=0

# Calabi-Yau threefolds in ${\mathbb{P}}^{n}$ and Gorenstein rings

Hal Schenck  Schenck: Mathematics Department
Auburn University
Auburn
AL 36849
USA<EMAIL_ADDRESS>,  Mike Stillman  Stillman: Mathematics Department
Cornell University
Ithaca
NY 14850
USA<EMAIL_ADDRESS>and  Beihui Yuan  Yuan: Mathematics Department
Cornell University
Ithaca
NY 14850
USA<EMAIL_ADDRESS>
###### Abstract.

A projectively normal Calabi-Yau threefold $X\subseteq{\mathbb{P}}^{n}$ has an
ideal $I_{X}$ which is arithmetically Gorenstein, of Castelnuovo-Mumford
regularity four. Such ideals have been intensively studied when $I_{X}$ is a
complete intersection, as well as in the case where $X$ is codimension three.
In the latter case, the Buchsbaum-Eisenbud theorem shows that $I_{X}$ is given
by the Pfaffians of a skew-symmetric matrix. A number of recent papers study
the situation when $I_{X}$ has codimension four. We prove there are 16
possible betti tables for an arithmetically Gorenstein ideal $I$ with
$\mathop{\rm codim}\nolimits(I)=4=\mathop{\rm regularity}\nolimits(I)$, and
that exactly 8 of these occur for smooth irreducible nondegenerate threefolds.
We investigate the situation in codimension five or more, obtaining examples
of $X$ with $h^{p,q}(X)$ not among those appearing for $I_{X}$ of lower
codimension or as complete intersections in toric Fano varieties. A key tool
in our approach is the use of inverse systems to identify possible betti
tables for $X$.

###### Key words and phrases:

Calabi-Yau, Gorenstein, Inverse System

###### 2000 Mathematics Subject Classification:

Primary 14J32, Secondary 13D02

Schenck supported by NSF 1818646

Stillman and Yuan supported by NSF 1502294

##  1\. Introduction

In their 1985 paper [6], Candelas-Horowitz-Strominger-Witten showed that
Calabi-Yau threefolds play a central role in string theory. This was further
developed in works by Candelas-Lynker-Schimmrigk [8] and Candelas-de la Ossa-
Green-Parkes [7]; the book of Cox-Katz [10] gives a comprehensive overview of
the field.

######  Definition 1.1.

A smooth variety $X$ of dimension $n$ is Calabi-Yau if
$K_{X}\simeq{\mathcal{O}}_{X}$ and

$H^{1}({\mathcal{O}}_{X}),\cdots,H^{n-1}({\mathcal{O}}_{X})=0.$

From the perspective of physics, the case $n=3$ is of paramount interest, and
a first example of a CY threefold is a quintic hypersurface in
${\mathbb{P}}^{4}$. Generalizing the hypersurface case, when $X$ is a complete
intersection (CI) of type
$\\{d_{1},\ldots,d_{n-3}\\}\subseteq{\mathbb{P}}^{n}$ we have

$K_{X}\simeq{\mathcal{O}}_{X}(-n-1+\sum d_{i}).$

So a complete intersection Calabi-Yau (CICY) threefold in ${\mathbb{P}}^{n}$
must have $\\{d_{1},\ldots,d_{n-3}\\}$ satisfying

$\begin{array}[]{ccc}\\{5\\}&\mbox{ in }&{\mathbb{P}}^{4}\\\ \\{2,4\\}&\mbox{
in }&{\mathbb{P}}^{5}\\\ \\{3,3\\}&\mbox{ in }&{\mathbb{P}}^{5}\\\
\\{2,2,3\\}&\mbox{ in }&{\mathbb{P}}^{6}\\\ \\{2,2,2,2\\}&\mbox{ in
}&{\mathbb{P}}^{7}\end{array}$

Green-Hübsch-Lütken characterize CICY’s
$X\subset\prod_{i=1}^{m}{\mathbb{P}}^{n_{i}}$ in [17]; when $m=1$,
$h^{1,1}(X)=1$ and $h^{1,2}(X)\in\\{65,73,89,101\\}$. Projective space is the
simplest complete toric variety [11], and in [1], Batyrev shows how to obtain
CY’s as hypersurfaces in toric varieties corresponding to reflexive polytopes.
Much activity over the last three decades has been devoted to this situation.
A complete intersection is the first avatar of a Gorenstein ring; a Gorenstein
ideal of codimension two is a complete intersection, and Buchsbaum-Eisenbud
[5] show that a codimension three Gorenstein ideal is generated by the
Pfaffians of a skew-symmetric matrix. From the CY perspective, this is
investigated in [28], [31] and subsequent papers. The codimension four case
was first studied systematically by Bertin in [2]; in [9] Coughlan-
Golebiowski-Kapustka-Kapustka list 11 Gorenstein Calabi-Yau (GoCY) threefolds
in ${\mathbb{P}}^{7}$ and ask if the list is complete. For other recent work
on GoCY’s, see [3], [4], and [27].

###  1.1. Preliminaries

For algebraic background, we refer to [12]. The first observation to make is
that if $S={\mathbb{K}}[x_{0},\ldots x_{n}]$ and $I$ is a nondegenerate (i.e.
containing no linear form) homogeneous ideal in $S$ such that $R=S/I$ is
arithmetically Cohen-Macaulay, then the canonical module $\omega_{R}$ is
isomorphic to a shift $R(a)$ exactly when $R$ is arithmetically Gorenstein
(henceforth Gorenstein). In general,

$\omega_{R}\simeq R(-n-1+\mathop{\rm regularity}\nolimits(R)+\mathop{\rm
codim}\nolimits(R)),\mbox{ so we have}$

######  Lemma 1.2.

If $X=Proj(R)$ is a arithmetically Cohen-Macaulay threefold, then

$K_{X}={\mathcal{O}}_{X}\longleftrightarrow-n-1+n-3+\mathop{\rm
regularity}\nolimits(R)=0\longleftrightarrow\mathop{\rm
regularity}\nolimits(R)=4.$ (1.1)

For $R$ Gorenstein, we may quotient by a regular sequence of linear forms,
reducing to an Artinian Gorenstein ring with the same homological behavior,
described below. Any Artinian Gorenstein ring arises ([12], §21) via
Macaulay’s inverse system construction: Let $F$ be a homogeneous polynomial in
$S$ of degree $d$ over a field ${\mathbb{K}}$ of characteristic zero. The set
of differential operators $P(\frac{\partial}{\partial
x_{0}},\ldots,\frac{\partial}{\partial x_{n}})$ which annihilate $F$ generates
an ideal $I_{F}$ in $T={\mathbb{K}}[\frac{\partial}{\partial
x_{0}},\ldots,\frac{\partial}{\partial x_{n}}]$, which is called the inverse
system $I_{F}\subseteq T$. The corresponding ring $T/I_{F}$ is an Artinian
Gorenstein ring of regularity $d$.

######  Definition 1.3.

For $I\subseteq S={\mathbb{K}}[x_{0},\ldots,x_{n}]$ homogeneous, the graded
betti numbers are

$b_{ij}=\dim_{{\mathbb{K}}}Tor_{i}(S/I,{\mathbb{K}})_{j}.$

In betti table notation [13], these numbers are displayed as an array with top
left entry in position $(0,0)$ and position $(i,j)$ equal to $b_{i,i+j}$. The
reason for this indexing is so that

$\mathop{\rm regularity}\nolimits(S/I)=\sup\\{j\mid b_{i,i+j}\neq 0\\}$

is given by the index of the bottom row of the betti table.

######  Example 1.4.

For a GoCY threefold $X\subseteq{\mathbb{P}}^{6}$ given by the Pfaffians of a
skew-symmetric $7\times 7$ matrix $M$ of generic linear forms, Rødland [28]
shows that $h^{1,1}(X)=1$ and $h^{1,2}(X)=50$. By [5] and [15], for a generic
quartic $F$ in ${\mathbb{K}}[x_{0},x_{1},x_{2}]$, $I_{F}$ is the Pfaffians of
a skew $7\times 7$ matrix $M$ of linear forms, with betti table below. So for
this example, the betti table of $I_{F}$ can be realized by a GoCY.

total 1 7 7 1 0 1 – – – 1 – – – – 2 – 7 7 – 3 – – – – 4 – – – 1

In [31], Tonoli finds smooth CY’s $X\subseteq{\mathbb{P}}^{6}$ with
$12\leq\deg(X)\leq 17$. By Lemma 1.5 below, only those with $\deg(X)\leq 14$
are GoCY’s; all have $h^{1,1}(X)=1$. Ordered by increasing degree of $X$ from
$\\{12,\ldots,17\\}$, the respective $h^{1,2}(X)$ are
$\\{73,61,50,40,31,23\\}$.

For $S/I$ Artin Gorenstein of regularity 4, the Hilbert function of $S/I$ is

$HF(S/I,t)=\Big{(}1,n+1,h_{2},n+1,1\Big{)},\mbox{ with }h_{2}\leq{n+2\choose
2}.$

Migliore-Zanello [26] show that Stanley’s example of a Gorenstein ring with
non-unimodal $H$-vector $(1,13,12,13,1)$ is minimal, so for $n\leq 11$,
$n+1\leq h_{2}$. If $I$ can be lifted to a prime ideal in four more variables
(Example 2.1 shows this can occur), the corresponding threefold $X$ will have
degree

$\deg(X)=\sum_{t}HF(S/I,t)=2n+4+h_{2}.$ (1.2)

######  Lemma 1.5.

A GoCY $X\subseteq{\mathbb{P}}^{n}$ with $n\leq 15$ has

$3n-7\leq\deg(X)\leq\frac{n^{2}-n-2}{2}.$ (1.3)

###### Proof.

Apply Equations 1.1 and 1.2 and the result of [26] to the Artinian reduction
of $S/I_{X}$. ∎

This generalizes Lemma 2.1 of [9], which shows that a GoCY
$X\subseteq{\mathbb{P}}^{7}$ has $14\leq\deg(X)\leq 20$.

##  2\. Gorenstein codimension four

There have been several recent works on GoCY threefolds in ${\mathbb{P}}^{7}$.
In [2], Bertin finds four distinct families. In addition to Reid’s work [27]
on a structure theory for Gorenstein codimension four ideals, work on GoCY’s
in ${\mathbb{P}}^{7}$ appears in [3], [4], and [9], where Coughlan-
Golebiowski-Kapustka–Kapustka give a table with 11 types of GoCY threefolds in
${\mathbb{P}}^{7}$. They ask if the list is complete; on their list
$h^{1,1}(X)\in\\{1,2\\}$ and
$h^{1,2}(X)\in\\{34,36,37,45,46,54,55,58,65,76,86\\}$. In this section, we use
Betti tables to explore their question. The $h^{p,q}(X)$ cannot be read from
the betti table of $S/I_{X}$: the two families of degree 18 GoCY’s appearing
in [9] have the same betti table, but $h^{1,2}(X)$ is either $45$ or $46$.

######  Example 2.1.

The inverse system of a generic quartic in four variables yields an ideal with
betti table labelled CGKK 11 below. For an $n\times n$ matrix $M$ with
$M_{i,j}=x_{ij}$, Gulliksen-Negȧrd [18] determine the resolution of the ideal
$I_{n-1}$ of $n-1\times n-1$ minors: it is Gorenstein of codimension four, and
has regularity $2n-4$. Hence if $n=4$, this yields a Gorenstein codimension
four ideal in ${\mathbb{P}}^{15}$. Quotienting with a regular sequence of
eight linear forms yields a smooth GoCY threefold in ${\mathbb{P}}^{7}$, with
betti diagram equal to that of CGKK 11. The Hodge numbers are $h^{1,1}=2$ and
$h^{1,2}=34$; this example was first identified by Bertin in [1].

######  Theorem 2.2.

An Artin Gorenstein algebra $A=S/I$ with $\mathop{\rm
regularity}\nolimits(A)=4=\mathop{\rm codim}\nolimits(A)$ and $I$
nondegenerate has one of the 16 betti diagrams below. Table 1 below
corresponds to the 11 classes of GoCY in [9], and Table 2 to the remaining
classes. We defer the proof until the end of this section.

    ΨΨΨCGKK 1             1 9 16 9 1              CGKK 5,6      1 9 16 9 1
    ΨΨΨ                0: 1 .  . . .                         0: 1 . .  . .
    ΨΨΨ                1: . 6  8 3 .                         1: . 3 2  . .
    ΨΨΨ                2: . .  . . .                         2: . 6 12 6 .
    ΨΨΨ                3: . 3  8 6 .                         3: . . 2  3 .
    ΨΨΨ                4: . .  . . 1                         4: . . .  . 1
    ΨΨΨ
    ΨΨΨCGKK 2             1 6 10 6 1              CGKK 7,8      1 10 18 10 1
    ΨΨΨ                0: 1 .  . . .                         0: 1  .  .  . .
    ΨΨΨ                1: . 5  5 . .                         1: .  2  .  . .
    ΨΨΨ                2: . 1  . 1 .                         2: .  8 18  8 .
    ΨΨΨ                3: . .  5 5 .                         3: .  .  .  2 .
    ΨΨΨ                4: . .  . . 1                         4: .  .  .  . 1
    ΨΨΨ
    ΨΨΨCGKK 3             1 4 6 4 1               CGKK 9,10     1 13 24 13 1
    ΨΨΨ                0: 1 . . . .                          0: 1  .  .  . .
    ΨΨΨ                1: . 4 . . .                          1: .  1  .  . .
    ΨΨΨ                2: . . 6 . .                          2: . 12 24 12 .
    ΨΨΨ                3: . . . 4 .                          3: .  .  .  1 .
    ΨΨΨ                4: . . . . 1                          4: .  .  .  . 1
    ΨΨΨ
    ΨΨΨCGKK 4             1 7 12 7 1              CGKK 11       1 16 30 16 1
    ΨΨΨ                0: 1 .  . . .                         0: 1  .  .  . .
    ΨΨΨ                1: . 3  . . .                         1: .  .  .  . .
    ΨΨΨ                2: . 4 12 4 .                         2: . 16 30 16 .
    ΨΨΨ                3: . .  . 3 .                         3: .  .  .  . .
    ΨΨΨ                4: . .  . . 1                         4: .  .  .  . 1
    ΨΨΨ
    ΨΨ

Table 1. Betti table for GoCY’s in Coughlan-Golebiowski-Kapustka–Kapustka

The first four columns on the left have degrees $\\{14,15,16,17\\}$, while the
columns on the right have degrees $\\{17,18,19,20\\}$, as noted above the
corresponding GoCY’s can have different Hodge numbers.

The other 8 betti tables for an Artin Gorenstein algebra $A$ with $\mathop{\rm
regularity}\nolimits(A)=4=\mathop{\rm codim}\nolimits(A)$ are:

    ΨΨΨType 2.1           1 11 20 11 1            Type 2.5      1 11 20 11 1
    ΨΨΨ                0: 1  . .  .  .                       0: 1 .  .  .  .
    ΨΨΨ                1: .  2 1  .  .                       1: . 3  3  1  .
    ΨΨΨ                2: .  9 18 9  .                       2: . 7  14 7  .
    ΨΨΨ                3: .  . 1  2  .                       3: . 1  3  3  .
    ΨΨΨ                4: .  . .  .  1                       4: . .  .  .  1
    ΨΨΨ
    ΨΨΨType 2.2           1 8 14 8 1              Type 2.6      1 9 16 9 1
    ΨΨΨ                0: 1 . .  . .                         0: 1 . .  . .
    ΨΨΨ                1: . 3 1  . .                         1: . 4 4  1 .
    ΨΨΨ                2: . 5 12 5 .                         2: . 4 8  4 .
    ΨΨΨ                3: . . 1  3 .                         3: . 1 4  4 .
    ΨΨΨ                4: . . .  . 1                         4: . . .  . 1
    ΨΨΨ
    ΨΨΨType 2.3           1 7 12 7 1              Type 2.7      1 7 12 7 1
    ΨΨΨ                0: 1 . . . .                          0: 1 .  . . .
    ΨΨΨ                1: . 4 3 . .                          1: . 5  5 1 .
    ΨΨΨ                2: . 3 6 3 .                          2: . 1  2 1 .
    ΨΨΨ                3: . . 3 4 .                          3: . 1  5 5 .
    ΨΨΨ                4: . . . . 1                          4: . .  . . 1
    ΨΨΨ
    ΨΨΨType 2.4           1 6 10 6 1              Type 2.8      1 9 16 9 1
    ΨΨΨ                0: 1 . .  . .                         0: 1 . .  . .
    ΨΨΨ                1: . 4 2  . .                         1: . 5 6  2 .
    ΨΨΨ                2: . 2 6  2 .                         2: . 2 4  2 .
    ΨΨΨ                3: . . 2  4 .                         3: . 2 6  5 .
    ΨΨΨ                4: . . .  . 1                         4: . . .  . 1
    ΨΨΨ
    ΨΨ

Table 2. Betti tables for the remaining 8 Artin Gorenstein algebras.

######  Theorem 2.3.

No diagram in Table 2 occurs for a smooth, irreducible nondegenerate threefold
in ${\mathbb{P}}^{7}$.

###### Proof.

In §3.4 we apply results of [33] to prove a structure theorem for any
irreducible nondegenerate threefold in ${\mathbb{P}}^{7}$ with betti diagram
of Type 2.4, and show the resulting variety cannot be smooth. For the other
betti tables, we apply a result of [30]. A matrix of linear forms is 1-generic
if no entry can be reduced to zero by (scalar) row or column operations; a
linear $n^{th}$ syzygy is an element of
$Tor^{S}_{n+1}(S/I,{\mathbb{K}})_{n+2}$. For a nondegenerate prime ideal $P$,
Theorem 1.7 of [30] shows:

  1. (1)

$P$ cannot have a linear $n^{th}$ syzygy of rank $\leq n+1$, or $P$ is not
prime.

  2. (2)

If $P$ has a linear $n^{th}$ syzygy of rank $n+2$, then $P$ contains the
$2\times 2$ minors of a 1-generic $2\times(n+2)$ matrix.

  3. (3)

If $P$ has a linear $n^{th}$ syzygy of rank $n+3$, then $P$ contains the
$4\times 4$ Pfaffians of a skew-symmetric 1-generic $n+4\times n+4$ matrix.

A betti table of Type 2.1 is ruled out by $(1)$, and a betti table of Type 2.2
is ruled out by $(2)$, since the $2\times 2$ minors of a $2\times 3$ matrix
have two independent linear syzygies. For the three betti tables having top
row of the form $(c,c,1)$, we argue as follows. When $c=3$, the linear second
syzygy can have rank at most 3, since it involves the 3 first syzygies. Hence
by $(1)$, the ideal cannot be prime. When $c=4$, the linear second syzygy can
have rank at most 4, and in this case by $(2)$ it contains the $2\times 2$
minors of a 1-generic $2\times 4$ matrix, which would yield a top row of the
betti table with entries $(6,8,3)$. When $c=5$, $(3)$ implies that $P$
contains the Pfaffians, and since there are only five quadrics, the quadratic
part of the idea is exactly the Pfaffians, which do not have a linear second
syzygy.

For Type 2.3, we will show that a prime non-degenerate ideal $P$ cannot have
top row of the betti table equal to $(4,3,0)$. Let $J_{2}$ be the subideal of
$P$ generated by quadrics in $P$. By $(1)$ and $(3)$ the first syzygies all
have rank three; take a subideal $I\subseteq J_{2}$ consisting of three
elements, which by $(2)$ is generated by the $2\times 2$ minors of a $2\times
3$ matrix, and let $F$ denote the remaining quadric, so $J_{2}=I+\langle
F\rangle$. Consider the mapping cone resolution of $S/J_{2}$ from the short
exact sequence

$0\longrightarrow S(-2)/I:F\longrightarrow S/I\longrightarrow
S/I+F\longrightarrow 0.$ (2.1)

It follows that $I:F$ must have a linear generator $L$, so $LF\in I$. If $I$
is prime, then either $L\in I$ or $F\in I$, a contradiction. So suppose $I$ is
not prime, and take a primary decomposition

$I=\cap_{i=1}^{m}Q_{i},\mbox{ with }\sqrt{Q_{i}}=P_{i}\mbox{ all codimension
two.}$

Since $I$ is codimension two and Cohen-Macaulay and $\deg(I)=3$, we must have
$m\leq 3$.

  1. (1)

Case 1: $m=3$. Then $Q_{i}=P_{i}$ and $I=\cap_{i=1}^{3}P_{i}$ with $P_{i}$
generated by two linear forms.

  2. (2)

Case 2: $m=2$. Then $\deg(Q_{1})=1$, $\deg(Q_{2})=2$, so $\sqrt{Q_{1}}$ is
generated by two linear forms.

  3. (3)

Case 3: $m=1$. Then $\sqrt{Q_{1}}=P_{1}$, with $\deg(P_{1})\in\\{1,2,3\\}$. If
$\deg(P_{1})=3$, then $I$ is prime, and if $\deg(P_{1})=1$ or $2$, $P_{1}$
contains a linear form.

In particular, we see that $P$ is degenerate. For Type 2.8, both second
syzygies must have rank six, because if either had lower rank, then we would
be in one of the cases $(1),(2),(3)$, all of which are inconsistent with a
betti table having top row $(5,6,2)$. Let $M$ denote the corresponding
$6\times 2$ matrix of linear second syzygies; $M$ is 1-generic: if not, there
is a second syzygy of rank $\leq 5$, a contradiction. We claim that $M^{t}$
has no linear first syzygies (notice that kernel of $M^{t}$ will contain any
linear first syzygies on $J_{2}$, which is the subideal of $P$ generated by
quadrics in $P$). This follows because $\mathop{\rm coker}\nolimits(M^{t})$
has a Buchsbaum-Rim resolution [12], Theorem A2.10. The Buchsbaum-Rim complex
is a resolution for $\mathop{\rm coker}\nolimits(M^{t})$ iff the $2\times 2$
minors of $M$ have depth $6-2+1=5$; since $M$ is 1-generic, the $2\times 2$
minors are Cohen-Macaulay with an Eagon-Northcott resolution; in particular
$0pt(I_{2}(M))=5$. As the first syzygies in the Buchsbaum-Rim complex for
$M^{t}$ come from $\Lambda^{3}(S^{6})$ via the splice map described in [12],
they are quadratic. We conclude there are no linear syzygies on $\mathop{\rm
coker}\nolimits(M^{t})$, hence no linear first syzygies on $J_{2}$, a
contradiction. ∎

For the proof of Theorem 2.2, we will need the theorems of Macaulay and
Gotzmann [29]: for a graded algebra $S/I$ with Hilbert function $h_{i}$, write

$h_{i}=\binom{a_{i}}{i}+\binom{a_{i-1}}{i-1}+\cdots\mbox{ and }h_{i}^{\langle
i\rangle}=\binom{a_{i}+1}{i+1}+\binom{a_{i-1}+1}{i}+\cdots,\mbox{ with
}a_{i}>a_{i-1}>\cdots$

Macaulay proved that $h_{i+1}\leq h_{i}^{\langle i\rangle}$, and Gotzmann
proved if $I$ is generated in a single degree $t$ and equality holds in
Macaulay’s formula in the first degree $t$, then

$h_{t+j}=\binom{a_{t}+j}{t+j}+\binom{a_{t-1}+j-1}{t+j-1}+\cdots$

We also need the following lemma

######  Lemma 2.4.

Let $I_{2}$ be the subideal of $I$ generated by the quadrics in $I$, and let
$v=(b_{23},b_{24})$, which are the number of linear first and second syzygies
on $I_{2}$. Then

  1. (a)

$v\neq(2,1)$.

  2. (b)

if $a=b_{12}\geq 4$, then $v\neq(3,1)$.

###### Proof.

To see that $v=(2,1)$ cannot occur, observe that if it did then there would be
a unique relation $L_{1}\cdot V_{1}+L_{2}\cdot V_{2}=0$ where $L_{1},L_{2}$
are linear forms, and $V_{i}$ are vectors of linear first syzygies. Changing
variables so $L_{1}=x_{1}\mbox{ and }L_{2}=x_{2}$, we have that $x_{1}\cdot
V_{i1}+x_{2}\cdot V_{i2}=0$ for all $i$, implying $V_{1}$ is $x_{2}\cdot C$
and $V_{2}$ is $-x_{1}\cdot C$, with $C$ a vector of constants, a
contradiction. So $v=(2,1)$ is impossible.
To prove part (b), the key point is that $v=(3,1)$ implies that $I_{2}$
contains $\\{Lx_{1},Lx_{2},Lx_{3}\\}$ with $L$ a linear form. When $a\geq 4$
the mapping cone construction implies $I_{2}$ is inconsistent with the
Gorenstein hypothesis (IGH). If $v=(3,1)$ then the unique linear second syzygy
$S$ must have rank 3, otherwise the argument showing that $v=(2,1)$ is
impossible applies. After change of variables, we may write $S$ as below, with
$a_{i},b_{i},c_{i}$ linear forms:

$\left[\\!\begin{array}[]{ccc}a_{1}&b_{1}&c_{1}\\\ a_{2}&b_{2}&c_{2}\\\
a_{3}&b_{3}&c_{3}\\\
a_{4}&b_{4}&c_{4}\end{array}\\!\right]\cdot\left[\\!\begin{array}[]{c}x_{1}\\\
x_{2}\\\ x_{3}\end{array}\\!\right]=0$

So the rows of the matrix of linear first syzygies on $I_{2}$ are Koszul
syzygies on $[x_{1},x_{2},x_{3}]^{t}$, that is to say

$\displaystyle\begin{bmatrix}a_{1}&b_{1}&c_{1}\\\ a_{2}&b_{2}&c_{2}\\\
a_{3}&b_{3}&c_{3}\\\
a_{4}&b_{4}&c_{4}\end{bmatrix}=C\begin{bmatrix}x_{2}&-x_{1}&0\\\
-x_{3}&0&x_{1}\\\ 0&x_{3}&-x_{2}\end{bmatrix}$

where $C$ is a full rank $4\times 3$ scalar matrix. This forces $I_{2}$ to
contain $\\{Lx_{1},Lx_{2},Lx_{3}\\}$.
If $a\geq 4$, $I_{2}$ must contain a quadric $Q$ which is a nonzero divisor on
$\\{Lx_{1},Lx_{2},Lx_{3}\\}$. To see this, note that if $Q\in\langle L\rangle$
then $\mathop{\rm codim}\nolimits(I_{2})=1$. After a change of variables
$I_{2}$ consists of a linear form times a subset of the variables, so that
$I_{2}$ has a Koszul resolution, hence $b_{45}(I_{2})\neq 0$ which is IGH; if
$Q\in\langle x_{1},x_{2},x_{3}\rangle$ then there is at least one additional
linear first syzygy, so $b\geq 4$. Now we know $Q$ must be a non-zero divisor
on $\\{Lx_{1},Lx_{2},Lx_{3}\\}$. This implies that if $v=(3,1)$, then $I_{2}$
has mapping cone betti table

$\left[\\!\begin{array}[]{ccccc}1&0&0&0&0\\\ 0&4&3&1&0\\\
0&0&3&3&1\end{array}\\!\right].$

This is IGH, because $Tor_{4}(R/I_{2},{\mathbb{K}})_{6}\neq 0$, and adding
additional generators to $I_{2}$ cannot force cancellation: for a cubic $F$,
we have the short exact sequence

$0\longrightarrow R(-3)/I_{2}:F\longrightarrow R/I_{2}\longrightarrow
R/I_{2}+F\longrightarrow 0$ (2.2)

and the associated long exact sequence gives exact sequence of vector spaces

$\displaystyle 0\to Tor_{4}(R(-3)/I_{2}:F,{\mathbb{K}})_{6}\to
Tor_{4}(R/I_{2},{\mathbb{K}})_{6}\to Tor_{4}(R/I_{2}+F,{\mathbb{K}})_{6}.$

Note that

$\displaystyle
Tor_{4}(R(-3)/I_{2}:F,{\mathbb{K}})_{6}=Tor_{4}(R/I_{2}:F,{\mathbb{K}})_{3}=0.$

Hence $Tor_{4}(R/I_{2},{\mathbb{K}})_{6}\neq 0$ implies
$Tor_{4}(R/I_{2}+F,{\mathbb{K}})_{6}\neq 0$. Therefore we conclude $v=(3,1)$
is IGH. ∎

######  Remark 2.5.

When $a=3$, $v=(3,1)$ occurs.

Now we can prove Theorem 2.2:

###### Proof of Theorem 2.2:.

We use the Hilbert function to establish the possible shape of the betti
table, combined with an analysis of the structure of the subideal $I_{2}$
generated by the quadrics in $I$ and subideal $C_{3}$ generated by the
quadrics and cubics in $I$. Let $a=b_{12}(I)$ be the number of quadratic
generators of $I\subseteq R={\mathbb{K}}[x_{1},\ldots,x_{4}]$, and let
$v=(b_{23},b_{34})=(b,c)$, which are the number of linear first and second
syzygies on $I_{2}$. Note that $b_{45}(I_{2})\neq 0$ is inconsistent with the
Gorenstein hypothesis (IGH), so cannot occur.

For an Artinian Gorenstein ideal $I$ with $\mathop{\rm
codim}\nolimits(I)=4=\mathop{\rm regularity}\nolimits(4)$ and fixed Hilbert
series, $v$ determines the entire betti table. If $\mathop{\rm
codim}\nolimits(I_{2})=1$ then after a change of variables $I_{2}$ consists of
a linear form times a subset of the variables, so that $I_{2}$ has a Koszul
resolution; in particular for $a\geq 4$ this is IGH. Similarly, $\mathop{\rm
codim}\nolimits(I_{2})=4$ can only occur if $I_{2}$ contains a complete
intersection. When $a\in\\{0,1,2\\}$ the analysis is straightforward, so we
begin with $a=3$.

  1. (1)

$a=3$: The Hilbert function is $(1,4,7,4,1)$ and a computation shows the betti
table must be (dropping the $1$ in upper left and lower right corners)

$\left[\\!\begin{array}[]{ccc}3&b&c\\\ b+4&2c+12&b+4\\\
c&b&3\end{array}\\!\right].$

By Macaulay’s theorem

$h_{2}=7=\binom{4}{2}+\binom{1}{1},\mbox{ so }h_{2}^{\langle 2\rangle}=11\geq
h_{3}=20-3\cdot 4+b,\mbox{ so }b\leq 3.$

A direct computation shows that for an ideal generated by three quadratic
monomials in $R$, $v\in\\{(0,0),(1,0),(2,0),(3,1)\\}$, all of which occur in
Tables 1 and 2. By uppersemicontinuity, $I_{2}$ must have
$v=(b,c)\leq(b^{\prime},c^{\prime})$ for $(b^{\prime},c^{\prime})$ in the list
above, so we need only show that $v\in\\{(3,0),(2,1)\\}$ do not occur. If
$b=3$ then we are in the situation where Gotzmann’s theorem applies, and we
compute

$h_{3}^{\langle 3\rangle}=h_{4}=16=35-3\cdot 10+3\cdot 4-c+b_{24}(I_{2}).$

In particular, $c\geq 1+b_{24}(I_{2})$, so $c\geq 1$ and $v=(3,0)$ does not
occur. By Lemma 2.4, $v=(2,1)$ is impossible. When $a\geq 4$, the set of betti
tables possible for quadratic monomial ideals has an element that is so large
that a similar analysis via the initial ideal becomes cumbersome.

  2. (2)

$a=4$: The Hilbert function is $(1,4,6,4,1)$ and the betti table is:

$\left[\\!\begin{array}[]{ccc}4&b&c\\\ b&2c+6&b\\\
c&b&4\end{array}\\!\right].$

Values for $v$ which actually occur are $v\in\\{(0,0),(2,0),(3,0),(4,1)\\}$.
Applying Macaulay’s theorem to the ideal $I_{2}$ generated by the quadrics in
$I$ shows $b\leq 6$. Now let $C_{3}$ denote the ideal generated by the
quadrics and cubics in $I$.

$h_{3}(C_{3})=h_{3}(I)=4=\binom{4}{3}\mbox{ so }h_{3}^{\langle
3\rangle}(C_{3})=5\geq h_{4}(C_{3})=c+1.$

Hence $c\leq 4$. The case $b=6$ is extremal, and applying Gotzmann’s theorem
we find

$h_{4}(I_{2})=35-4\cdot 10+6\cdot 4+b_{24}(I_{2})-c=15,\mbox{ so
}c=4+b_{24}(I_{2}).$

Combined with our work above, this shows $b=6\Rightarrow c=4$. As
$h_{4}(C_{3})=h_{4}(I)+4=5$, we have

$h_{4}^{\langle 4\rangle}(C_{3})=6\geq h_{5}(C_{3})=56-80+40+b_{25}(C_{3})-6,$

we conclude $b_{25}(C_{3})\leq-4$, which is impossible. Thus,
$b\in\\{0,\ldots,5\\}$. If $b\in\\{0,1\\}$ then $c=0$; clearly $v=(0,0)$
yields a complete intersection, which occurs, while $v=(1,0)$ leads to an
almost complete intersection (ACI), and by [23] there are no Gorenstein ACI’s.
Henceforth we assume $b\in\\{2,3,4,5\\}$. We saw above that $c\leq 4$; we now
show that $c\in\\{2,3,4\\}$ is IGH.

$h_{5}(C_{3})=56-80+4(c+6)+b_{25}(C_{3})-b.$

So

$\begin{array}[]{ccc}c=2\implies h_{4}(C_{3})=3\implies h_{4}^{\langle
4\rangle}(C_{3})=3&\geq&h_{5}(C_{3})=8+b_{25}(C_{3})-b\\\ c=3\implies
h_{4}(C_{3})=4\implies h_{4}^{\langle
4\rangle}(C_{3})=4&\geq&h_{5}(C_{3})=12+b_{25}(C_{3})-b\\\ c=4\implies
h_{4}(C_{3})=5\implies h_{4}^{\langle
4\rangle}(C_{3})=6&\geq&h_{5}(C_{3})=16+b_{25}(C_{3})-b\end{array}$

As $b\leq 5$, only the case $b=5,c=2,b_{25}(C_{3})=0$ is possible; this has
betti table

$\left[\\!\begin{array}[]{ccc}4&5&2\\\ 5&10&5\\\ 2&5&4\end{array}\\!\right].$

Computing, we find that in this situation $h_{5}(C_{3})=3$, so

$h_{5}^{\langle 5\rangle}(C_{3})=3\geq
h_{6}(C_{3})=84-140+80-20+b_{26}(C_{3})-b_{36}(C_{3}).$

In particular, $b_{36}(C_{3})\geq 1+b_{26}(C_{3})$, which means the $5\times
4$ submatrix $M$ of $d_{3}$ representing the “bottom right corner” of the
table for $I$, one of the four columns of $M$ is zero. By symmetry of the free
resolution this means that one of the four rows of the matrix $M^{t}$ of
linear first syzygies on $I_{2}$ is zero. Hence the five linear first syzygies
on $I_{2}$ only involve a subideal $J\subseteq I_{2}$ generated by 3 quadrics,
which is impossible.

It remains to deal with $c\in\\{0,1\\}$. When $c=0$, we know
$v\in\\{(0,0),(2,0),(3,0)\\}$ occur, and we have already shown that $v=(1,0)$
is IGH. As $b\leq 5$, we need to show $v\in\\{(4,0),(5,0)\\}$ are IGH. To do
this, we use the ideal $I_{2}$ of four quadrics; $h_{3}(I_{2})=20-16+b=4+b$,
so we have

$\begin{array}[]{ccc}b=4\Rightarrow h_{3}(I_{2})=8\Rightarrow h_{3}^{\langle
3\rangle}(I_{2})=10&\geq
h_{4}(I_{2})=35-40+16+b_{24}(I_{2})=11+b_{24}(I_{2})\\\ b=5\Rightarrow
h_{3}(I_{2})=9\Rightarrow h_{3}^{\langle 3\rangle}(I_{2})=12&\geq
h_{4}(I_{2})=35-40+20+b_{24}(I_{2})=15+b_{24}(I_{2}),\end{array}$

both of which force $b_{24}(I_{2})\leq-1$, which is impossible. When $c=1$,
the only change to the second equation above is to subtract one (because
$c=1$) from the right hand side, so $h_{4}(I_{2})=14+b_{24}(I_{2})$, forcing
$b_{24}\leq-2$, which is impossible.

  3. (3)

$a=5$: The Hilbert function is $(1,4,5,4,1)$ so the betti table is

$\left[\\!\begin{array}[]{ccc}5&b&c\\\ b-4&2c&b-4\\\
c&b&5\end{array}\\!\right].$

Note that $h_{3}=20-5\cdot 4+b$, so $h_{3}=b$. By Macaulay’s theorem

$h_{2}=5=\binom{3}{2}+\binom{2}{1},\mbox{ so }h_{2}^{\langle 2\rangle}=7\geq
h_{3}=b,\mbox{ so }7\geq b.$

If $b=7$, applying Gotzmann’s theorem gives $c=b_{24}(I_{2})+4$. Let $C_{3}$
denote the subideal of $I$ generated in degrees two and three; applying
Macaulay’s theorem to $h_{3}(C_{3})=4$ yields

$5\geq h_{4}(C_{3})=35-5\cdot 10+4\cdot 4+c,$

so $c\leq 4$; combined with $c=b_{24}(I_{2})+4$ this forces $c=4$. Since
$h_{4}^{\langle 4\rangle}(C_{3})=6$, we find

$6\geq h_{5}(C_{3})=56-5\cdot 20+4\cdot 10+4\cdot
4+b_{25}(C_{3})-3=9+b_{25}(C_{3}).$

This shows $b_{25}(C_{3})\leq-3$, hence $b=7$ is IGH, and $b\in\\{4,5,6\\}$.
Case 1: Suppose $b=4$. This means there are no cubics in the ideal, and

$h_{3}=4=\binom{4}{3}\mbox{ so }h_{3}^{\langle 3\rangle}=5\geq h_{4}=35-5\cdot
10+4\cdot 4+c.$

We conclude $c\leq 4$. We can immediately rule out $c=0$, as then $I$ would be
an ACI, which is IGH. The possibilities $c\in\\{2,3,4\\}$ are also ruled out
by Macaulay; we illustrate for $c=2$:

$h_{4}=35-5\cdot 10+4\cdot 4+2=3,\mbox{ so }h_{4}^{\langle 4\rangle}=3\geq
h_{5}=4+b_{25}(I_{2}),$

which would force $b_{25}(I_{2})\leq-1$. Finally, suppose $c=1$, so
$I=I_{2}+q$ for a single quartic $q$. Since $I_{2}+q$ has codimension four,
the codimension of $I_{2}$ must be three or four, and if $\mathop{\rm
codim}\nolimits(I_{2})=4$ then $I_{2}$ contains a complete intersection $C$.
We claim this is impossible: write $I_{2}=C+f$ with $f\in I_{2}\setminus C$.
Since $b_{23}(C)=0$ the fact that $b_{23}(I_{2})=4$ means that $C:f=\langle
x_{1},x_{2},x_{3},x_{4}\rangle$, whose mapping cone is inconsistent with the
betti table for $I_{2}$. Hence $\mathop{\rm codim}\nolimits(I_{2})=3$, and $q$
is a nonzero divisor on the codimension three associated primes of $I_{2}$.
Since $h_{4}(I_{2})=2$, Macaulay’s theorem implies the degree of $I_{2}$ is
one or two. Observe that the rank of the linear second syzygy $S$ cannot be 4;
if it was then $S=[x_{1},x_{2},x_{3},x_{4}]^{t}$. By the symmetry of the
differentials in the free resolution, this means that $I_{2}:q=\langle
x_{1},\ldots,x_{4}\rangle$. By additivity of the Hilbert polynomials on the
short exact sequence

$0\longrightarrow R(-4)/(I_{2}:q)\longrightarrow R/I_{2}\longrightarrow
R/I\longrightarrow 0,$

this is impossible. Hence $\mathop{\rm rank}\nolimits(S)=3$, and as in the
proof that $v=(3,1)$ is impossible for $a=4$, $I_{2}$ must contain, after a
change of variables, $\\{L\cdot x_{1},L\cdot x_{2},L\cdot x_{3}\\}$ for a
linear form $L$. Since $\mathop{\rm codim}\nolimits(I_{2})=3$, this forces
$L,q_{4},q_{5}$ to be a regular sequence. In particular, $\deg(I_{2})=4$, a
contradiction.
Case 2: Suppose $b=5$. The cases $v\in\\{(5,0),(5,1)\\}$ do occur.

$h_{3}=5=\binom{4}{3}+\binom{2}{2}\mbox{ so }h_{3}^{\langle 3\rangle}=6\geq
h_{4}=35-5\cdot 10+4\cdot 5+b_{24}(I_{2})-c.$

So $c+1\geq b_{24}(I_{2})$. Let $C_{3}$ denote the subideal of $I$ generated
in degrees two and three.

$h_{3}(C_{3})=4\mbox{ so }h_{3}^{\langle 3\rangle}=5,\mbox{ thus }5\geq
h_{4}=35-50+16+c,$

implying $c\leq 4$. Since $c\in\\{0,1\\}$ does occur, we need to rule out
$c\in\\{2,3,4\\}$. Computing values for $h_{4}$, we find

$\begin{array}[]{ccc}c=2\mbox{ implies }&h_{4}=3&\mbox{ hence }h_{5}\leq 3\\\
c=3\mbox{ implies }&h_{4}=4&\mbox{ hence }h_{5}\leq 4\\\ c=4\mbox{ implies
}&h_{4}=5&\mbox{ hence }h_{5}\leq 6\end{array}$

Since $h_{5}=56-100+40+4c-1+b_{25}(C_{3})$, combining this with the above
shows

$\begin{array}[]{ccc}c=2\mbox{ implies }&h_{5}=3+b_{25}(C_{3})&\leq 3\\\
c=3\mbox{ implies }&h_{5}=7+b_{25}(C_{3})&\leq 3\\\ c=4\mbox{ implies
}&h_{5}=11+b_{25}(C_{3})&\leq 6\end{array}$

This rules out $c\in\\{3,4\\}$, and shows if $c=2$ then $b_{25}(C_{3})=0$. So
in this case $h_{5}(C_{3})=3$, and

$h_{5}^{\langle 5\rangle}(C_{3})=3\geq
h_{6}(C_{3})=84-175+80+20-4+b_{26}(C_{3})-b_{36}(C_{3}).$

In particular, we have $3\geq 5+b_{26}(C_{3})-b_{36}(C_{3})$, hence
$b_{36}(C_{3})\geq 2$, so the betti table for $C_{3}$ is at least

$\left[\\!\begin{array}[]{ccc}5&5&2\\\ 1&4&1\\\ 0&0&2\end{array}\\!\right].$

Hence in the $5\times 5$ submatrix $M$ of $d_{3}$ representing the “bottom
right corner” of the table for $I$, two of the five columns of $M$ are zero,
which by symmetry of the betti table means that two of the five rows of the
matrix $M^{t}$ of linear first syzygies on $I_{2}$ are zero. Hence the five
linear first syzygies on $I_{2}$ only involve a subideal $J\subseteq I_{2}$
generated by 3 quadrics, which is impossible.
Case 3: Suppose $b=6$; the only case that actually occurs is $v=(6,2)$.

$h_{3}=6=\binom{4}{3}+\binom{2}{2}+\binom{1}{1}\mbox{ so }h_{3}^{\langle
3\rangle}=7\geq h_{4}=35-5\cdot 10+4\cdot 6+b_{24}(I_{2})-c.$

So $c\geq b_{24}(I_{2})+2$. Let $C_{3}$ denote the subideal of $I$ generated
in degrees two and three.

$h_{3}(C_{3})=4=\binom{4}{3}\mbox{ so }5\geq h_{4}=35-50+16+c$

Thus, $c\leq 4$. To show that $c\in\\{3,4\\}$ do not occur, we compute

$\begin{array}[]{ccc}\mbox{If }c=4,\mbox{ then }&h_{4}(C_{3})=5\mbox{ and
}&h_{5}(C_{3})\leq 6\\\ \mbox{If }c=3,\mbox{ then }&h_{4}(C_{3})=4\mbox{ and
}&h_{5}(C_{3})\leq 4\end{array}$

Since $h_{5}(C_{3})=56-100+40+4c+b_{25}(C_{3})-2$, we see that

$\begin{array}[]{ccc}\mbox{If }c=4&\mbox{ then }h_{5}=10+b_{25}&\leq 6,\mbox{
so }b_{25}(C_{3})\leq-4\\\ \mbox{If }c=3&\mbox{ then }h_{5}=6+b_{25}&\leq
4,\mbox{ so }b_{25}(C_{3})\leq-2\end{array}$

We have shown that when $b=6$, the only value possible for $v$ is $(6,2)$.

  4. (4)

$a=6$: The Hilbert function is $(1,4,4,4,1)$ so the betti table is

$\left[\\!\begin{array}[]{ccc}6&b&c\\\ b-8&2c-6&b-8\\\
c&b&6\end{array}\\!\right].$

As

$h_{2}=4=\binom{3}{2}+\binom{1}{1},\mbox{ Macaulay's theorem shows
}h_{2}^{\langle 2\rangle}=5\geq h_{3}=20-6\cdot 4+b.$

So $b\leq 9$. If $b=9$ there is a unique cubic $F\in I$; since $b=9$ is
extremal we may apply Gotzmann’s theorem, yielding $b_{24}(I_{2})=c-5$. Since
$(2c-6)-(c-5)=c-1$ and $c\geq 5$, this means there are always at least four
independent syzygies which are linear on $F$ and quadratic on elements of
$I_{2}$. Hence $I_{2}:F=\langle x_{1},\ldots x_{4}\rangle$ and the mapping
cone arising from short exact sequence

$0\longrightarrow S(-3)/I_{2}:F\longrightarrow S/I\longrightarrow
S/I_{2}+F\longrightarrow 0,$

gives a resolution of $S/I$. The top row of the mapping cone is simply the
Koszul complex on the variables, and a check of the degrees shows the second
syzygies involve a summand $R^{6}(-5)$ which cannot cancel. This would imply
$b_{35}(I)=b-8\geq 6$, which is impossible since $b=9$.

Finally, we need to show that when $b=8$ we must have $c=3$. From the Hilbert
function constraint on the betti table, $c\geq 3$. When $b=8$, there are no
cubics in $I$; this means

$b_{24}(I_{2})-c=c-6.$

We compute

$h_{3}=4=\binom{4}{3}\mbox{ so }h_{3}^{\langle 3\rangle}=5\geq h_{4}=35-6\cdot
10+8\cdot 4+c-6,$

hence $c\leq 4$. Finally, if $c=4$, then $h_{4}=5$ and $h_{4}^{\langle
4\rangle}=6$. So

$6\geq h_{5}=56-6\cdot 20+8\cdot 10-2\cdot 4+b_{25}(I_{2}).$

This would force $b_{25}(I_{2})\leq-2$. We have shown that the only betti
table possible for $a=6$ is

$\left[\\!\begin{array}[]{ccc}6&8&3\\\ 0&0&0\\\ 3&8&6\end{array}\\!\right].$

Hence there are 16 betti tables for an Artin Gorenstein algebra $A$ with
$\mathop{\rm regularity}\nolimits(A)=4=\mathop{\rm codim}\nolimits(A)$. All
diagrams in Table 1 and Table 2 do occur, which can be checked via a Macaulay2
search. ∎

##  3\. Gorenstein deviation two

The deviation of an ideal $I$ is the number of generators of $I$ minus the
codimension of $I$. Complete intersections are the simplest Gorenstein rings,
and have deviation zero; in [23], Kunz shows a Gorenstein ring cannot have
deviation one. In this section, we study Gorenstein rings of deviation two.
This is similar to the codimension three case, where the classification of [5]
shows that such ideals come from Pfaffians. In [20], Huneke-Ulrich give a
construction for Gorenstein rings of deviation two; Let $Y$ be a $2n\times 2n$
skew symmetric matrix of variables, and $X$ a $1\times 2n$ vector of
variables. Then the ideal generated by the quadrics in $Y\cdot X$ plus the
Pfaffian of $Y$ is Gorenstein deviation two. Such an ideal will have
regularity four iff $n=3$, and we analyze this case in §3.2. The corresponding
GoCY threefold $X\subseteq{\mathbb{P}}^{8}$ has Hodge numbers different from
the $h^{p,q}(X)$ for any $X\subseteq{\mathbb{P}}^{n}$ with $n\leq 7$.

By the Buchsbaum-Eisenbud theorem, the Pfaffians of a skew $5\times 5$ matrix
$M$ also have deviation two, and quotienting such an ideal by a regular
sequence preserves the deviation two property. If $M$ is a matrix of linear
forms, in order to have regularity four, the regular sequence must consist of
two quadrics or a single cubic; if $M$ has linear and quadratic entries, the
regular sequence is a single quadric. We analyze these ideals in §3.3.

###  3.1. Huneke-Ulrich ideals

Let $I$ be the ideal consisting of the six quadrics of $Y\cdot X$

$\left[\\!\begin{array}[]{cccccc}0&-y_{12}&\cdots&\cdots&\cdots&-y_{16}\\\
y_{12}&0&-y_{23}&\cdots&\cdots&\cdots\\\
\cdots&y_{23}&0&-y_{34}&\cdots&\cdots\\\
\cdots&\cdots&y_{34}&0&-y_{45}&\cdots\\\
\cdots&\cdots&\cdots&y_{45}&0&-y_{56}\\\
y_{16}&\cdots&\cdots&\cdots&y_{56}&0\end{array}\\!\right]\cdot\left[\\!\begin{array}[]{c}x_{1}\\\
x_{2}\\\ x_{3}\\\ x_{4}\\\ x_{5}\\\ x_{6}\end{array}\\!\right]=0$

along with the cubic Pfaffian of $Y$. By [20], $I$ is Gorenstein, with $V(I)$
of dimension 15 in ${\mathbb{P}}^{20}$, and is singular in dimension 8.
Therefore quotienting by a regular sequence of 12 linear forms yields a smooth
GoCY threefold in ${\mathbb{P}}^{8}$ with betti diagram identical to that of
$I$:

total | 1 | 7 | 22 | 22 | 7 | 1 |
---|---|---|---|---|---|---|---
0 | 1 | – | – | – | – | – |
1 | – | 6 | 1 | – | – | – |
2 | – | 1 | 21 | 21 | 1 | – |
3 | – | – | – | 1 | 6 | – |
4 | – | – | – | – | – | 1 |

This yields a GoCY threefold in ${\mathbb{P}}^{8}$, whose Artinian reduction
has $H$-vector $(1,5,9,5,1)$. So $\deg(X)=21$, and a Macaulay2 computation
shows that for a generic $X$ the Hodge numbers are

$\begin{array}[]{ccccccc}&&&1&&&\\\ &&0&&0&&\\\ &0&&1&&0&\\\ 1&&52&&52&&1\\\
&0&&1&&0&\\\ &&0&&0&&\\\ &&&1&&&\end{array}$

These are not the $h^{p,q}(X)$ for any GoCY $X$ of codimension $\leq 4$, and
correspond to a point on the line connecting the Hodge numbers of the degree
13 and 14 examples of Tonoli–see Figure 1 in [31]. While $X$ can be projected
to a smooth CY in ${\mathbb{P}}^{7}$, it is no longer projectively normal
(hence not a GoCY).

###  3.2. Quotients of Pfaffians, I

A second class of Gorenstein ideals of deviation two and regularity four come
from quotienting the Pfaffians $Pf(M)$ of a $5\times 5$ matrix $M$ of linear
forms by a regular sequence. If $M$ is generic, then $Pf(M)$ is codimension 3
in ${\mathbb{P}}^{9}$, and is smooth. Quotienting by a regular sequence of
degrees $\\{2,2,1\\}$ yields a smooth GoCY threefold
$X\subseteq{\mathbb{P}}^{8}$ of degree 20, with betti table

total | 1 | 7 | 16 | 16 | 7 | 1 |
---|---|---|---|---|---|---|---
0 | 1 | – | – | – | – | – |
1 | – | 7 | 5 | – | – | – |
2 | – | – | 11 | 11 | – | – |
3 | – | – | – | 5 | 7 | – |
4 | – | – | – | – | – | 1 |

For the generic case, $h^{1,1}(X)=1$ and $h^{1,2}(X)=61$, which are attained
by Tonoli’s degree 13 example. On the other hand, if we quotient the pfaffians
of $M$ by a generic cubic, this yields an ideal $I$ with betti table CGKK 2.
Quotienting with two generic linear forms yields a smooth GoCY
$X\subseteq{\mathbb{P}}^{7}$ of degree $15$, discovered in [32], which has
$h^{1,1}(X)=1$ and $h^{1,2}(X)=76$.

###  3.3. Quotients of Pfaffians, II

We now show that the betti diagram of Type 2.4 corresponds to a mapping cone,
and that any nondegenerate irreducible GoCY in ${\mathbb{P}}^{7}$ with betti
table of Type 2.4 must be singular. Recall Type 2.4 has betti diagram

total | 1 | 6 | 10 | 6 | 1 |
---|---|---|---|---|---|---
0 | 1 | – | – | – | – |
1 | – | 4 | 2 | – | – |
2 | – | 2 | 6 | 2 | – |
3 | – | – | 2 | 4 | – |
4 | – | – | – | – | 1 |

A key tool in our analysis is a result of Vasconcelos-Villereal [33], which
shows that if $R$ is a Gorenstein local ring and $2\in R$ is a unit, then if
$I$ is a Gorenstein ideal of codimension $4$ and deviation two, such that $I$
is a generic complete intersection (the localization at all minimal primes is
a complete intersection), then $I$ is a hypersurface section of a Gorenstein
ideal of height $3$. Table 1 and Table two contain two examples of Gorenstein
ideals of codimension four and deviation two: CGKK 2, and Type 2.4; the first
can be obtained by quotienting the Pfaffians of a skew matrix of linear forms
by a cubic.

######  Theorem 3.1.

A nondegenerate irreducible threefold in ${\mathbb{P}}^{7}$ with betti diagram
of Type 2.4 is singular.

We start with several preparatory lemmas. Note that a betti diagram of Type
2.4 cannot arise as the mapping cone of a cubic, so will arise from
quotienting the Pfaffians by a quadric.

######  Lemma 3.2.

There is a prime subideal $J\subseteq I_{2}$ generated by three quadrics, such
that $J$ consists of the $2\times 2$ minors of a 1-generic $2\times 3$ matrix
$M$, and the quadric $q_{4}\in I_{2}\setminus J$ is a nonzero divisor on
$R/J$.

###### Proof.

By Theorem 1.7 of [30], a linear first syzygy on $I_{2}$ of rank four would
imply that $I_{2}$ contains the Pfaffians of a $5\times 5$ skew matrix of
linear forms, while if there was a linear first syzygy on $I_{2}$ of rank two,
$I$ would not be prime. So Theorem 1.7 implies that $I_{2}$ contains a
subideal $J$ of $2\times 2$ minors of a 1-generic $2\times 3$ matrix of linear
forms. The ideal $J$ must be prime, for if not, it would have a primary
decomposition into components of degrees one or two, which would force $I$ to
be degenerate. Finally, $q_{4}$ is regular on $J$, for if not, then
$\mathop{\rm codim}\nolimits(I_{2})=2$ and degree one or two; the two cubics
in $I$ must be nonzero divisors on the codimension two primary component,
because $\mathop{\rm codim}\nolimits(I)=4$. But this would imply that
$\deg(I)$ is $9$ or $18$, contradicting the fact that $\deg(I)=16$. ∎

In what follows, we use the notation of Lemma 3.2, so $J$ is the ideal of
$2\times 2$ minors of the one-generic matrix $M$. The entries of $M$ are
linear forms, because $J$ is prime the linear forms span a space of dimension
$\\{4,5,6\\}$. This means $V(J)$ is a cone, with singular locus of dimension
(respectively) $\\{3,2,1\\}$. Let $C$ be the ideal generated by $q_{4}$ and
the two cubic generators of $I$; intersecting $V(J)$ with $V(C)$ drops the
dimension by two, so if the linear forms of $M$ span a space of dimension four
or five, $V(I)$ is singular. It remains to deal with the case that the span of
the linear forms has dimension six; after a change of variables we may assume

$\displaystyle M=\begin{bmatrix}x_{1}&x_{2}&x_{3}\\\
x_{4}&x_{5}&x_{6}\end{bmatrix}$

######  Lemma 3.3.

Let $I$ be a codimension four Gorenstein prime ideal with betti diagram Type
2.4. If $I_{2}$ contains an ideal $J$ consisting of the $2\times 2$ minors of
$M$ as above, then $I=I^{\prime}+\langle F\rangle,$ with $\mathop{\rm
codim}\nolimits(I^{\prime})=3$ and $I^{\prime}$ Gorenstein, and $F$ a nonzero
divisor on $R/I^{\prime}$. Hence $R/I$ has a mapping cone resolution.

###### Proof.

Because the two linear first syzygies on $I_{2}$ are of the form
$[x_{1},x_{2},x_{3}]^{t}$ and $[x_{4},x_{5},x_{6}]^{t}$ and $I$ is
nondegenerate, $I$ contains no linear form, so $\\{x_{1},\ldots,x_{6}\\}$ are
all units when $R/I$ is localized at $I$. Thus, in the localization, two of
the generators for $J$ are redundant, and therefore $I$ is a generic complete
intersection, of deviation two, so the result of [33] applies (we assume
henceforth that $2$ is a unit). ∎

######  Lemma 3.4.

Assume $Y$ is an arithmetically Gorenstein variety of codimension $3$ and $X$
is a nondegenerate hypersurface section of $Y$ with betti diagram of Type 2.4.
Then $Y$ must have betti diagram:

total 1 5 5 1 0 1 – – – 1 – 3 2 – 2 – 2 3 – 3 – – – 1

###### Proof.

The Hilbert series of $X$ is

$\displaystyle h_{t}(X)=\frac{1}{(1-t)^{n}}(1-t^{2})^{4}.$

Assume $h_{t}(Y)=\frac{1}{(1-t)^{n}}f(t)$. Then

$f(t)(1-t^{d})=(1-t^{2})^{4}.$

So $d\in\\{1,2\\}$. But $X$ does not lie in any hyperplane. Therefore $d$ must
be $2$ and $Y$ has the desired betti table. ∎

######  Proposition 3.5.

Let $V(I)$ be GoCY in ${\mathbb{P}}^{7}$ with betti diagram of Type 2.4. If
the linear forms of the matrix $M$ span a space of dimension six, then up to a
change of basis, $I$ is generated by the Pfaffians of a $5\times 5$ skew
symmetric matrix $N$ as below, along with a quadric $q_{4}$ which is a nonzero
divisor on $R/\mathop{\rm Pfaff}\nolimits(N)$. The ideal $\mathop{\rm
Pfaff}\nolimits(N)$ is singular along a ${\mathbb{P}}^{1}$, and so $V(I)$ has
at least two singular points.

$\displaystyle N=\begin{bmatrix}0&x_{1}&x_{2}&x_{3}&0\\\
-x_{1}&0&q_{1}&q_{2}&x_{4}\\\ -x_{2}&-q_{1}&0&q_{3}&x_{5}\\\
-x_{3}&-q_{2}&-q_{3}&0&x_{6}\\\ 0&-x_{4}&-x_{5}&-x_{6}&0\end{bmatrix}$

where the $q_{j}$’s are quadrics.

###### Proof.

Combining Lemmas 3.2, 3.3, and 3.4 and the results of [33] shows that $I$ is
of the form above. To see that the singular locus is as claimed, we compute
that

$\mathop{\rm Pfaff}\nolimits(N)=J+\langle
x_{3}q_{1}-x_{2}q_{2}+x_{1}q_{3},x_{6}q_{1}-x_{5}q_{2}+x_{4}q_{3}\rangle,$

where $J$ is the ideal of the minors of the matrix $M$ above Lemma 3.3. In
particular,

$V(x_{1},\ldots x_{6})\simeq{\mathbb{P}}^{1}\subseteq V(\mathop{\rm
Pfaff}\nolimits(N)),$

and $V(\mathop{\rm Pfaff}\nolimits(N))$ is singular along this
${\mathbb{P}}^{1}$, because the Jacobian matrix of $\mathop{\rm
Pfaff}\nolimits(N)$ is

$\displaystyle\mathrm{Jac}(\mathop{\rm
Pfaff}\nolimits(N))=\begin{bmatrix}x_{5}&x_{6}&0&*&*\\\ -x_{4}&0&x_{6}&*&*\\\
0&-x_{4}&-x_{5}&*&*\\\ -x_{2}&-x_{3}&0&*&*\\\ x_{1}&0&-x_{3}&*&*\\\
0&x_{1}&x_{2}&*&*\\\ 0&0&0&*&*\\\ 0&0&0&*&*\end{bmatrix}$

where $*$ are quadrics. Hence when $\\{x_{1},\ldots,x_{6}\\}$ vanish,
$\mathrm{Jac}(\mathop{\rm Pfaff}\nolimits(N))$ has rank $\leq 2$, so is
singular along the ${\mathbb{P}}^{1}$. Intersecting $V(\mathop{\rm
Pfaff}\nolimits(N))$ with the hypersurface $V(q_{4})$, we find that $V(I)$
must be singular (at least) at a degree two zero scheme. ∎

##  4\. Computational aspects and Ideals with mostly quadratic generators

As noted earlier, GoCY’s were first investigated systemically from a
computational standpoint by Bertin in [2]. Below we describe algorithms which,
in certain situations, offer a substantial speed up in processing; in some
cases, we have seen an improvement in runtime by a factor of 500.

Let $I$ be the ideal sheaf of a smooth Calabi-Yau threefold in
${\mathbb{P}}^{n}$. This implies that

$H^{1}(\mathcal{O}_{X})=H^{2}(\mathcal{O}_{X})=0,$

and Kodaira vanishing and Serre duality give that

$H^{1}(\mathcal{O}_{X}(-1))=H^{2}(\mathcal{O}_{X}(-1))=0.$

From the fundamental short exact sequence

$0\longrightarrow
I/I^{2}\longrightarrow\Omega^{1}_{{\mathbb{P}}^{n}}\otimes\mathcal{O}_{X}\longrightarrow\Omega^{1}_{X}\longrightarrow
0,$ (4.1)

we have that

$\chi(\Omega^{1}_{X})=\chi(\Omega^{1}_{{\mathbb{P}}^{n}}\otimes\mathcal{O}_{X})-\chi(I/I^{2}).$

The Euler characteristic of both sheaves can be computed from the
corresponding modules of global sections via Gröbner bases; it turns out we
actually only need one computation. Tensoring the short exact sequence

$0\longrightarrow\Omega^{1}_{{\mathbb{P}}^{n}}\longrightarrow\mathcal{O}^{n+1}_{{\mathbb{P}}^{n}}(-1)\longrightarrow\mathcal{O}_{{\mathbb{P}}^{n}}\longrightarrow
0$ (4.2)

with $\mathcal{O}_{X}$, and using the cohomology vanishings for
$H^{1}(\mathcal{O}_{X})$ and $H^{2}(\mathcal{O}_{X})$ yields that

$\chi(\Omega^{1}_{{\mathbb{P}}^{n}}\otimes\mathcal{O}_{X})=(n+1)HP(S/I_{X},-1).$

In particular, all the computational expense to compute $\chi(\Omega^{1}_{X})$
comes from computing $in(I/I^{2})$. To compute $h^{1,1}(X)$, we use the long
exact sequence in cohomology of Equation 4.1. Vanishing of
$H^{0}(\Omega^{1}_{X})$ due to the Calabi-Yau property and the long exact
sequence in cohomology yield

$0\longrightarrow H^{1}(I/I^{2})\longrightarrow
H^{1}(\Omega^{1}_{{\mathbb{P}}^{n}}\otimes\mathcal{O}_{X})\longrightarrow
H^{1}(\Omega_{X})\longrightarrow H^{2}(I/I^{2})\longrightarrow
H^{2}(\Omega^{1}_{{\mathbb{P}}^{n}}\otimes\mathcal{O}_{X})\longrightarrow\cdots$

Now, $H^{2}(\Omega^{1}_{{\mathbb{P}}^{n}}\otimes\mathcal{O}_{X})=0$, because
it sits in the exact sequence

$\cdots\longrightarrow H^{1}(\mathcal{O}_{X})\longrightarrow
H^{2}(\Omega^{1}_{{\mathbb{P}}^{n}}\otimes\mathcal{O}_{X})\longrightarrow
H^{2}(\mathcal{O}_{X}^{n+1}(-1))\longrightarrow\cdots,$

and from Equation 4.2 we have
$h^{1}(\Omega^{1}_{{\mathbb{P}}^{n}}\otimes\mathcal{O}_{X})=1$. Hence

$h^{1,1}(X)=h^{2}(I/I^{2})-h^{1}(I/I^{2})+1.$

Writing $I/I^{2}$ for both the graded $S$-module and the sheaf, by local
duality (see [12])

$h^{1}(I/I^{2})=\dim Ext^{n-1}(I/I^{2},S)_{-n-1}\mbox{ and
}h^{2}(I/I^{2})=\dim Ext^{n-2}(I/I^{2},S)_{-n-1}$ (4.3)

If the projective dimension of the $S$-module $I/I^{2}$ is less than $n-2$,
then the vanishing of the Ext modules above is automatic, and $h^{1,1}(X)=1$.
If not, we can compute $h^{1,1}(X)$ from the formula. Projective dimension can
be computed quickly using the Macaulay2 command minimalBetti, which we
illustrate below.

######  Example 4.1.

We revisit the Huneke-Ulrich ideal appearing in §3, defining a smooth Calabi-
Yau 3-fold in ${\mathbb{P}}^{8}$ of degree 21.

    -- Input: a homogeneous ideal I defining a dim 3 smooth Calabi-Yau variety X
    -- Output: Euler characteristic of Omega^1_X

    CYEuler = (I) -> (
    M := coker prune presentation (I/I^2);
    MM := coker leadTerm gb presentation M;
    HP1 := hilbertPolynomial MM;
    F := source vars ring I;
    HP2 := hilbertPolynomial(F ** coker gens I);
    euler HP2 - euler HP1)

    i47 : time CYEuler HunekeUlrich
    -- used 28.3018 seconds

    o47 = 51

    i48 : time minimalBetti M
    -- used 2.71652 seconds

    0  1   2   3  4  5
    o48 = total: 7 50 140 154 78 21
    2: 6  1   .   .  .  .
    3: 1 42  27   1  .  .
    4: .  6 107 132 21  .
    5: .  1   6  21 57 21

By Equation 4.3, $h^{1,1}(X)=1$, because $Ext^{n-2}(I/I^{2},S)=0$, so in
particular $Ext^{n-2}(I/I^{2},S)_{-n-1}=0$. Since the Euler characteristic is
51, $h^{1,2}(X)=52$.

Remarks and Open Questions:

  1. (1)

For codimension four and regularity two, only one betti table occurs: for the
$2\times 2$ minors of a generic $3\times 3$ matrix, and for regularity three,
there are three betti tables. The calculations are the same as in the proof
for Theorem 2.2, and we leave them to the interested reader.

  2. (2)

For regularity four and codimension five, and regularity five and codimension
four there are, respectively, 68 and 36 possibilities, which can be tackled
with the same approach.

  3. (3)

Relate Theorem 2.2 to the parameter space $Gor_{T}$, studied in depth in [21].

  4. (4)

Theta characteristics are Gorenstein of regularity 4. Is there a connection to
GoCY’s?

  5. (5)

Gorenstein point configurations [14] have Gale duality. Does this lead to a
mirror construction?

## References

  * [1] V. Batyrev, Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties, J. Algebraic Geom. 3 (1994), 493-535.
  * [2] M. Bertin, Examples of Calabi–Yau 3-folds of ${\mathbb{P}}^{7}$ with $\rho=1$, Canad. J. Math. 61, (2009), 1050-1072.
  * [3] G. Brown, K. Georgiadis, Polarized Calabi-Yau 3-folds in codimension 4, Math. Nachr. 290 (2017), 710-725.
  * [4] G. Brown, A. Kasprzyk, L. Zhu, Gorenstein formats, Canonical and Calabi-Yau threefolds, Experimental Mathematics (2019) 46–74.
  * [5] D. Buchsbaum, D. Eisenbud, Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension three, Amer. J. Math. 99 (1977), 447-485.
  * [6] P. Candelas, G. Horowitz, A. Strominger, E. Witten, Vacuum configurations for superstrings, Nuclear Physics B. 258 (1985) 46–74.
  * [7] P. Candelas, X. de la Ossa, P. Green, L. Parkes, A pair of Calabi–Yau manifolds as an exactly soluble superconformal field theory, Nuclear Physics B. 359 (1991): 21–74.
  * [8] P. Candelas, M. Lynker, R. Schimmrigk, Calabi–Yau manifolds in weighted ${\mathbb{P}}^{4}$, Nuclear Physics B. 341 (1990), 383–402.
  * [9] S. Coughlan, L. Golebiowski, G. Kapustka, M. Kapustka, Arithmetically Gorenstein Calabi-Yau threefolds in ${\mathbb{P}}^{7}$, Electron. Res. Announc. Math. Sci. 23 (2016), 52–68.
  * [10] D. Cox, S. Katz, Mirror Symmetry and Algebraic Geometry, AMS Math Surveys and Monographs, (1999).
  * [11] D. Cox, J. Little, H. Schenck, Toric Varieties, AMS Graduate Studies in Mathematics, (2010).
  * [12] D. Eisenbud, Commutative Algebra with a view towards Algebraic Geometry, Graduate Texts in Mathematics, vol. 150, Springer, Berlin-Heidelberg-New York, 1995.
  * [13] D. Eisenbud, The geometry of syzygies, Graduate Texts in Mathematics, vol. 229, Springer, Berlin-Heidelberg-New York, 2005.
  * [14] D. Eisenbud, S. Popescu, The projective geometry of the Gale transform, J. Algebra 230 (2000), 127-173.
  * [15] J. Emsalem, A. Iarrobino, Some zero dimensional generic singularities: finite algebras having small tangent space, Compositio Math. 36 (1978), 145-188.
  * [16] D. Grayson, M. Stillman, Macaulay 2: a software system for algebraic geometry and commutative algebra, http://www.math.uiuc.edu/Macaulay2
  * [17] P. Green, T. Hübsch, C. Lütken, All the Hodge numbers for all Calabi-Yau complete intersections, Class. Quantum Grav. 6 (1989), 105-124.
  * [18] T. Gulliksen, O. Negȧrd, Un complexe résolvant pour certains idéaux déterminantiels, C. R. Acad. Sci. Paris 274, (1972), 16-18.
  * [19] Y. H. He, The Calabi-Yau landscape: from geometry to physics to machine learning, preprint, 2018, arXiv 1812.02893v1
  * [20] C. Huneke, B. Ulrich, Divisor class groups and deformations, Amer. J. Math. 107 (1985), 1265-1303.
  * [21] T. Iarrobino, V. Kanev, Power sums, Gorenstein algebras, and determinantal loci, Lecture Notes in Mathematics, 1721. Springer-Verlag, Berlin, 1999.
  * [22] Kapustka, M., Kapustka, G., A cascade of determinantal Calabi-Yau threefolds, Math. Nachr. 283 (2010), 1795-1809.
  * [23] E. Kunz, Almost complete intersections are not Gorenstein rings, J. Algebra 28 (1974), 111-115.
  * [24] M. Mastroeni, H. Schenck, M. Stillman, Quadratic Gorenstein rings and the Koszul property II, preprint, arXiv 1903.08273
  * [25] J. Migliore, U. Nagel, Gorenstein algebras presented by quadrics, Collect. Math. 64 (2013), 211-233.
  * [26] J. Migliore, F. Zanello, Stanley’s nonunimodal Gorenstein h-vector is optimal., Proc. Amer. Math. Soc. 145 (2017), 1-9.
  * [27] M. Reid, Gorenstein in codimension 4: the general structure theory Adv. Stud. Pure Math, 65 (2015), 201-227.
  * [28] E. Rødland, The Pfaffian Calabi-Yau, its mirror, and their link to the Grassmannian G(2,7), Compositio Math. 122 (2000), 135-149.
  * [29] H. Schenck, Computational Algebraic Geometry, Cambridge Univ. Press, Cambridge, (2003).
  * [30] H. Schenck, M. Stillman, High rank linear syzygies on low rank quadrics, American Journal of Mathematics, 134 (2012), 561-579.
  * [31] F. Tonoli, Construction of Calabi-Yau 3-folds in ${\mathbb{P}}^{6}$, J. Algebraic Geom. 13 (2004), 209-232.
  * [32] C. van Enckevort, D. van Straten, Monodromy calculations of fourth order equations of Calabi-Yau type, Mirror Symmetry, AMS/IP Advanced Studies in Math 38 (2006), 539-559.
  * [33] Vasconcelos, WV and Villarreal, R, On Gorenstein ideals of codimension four, Proceedings of the American Mathematical Society 98 no. 2 (1986), 205 - 210.


I_{t}^{2}\setminus\Delta(\alpha_{t}^{2})}\rangle=\langle(\mathcal{L}^{1}\nabla^{1}_{\mathcal{L}})_{\Phi_{p}\rightarrow\Delta(\alpha_{p}^{1})}^{\Phi_{p}\rightarrow\Delta(\alpha_{p}^{1})},(\mathcal{L}^{2}\nabla^{2}_{\mathcal{L}})_{K_{t}^{2}\setminus\Phi_{t}^{2}\rightarrow
I_{t}^{2}\setminus\Delta(\alpha_{t}^{2})}^{K_{t}^{2}\setminus\Phi_{t}^{2}\rightarrow
I_{t}^{2}\setminus\Delta(\alpha_{t}^{2})}\rangle=0,$

for if $\Delta(\alpha_{p}^{1})$ is nontrivial, it is the leftmost $X$-run, and
since we cut out an $X$-run in the second matrix, the result is zero.

  * •

$(\nabla^{1}_{\mathcal{L}}\mathcal{L}^{1})_{\bar{L}_{u}^{1}\setminus\Psi_{u+1}^{1}\rightarrow\bar{J}_{u}^{1}\setminus\bar{\Delta}(\bar{\beta}_{u}^{1})}^{\bar{L}_{u}^{1}\setminus\Psi_{u+1}^{1}\rightarrow\bar{J}_{u}^{1}\setminus\bar{\Delta}(\bar{\beta}_{u}^{1})}=0$
for all $u$;

  * •

All terms that involve the blocks of the first function with indices $u<p-1$
vanish;

Additionally, one term requires a special care. Observe that

$(\mathcal{L}^{1}\nabla^{1}_{\mathcal{L}})_{K_{p}^{1}\setminus\Phi_{p}^{1}\rightarrow
I_{p}^{1}\setminus\Delta(\alpha_{p}^{1})}^{K_{p}^{1}\setminus\Phi_{p}^{1}\rightarrow
I_{p}^{1}\setminus\Delta(\alpha_{p}^{1})}=\begin{cases}e_{ii},\ \
i\notin\Delta(\alpha_{p}^{1})\\\ 0,\ \ \text{otherwise}\end{cases}$

We argue that

$\langle(\mathcal{L}^{1}\nabla^{1}_{\mathcal{L}})_{K_{p}^{1}\setminus\Phi_{p}^{1}\rightarrow
I_{p}^{1}\setminus\Delta(\alpha_{p}^{1})}^{K_{p}^{1}\setminus\Phi_{p}^{1}\rightarrow
I_{p}^{1}\setminus\Delta(\alpha_{p}^{1})},\pi_{\Gamma_{1}^{r}}(\mathcal{L}^{2}\nabla^{2}_{\mathcal{L}})_{K_{t}^{2}\setminus\Phi_{t}^{2}\rightarrow
I_{t}^{2}\setminus\Delta(\alpha_{t}^{2})}^{K_{t}^{2}\setminus\Phi_{t}^{2}\rightarrow
I_{t}^{2}\setminus\Delta(\alpha_{t}^{2})}\rangle=\langle\pi_{\Gamma_{1}^{r}}e_{ii},\begin{bmatrix}0&0\\\
0&(\mathcal{L}^{2}\nabla^{2}_{\mathcal{L}})_{K_{t}^{2}\setminus\Phi_{t}^{2}}^{K_{t}^{2}\setminus\Phi_{t}^{2}}\end{bmatrix}\rangle.$

Indeed, the formula follows right away if $i\notin\Delta(\alpha_{p}^{1})$. On
the contrary, if $i\in\Delta(\alpha_{p}^{1})$, then the LHS is zero; such is
the RHS as well, for we remove an $X$-run from the second matrix.

With all the above observations, Formula (8.22) expands as

$\begin{split}\\{\log
y(g_{ii}),\log\psi\\}&=-\sum_{\beta_{t}^{2}<\beta_{p}^{1}}\langle(\mathcal{L}^{1}\nabla^{1}_{\mathcal{L}})_{\rho(K_{t}^{2})}^{\rho(K_{t}^{2})}(\mathcal{L}^{2}\nabla^{2}_{\mathcal{L}})_{K_{t}^{2}}^{K_{t}^{2}}+\sum_{\beta_{t}^{2}<\beta_{p}^{1}}\langle(\nabla^{1}_{\mathcal{L}}\mathcal{L}^{1})_{\rho(L_{t}^{2})}^{\rho(L_{t}^{2})}(\nabla^{2}_{\mathcal{L}}\mathcal{L}^{2})_{L_{t}^{2}}^{L_{t}^{2}}\rangle+\\\
&+\sum_{t=1}^{s^{2}}\langle\pi_{\Gamma_{1}^{r}}e_{ii},\begin{bmatrix}0&0\\\
0&(\mathcal{L}^{2}\nabla^{2}_{\mathcal{L}})_{K_{t}^{2}\setminus\Phi_{t}^{2}}^{K_{t}^{2}\setminus\Phi_{t}^{2}}\end{bmatrix}\rangle+\sum_{t=1}^{s^{2}}\langle(\nabla^{1}_{\mathcal{L}}\mathcal{L}^{1})_{L_{p}^{1}\rightarrow
J_{p}^{1}}^{L_{p}^{1}\rightarrow
J_{p}^{1}},\gamma_{c}^{*}(\nabla^{2}_{\mathcal{L}}\mathcal{L}^{2})_{\bar{L}_{t}^{2}\setminus\Psi_{t+1}^{2}\rightarrow\bar{J}_{t}^{2}\setminus\bar{\Delta}(\bar{\beta}_{t}^{2})}^{\bar{L}_{t}^{2}\setminus\Psi_{t+1}^{2}\rightarrow\bar{J}_{t}^{2}\setminus\bar{\Delta}(\bar{\beta}_{t}^{2})}+\\\
&+\sum_{\beta_{t}^{2}<\beta_{p}^{1}}\langle(\mathcal{L}^{1}\nabla^{1}_{\mathcal{L}})_{\rho(\Phi_{t}^{2})}^{\rho(\Phi_{t}^{2})}(\mathcal{L}^{2}\nabla^{2}_{\mathcal{L}})_{\Phi_{t}^{2}}^{\Phi_{t}^{2}}\rangle-\sum_{\beta_{t}^{2}=\beta_{p}^{1}}\langle(\mathcal{L}^{1}\nabla^{1}_{\mathcal{L}})_{\rho(K_{t}^{2}\setminus\Phi_{t}^{2})}^{\rho(K_{t}^{2}\setminus\Phi_{t}^{2})}(\mathcal{L}^{2}\nabla^{2}_{\mathcal{L}})_{K_{t}^{2}\setminus\Phi_{t}^{2}}^{K_{t}^{2}\setminus\Phi_{t}^{2}}\rangle+\\\
&+\sum_{\beta_{t}^{2}=\beta_{p}^{1}}\langle(\nabla^{1}_{\mathcal{L}}\mathcal{L}^{1})_{L_{p}^{1}}^{L_{p}^{1}}(\nabla^{2}_{\mathcal{L}}\mathcal{L}^{2})_{L_{t}^{2}}^{L_{t}^{2}}\rangle+\sum_{t=1}^{s^{2}}\langle\pi_{\hat{\Gamma}_{1}^{r}},\begin{bmatrix}0&0\\\
0&(\mathcal{L}^{2}\nabla^{2}_{\mathcal{L}})_{K_{t}^{2}\setminus\Phi_{t}^{2}}^{K_{t}^{2}\setminus\Phi_{t}^{2}}\end{bmatrix}\rangle-\\\
&-\sum_{t=1}^{s^{2}}\langle
e_{i-1,i-1},\begin{bmatrix}(\mathcal{L}^{1}\nabla^{1}_{\mathcal{L}})_{L_{t}^{2}}^{L_{t}^{2}}&0\\\
0&0\end{bmatrix}\rangle-\sum_{t=1}^{s^{2}}\langle
e_{i-1,i-1},\gamma_{c}^{*}\begin{bmatrix}0&0\\\
0&(\nabla^{1}_{\mathcal{L}}\mathcal{L}^{1})_{\bar{L}_{t}^{2}\setminus\Psi_{t+1}^{2}}^{\bar{L}_{t}^{2}\setminus\Psi_{t+1}^{2}}\end{bmatrix}\rangle.\end{split}$

Now let’s combine these terms together. The conditions under the second and
the seventh sums combine into $\beta_{t}^{2}\leq\beta_{p}^{1}$, which is
satisfied for all blocks due to our assumption; hence, these terms cancel out
with the ninth sum. The fourth term cancels out with the last one. The third
and the eighth term combine into

$\begin{split}\sum_{t=1}^{s^{2}}\langle\pi_{\Gamma_{1}^{r}}e_{ii},\begin{bmatrix}0&0\\\
0&(\mathcal{L}^{2}\nabla^{2}_{\mathcal{L}})_{K_{t}^{2}\setminus\Phi_{t}^{2}}^{K_{t}^{2}\setminus\Phi_{t}^{2}}\end{bmatrix}\rangle+\sum_{t=1}^{s^{2}}\langle\pi_{\Gamma_{1}^{r}}e_{ii}&,\begin{bmatrix}0&0\\\
0&(\mathcal{L}^{2}\nabla^{2}_{\mathcal{L}})_{K_{t}^{2}\setminus\Phi_{t}^{2}}^{K_{t}^{2}\setminus\Phi_{t}^{2}}\end{bmatrix}\rangle=\\\
&=\sum_{t=1}^{s^{2}}\langle e_{ii},\begin{bmatrix}0&0\\\
0&(\mathcal{L}^{2}\nabla^{2}_{\mathcal{L}})_{K_{t}^{2}\setminus\Phi_{t}^{2}}^{K_{t}^{2}\setminus\Phi_{t}^{2}}\end{bmatrix}\rangle.\end{split}$
(8.23)

The remaining terms (the first, the fifth and the sixth) add up to

$\begin{split}-\sum_{\beta_{t}^{2}<\beta_{p}^{1}}\langle(\mathcal{L}^{1}\nabla^{1}_{\mathcal{L}})_{\rho(K_{t}^{2})}^{\rho(K_{t}^{2})}(\mathcal{L}^{2}\nabla^{2}_{\mathcal{L}})_{K_{t}^{2}}^{K_{t}^{2}}\rangle&+\sum_{\beta_{t}^{2}<\beta_{p}^{1}}\langle(\mathcal{L}^{1}\nabla^{1}_{\mathcal{L}})_{\rho(\Phi_{t}^{2})}^{\rho(\Phi_{t}^{2})}(\mathcal{L}^{2}\nabla^{2}_{\mathcal{L}})_{\Phi_{t}^{2}}^{\Phi_{t}^{2}}\rangle-\\\
&-\sum_{\beta_{t}^{2}=\beta_{p}^{1}}\langle(\mathcal{L}^{1}\nabla^{1}_{\mathcal{L}})_{\rho(K_{t}^{2}\setminus\Phi_{t}^{2})}^{\rho(K_{t}^{2}\setminus\Phi_{t}^{2})}(\mathcal{L}^{2}\nabla^{2}_{\mathcal{L}})_{K_{t}^{2}\setminus\Phi_{t}^{2}}^{K_{t}^{2}\setminus\Phi_{t}^{2}}\rangle=\\\
=&-\sum_{t=1}^{s^{2}}\langle(\mathcal{L}^{1}\nabla^{1}_{\mathcal{L}})_{\rho(K_{t}^{2}\setminus\Phi_{t}^{2})}^{\rho(K_{t}^{2}\setminus\Phi_{t}^{2})}(\mathcal{L}^{2}\nabla^{2}_{\mathcal{L}})_{K_{t}^{2}\setminus\Phi_{t}^{2}}^{K_{t}^{2}\setminus\Phi_{t}^{2}}\rangle,\end{split}$
(8.24)

where we used the fact that $\beta_{t}^{2}\leq\beta_{p}^{1}$ is satisfied for
all blocks under the stated assumption. Now notice that the total contribution
of (8.23) and (8.24) is zero. Thus the result follows. ∎

## References

  * [BD82] A. Belavin and V. Drinfeld  ‘‘Solutions of the classical Yang–Baxter equation for simple Lie algebras’’  In _Funktsional’nyi Analiz i ego Prilozheniya_ 16.3 Russian Academy of Sciences, Steklov Mathematical Institute of Russian Academy of Sciences, 1982, pp. 1–29
  * [BFZ05] A. Berenstein, S. Fomin and A. Zelevinsky  ‘‘Cluster algebras III: Upper bounds and double Bruhat cells’’  In _Duke Mathematical Journal_ 126.1 Duke University Press, 2005, pp. 1–52
  * [BZ05] A. Berenstein and A. Zelevinsky  ‘‘Quantum cluster algebras’’  In _Advances in Mathematics_ 195.2 Elsevier, 2005, pp. 405–455
  * [CP95] V. Chari and A. Pressley  ‘‘A guide to quantum groups’’  Cambridge university press, 1995
  * [CS14] L. Chekhov and M. Shapiro  ‘‘Teichmüller spaces of Riemann surfaces with orbifold points of arbitrary order and cluster variables’’  In _International Mathematics Research Notices_ 2014.10 Oxford University Press, 2014, pp. 2746–2772
  * [Eis14] I. Eisner  ‘‘Exotic cluster structures on $\operatorname{SL}_{5}$’’  In _Journal of Physics A: Mathematical and Theoretical_ 47.47 IOP Publishing, 2014, pp. 474002
  * [ES98] P. Etingof and O. Schiffmann  ‘‘Lectures on quantum groups’’  International press Boston, MA, 1998
  * [FG06] V. Fock and A. Goncharov  ‘‘Cluster $\mathcal{X}$-varieties, amalgamation, and Poisson-Lie groups’’  In _Algebraic geometry and number theory_ Springer, 2006, pp. 27–68
  * [FP16] S. Fomin and P. Pylyavskyy  ‘‘Tensor diagrams and cluster algebras’’  In _Advances in Mathematics_ 300 Elsevier, 2016, pp. 717–787
  * [FZ02] S. Fomin and A. Zelevinsky  ‘‘Cluster algebras I: foundations’’  In _Journal of the American Mathematical Society_ 15.2, 2002, pp. 497–529
  * [GSV10] M. Gekhtman, M. Shapiro and A. Vainshtein  ‘‘Cluster algebras and Poisson geometry’’  American Mathematical Soc., 2010
  * [GSV12] M. Gekhtman, M. Shapiro and A. Vainshtein  ‘‘Cluster structures on simple complex Lie groups and Belavin–Drinfeld classification’’  In _Moscow Mathematical Journal_ 12.2, 2012, pp. 293–312
  * [GSV17] M. Gekhtman, M. Shapiro and A. Vainshtein  ‘‘Exotic cluster structures on $\operatorname{SL}_{n}$: the Cremmer–Gervais case’’  American Mathematical Society, 2017
  * [GSV18] M. Gekhtman, M. Shapiro and A. Vainshtein  ‘‘Drinfeld double of $\operatorname{GL}_{n}$ and generalized cluster structures’’  In _Proceedings of the London Mathematical Society_ 116.3 Wiley Online Library, 2018, pp. 429–484
  * [GSV19] M. Gekhtman, M. Shapiro and A. Vainshtein  ‘‘Periodic staircase matrices and generalized cluster structures’’  In _International Mathematics Research Notices_ , 2019
  * [GSV19a] M. Gekhtman, M. Shapiro and A. Vainshtein  ‘‘Plethora of cluster structures on $\operatorname{GL}_{n}$’’, 2019  arXiv:1902.02902
  * [RS03] A. Reyman and M. Semenov-Tian-Shansky  ‘‘Integrable systems’’  In _Institute of Computer Studies, Moscow_ , 2003
  * [SS19] G. Schrader and A. Shapiro  ‘‘A cluster realization of $U_{q}(\operatorname{sl}_{n})$ from quantum character varieties’’  In _Inventiones mathematicae_ 216.3 Springer, 2019, pp. 799–846