $Z$ is $-cH+2F$ where
$H$ is a hyperplane section and $F$ is the class of a ruling in the scroll. So
by adjunction, $X$ is a Calabi-Yau threefold if $c=4$ and $X$ is a divisor of
type $4H-2F$ on $Z$. It is bilinked on $Z$ to a quadric threefold, a divisor
of type $H-2F$. A quadric threefold is arithmetic Gorenstein, so therefore
also, by bilinkage, the threefold $X$ is.

Type $[550]$:

From the Betti table we deduce that the space of quadrics in $F^{\perp}$
defines a finite AG scheme which is a proper linear section of $G(2,5)$. The
ideal $F^{\perp}$ is generated by the quadrics and an additional cubic form,
so the $A_{F}$ lifts to the intersection of a cone over $G(2,5)$ with a cubic
hypersurface.

Type $[400]$:

From the Betti table we deduce that $F^{\perp}$ is a complete intersection of
quadrics and hence lifts to any dimension as a smooth intersection of four
quadrics.

Type $[300]$:

There are three types of arithmetic Gorenstein varieties $X$ of degree $17$ in
a complete intersection of three quadrics with this Betti table. They are
distinguished by the rank of the apolar quartic of a generic Artinian
reduction $A_{F}$.

Type $[300a]$:

In the case where the quartic $F$ is of rank $8$, the ideal $F^{\perp}$ is
bilinked in a complete intersection $Z$ of three quadrics to an ideal of
colength $1$. This ideal lifts to a linear space in any dimension. The
bilinkage to $P+2H$ of the linear space $P$ in a complete intersection $Z$ of
three quadrics provides liftings of general schemes of type $A_{F}$ to AG-
varieties. In fact, this family of liftings is unique by Remark 8.1.

Type $[300b]$:

When $Q_{F}$ is a complete intersection $F$ has rank $7$ the ideal $F^{\perp}$
contains the ideal of seven among the eight points defined by the complete
intersection $Q_{F}$ of three quadrics $F^{\perp}$. These seven points can be
lifted to a variety of degree $7$ obtained as the residual component $Y$ of
the intersection of quadrics containing a linear space $P$ of codimension $3$.
The intersection $X_{0}=Y\cap P$ is a cubic hypersurface in $P$ of codimension
4. A lifting of $F$ may be found as a variety $X\in|X_{0}+2H|_{Y}|$ obtained
by bilinkage of $X_{0}$ in $Y$.

A general $F$ lifts smoothly to a surface $X$ by this construction. The
fourfold variety $Y$ in ${\mathbb{P}}^{7}$ has isolated singularities, among
them six on $X_{0}$ and a threefold lifting of $F$ has isolated singularities.

Type $[300c]$:

When the quadric ideal $Q_{F}$ in $F^{\perp}$ defines a line and four points,
then a lifting of $A_{F}$ may be found as a reducible arithmetic Gorenstein
threefold $X$ of degree $17$ in $\mathbb{P}^{7}$. The threefold $X$ lies in
three quadrics, a lifting of $Q_{F}$, that define a ${\mathbb{P}}^{5}$ and a
rational fourfold scroll $Y_{4}$ of degree $4$, and have Betti table of type
$[300]$. A lifting of a set $\Gamma$ of seven points apolar to $F$, to a
fourfold, is the union of $Y_{4}$ and a cubic fourfold $Y_{3}$ in
${\mathbb{P}}^{5}$. The fourfolds $Y_{4}$ and $Y_{3}$ intersect in a threefold
that spans ${\mathbb{P}}^{5}$, so a Segre cubic threefold scroll $X_{3}$. A
reducible threefold $X$ of degree $17$ may now be constructed by a doubling of
$Y_{4}\cup Y_{3}$ as in Example 2.21: Consider $Y_{16}$ a general complete
intersection $(2,2,4)$ that contains $Y_{3}\cup Y_{4}.$ Then $Y_{16}$ is
Gorenstein, with trivial canonical bundle, and $X=(Y_{3}\cup Y_{4})\cap
Y_{9}$, where $Y_{3}\cup Y_{4}\cup Y_{9}=Y_{16}$. In this case $X=X_{6}\cup
X_{11}$, where $X_{6}=X\cap Y_{3}$ is a determinantal threefold of degree $6$,
linked $(3,3)$ to the cubic scroll $X_{3}$ in ${\mathbb{P}}^{5}$, while
$X_{11}=X\cap Y_{4}$ is a threefold of degree $11$, residual to a plane in the
intersection of $Y_{4}$ with a cubic hypersurface.

A particular phenomenon appears in $X$: The intersection $X_{6}\cap X_{11}$ is
an anticanonical divisor on both $X_{6}$ and on $X_{11}$. A general
anticanonical divisor on $X_{6}$ is a complete intersection $K3$ surface of
degree $8$. The intersection $X_{6}\cap X_{11}$, however, is not. It coincides
with the $X_{6}\cap X_{3}$ and is an anticanoncial divisor also on $X_{3}$ ;
it is a $K3$ surface of degree $8$ that is not a complete intersection.

Note that $[300c]$ cannot be constructed by bilaison of height 2 from a
(possibly reducible) cubic threefold on $Y_{4}\cup Y_{3}$ as in $[300b]$.
Indeed, such a biliaison would give an effective divisor in $|-K_{Y_{3}}-2H|$,
but this system is empty. However, we can still find an effective divisor in
$|-K_{Y_{3}}-H|$ (for example using the construction in Example 2.17) and
perform bilinkage from a degree 10 threefold being an effective divisor in
that system.

Type $[320]$:

When $F$ is a general form of type $[320]$, then it is apolar to a pencil of
sets $\Gamma$ of seven points on a twisted cubic curve $C$. On one hand, this
means that $F^{\perp}$ is the sum of ideals of two such sets $\Gamma$. On the
other hand, we show that $F^{\perp}$ is also generated by the $2\times 2$
minors of a $3\times 3$ matrix, with two rows of linear forms and one row of
quadratic forms.

Indeed, we may assume the twisted cubic curve $C$ is smooth and that its ideal
$I_{C}$ is generated by the $2\times 2$ minors of the top $2\times 3$
submatrix of the $3\times 3$ matrix

$M=\begin{pmatrix}x_{0}&x_{1}&x_{2}\\\ x_{1}&x_{2}&x_{3}\\\
q_{1}&q_{2}&q_{3}\end{pmatrix},$

where the $q_{i}$ are quadratic forms. Let
$p_{1}={\mathbb{V}}(x_{0},x_{1},x_{2})$,
$p_{2}={\mathbb{V}}(x_{1},x_{2},x_{3})$, and let $I_{ij}$ be the ideal
generated by the $2\times 2$ minors of the submatrix consisting of the columns
$i$ and $j$ in $M$. Consider the two ideals $J_{12}=(I_{C}+I_{12}):I_{p_{1}}$
and $J_{23}=(I_{C}+I_{23}):I_{p_{2}}$. If none of the quadrics
$q_{1},q_{2},q_{3}$ vanish in $p_{1}$ or $p_{2}$, then both $J_{12}$ and
$J_{23}$ define a scheme of length $7$ on $C$, and if these schemes are
disjoint, the ideal $J_{12}+J_{23}$ coincides with the ideal $I_{M}$ generated
by the $2\times 2$ minors of $M$. Varying $q_{1}$ and $q_{2}$, the ideal
$J_{12}$ moves in a $7$-dimensional family, while fixing $q_{2}$ and varying
$q_{3}$ the ideal $J_{23}$ moves in a $6$-dimensional family. So if $F$ is a
general quartic apolar to $C$, then we may choose the $q_{i}$ such that
${\mathbb{V}}(J_{13})$ and ${\mathbb{V}}(J_{23})$ generate the pencil of
schemes of length $7$ on $C$ that are apolar to $F$. But by the first
characterization, $J_{12}+J_{23}=F^{\perp}$ when $F$ is of type $[320]$, on
the other hand $J_{12}+J_{23}=I_{M}$, so $F^{\perp}=I_{M}$.

In the first characterization, the ideal $F^{\perp}$ lifts to a complete
intersection of divisors of type $(3,2),(1,3)$ in
$\mathbb{P}^{1}\times\mathbb{P}^{2}$ in ${\mathbb{P}}^{5}$. Each of the
divisors $(3,2)$ and $(1,3)$ are surfaces $S$ of degree $7$. For a general $F$
of type $[320]$, there is a surface $S$ of type $(3,2)$, and a surface
$S^{\prime}$ of type $(1,3)$ such that $A_{F}$ is the Artinian reduction of a
curve in the system $|2H-K_{S}|$ respectively $|2H-K_{S^{\prime}}|$.

In the second characterization, the ideal $F^{\perp}$ clearly lifts to the
ideal generated by the $2\times 2$ minors of a $3\times 3$ matrix with two
rows of linear forms and one row of quadratic forms in ${\mathbb{P}}^{7}$ (cf.
[CGKK]*§4.5) .

Type $[200]$:

The ideal $F^{\perp}$ is bilinked in a complete intersection of type $(2,2,3)$
to an Artinian Gorenstein ideal of length $6$, that lifts to any del Pezzo
variety $Y$ of degree $6$. The higher dimensional del Pezzo variety of degree
$6$ split in two kinds: either to
$\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}\subset\mathbb{P}^{7}$
(called Jerry) or to $\mathbb{P}^{2}\times\mathbb{P}^{2}\subset\mathbb{P}^{8}$
(called Tom) each having the same Betti table. By projective normality the
bilinkage extends to these dimensions producing a lifting $X$ of $F$.

######  Proposition 8.2.

A general Artinian Gorenstein ring $A_{F}$ for a quaternary quartic $F$ of
rank $8$ lifts to a halfcanonical curve $C_{F}$ in ${\mathbb{P}}^{5}$ such
that $C_{F}$ is an element of the system $|2H-K_{X}|$ for a surface $X$ of
degree 8 in $\mathbb{P}^{5}$.

###### Proof.

By assumption $F^{\perp}$ contains the ideal $I_{\Gamma}$ of a scheme $\Gamma$
consisting of eight general points lying on a pencil of quadric surfaces. Let
us choose one of these surfaces and denote it by $Q$. Then the ideal of
$\Gamma$ is generated by five bihomogeneous forms of bidegrees
$(2,2),(3,2),(3,2),(2,3),(2,3)$ admitting four syzygies in degree $(3,3)$.
Hence $\Gamma$ arises as the corank 1 locus of a map $4\mathcal{O}_{Q}\to
2\mathcal{O}_{Q}(1,0)\oplus 2\mathcal{O}_{Q}(0,1)\oplus\mathcal{O}_{Q}(1,1)$.
The latter extends to a map
$4\mathcal{O}_{G}\to\mathcal{S}\oplus\mathcal{O}_{G}(1)$ degenerating over a
surface $Y_{2}$ of degree 8 on the Grassmannian $G=G(2,4)$ in its Plücker
embedding equipped with the universal quotient bundle $\mathcal{Q}$. The
surface $Y_{2}$ is described in [Gross]*Thm. 4.1 (g) as $\mathbb{P}^{2}$ blown
up in ten points embedded via the system of proper transforms of sextic curves
passing through six of the points with multiplicity $2$ and the four remaining
ones with multiplicity $1$.

Consider a curve $C$ in the system $|2H-K_{Y_{2}}|$ with $H$ the class of the
hyperplane section on $Y_{2}$. Note that $C$ is then an arithmetic Gorenstein
curve of degree $18$. The resolution of its ideal is a doubling of the
resolution of the ideal of $Y_{2}$ and the Artinian reductions of $C$ are of
type $[200]$. To prove that $A_{F}$ is an Artinian reduction of $C$ it is
enough to prove that the restriction map:

$H^{0}({\mathcal{O}_{Y_{2}}(2H-K_{Y_{2}})})\to
H^{0}({\mathcal{O}_{\Gamma}(2H-K_{Y_{2}})})$

is surjective. We proceed step by step. Let $Y_{1}$ be a hyperplane section of
$Y_{2}$ containing $\Gamma$. Then $Y_{1}$ is a curve of genus $4$ and the
elements of the system $|2H-K_{Y_{1}}|$ are of degree $10$ hence
$H^{1}(\mathcal{O}(2H-K_{Y_{1}}))=H^{0}(2K_{Y_{1}}-2H)=0$. This together with
the sequence

$0\to{\mathcal{O}_{Y_{1}}(2H-K_{Y_{1}})}\to{\mathcal{O}_{Y_{1}}(2H-K_{Y_{2}})}\to{\mathcal{O}_{\Gamma}(2H-K_{Y_{1}})}\to,0$

yields surjectivity of the restriction

$H^{0}({\mathcal{O}_{Y_{1}}(2H-K_{Y_{2}})})\to
H^{0}({\mathcal{O}_{\Gamma}(2H-K_{Y_{2}})}).$

The remaining step is the surjectivity of the map

$H^{0}({\mathcal{O}_{Y_{2}}(2H-K_{Y_{2}})})\to
H^{0}({\mathcal{O}_{Y_{1}}(2H-K_{Y_{2}})}),$

which is implied by the vanishing of
$H^{1}({\mathcal{O}_{Y_{2}}(H-K_{Y_{2}})})$. The latter system is the system
of strict transforms of curve of degree $9$ with six triple points and four
double points in the ten blown up points of $\mathbb{P}^{2}$. This has
vanishing first cohomology for a general configuration of points, which
concludes the proof. ∎

Type $[100]$:

Two families of Calabi-Yau threefold $X$ of degree $19$ with this Betti table
are constructed as a Lagrangian degeneracy locus of a map between two
Lagrangian subbundles of an orthogonal bundle on a smooth quadric of dimension
6 in Proposition 7.12 in Section 7.3.

######  Proposition 8.3.

A general Artinian Gorenstein ring $A_{F}$ for a quaternary quartic $F$ of
rank $9$ lifts to a halfcanonical curve $C_{F}$ in ${\mathbb{P}}^{5}$, a
deformation of any smooth ${\mathbb{P}}^{5}$ section of an AG Calabi-Yau
threefold $Y$ of degree $19$ in ${\mathbb{P}}^{7}$. Moreover, $C_{F}$ is an
element of the system $|2H-K_{X}|$ for a surface $X$ of degree 9 in
$\mathbb{P}^{5}$.

###### Proof.

To show that a general $A_{F}$ has a lifting to a halfcanonical curve in
${\mathbb{P}}^{5}$, we go step by step. By assumption $F^{\perp}$ contains the
ideal $I_{\Gamma}$ of a scheme $\Gamma$ consisting of nine general points
lying on a quadric surface $Q$. The space of $(3,2)$ forms as well as that of
$(2,3)$ forms on $Q$ that vanish on $\Gamma$ are $3$-dimensional, whereas the
space of $(3,3)$ forms vanishing on $\Gamma$ is 7 dimensional. It follows that
there are five syzygies in degree $(3,3)$ between six generators of degree
$(3,2)$ and $(2,3)$. We conclude that $\Gamma$ is the corank 1 locus of a map
$5\mathcal{O}_{Q}\to 3\mathcal{O}_{Q}(0,1)\oplus 3\mathcal{O}_{Q}(1,0)$. Now
$Q$ can be seen as a codimension $2$ linear section of a quadric fourfold. The
latter can be interpreted as a Grassmannian $G=G(2,4)$ in its Plücker
embedding. Clearly the map $5\mathcal{O}_{Q}\to 3\mathcal{O}_{Q}(0,1)\oplus
3\mathcal{O}_{Q}(1,0)$ extends to a map $5\mathcal{O}_{G}\to
3\mathcal{Q}_{G}$. The corank 1 locus of the latter map is a surface of degree
9 that was described by Gross in [Gross]*Thm. 4.2 and provides a lifting of
$\Gamma$.

Denote the surface by $Y_{2}$. It is rational, the embedding of
${\mathbb{P}}^{2}$ blown up in a set of ten points
$\Delta\subset{\mathbb{P}}^{2}$ embedded by the linear system of strict
transforms of plane curves of degree $7$ with double points at $\Delta$. (
loc.cit.). A halfcanonical curve $C$ on $Y_{2}$ in the system $|2H-K_{Y_{2}}|$
is the strict transform of a curve of degree $17$ with multiplicity $5$ along
$\Delta$.

First, on a hyperplane section $Y_{1}$ of $Y_{2}$ there exists
$C_{1}\in|3H-K_{Y_{1}}|$ which is a lifting of $A_{F}$ if on $Y_{1}$ the
global sections of the line bundle ${\mathcal{O}_{Y_{1}}(3H-K_{Y_{1}})}$ map
surjectively to ${\mathcal{O}_{Y_{0}}(|3H-K_{Y_{1}})}$, where $Y_{0}$ is a
${\mathbb{P}}^{3}$ section of $Y$: The global sections
$H^{0}({\mathcal{O}_{Y_{0}}(3H-K_{Y_{1}})})$ is naturally identified with the
space of quaternary forms apolar to $Y_{0}$. Consider now, the cohomology of
the sequence

$0\to{\mathcal{O}}_{Y_{1}}(2H-K_{Y_{1}})\to{\mathcal{O}_{Y_{1}}(3H-K_{Y_{1}})}\to{\mathcal{O}_{Y_{0}}(3H-K_{Y_{1}})}\to
0.$

The curve $Y_{1}$ has genus $5$ and the divisor $2H-K_{Y_{1}}$ on $Y_{1}$ has
degree $10$, so is nonspecial. Therefore the above restriction map on global
sections is surjective and $A_{F}$ lifts to points on $Y_{1}$.

Similarly the cohomology of the exact sequence

$0\to{\mathcal{O}}_{Y_{2}}(H-K_{Y_{2}})\to{\mathcal{O}_{Y_{2}}(2H-K_{Y_{2}})}\to{\mathcal{O}_{Y_{1}}(2H-K_{Y_{2}})}\to
0.$

gives a lifting of $C_{1}$ to $C_{2}\in|2H-K_{Y_{2}}|$. In fact $H-K_{Y_{2}}$
is the class of the strict transform of a plane curve of degree $10$ with
multiplicity $3$ along $\Delta$. For general $\Delta$ the cohomology
$h^{1}({\mathcal{O}}_{Y_{2}}(H-K_{Y_{2}}))=0$. So again the lifting follows. ∎

Type $[000]$:

There are several families of Calabi-Yau threefolds of degreee $20$ whose
general Artinian reduction are quartic forms $F$ of rank $10$. We first
describe briefly a well known complete family of smooth Calabi-Yau threefolds
of degree 20, and then construct another family of nodal non-smoothable
Calabi-Yau threefolds of degree 20. The latter are rationally fibred in
elliptic curves over ${\mathbb{P}}^{2}$.

For the first family, consider the $3\times 3$ minors of a general $4\times 4$
matrix of linear forms in $\mathbb{P}^{7}$. These minors define a Calabi-Yau
threefold of degree $20$. We call these determinantal threefolds. They form a
complete family [CGKK]. But their Artinian reductions are Artinian Gorenstein
rings of regularity $4$ that do not form a complete family. The corresponding
family of quartic forms has codimension $1$ in the space of quartic forms.

The other family consists of threefolds obtained as anticanonical divisors in
fourfolds of degree $10$.

The $3\times 3$ minors of a general $3\times 5$ matrix $A$ of linear forms in
$\mathbb{P}^{7}$ generate the ideal of a fourfold $Y$ of degree 10. This
fourfold is a ${\mathbb{P}}^{2}$-scroll over ${\mathbb{P}}^{2}$. An effective
anticanonical divisor $X$ on $Y$ would be a Calabi-Yau threefold of degree
$20$ with Betti table of type $[000]$. The class of the anticanonical divisor
is $3H-2F$ where $H$ is a hyperplane section and $F$ is the pullback of a line
from ${\mathbb{P}}^{2}$. So any anticanonical divisor would be residual in a
cubic hypersurface section to a scroll in $Y$, the pullback in $Y$ of a conic
in ${\mathbb{P}}^{2}$. If the conic degenerates to two lines, the scroll is
the union of two scrolls over lines in $Y$. Each such scroll over a line is
the rank $1$ locus of a pencil of rows in $A$. It is now an easy
Macaulay2-computation, cf. [M2]*Package QuaternaryQuartics, to check that a
scroll over a conic is not in any cubic hypersurface section. In fact the
space of cubic forms in the ideal of such a scroll is exactly
$10$-dimensional, as expected by Riemann Roch. So these cubic forms coincide
with the $3\times 3$ minors of $A$. Therefore, the anticanonical divisor of
$Y$ is not effective.

If, however $A$ has a rank $1$ point $p$ in ${\mathbb{P}}^{7}$, then $Y$ is
singular, and any ${\mathbb{P}}^{2}$-scroll over a conic would have a non-
normal double point at $p$. By Riemann-Roch it is then expected to lie in $11$
cubic hypersurfaces in ${\mathbb{P}}^{7}$, and hence $Y$ has an anticanonical
divisor in this case. In computations with Macaulay2 ([M2]*Package
QuaternaryQuartics ) we find $Y$ with a single rank $1$ point, and a unique
anticanonical divisor $X$ with two quadratic singularities, one at $p$ and one
at some other point $q$, a smooth point on $Y$. A general
${\mathbb{P}}^{2}$-scroll $Z$ over a line in $Y$ passes through $p$. The
surface of intersection $Z\cap X$ has a quadratic singularity at $p$ if it
does not pass through $q$, and is smooth at $p$ and has a quadratic
singularity at $q$ if it passes through $q$.

We conclude with the following.

######  Proposition 8.4.

Let $Y\subset{\mathbb{P}}^{7}$ be a fourfold defined by the maximal minors of
a $3\times 5$ matrix $A$ of linear forms. If $A$ has rank $\geq 2$ at every
point in ${\mathbb{P}}^{7}$, then the anticanonical divisor of $Y$ is not
effective. If $A$ has rank $1$ at some point, then $Y$ has an effective
anticanonical divisor. For a general $A$ with a rank $1$ point, the
anticanonical system consists of a unique divisor $X$ that is a $2$-nodal AG
Calabi-Yau threefold of degree $20$. Furthermore, $X$ is rationally fibred in
plane elliptic curves, and $X$ is not smoothable.

A general Artinian Gorenstein ring $A_{F}$ of a quaternary quartic lifts to a
halfcanonical curve in ${\mathbb{P}}^{5}$, a deformation of any smooth
${\mathbb{P}}^{5}$ section of the nodal AG Calabi-Yau threefold of degree 20
in ${\mathbb{P}}^{7}$.

###### Proof.

We observe that the net of rows in $A$ defines $Y$ as a ${\mathbb{P}}^{2}$
scroll over ${\mathbb{P}}^{2}$. The rank $1$ point is a common point of a
pencil of planes in the scroll, and the planes intersect the anticanonical
divisor in cubic curves, so $X$ has a rational ellipic fibration over
${\mathbb{P}}^{2}$. Furthermore, if $Z$ is a general scroll over a line that
passes through $q$, then the surface of intersection $S=X\cap Z$ is smooth at
the rank $1$ point and has a quadratic singularity at $q$, so it is not
Cartier as a divisor at the rank $1$ point on $X$. It is Cartier elsewhere, so
it follows by a result by R. Friedman [Friedman]*Rmk. 4.5, that $X$ is not
smoothable.

It remains then to check liftings of $A_{F}$ for general quartics $F$. An
Artinian reduction of the nodal threefold $X$ is $A_{F}$ for a quartic form
$F$ that is apolar to a determinantal scheme of length $10$ defined by the
cubic minors of a $3\times 5$ matrix of linear forms. The locus of these
schemes in the Hilbert scheme has dimension $27$. There is no quartic singular
along a general length $10$ scheme of this kind, so by Terracini the set of
quartic forms that are apolar to such a scheme is dense in the space of
quartics. In particular a general quartic is apolar to a surface of
$10$-tuples of points that are defined by the cubic minors of a $3\times 5$
matrix of linear forms.

Let $Y_{0}$ be a ${\mathbb{P}}^{3}$ section of $Y$, then
$H^{0}({\mathcal{O}_{Y_{0}}(X)})$ is naturally identified with the space of
quaternary quartic forms apolar to $Y_{0}$. The Artinian reduction of $X$ as a
divisor in $Y$ is then the image of the section $s\in
H^{0}({\mathcal{O}_{Y}(X)})$, that defines $X$, by the restriction map

$H^{0}({\mathcal{O}_{Y}(X)})\to H^{0}({\mathcal{O}_{Y_{0}}(X)}).$

Now any determinantal scheme of length $10$ defined by the cubic minors of a
$3\times 5$ matrix is a ${\mathbb{P}}^{3}$ section of a determinantal variety
$Y\subset{\mathbb{P}}^{N}$ for any $N>3$. Therefore, to show that a general
$A_{F}$ has a lifting to a halfcanonical curve, we show that the restriction
map

$H^{0}({\mathcal{O}_{Y_{2}}(C)})\to H^{0}({\mathcal{O}_{Y_{0}}(C)})$

is surjective for a general determinantal surface $Y_{2}$, a smooth
intersection of $Y$ with a ${\mathbb{P}}^{5}$ that contains $Y_{0}$, where
$C\subset Y_{2}$ is a halfcanonical curve, i.e. $C\equiv 2H-K_{Y_{2}}$ on
$Y_{2}$.

The surjectivity of the restriction map is best analyzed step by step. If
$Y_{1}\subset Y_{2}$ is a smooth hyperplane section that contains $Y_{0}$,
then $H^{0}({\mathcal{O}_{Y_{2}}(C)})\to H^{0}({\mathcal{O}_{Y_{0}}(C)})$ is
surjective, if both $H^{0}({\mathcal{O}_{Y_{2}}(C)})\to
H^{0}({\mathcal{O}_{Y_{1}}(C)})$ and $H^{0}({\mathcal{O}_{Y_{1}}(C)})\to
H^{0}({\mathcal{O}_{Y_{0}}(C)})$ are surjective.

By the cohomology of the exact sequences

$0\to{\mathcal{O}}_{Y_{2}}(C-H)\to{\mathcal{O}_{Y_{2}}(C)}\to{\mathcal{O}_{Y_{1}}(C)}\to
0$

and

$0\to{\mathcal{O}}_{Y_{1}}(C-H)\to{\mathcal{O}_{Y_{1}}(C)}\to{\mathcal{O}_{Y_{0}}(C)}\to
0,$

the surjectivity then follows if both

$H^{1}({\mathcal{O}_{Y_{2}}(C-H)})=0\quad{\rm and}\quad
H^{1}({\mathcal{O}_{Y_{1}}(C-H)})=0.$

Now, the surface $Y_{2}$ is the rank $2$ locus of a $3\times 5$ matrix with
entries in the $6$-dimensional vector space $W$ of linear forms on
${\mathbb{P}}^{5}$. Interpreting each point on $Y_{2}$ as map
$\phi:{\mathbb{C}}^{3}\to{\mathbb{C}}^{5}$, there is a birational morphism
$\phi\mapsto{\rm ker}(\phi)$ to ${\mathbb{P}}^{2}$. The inverse of this
morphism is the map on ${\mathbb{P}}^{2}$ defined by the $5\times 5$ minors of
a $5\times 6$ matrix with linear forms. The map extends to a morphism on
${\mathbb{P}}^{2}$ blown up in the fifteen points
$\Gamma\subset{\mathbb{P}}^{2}$, the common zeros of the $5\times 5$ minors.

A halfcanonical curve $C$ on $Y_{2}$ is the strict transform of a plane curve
of degree $13$ with multiplicity $3$ along $\Gamma$. Therefore a section
$H^{0}({\mathcal{O}_{Y_{2}}(C-H)})$ defines the strict transform of a plane
curve of degree $8$ with multiplicity $2$ along $\Gamma$. But, by Riemann
Roch, such a curve exists if and only if
$H^{1}({\mathcal{O}_{Y_{2}}(C-H)})\not=0$. The existence of such a curve is
therefore a closed condition on $\Gamma$. A computation in Macaulay2,
[M2]*Package QuaternaryQuartics, shows that there are no plane curve of degree
$8$ with multiplicity $2$ along a random determinantal $\Gamma$, so we
conclude that $H^{1}({\mathcal{O}_{Y_{2}}(C-H)})=0$ for a general $\Gamma$.

It remains to show that $H^{1}({\mathcal{O}_{Y_{1}}(C-H)})=0$ for a general
hyperplane section $Y_{1}$ of $Y_{2}$. The curve $Y_{1}$ has genus $6$ and the
divisor $C-H$ on $Y_{1}$ has degree $10$, so
$H^{1}({\mathcal{O}}_{Y_{1}}(C-H))\not=0$ if and only if
${\mathcal{O}}_{Y_{1}}(C-H)$ is the canonical sheaf on $Y_{1}$. But $Y_{1}$ is
the strict transform on $Y_{2}$ of a smooth plane quintic $\bar{Y}_{1}$ curve
that contains $\Gamma$. So a the canonical sheaf on $Y_{1}$ is the restriction
of ${\mathcal{O}}_{{\mathbb{P}}^{2}}(2L)$ to $Y_{1}$, where $L$ is the class
of a line in ${\mathbb{P}}^{2}$. In particular, ${\mathcal{O}}_{Y_{1}}(C-H)$
is canonical if and only if
${\mathcal{O}}_{Y_{1}}(C-H-2L)={\mathcal{O}}_{Y_{1}}$. Curves in the class of
$C-H-2L$ are strict transforms on $Y_{2}$ of plane curves of degree $6$ with
multiplicity $2$ along $\Gamma$. Their restriction to $\bar{Y}_{1}$ has
multiplicity $2$ at the points of $\Gamma$, so the line bundle
${\mathcal{O}}_{Y_{1}}(C-H-2L)$ is trivial, i.e. has a nonzero section if and
only if there is a plane curve of degree $6$ tangent to $\bar{Y}_{1}$ along
$\Gamma$. This occurs, only if the $5\times 6$ matrix defining the map into
${\mathbb{P}}^{5}$ has a symmetric $5\times 5$ submatrix that defines $Y_{1}$,
or equivalently ${\mathcal{O}}_{Y_{1}}(3L-\Gamma)$ is a $2$-torsion element in
the Picard group of $Y_{1}$. For a general, hyperplane section $Y_{1}$ this is
not the case. We may conclude that a general $A_{F}$ lifts to points on
$Y_{1}$, and hence also to a halfcanonical curve on $Y_{2}.$ ∎

Note that Calabi-Yau threefolds defined by $3\times 3$ minors of a $4\times 4$
matrix with linear forms are lifting of special AG rings with Betti table
$[000]$.

######  Question 8.5.

Describe the quartics corresponding to AG rings of type $[000]$ that lift to
smooth threefolds.

In this section we have considered liftings of apolar rings $A_{F}$ to smooth
AG varieties, in particular to smooth threefolds. In some cases, the liftings
extends to higher dimensions. We first summarize the possible liftings of
$A_{F}$ to curves $C_{F}$:

######  Proposition 8.6.

Let $F$ be a quartic form of a type appearing in Appendix A, Table 6. Assume
that $F$ is general in its family. Then there exists a configuration of
distinct points $\Gamma$ apolar to $F$ with the following property: The set of
points $\Gamma$ admits a lifting to a surface $S_{\Gamma}$ that contains a
curve $C_{F}\subset S_{\Gamma}$ such that $A_{F}$ is an Artinian reduction of
$C_{F}$. Moreover, we have two possibilities:

  1. (1)

If $F$ is not of type $[300a]$ then $C_{F}\in|2H-K_{S_{\Gamma}}|$.

  2. (2)

If $F$ is of type $[300a]$, then $S_{\Gamma}$ is a K3 surface complete
intersection of three quadrics containing a line $l$ and
$C_{F}\in|2H-K_{S_{\Gamma}}+l|=|2H+l|$.

###### Proof.

We let $\Gamma$ be a finite set of points that computes the rank of $F$ as
listed in Table 3. The Betti table of $S/\Gamma$, is then listed in Table 1,
from which a lifting of $\Gamma$ to a surface $S_{\Gamma}$ is straightforward
and may be summarized as follows:

Type $[683]$:

$S_{\Gamma}$ is a rational scroll of degree $4$.

Type $[550]$:

$S_{\Gamma}$ is an anticanonical del Pezzo surface of degree $5$.

Type $[400]$:

$S_{\Gamma}$ is a complete intersection of three quadrics.

Type $[300a]$:

$S_{\Gamma}$ is a complete intersection of three quadrics containing a line.

Type $[300b]$:

$S_{\Gamma}$ is a scroll of degree $7$ obtained as the residual component of
the intersection of three quadrics contianing a $\mathbb{P}^{2}$.

Type $[320]$:

$S_{\Gamma}$ is a surface of degree $7$ described as a divisor of type $(3,2)$
on ${\mathbb{P}}^{1}\times{\mathbb{P}}^{2}$ in its Segre embedding.

Type $[200]$:

$S_{\Gamma}$ is a rational surface of degree $8$ described as the image of
$\mathbb{P}^{2}$ through the map induced by the system of sextics with six
fixed double points and passing through additional four fixed points.

Type $[100]$:

$S_{\Gamma}$ is a rational surface of degree $9$ described as the image of
$\mathbb{P}^{2}$ through the map induced by the system of septics with $10$
fixed double points.

Type $[000]$:

$S_{\Gamma}$ is a surface defined as the corank $1$ locus of a $5\times 3$
matrix of linear forms on $\mathbb{P}^{5}$. It is a rational surface of degree
$10$ described as the image of $\mathbb{P}^{2}$ through the map induced by the
system of $5\times 5$ minors of a $5\times 6$ matrix of linear forms on
$\mathbb{P}^{2}$.

In each of these cases the system $|2H-K_{S_{\Gamma}}|$ is base point free and
its general elements are arithmetic Gorenstein manifolds whose resolution is a
doubling of the resolution of $S_{\Gamma}$. The fact that a general quartic in
the families are obtained follows from the surjectivity of the restriction
maps as in the cases type $[100]$ and type $[200]$ computed explicitly above.

In the case of type $[300b]$ we check by the adjunction formula that $C_{F}$
is an AG curve and since it is bilinked to $l$ in a complete intersection of
quadrics it is of type $[300b]$. To see that in this way we obtain a general
$F$ in the family it is enough to observe that the bilinkage construction of
$A_{F}$ can be lifted to a bilinkage construction of $C_{F}$. ∎

The lifting of $A_{F}$ to smooth varieties of higher dimensions is summarized
in the following proposition.

######  Corollary 8.7.

Let $F$ be a quartic form of type $[683]$, $[550]$, $[320]$, $[300a]$,
$[300b]$ or $[200]$. Assume that $F$ is general in its family. Then there
exists a smooth AG variety $X$ of dimension $d$ such that $A_{F}$ is an
Artinian reduction of $X_{F}$, where $d=3,6,4,3,2,4$ respectively.

###### Proof.

We consider the constructions of surfaces $S_{\Gamma}$ and curves $C_{F}$ of
Proposition 8.6. In cases $[683]$, $[550]$ and $[320]$ the surface
$S_{\Gamma}$ and the curve $C_{F}$ are either determinantal varieties or
divisors in determinantal varieties. The obstruction to smooth liftings to
higher dimensions is the singular locus of the determinantal varieties. In
cases $[683]$, $[550]$, $[320]$, the singularities of the determinantal
variety allow smooth liftings of $A_{F}$ to dimensions $3,6,4$, respectively.

In the cases $[300a]$ and $[300b]$ we may lift $\Gamma$ further to a complete
intersection $Y_{8}$ of type $(2,2,2)$ and a codimension $3$ variety $Y_{7}$
linked to a codimension $3$ linear space $P$ in a complete intersection
$(2,2,2)$, respectively. A lifting of $A_{F}$ is obtained by bilinkage of a
codimension $4$ linear space $P^{\prime}$ in $Y_{8}$, and the variety
$Y_{7}\cap P$, respectively. Obstructions to a smooth lifting, are the
singularities of $Y_{8}$ in $P^{\prime}$ and of $Y_{7}$ in $P$, respectively.
Smooth liftings to a threefold of $F$ of type $[300a]$ was found in [CGKK],
while a lifting of $F$ of type $[300b]$ to a threefold has isolated
singularities, as seen above.

Similarly, in case $[200]$ a general $A_{F}$ lifts to a Fano fourfold bilinked
to $\mathbb{P}^{2}\times\mathbb{P}^{2}$ in a complete intersection $(2,2,3)$
and the same types of obstructions to further lifting arise. ∎

##  9\. Reducible liftings

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. We recover these liftings in the following constructions of liftings to AG
threefolds in ${\mathbb{P}}^{7}$.

Any lifting of a ring $A_{F}$ of type $[300c]$ is reducible; this case was
treated in the previous section. For liftings of rings $A_{F}$ considered in
this section we have:

######  Proposition 9.1.

Every Betti table appearing in Table 7 can be realized as the Betti table of
an AG threefold in $\mathbb{P}^{7}$. The threefold is irreducible but singular
for the type $[420]$, in all other cases it is reducible.

###### Proof.

In Theorem 2.3 of [SSY], it is shown that every Betti table in Table 7 of
Appendix A (except for type $[420]$) cannot be the Betti table of an
irreducible threefold. We construct the liftings case by case, and confirm the
constructions by checking a random example using Macaulay2, cf. [M2]*Package
QuaternaryQuartics .

Type $[210]$:

There exists an AG variety $X$ of codimension 4 in $\mathbb{P}^{7}$ that lifts
$A_{F}$ for quartic forms $F$ with Betti table of type $[210]$. The variety
$X$ is reducible with three components, $X_{1},X_{6}$ and $X_{11}$ of degrees
$1,6$ and $11$ respectively. The variety $X_{11}$ is linked to a threefold
$X^{\prime}_{7}$ in a complete intersection of degrees $(1,2,3,3)$, and
$X^{\prime}_{7}$ is linked to a $\mathbb{P}^{3}$ in a complete intersection of
degrees $(1,2,2,2)$. The threefold $X_{6}$ is a complete intersection
$(1,1,2,3)$ and $X_{1}$ is a $\mathbb{P}^{3}$. The space $X_{1}$ intersects
both $X_{6}$ and $X_{11}$ in a quadric surface, while $X_{11}$ intersect
$X_{6}$ in a complete intersection surface of degrees $(1,1,1,2,2)$.

Here is a more detailed description of a lifting of $A_{F}$ to an AG curve. A
general ${\mathbb{P}}^{5}$ intersects the three components of the threefold
$X$ in curves $L,C_{6}$ and $C_{11}$ respectively. The curve $C_{11}$ is
linked $(3,3)$ on a quadric threefold $Q$ to a curve of degree $7$ which again
is linked $(2,2)$ on $Q$ to a line. The curve $C_{11}$ lie in five cubics on
$Q$. The curve $C_{6}$ is cubic section of a quadric surface $S_{2}$. The
curves $C_{11}$ and $C_{6}$ intersect in four coplanar points. The line $L$
lies in this plane $S_{1}\subset Q$, and intersects $S_{2}$ in a conic. The
line $L$ is a secant line to both $C_{11}$ and $C_{6}$. There is a pencil of
cubic sections of $Q$ that contain $C_{11}\cup S_{1}$. These sections are
reducible, in a quintic surface and the plane $S_{1}$. Let $S_{5}$ be one of
the quintic surfaces. Then $C_{11}\subset S_{5}$, $L\subset S_{1}$ and
$C_{6}\subset S_{2}$. Furthermore, $S_{5}\cap S_{1}=C$ is a conic and
$S_{1}\cap S_{2}=C^{\prime}$ is a conic. The curve $C_{11}$ is linearly
equivalent to $2H-K_{S_{5}}-C$ on $S_{5}$. The line $L$ is linearly equivalent
to $2H-K_{S_{1}}-C-C^{\prime}$ on $S_{1}$. The curve $C_{6}$ is linearly
equivalent to $2H-K_{S_{2}}-C^{\prime}$ on $S_{2}$. And $S_{\Gamma}=S_{5}\cup
S_{1}\cup S_{2}$ is an arithmetic Cohen-Macaulay surface of degree $8$ that
contain the AG curve $C_{F}=C_{11}\cup L\cup C_{6}$. The curve $C_{F}$ has
Artinian reduction $A_{F}$ for a quartic form $F$ of type $[210]$.

Type $[310]$:

There exists an AG variety $X$ of codimension 4 in $\mathbb{P}^{7}$ that lifts
$A_{F}$ for quartics $F$ with Betti table of type $[310]$. The variety $X$ is
reducible with two components, $X_{6}$ and $X_{11}$ of degrees $6$ and $11$
respectively. The variety $X_{11}$ is linked to a threefold $Y_{7}$ in a
complete intersection of degrees $(1,2,3,3)$, and $Y_{7}$ is linked to a
$\mathbb{P}^{3}$ in a complete intersection of degrees $(1,2,2,2)$. The
threefold $X_{6}$ is a complete intersection $(1,1,2,3)$ and intersects
$X_{11}$ in a hyperplane section of $X_{6}$.

Here is a more detailed description of a lifting of $A_{F}$ to an AG curve.
Consider a rational surface $S_{5}$ of degree $5$ in a ${\mathbb{P}}^{4}$
linked to a plane in the complete intersection $(2,3)$ and a quadric surface
$S_{2}$ in a ${\mathbb{P}}^{3}$, so that $S_{5}\cup S_{2}$ spans a
${\mathbb{P}}^{5}$, and $S_{5}\cap S_{2}$ is a conic section $C$. Assume
furthermore that the conic $C$ moves in a pencil on $S_{5}$. Let $C_{11}$ be a
curve linearly equivalent to $2H-K_{S_{5}}-C$ on $S_{5}$ that intersects $C$
properly, i.e. in a scheme of length $6$. Let $C_{6}$ be a curve linearly
equivalent to $2H-K_{S_{2}}-C$ such that $C_{6}\cap C=C_{11}\cap C$. Then
$S_{\Gamma}=S_{5}\cup S_{2}$ is arithmetic Cohen-Macaulay and
$C_{F}=C_{11}\cup C_{6}\subset S_{\Gamma}$ is arithmetic Gorenstein and has
Artinian reduction $A_{F}$ for a quartic form $F$ of type $[310]$.

######  Proposition 9.2.

Type $[310]$ gives a negative answer to Reid’s question from [ReidGeneral]:
“For odd $k$, are hypersurfaces in a codimension 3 Gorenstein variety the only
cases?”

###### Proof.

If these were the only cases, then the hypersurface ${\mathbb{V}}(f)$ would
have to have $f$ a nonzero divisor of degree $2$ or $3$.

In the former case, the free resolution of $I\setminus f$ has Betti table of
the form below, where $*$ denotes an unknown entry:

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

The linear syzygy on the two remaining quadrics must persist, for if not, this
would imply that $f$ is a zero divisor. Since the Pfaffians are unmixed of
codimension $3$, quotienting by a zero divisor $f$ cannot yield a codimension
$4$ ideal. Taking the mapping cone would then force $b_{25}\geq 5$, which is
inconsistent with the Betti table of type $[310]$. Similarly, if $f$ is a
cubic, the Betti table of $I\setminus f$ is

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

The mapping cone construction yields $b_{25}\geq 4$, which is again
inconsistent with a Betti table of type $[310]$. For an irreducible threefold
providing a negative answer to Reid’s question, reason as above with type
$[200]$. ∎

Type $[430]$:

There exists an AG variety $X$ of codimension 4 in $\mathbb{P}^{7}$ that lifts
$A_{F}$ for quartic forms $F$ with Betti table of type $[430]$. The variety
$X$ is reducible with three components, $X_{1},X_{6}$ and $X_{9}$ of degrees
$1,6$ and $9$ respectively. The union $X_{1}\cup X_{9}$ is linked to a quadric
threefold $Y_{2}$ in a complete intersection of degrees $(1,2,2,3)$, such that
$X_{1}\cap Y_{2}$ is a quadric surface. The threefold $X_{6}$ is a complete
intersection $(1,1,2,3)$ and intersects $X_{1}\cup X_{9}$ in a hyperplane
section of $X_{6}$.

Here is a more detailed description of a lifting of $A_{F}$ to an AG curve.
Consider a plane $S_{1}$ and a ${\mathbb{P}}^{4}$ and a ${\mathbb{P}}^{3}$ in
a ${\mathbb{P}}^{5}$ that intersect in $S_{1}$. Let
$S_{3}\subset{\mathbb{P}}^{4}$ be a cubic surface scroll that intersect
$S_{1}$ in a conic $C$. Let $C_{9}\subset S_{3}$ the linked to $C\cup L$ in a
cubic section of $S_{3}\cup S_{1}$. Let $S_{2}\subset{\mathbb{P}}^{3}$ be a
quadric surface, and let $C^{\prime}=S_{1}\cap S_{2}$. Then $C^{\prime}\cap
L\cup C_{9}$ is a complete intersection $(2,3)$ in $S_{1}$. Let $S_{6}$ be a
general cubic section of $S_{2}$ that contains $C^{\prime}\cap L\cup C_{9}$.
Then $S_{1}\cap(S_{3}\cup S_{2})=C\cup C^{\prime}$, $S_{3}\cap(S_{1}\cup
S_{2})=C$ and $S_{2}\cap(S_{3}\cup S_{1})=C^{\prime}$, and therefore $C_{1}$
is linearly equivalent to $2H-K-C-C^{\prime}$ on $S_{1}$, $C_{9}$ is linearly
equivalent to $2H-K-C$ on $S_{3}$, and $C_{1}$ is linearly equivalent to
$2H-K-C^{\prime}$ on $S_{2}$.

Furthermore $S_{\Gamma}=S_{3}\cup S_{1}\cup S_{2}$ is arithmetic Cohen-
Macaulay and $C_{F}=C_{9}\cup L\cup C_{6}\subset S_{\Gamma}$ is AG and has
Artinian reduction $A_{F}$ for a quartic form $F$ of this type.

Type $[420]$:

There exists an AG variety $X$ of codimension 4 in $\mathbb{P}^{7}$ that lifts
$A_{F}$ for quartic forms $F$ with Betti table of type $[420]$. The variety
$X$ is irreducible of degree $16$. It is linked to a quadric threefold in a
complete intersection $(2,3)$ on a fivefold cubic scroll. $X$ is singular in
its intersection with the vertex of the scroll.

Here is a more detailed description of a lifting of $A_{F}$ to an AG curve.
Let $S_{6}$ be a general quadric hypersurface section of
${\mathbb{P}}^{1}\times{\mathbb{P}}^{2}$ in its Segre embedding in
${\mathbb{P}}^{5}$. It is a conic bundle over ${\mathbb{P}}^{1}$. Let
$C\subset S_{6}$ be a conic, and let $C_{16}$ be the residual curve to $C$ in
a general cubic section of $S_{6}$ that contains $C$. Then $C_{16}$ is
linearly equivalent to $2H-K_{S_{6}}$ on $S_{6}$. Furthermore $S_{6}$ is
arithmetic Cohen-Macaulay and $C_{16}\subset S_{6}$ is arithmetic Gorenstein
and has Artinian reduction $A_{F}$ for a quartic form $F$ of this type.

Type $[331]$:

There exists an AG variety $X$ of codimension 4 in $\mathbb{P}^{7}$ that lifts
$A_{F}$ for quartic forms $F$ with Betti table of type $[331]$. The variety
$X$ is reducible with two components, $X_{4}$ and $X_{13}$ of degrees $4$ and
$13$ respectively. The threefold $X_{13}$ is linked to a $\mathbb{P}^{3}$ in
the cubic pfaffians of a $7\times 7$ skew matrix, and $X_{4}$ is a quartic
threefold in a $\mathbb{P}^{4}$. The two components of $X$ intersect in a
hyperplane section of $X_{4}$.

Here is a more detailed description of a lifting of $A_{F}$ to an AG curve.
Let $C_{13}\subset{\mathbb{P}}^{4}$ be a smooth curve of degree $13$ residual
to a line $L$ in the zero locus of the cubic pfaffians of a $7\times 7$ skew
symmetric matrix $M$ with linear forms. Then $C_{13}$ is contained in a
surface $S_{6}$ of degree $6$ defined by the $3\times 3$ minors of a $3\times
4$ submatrix of $M$. Furthermore, $L\cap C_{13}$ is a scheme of length $4$.
Let $C_{4}$ be a plane quartic curve that contains $L\cap C_{13}$, and let
$S_{1}$ be the plane of $C_{4}$. Then $C_{F}=C_{13}\cup C_{4}$ is a curve in
the reducible surface $S_{\Gamma}=S_{6}\cup S_{1}$ that spans
${\mathbb{P}}^{5}$.

Then $C_{13}$ is linearly equivalent to $2H-K_{S_{6}}-L$ on $S_{6}$, while
$C_{4}$ is linearly equivalent to $2H-K_{S_{1}}-L$ on $S_{1}$. Furthermore
$S_{\Gamma}$ is arithmetic Cohen-Macaulay and $C_{F}\subset S_{\Gamma}$ is
arithmetic Gorenstein and has Artinian reduction $A_{F}$ for a quartic form
$F$ of this type.

Type $[441a]$:

There exists an AG variety $X$ of codimension 4 in $\mathbb{P}^{7}$ that lifts
$A_{F}$ for quartic forms $F$ with Betti table of type $[441a]$. The quadrics
in $F^{\perp}$ define a conic and a point.

The threefold $X$ is reducible with two components, $X_{4}$ and $X_{12}$ of
degrees $4$ and $12$ respectively. The threefold $X_{12}$ is the rank $1$
locus of a $2\times 4$ matrix $M$ with two columns of linear forms and two
columns of quadratic forms in a $\mathbb{P}^{6}$. The quadric minor defines a
quadric hypersurface that reduces to a conic for $F$. The component $X_{4}$ is
a quartic threefold in a $\mathbb{P}^{4}$ that intersects $X_{12}$ in a
hyperplane section of $X_{4}$. The $\mathbb{P}^{4}$ reduces to a point for
$F$. We check using Macaulay2, [M2]*Package QuaternaryQuartics, on a random
example that the Betti table of $X$ is of this type.

Restricted to a general ${\mathbb{P}}^{5}$, a submatrix of $M$ consisting of
two columns of linear forms and one column of quadratic forms has rank $1$ on
a surface $S_{5}$ of degree $5$ in a ${\mathbb{P}}^{4}$. The surface $S_{5}$
is a conic bundle with some reducible conics. The ${\mathbb{P}}^{4}$ of the
quartic threefold $X_{4}$ intersects $S_{5}$ in a line $L$, and the
${\mathbb{P}}^{5}$ in a plane $S_{1}$. The restriction of $X_{12}$ to $S_{5}$
is a curve $C_{12}$ linearly equivalent to $2H-K_{S_{5}}-L$ on $S_{5}$, while
the restriction of $X_{4}$ to $S_{1}$ is a curve $C_{4}$, linearly equivalent
to $2H-K_{S_{1}}-L$ on $S_{1}$.

Type $[441b]$:

There exists an AG variety $X$ of codimension 4 in $\mathbb{P}^{7}$ that lifts
$A_{F}$ for quartic forms $F$ with Betti table of type $[441b]$. The quadrics
in $F^{\perp}$ define two skew lines.

The threefold $X$ is reducible with two components, $X_{8}$ and $Y_{8}$ both
of degrees $8$. Both components are linked to a common $\mathbb{P}^{3}$ in
complete intersections of degrees $(1,1,3,3)$, and they intersect along a
quartic surface in the common $\mathbb{P}^{3}$. The two linear forms define
${\mathbb{P}}^{5}$’s for each component that reduces to the two skew lines for
$F$.

Restricted to a general ${\mathbb{P}}^{5}$, each of the two components $X_{8}$
and $Y_{8}$, restrict to curves $C_{8}$ and $C^{\prime}_{8}$ respectively of
degree $8$, each in a ${\mathbb{P}}^{3}$. The two ${\mathbb{P}}^{3}$’s
intersect in a line $L$, and each of the two curves lie in a pencil of cubic
surfaces that contain that line. Let $S_{3}$ and $S^{\prime}_{3}$ be two such
surfaces, such that $C_{F}=C_{8}\cup C^{\prime}_{8}$ is contained in
$S_{F}=S_{3}\cup S^{\prime}_{3}$. Then $S_{3}\cap S^{\prime}_{3}=L$ and
$C_{8}$ is linearly equivalent to $2H-K_{S_{3}}-L$ on $S_{3}$, while
$C^{\prime}_{8}$ is linearly equivalent to $2H-K_{S^{\prime}_{3}}-L$ on
$S^{\prime}_{3}$.

Type $[551]$:

There exists an AG variety $X$ of codimension $4$ in $\mathbb{P}^{n}$ that
lifts $A_{F}$ for quartic forms $F$ with Betti table of this type.

Consider the variety $Y$ that is the intersection of five quadrics and
decomposes as the union of a linear space $P$ of codimension $3$ and a
complete intersection $Q$ of type $(2,2)$ in a hyperplane $H$, such that
$Q\cap P$ is the linear space $H\cap P$ of codimension $4$. Take a general
cubic hypersurface $C$ containing $P$. Then $Z=\overline{Y\cap C\setminus P}$
is a variety of codimension $4$ and degree $11$. Let finally $D$ be a quartic
hypersurface containing $Z$, then $X=Y\cap C\cap D$ is a variety of degree
$15$ decomposing as the union of a quartic hypersurface $D\cap P$ and the
variety $Z$ of degree $11$.

A general ${\mathbb{P}}^{5}$ intersects the fourfold $Q$ in a surface $S_{4}$
in a ${\mathbb{P}}^{4}$ and $P$ in a plane $S_{1}$, such that $S_{1}\cap
S_{4}$ is a line $L$. In a cubic section containing $L$, the residual curve
$C_{11}$ on $S_{4}$ has degree $11$. The quartic hypersurface $D$ intersects.
$S_{1}$ in a curve $C_{4}$ of degree $4$. Thus $C_{F}=C_{11}\cup C_{4}$ is
contained in $S_{F}=S_{4}\cup S_{1}$ and $C_{11}$ is linearly equivalent to
$2H-K_{S_{4}}-L$ on $S_{4}$, while $C_{4}$ is linearly equivalent to
$2H-K_{S_{1}}-L$ on $S_{1}$.

Type $[562]$:

There exist two families (I and II) of AG varieties $X$ of codimension $4$ in
$\mathbb{P}^{7}$ that lift $A_{F}$ for quartic forms $F$ with Betti table of
type $[562]$. Note that the quadrics in $F^{\perp}$ define a line and two
points that altogether span ${\mathbb{P}}^{3}$, and that $F$ is apolar to the
union of a set of three points on the line and the two points outside.
Therefore the forms of this family are projectively equivalent. So the
different AG-varieties $X$ all have the same Artinian reduction.

In this case a general Artinian algebra $A_{F}$ with this Betti table lift to
AG-point sets in both families.

The general member of the first family (I) is a reducible threefold

$X=X_{7}\cup X_{8}$

where the components $X_{7}$ and $X_{8}$ have degree $7$ and $8$ respectively.
The threefold $X_{8}$ is linked to a $\mathbb{P}^{3}$ in a complete
intersection $(1,1,3,3)$, while $X_{7}$ is linked to a $\mathbb{P}^{3}$ in a
complete intersection $(1,1,2,4)$.

The two components intersect in a quartic surface in the intersection of their
spans.

The general member of the second family (II) is a reducible threefold

$X=X_{7}\cup X_{4}\cup X^{\prime}_{4},$

where the components $X_{7},X_{4}$ and $X^{\prime}_{4}$ have degrees $7,4$ and
$4$ respectively. The threefold $X_{7}$ is linked to a reducible quadric
$Y_{2}=Y_{1}\cup Y^{\prime}_{1}$ in a complete intersection $(1,1,3,3)$, while
$X_{4}$ is a quartic threefold in a ${\mathbb{P}}^{4}$ that contains the
quartic surface $(X_{7}\cap Y_{1})\cup(Y_{1}\cap Y^{\prime}_{1})$, and
$X^{\prime}_{4}$ is a quartic threefold in a ${\mathbb{P}}^{4}$ and contains
the quartic surface $(X_{7}\cap Y^{\prime}_{1})\cup(Y_{1}\cap
Y^{\prime}_{1})$. The span of $X_{7}$ is a ${\mathbb{P}}^{5}$, while the two
${\mathbb{P}}^{4}$’s of $X_{4}$ and $X^{\prime}_{4}$ each intersect
${\mathbb{P}}^{5}$ in a ${\mathbb{P}}^{3}$ and each other in a
${\mathbb{P}}^{2}$. The threefold $X$, of course, spans $\mathbb{P}^{7}$.

It is instructive to see the two distinct types of liftings (I and II) of type
$[562]$ to AG sets of points $X\subset{\mathbb{P}}^{4}$. A general member of
the first family (I) restricts to a general ${\mathbb{P}}^{4}$ as the union
$\Gamma=\Gamma_{1}\cup\Gamma_{2}$ of two planar sets of points. The set
$\Gamma_{1}$ in one plane is eight points that are linked to the intersection
point of the two planes in a complete intersection of two cubics. The set
$\Gamma_{2}$ in the other plane is seven points that are linked to the
intersection point in a conic and a quartic. In particular $\Gamma$ lies in
the union of a plane cubic curve and a plane conic that meet at a point, the
eight points $\Gamma_{1}$ on the cubic, the seven points $\Gamma_{2}$ on the
conic. The eight points together with the intersection point is the complete
intersection with another cubic.

The general member of the other family (II) restricts to a general
${\mathbb{P}}^{4}$ as a set of fifteen points
$\Gamma=\Gamma_{1}\cup\Gamma_{2}\cup\Gamma_{3}$. The set $\Gamma_{1}$ is seven
points that lie in a plane. Each of the two sets $\Gamma_{2}$ and $\Gamma_{3}$
consists of four points that lie in a line. The two lines are skew and meet
the plane of $\Gamma_{1}$ in two points that are linked in the plane to
$\Gamma_{1}$ in a complete intersection of two cubics.

In particular $\Gamma$ lies in the union of a plane cubic curve and two skew
lines that both intersect the plane cubic curve. The seven points on the cubic
curve together with the intersection points with the lines is the complete
intersection with another cubic.

A Macaulay2 computation, cf. [M2]*Package QuaternaryQuartics, shows that the
two families of liftings have a common degeneration where two of the fifteen
points come together as a scheme of length $2$ supported at a point. This
corresponds to the quintic curves, in the two liftings above, degenerating to
a plane cubic and two lines that meet at a point on the plane cubic.

∎

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