# Stable Ulrich bundles on cubic fourfolds

Hoang Le Truong  Institute of Mathematics, VAST, 18 Hoang Quoc Viet Road,
10307 Hanoi, Viet Nam  Thang Long Institute of Mathematics and Applied
Sciences, Hanoi, Vietnam<EMAIL_ADDRESS>
<EMAIL_ADDRESS>and  Hoang Ngoc Yen  The Department of Mathematics,
Thai Nguyen University of education. 20 Luong Ngoc Quyen Street, Thai Nguyen
City, Thai Nguyen Province, Viet Nam<EMAIL_ADDRESS>
###### Abstract.

In this paper, we give necessary and sufficient conditions for the existence
of Ulrich bundles on cubic fourfold $X$ of given rank $r$. As consequences, we
show that for every integer $r\geq 2$ there exists a family of indecomposable
rank $r$ Ulrich bundles on the certain cubic fourfolds, depending roughly on
$r$ parameters, and in particular they are of wild representation type;
special surfaces on the cubic fourfolds are explicitly constructed by
Macaulay2; a new $19$-dimensional family of projective ten-dimensional
irreducible holomorphic symplectic manifolds associated to a certain cubic
fourfold is constructed; and for certain cubic fourfold $X$, there exist
arithmetically Cohen-Macaulay smooth surface $Y\subset X$ which are not an
intersection $X\cap T$ for a codimension two subvariety
$T\subset\mathbb{P}^{5}$.

###### Key words and phrases:

Cubic fourfolds, Ulrich bundles, Matrix factorizations, Arithmetically Cohen-
Macaulay

2020 Mathematics Subject Classification: 13C14, 13D02, 14M05, 14M06, 13C40:
(Primary), 14J25, 14J60 (Secondary) The first author was partially supported
by the Alexander von Humboldt Foundation and the Vietnam National Foundation
for Science and Technology Development (NAFOSTED) under grant number
101.04-2019.309. The second author was partially supported by Grant number
ICRTM02-2020.05, awarded in the internal grant competition of International
Center for Research and Postgraduate Training in Mathematics, Hanoi.

##  1\. Introduction

The study of moduli spaces of stable vector bundles of given rank and Chern
classes on algebraic varieties is a very active topic in algebraic geometry.
In recent years attention has focused on ACM bundles, that is vector bundles
without intermediate cohomology. Some of the reasons, why the study of ACM
bundles is important are:

  1. $\bullet$

ACM bundles on the projective $n$-dimensional space are precisely the bundles
which are direct sum of line bundles ([Hor64]).

  2. $\bullet$

The $i$-th syzygy of a resolution of any vector bundle on hypersurfaces by
split bundles, is an arithmetically Cohen–Macaulay bundle ([Eis80]).

  3. $\bullet$

These sheaves correspond to maximal Cohen–Macaulay modules over the associated
coordinate ring of hypersurfaces ([Bea00]).

  4. $\bullet$

In [KRR09], ACM bundles on hypersurfaces have been used to provide
counterexamples to a conjecture of Griffiths and Harris about whether
subvarieties of codimension two of a hypersurface can be obtained by
intersecting with a subvariety of codimension two of the ambient space.

Note that there have been numerous studies of rank $2$ ACM bundles on surfaces
and threefolds (see ( [ArM09, BrF09, ChF09, ChM05, Mad00]) and the references
in those papers), and a few studies of ACM bundles of higher rank [ArG99,
ArM09, Mad05, Tru19a, TrY20a]. There have also been a few examples of
indecomposable ACM bundles of arbitrarily high rank [PoT09]. But as far as we
can tell, examples of stable ACM bundles of higher ranks are essentially
unknown.

Very often the ACM bundles that we will construct and study will share another
stronger property, namely they have the maximal possible number of global
sections; they will be the so-called Ulrich bundles. Notice that Ulrich
bundles exist on curves, linear determinantal varieties, hypersurfaces, and
complete intersections. Moreover, Ulrich bundles are semistable in the sense
of Gieseker, so once fix rank and Chern class, they may be parametrized by a
quasi-projective scheme. In the case that $X$ is a hypersurface in
$\mathbb{P}^{n}$ defined by a homogeneous form $f$, Ulrich bundles on $X$
correspond to linear determinantal descriptions of powers of $f$ ([Bea00]).
This has been generalized to the case of arbitrary codimension in [ESW03],
where Ulrich bundles are shown to correspond to linear determinantal
descriptions of powers of Chow forms. It is clear by now that the concept of
an Ulrich bundle encodes substantial algebraic information. We show in this
article that in the case of a cubic fourfold, it also encodes substantial
geometry. In particular, we prove our first main result:

######  Theorem 1.1.

[cf. Theorem 3.9] Let $\mathcal{F}$ be a vector bundle of rank $r\geq 3$ on a
cubic fourfold $X$ in $\mathbb{P}^{5}$. Then the following statements are
equivalent.

  1. $1)$

$\mathcal{F}$ is an Ulrich bundle of rank $r$.

  2. $2)$

$\mathcal{F}$ is isomorphic to a vector bundle obtained from a certain surface
$Y\subset X$ of degree $d=\frac{1}{2}(3r^{2}-r)$ and sectional genus
$r^{3}-2r^{2}+1$.

  3. $3)$

There exists a $3r\times 3r$ matrix $M$ of linear forms on $\mathbb{P}^{5}$
such that the sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{\mathbb{P}^{5}}^{3r}(-1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{M}$$\textstyle{\mathcal{O}_{\mathbb{P}^{5}}^{3r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

is exact.

Recall that varieties that admit only a finite number of indecomposable ACM
bundles (up to twist and isomorphism) are called of finite representation type
(see [DrG01] and references herein). Varieties of finite representation type
have been completely classified into a short list in [BGS87, Theorem C] and
[EiH88, p. 348]. They are three or less reduced points on $\mathbb{P}^{2}$, a
projective space, a smooth quadric hypersurface $X\subseteq\mathbb{P}^{n}$, a
cubic scroll in $\mathbb{P}^{4}$, the Veronese surface in $\mathbb{P}^{5}$ or
a rational normal curve.

On the other extreme, we would find the varieties of wild representation type
namely, varieties for which there exist $t$-dimensional families of non-
isomorphic indecomposable ACM sheaves for arbitrary large $t$. In the case of
dimension one, it is known that curves of wild representation type are exactly
those of genus larger than or equal to two. In dimension two and three,
Casanellas and Hartshorne showed in [CaH11] that cubic surfaces and threefolds
are of wild representation type. On the other hand, the first result for
varieties of arbitrary dimension were obtained in [MiP14], where the authors
showed that Fano blow-ups of points in $\mathbb{P}^{n}$ are of wild
representation type. In [MiP13] it had already been proven that Bordiga or
Castenuovo surfaces were of wild representation type. However, a broader
problem has been much less studied: which are the possible dimensions of
families of ACM irreducible sheaves on ACM projective schemes
$X\subset\mathbb{P}^{n}$. In [BHP18], authors showed that for every $r\geq 0$,
there exists a family of indecomposable ACM vector bundles of rank $r$,
depending roughly on $r$ parameters, on certain projective surfaces. The main
goal of this paper is to provide families of ACM vector bundles on a large
range of a certain cubic fourfold, and in particular they are of wild
representation type. Our source of examples will be the special cubic
fourfold. Recall that a smooth cubic fourfold $X$ in $\mathbb{P}^{5}$ is
special, if there is an embedding of a saturated rank-$2$ lattice

$L_{\delta}:=\langle h^{2},Y\rangle\hookrightarrow A(X),$

where $A(X)$ is the lattice of middle Hodge classes, $h\in{\rm Pic}(X)$ is the
hyperplane class, $Y$ is an algebraic surface not homologous to a complete
intersection, and $\delta$ is the determinant of the intersection matrix of
$\langle h^{2},Y\rangle$. Note that the locus $\mathcal{C}_{\delta}$ of
special cubic fourfolds $X$ of a discriminant $\delta$ is an irreducible
divisor which is nonempty if and only if $\delta>6$ and $\delta\equiv
0,2\pmod{6}$([Has00]). With the above notation, we have the following theorem.

######  Theorem 1.2.

There exists a cubic fourfold $X$ such that

  1. $1)$

$[X]\in\mathcal{C}_{14}\cap\mathcal{C}_{18}$.

  2. $2)$

For any $r\geq 2$, the moduli space of stable rank $r$ Ulrich bundles on a
cubic fourfold $X$ is nonempty and smooth of dimension $r^{2}+1$.

  3. $3)$

$X$ is of wild representation type.

Recently, the study of some geometric properties of special ACM bundles on
smooth cubic fourfolds has received considerable attention. In [LLMS18], the
authors constructed an irreducible holomorphic symplectic eightfold $Z$ from
the irreducible component of the Hilbert scheme of twisted cubic curves on
each cubic fourfold $X$ not containing a plane. Moreover, $Z$ is isomorphic to
an irreducible component of a moduli space of Gieseker stable torsion sheaves
or rank three torsion free sheaves. In [AdL17] it was shown that $Z$ is
deformation equivalent to a Hilbert scheme of four points on a K3 surface.
After that, Lahoz et al. ([LLMS18]) was shown that $Z$ is birational to an
irreducible component of a moduli space of Gieseker stable ACM bundles of rank
six. In [LMS17], Lahoz, Macrì and Stellari studied ACM bundles on cubic
fourfolds containing a plane exploiting the geometry of the associated quadric
fibration and Kuznetsov’s treatment of their bounded derived categories of
coherent sheaves. More precisely, they constructed the K3 surface naturally
associated to the fourfold as a moduli space of Gieseker stable ACM bundles of
rank four. In view of above results, we search for a moduli space of stable
Ulrich bundles of rank $3$, which is a ten-dimensional irreducible holomorphic
symplectic manifold. More precisely, we are able to prove the next result.

######  Theorem 1.3.

Let $X$ be a general special cubic fourfold in $\mathcal{C}_{18}$. Then the
moduli space of stable rank $3$ Ulrich bundles on $X$ with the Chern classes
$c_{1}=3$ and $c_{2}=12$ is a smooth ten-dimensional holomorphically
symplectic manifold.

Let us explain briefly how this paper is divided. This paper is divided into 4
sections. In Section 2 let us briefly note properties of ACM sheaves including
the theory of matrix factorizations due to Eisenbud. We shall prove Theorem
1.1 and its corollaries in Section 3. In Section 4, as corollaries to the main
theorem, we give to explicit construction of families of simple Ulrich vector
bundles on a special cubic fourfold and remarks on a conjecture of Griffiths
and Harris about whether subvarieties of codimension two of a hypersurface can
be obtained by intersecting with a subvariety of codimension two of the
ambient space. We construct a new $19$-dimensional family of projective ten-
dimensional irreducible holomorphic symplectic manifolds.

##  2\. Preliminaries

Throughout this section, assume that $X$ is a closed subscheme in
$\mathbb{P}^{n}$, $R$ is the polynomial ring $k[x_{0},\ldots,x_{n}]$ over an
algebraically closed field $k$ and $R_{X}$ is the coordinate ring of $X$. For
any coherent sheaf $\mathcal{F}$, we denote by
$H_{\bullet}^{i}(X,\mathcal{F})$ the sum
$\bigoplus_{l\in\mathbb{Z}}H^{i}(X,\mathcal{F}(l))$. Let us recall the
definition of maximal Cohen-Macaulay $R_{X}$-modules and arithmetically Cohen-
Macaulay coherent sheaves over $X$.

######  Definition 2.1.

A graded $R_{X}$-module $E$ is a maximal Cohen-Macaulay module (MCM), if ${\rm
depth}E=\dim E=\dim_{R}X$. A closed subscheme $X\subset\mathbb{P}^{n}_{k}$ is
arithmetically Cohen-Macaulay (ACM), if its homogeneous coordinate ring
$R_{X}=R/I_{X}$ (where $I_{X}$ is the saturated ideal of $X$) is a Cohen-
Macaulay ring. This is equivalent to saying
$H_{\bullet}^{1}(\mathcal{I}_{X,\mathbb{P}^{n}})=0$ and
$H_{\bullet}^{i}(\mathcal{O}_{X})=0$, for any $0<i<\dim X$. A coherent sheaf
$\mathcal{E}$ on an ACM $X$ is an ACM sheaf, if it is locally Cohen-Macaulay
on $X$ and $H_{\bullet}^{i}(\mathcal{E})=0$, for any $0<i<\dim X$.

Thanks to the graded version of the Auslander-Buchsbaum formula (for any
finitely generated $R$-module $M$)

${\rm pd}_{R}M=n+1-{\rm depth}M,$

where ${\rm pd}_{R}M$ denotes the projective dimension of the $R$-module $M$.
Moreover, we deduce that a subscheme $X\subseteq\mathbb{P}^{n}$ is ACM if and
only if ${\rm pd}(R_{X})={\rm codim}X$. Hence, if $X\subseteq\mathbb{P}^{n}$
is a codimension $c$ ACM subscheme, a graded minimal free $R$-resolution of
$I_{X}$ is of the form:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F_{c}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{c}}$$\textstyle{F_{c-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{c-1}}$$\textstyle{\ldots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{1}}$$\textstyle{F_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{R_{X}\to
0,}$

where $F_{0}=R$ and $F_{i}=\bigoplus_{j}R(-i-j)^{\beta_{ij}(X)}$, $1\leq i\leq
c$. The integers $\beta_{ij}(X)$ are called the graded Betti numbers of $X$
and they are defined by

$\beta_{ij}(X)=\dim_{k}{\rm Tor}^{i}(R_{X},k)_{i+j}.$

We construct the Betti diagram of $X$ writing in the $(i,j)$-th position the
Betti number $\beta_{ij}(X)$. In this setting, minimal means that ${\rm
Im}\varphi_{i}\subset{\mathfrak{m}}F_{i-1}$. Therefore, the free resolution of
$X$ is minimal if, after choosing a basis of $F_{i}$, the matrices
representing $\varphi_{i}$ do not have any non-zero scalar.

Notice that there is a one-to-one correspondence between ACM sheaves on an ACM
scheme $X$ and graded MCM $R_{X}$-modules that send $\mathcal{E}$ to
$H^{0}_{\bullet}(X,\mathcal{E})$(see [CaH11, 2.1]). In the algebraic context,
MCM modules have been extensively studied (see for example the book of Yoshino
[Yos90]), as they reflect relevant properties of the corresponding ring. There
has also been recent work on ACM bundles of small rank on particular varieties
such as Fano threefolds, quartic threefolds and Grassmann varieties (see
[ArM09] and the references therein). When $X$ is a non-singular variety, which
is going to be mainly our case, any coherent ACM sheaf on $X$ is locally free.
For this reason, we are going to speak often of ACM bundles (since we identify
locally free sheaves with their associated vector bundle).

It is well known that ACM sheaves provide a criterium to determine the
complexity of the underlying variety. Indeed, this complexity can be studied
in terms of the dimension and number of families of indecomposable ACM sheaves
that it supports. Recently, inspired by an analogous classification for
quivers and fork-algebras of finite type, the classification of any ACM
variety as being of finite, tame or wild representation type(cf. [DrG01] for
the case of curves and [CaH11] for the higher dimensional case) has been
proposed. Let us recall the definitions:

######  Definition 2.2.

An ACM scheme $X\subset\mathbb{P}^{n}$ is of finite representation type if it
has, up to twist and isomorphism, only a finite number of indecomposable ACM
sheaves. An ACM scheme $X\subseteq\mathbb{P}^{n}$ is of tame representation
type if for each rank $r$, the indecomposable ACM sheaves of rank $r$ form a
finite number of families of dimension at most one. On the other hand, $X$
will be of wild representation type if there exist $t$-dimensional families of
non-isomorphic indecomposable ACM sheaves for arbitrary large $t$.

Varieties of finite representation type have been completely classified into a
short list in [BGS87, Theorem C] and [EiH88, p. 348]. They are three or less
reduced points on $\mathbb{P}^{2}$, a projective space, a non-singular quadric
hypersurface $X\subseteq\mathbb{P}^{n}$, a cubic scroll in $\mathbb{P}^{4}$,
the Veronese surface in $\mathbb{P}^{5}$ or a rational normal curve. The only
known example of a variety of tame representation type is the elliptic curve.
On the other hand, so far only a few examples of varieties of wild
representation type are known: curves of genus $g\geq 2$ (cf. [CaH11]), del
Pezzo surfaces and Fano blow-ups of points in $\mathbb{P}^{n}$(the cases of
the cubic surface and the cubic threefold have also been handled in [CaH11])
and ACM rational surfaces on $\mathbb{P}^{4}$([MiP13]).

Very often the ACM bundles that we will construct will share another stronger
property, namely they have the maximal possible number of global sections;
they will be the so-called Ulrich bundles. Let us end this section recalling
the definition of Ulrich sheaves and summarizing the properties that they
share and that will be needed in the sequel.

######  Definition 2.3.

A coherent sheaf $\mathcal{F}$ on $X$ is said to be initialized, if

$H^{0}(X,\mathcal{F}(-1))=0\text{ and }H^{0}(X,\mathcal{F})\neq 0.$

If $\mathcal{F}$ is ACM, then there exists an integer $k$ such that
$\mathcal{F}_{init}:=\mathcal{F}(k)$ is initialized.

######  Definition 2.4.

Given a projective scheme $X\subseteq\mathbb{P}^{n}$ and a coherent sheaf
$\mathcal{F}$ on $X$. A vector bundle $\mathcal{F}$ is an Ulrich bundle, if
$\mathcal{F}$ is ACM on $X$ and the initialized twist $\mathcal{F}_{init}$ of
$\mathcal{F}$ satisfies $h^{0}(X,\mathcal{F}_{init})={\rm rank}(F)\deg(X)$.

When $X\subseteq\mathbb{P}^{n}$ is a hypersurface of degree $s$, the existence
of Ulrich bundles is related to classical problems in algebraic geometry
([Bea00]). If ${\rm rank}(\mathcal{F})=1$, then one has a determinantal
presentation of $X:=\\{\det(M)=0\\}$, where $M=(\ell_{ij})$, $1\leq i,j\leq s$
is a matrix of linear forms; a bundle $\mathcal{F}$ with ${\rm
rank}(\mathcal{F})=2$ corresponds to a Pfaffian equation of $X:\\{{\rm
pf}(M)=0\\}$, where $M$ is a $(2s)\times(2s)$ skew-symmetric linear matrix(see
[Bea00]). In this paper, we describe some criterions for determining when a
cubic fourfold $X$ has an Ulrich bundle of given rank $r$. It is clear that
each direct summand of an Ulrich bundle is also Ulrich. Thus, one can restrict
the attention to indecomposable bundles, i.e. bundles that do not split as
direct sum of bundles of smaller ranks. The study of indecomposable Ulrich
bundles is a particularly in problem that could give some suggestions on the
complexity of the embedding $X\subset\mathbb{P}^{N}$.

Now let $X=V(f)\subseteq\mathbb{P}^{n}$ be a hypersurface cut out by a
homogeneous form $f$ of degree $s$. It is well-known that a matrix
factorization $(A,B)$ of $f$ induces a maximal Cohen-Macaulay module supported
on $X$ by ${\rm coker}A$. Conversely, if we have a maximal Cohen-Macaulay
$R_{X}=R/(f)$-module, one has a matrix $A$ by reading off its length $1$
resolution. Indeed, it forms a part of a matrix factorization of $f$.
Therefore, there is a unique matrix $B$ such that $AB=BA=f\cdot\mathrm{Id}$.
As a conclusion, there is a bijection between the maximal Cohen-Macaulay
modules and the equivalence classes of matrix factorizations of $f$.

Now, let us briefly recall Shamash’s construction (see [Sha69]). Suppose that
$Y$ is a subscheme contained in hypersurface $X=V(f)\subseteq\mathbb{P}^{n}$.
Let $R_{Y}$ be the coordinate ring of $Y$, and $F^{*}$ the minimal free
$R$-resolution of $R_{Y}$. Since $Y\subseteq X$, we have a right exact
sequence

$\textstyle{\ldots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F_{1}\otimes_{R}R_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F_{0}\otimes_{R}R_{X}\cong
R_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{R_{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Hence, there is an $R$-free resolution of $R_{Y}$ (possibly non-minimal)

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
154.3553pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\ldots\to G_{4}\oplus G_{2}(-s)\oplus G_{0}(-2s)\to
G_{3}\oplus G_{1}(-s)\to G_{2}\oplus G_{0}(-s)\to G_{1}\to G_{0}\to R_{Y}\to
0}$}}}}}}}\ignorespaces}}}}\ignorespaces,$

where $G_{i}=F_{i}\otimes_{R}R_{X}$. It becomes eventually $2$-periodic after
the $(c-1)$-th step, where $c={\rm codim}R_{Y}$. Hence, it induces a matrix
factorization of $f$. Such a matrix factorization provides a presentation of
an ACM sheaf on $X$. Moreover, this construction allows us to control some
extent the degrees of the entries of the corresponding minimal matrix
factorization of $f$ induced by an $R_{X}$-module $R_{Y}$, if we know the
Betti numbers of $R_{Y}$ as an $R$-module. Thus, the Betti numbers of the
minimal periodic resolution is called the shape of the matrix factorization.

Let $X$ be a cubic fourfold containing a quintic Del Pezzo surface $T$. Then,
$T$ have the minimal free resolution

(1)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{\mathbb{P}^{5}}(-5)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{\mathbb{P}^{5}}^{5}(-3)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathcal{O}^{5}_{\mathbb{P}^{5}}(-2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{I}_{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

An easy way to produce matrix factorizations on a hypersurface $X=V(f)$ in
$\mathbb{P}^{5}$ is to consider $\Gamma_{\bullet}(\mathcal{O}_{T})$ as a
module over $R=K[x_{0},\ldots,x_{5}]$ annihilated by $f$. A matrix
factorization of $f$ is given by the periodic part of a minimal free
resolution of $\Gamma_{\bullet}(\mathcal{O}_{T})$ as a module over
$R_{X}=R/(f)$. In particular, the minimal resolution of
$\Gamma_{\bullet}(\mathcal{O}_{T})$ as a module over the homogeneous
coordinate ring of a cubic fourfold $X\supset T$ is eventually $2$-periodic
with Betti numbers

| $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$
---|---|---|---|---|---|---|---
0 | $1$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$
1 | $\cdot$ | $5$ | $6$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$
2 | $\cdot$ | $\cdot$ | $\cdot$ | $6$ | $6$ | $\cdot$ | $\cdot$
3 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $6$ | $6$

Hence, $X$ has a matrix factorization of type
$(\psi:\mathcal{O}^{6}_{X}(-3)\to\mathcal{O}^{6}_{X}(-1);\varphi:\mathcal{O}^{6}_{X}(-1)\to\mathcal{O}_{X}^{6})$.
Let $\mathcal{F}={\rm coker}\varphi$. Then $\mathcal{F}$ is an Ulrich bundle
of rank $2$.

To each $Y$ as above, we associate the vector bundle $\mathcal{F}$ by Serre’s
construction:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{X}(-2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\textstyle{\mathcal{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{I}_{Y/X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

where $\mathcal{I}_{Y/X}$ is the ideal sheaf of $Y$ in $X$, and $s$ is a
section of $\mathcal{F}$ such that $Y$ is its zero locus. Let $H$ be the class
of a hyperplane section of $X$. Then, we have $c_{1}(\mathcal{F})=2H$ and
$c_{2}(\mathcal{F})=5$. As follows from [Bea00, Bea02], the vector bundles
$\mathcal{F}$ of this type have several other equivalent characterizations:

######  Theorem 2.5.

Let $X$ be a cubic fourfold in $\mathbb{P}^{5}$ and $\mathcal{F}$ a rank $2$
vector bundle on $X$. Then the following statements are equivalent.

  * $1)$

$\mathcal{F}$ is an Ulrich bundle of rank $2$.

  * $2)$

$\mathcal{F}$ is isomorphic to a vector bundle obtained by Serre’s
construction from a quintic Del Pezzo surface $T\subset X$.

  * $3)$

There exists a skew-symmetric $6$ by $6$ matrix $M$ of linear forms on
$\mathbb{P}^{5}$ such that $\mathcal{F}$ considered as a sheaf on
$\mathbb{P}^{5}$ can be included in the following exact sequence:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{\mathbb{P}^{5}}^{6}(-1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{\mathcal{O}_{\mathbb{P}^{5}}^{6}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

##  3\. Generalities on Ulrich bundles

The goal of this section is to extend Theorem 2.5 from [Bea00, Bea02] to
Ulrich bundles of rank $r\geq 3$ on cubic fourfolds. First, we show how to
associate an ACM surface to an Ulrich bundle on cubic fourfolds from
$\mathcal{N}$-type resolutions (or so-called Bourbaki sequences, c.f.
[BHU87]). We recall the definition here.

######  Definition 3.1.

Let $X\subset\mathbb{P}^{n}$ be an equidimensional scheme, and let $Y\subset
X$ be a codimension $2$ subscheme without embedded components. A coherent
sheaf $\mathcal{L}$ on $X$ is dissocié if it is isomorphic to a direct sum
$\bigoplus\mathcal{O}_{X}(a_{i})$ for various $a_{i}\in\mathbb{Z}$. An
$\mathcal{N}$-type resolution of $Y$ is an exact sequence

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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
65.75pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{N}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
103.25pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
103.25pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{J}_{Y,X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
148.58554pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
148.58554pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{0}$}}}}}}}\ignorespaces}}}}\ignorespaces,$

with $\mathcal{L}$ is dissocié, and $\mathcal{N}$ is a coherent sheaf
satisfying $H_{\bullet}^{1}(\mathcal{N}^{\vee})=0$, and ${\rm
Ext}^{1}(\mathcal{N},\mathcal{O}_{X})=0.$

In the case that $X$ satisfies Serre’s condition $S_{2}$, and
$H_{\bullet}^{1}(\mathcal{O}_{X})=0$, then an $\mathcal{N}$-type resolution of
$Y$ exists (see [Har03, 2.12]). An $\mathcal{N}$-type resolution is not unique
but it is well known that any two $\mathcal{N}$-type resolutions of the same
subscheme are stably equivalent (see [Har03, 1.10]). In other words, if
$\mathcal{N}$, and $\mathcal{N}^{\prime}$ are two sheaves appearing in the
middle of an $\mathcal{N}$-type resolution of a subscheme $Y$, then there
exist dissocié sheaves $\mathcal{L}_{1}$, $\mathcal{L}_{2}$, and an integer
$a$ such that

$\mathcal{N}\oplus\mathcal{L}_{1}\cong\mathcal{N}^{\prime}(a)\oplus\mathcal{L}_{2}.$

The following theorem and its converse involve the relationship between Ulrich
bundles and ACM surfaces.

######  Theorem 3.2.

Let $\mathcal{F}$ be a Ulrich bundle of rank $r\geq 2$ on a cubic fourfold $X$
and generated by its global sections. Then, for every $a\geq 1$, there is a
surface $Y\subset X$ such that $Y$ has an $\mathcal{N}$-type resolution

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{X}^{r-1}(-a)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathcal{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{I}_{Y/X}(b)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

for some $b\in\mathbb{Z}$.

###### Proof.

We denote by $M$ the module $H_{\bullet}^{0}(\mathcal{F})$. Then, we have a
short exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{R^{3r}(-1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{R^{3r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Therefore, the Castelnuovo-Mumford regularity of $M$ is $0$. Let $N$ be a
graded submodule of $M$ generated by homogeneous elements of $M$ all of the
same degree $a-1$. It follows from the Castelnuovo-Mumford regularity of $M$
is $0$ and $a\geq 1$ that $M/N$ is of finite length over $R$. We now observe
that $N$ has a presentation

$\textstyle{R^{c}(-a)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{R^{b}(-a+1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

in which the presenting matrix $\phi$ has homogeneous entries with all the
entries having degree $1$. We then adjoin indeterminates $Y_{ij}$ to $R$ for
$1\leq i\leq b$, and $1\leq j\leq 5$ to obtain a new ring $A=R[Y_{ij}]$. We
put $M_{1}=A\otimes_{R}M$, and $N_{1}=A\otimes_{R}N$. We may consider the
matrix $\phi$ as an $A$-homomorphism $A^{c}(-a)\to A^{b}(-a+1)$ presenting
$N_{1}$, that is, we are identifying $A\otimes\phi$ with $\phi$. Let $\Phi$ be
the matrix with elements $Y_{ij}$, we obtain an $A$-homomorphism
$A^{c}(-a)\oplus A^{3r}(-a)\to A^{b}(-a+1)$ via the matrix $\Psi=[\phi|\Phi]$,
where the vertical line represents a matrix partitioning corresponding to the
two direct summands of $A^{c}(-a)\oplus A^{3r}(-a)$. Let $E$ be the image of
$A^{3r}(-a)$ in $N_{1}$ under the composition of the maps

$\textstyle{A^{c}(-a)\oplus
A^{3r}(-a)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad\quad\
\Psi}$$\textstyle{A^{b}(-a+1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{N_{1}.}$

Then, $E$ is a free submodule of $N_{1}$ of rank ${r}$, and also $E$ is a
graded submodule of $N_{1}$. Since $M$ is Cohen-Macaulay as $R_{X}$-module,
$M_{1}$ is Cohen-Macaulay as $A$-module. It follows from $M_{1}/E$ being a
graded $A$-module that its associated prime ideals are homogeneous. Since
$\dim A\geq\dim R=6$, and the $(x_{0},\ldots,x_{5})$-depth of $M$ over $R$ is
equal to the $(x_{0},\ldots,x_{5})$-depth of $M_{1}$ over $A$, we get that
${\rm depth}_{(x_{0},\ldots,x_{5})}(M_{1}/E)=4$. Hence, because our field is
infinite, there must be a form $g$ of degree one in $x_{0},...,x_{5}$ that is
not a zero divisor on $M_{1}/F$.

Next we show that $A_{g}\otimes_{A}(N_{1}/E)$ is isomorphic to a normal prime
ideal $I_{g}$ of $A_{g}$. Indeed, since degree of $g$ is one in
$x_{0},\ldots,x_{5}$, we have that $M_{g}$ at $N_{g}$ is a free
$A_{g}$-module. Hence, $(N_{1})_{g}=A_{g}\otimes_{A}N_{g}$ is also a free
$A_{g}$-module. Thus, the sequence

$\textstyle{A_{g}^{c}(-a)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi}$$\textstyle{A_{g}^{b}(-a+1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(N_{1})_{g}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

splits. After performing invertible row and column operations on $\Psi$, we
obtain a $b\times c$ matrix of the form

$\bf 0$ | $\bf{I}_{t}$
---|---
$\bf 0$ | $\bf 0$

where $\bf I_{t}$ is the $t\times t$ identity matrix for some $t$. The effect
of the row operations on the matrix $\Psi=[\phi|\Phi]$, is to obtain a matrix
$\Psi^{\prime}=[\phi^{\prime}|\Phi^{\prime}]$, where $\Phi^{\prime}$ consists
of entries $y^{\prime}_{ij}$ which are a new set of ${3r}\times b$ polynomial
indeterminates over $R_{g}$. After applying further invertible column
operations on $\Psi$, we obtain

$\Psi^{\prime}$=  $\bf 0$ $\bf{I}_{t}$ $\bf 0$ $\bf 0$ $\bf 0$
$y^{\prime\prime}_{ij}$

where the $y^{\prime\prime}_{ij}$ in the lower right-hand corner are a subset
of the $y^{\prime}_{ij}$. Let $I_{g}$ be the $A_{g}$-ideal of projective
dimension one, that is, the ideal generated by the maximal minors of the
matrix $(y^{\prime\prime}_{ij})$ in the lower right-hand corner of
$\Psi^{\prime}$. Then, $A_{g}\otimes_{A}(N_{1}/E)\cong I_{g}$. Since $g$ is
regular on $M_{1}/E$, we get that $M_{1}/E\cong I$. Since this matrix consists
of indeterminates over $R_{g}$, it follows that $I_{g}$ is a normal prime
ideal of $A_{g}$, and so $I$ is a normal prime ideal of $A$. Therefore, we
have an exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{A(-a)^{3r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

where $I$ is a homogeneous prime ideal of $A$ of height two, and $A/I$ is
Cohen-Macaulay. Then, it follows from Flenner’s version of Bertini’s Theorem
(see [Fle77, Section 5]) that there exists a linear regular sequence
$h_{1},\ldots,h_{{3r}b}$ on $A/I$. It follows that the sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(A/(h_{1},\ldots,h_{{3r}b})A)(-a)^{3r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{1}/(h_{1},\ldots,h_{{3r}b})M_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I/(h_{1},\ldots,h_{{3r}b})I\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

is exact. Since $M_{1}/(h_{1},\ldots,h_{{3r}b})M_{1}\cong M$ and
$A/(h_{1},\ldots,h_{{3r}b})A\cong R$, there is a codimension $2$ subscheme
$Y\subset X$ without embedded components such that $Y$ has an
$\mathcal{N}$-type resolution

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{X}^{r-1}(-a)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathcal{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{I}_{Y/X}(b)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

for some $b\in\mathbb{Z}$.

∎

######  Proposition 3.3.

Let $\mathcal{F}$ be an Ulrich bundle of rank $r\geq 3$ on the cubic fourfold
$X$ and generated by its global sections. Then there is an ACM surface
$Y\subset X$ of degree $d=\frac{1}{2}(3r^{2}-r)$ and sectional genus
$r^{3}-2r^{2}+1$ such that $Y$ has an $\mathcal{N}$-type resolution

(2)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{X}^{r-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathcal{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{I}_{Y/X}(r)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Moreover, $Y$ has the minimal free resolution

(3)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{\mathbb{P}^{5}}^{r-1}(-r-3)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{\mathbb{P}^{5}}^{3r}(-r-1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathcal{O}^{2r+1}_{\mathbb{P}^{5}}(-r)\oplus\mathcal{O}_{\mathbb{P}^{5}}(-3)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{I}_{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

###### Proof.

Since $\mathcal{F}$ is an Ulrich bundle of rank $r$, we can take $r-1$ general
sections of $\mathcal{F}$ so that the quotient of $\mathcal{F}$ by
$\mathcal{O}_{X}^{r-1}$ is torsion free. Therefore we have an exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{X}^{r-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathcal{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{I}_{Y/X}(D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where $D=c_{1}(\mathcal{F})=rH$ is a certain divisor on $X$ and $Y$ is a
surface of degree equal to $c_{2}(\mathcal{F})$. As $\mathcal{F}$ is generated
by its global sections, $\mathcal{I}_{Y/X}(r)$ is also generated by global
sections and $h^{0}(\mathcal{I}_{Y/X}(r))=2r+1$. On the other hand,
$\mathcal{F}$ has the following minimal free resolution in $\mathbb{P}^{5}$

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{\mathbb{P}^{5}}^{3r}(-1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{\mathcal{O}_{\mathbb{P}^{5}}^{3r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

by [CaH11, Proposition 3.7]. Using the mapping cone procedure with the free
resolutions of $\mathcal{O}_{X}$ and the exact sequence 2, $Y$ has the minimal
free resolution as required. Therefore $Y$ is an ACM surface. By [CaH11,
Proposition 3.7], we have $\deg Y=\frac{1}{2}(3r^{2}-r)$ and by Riemann–Roch,
sectional genus of $Y$ is $r^{3}-2r^{2}+1$.

∎

######  Remark 3.4.

There have been many studies about ACM bundles of rank two in dimensions two
and three in [ArM09, BrF09, ChF09, ChM05, Mad00]. The higher rank case has
been investigated in [ArG99, ArM09, Mad05, Tru19a, TrY20a]. The papers [MiP14]
and [PoT09] give a few examples of indecomposable ACM bundles of arbitrarily
high rank. The already mentioned papers [CaH11, CaH12] contain a systematic
study of stable ACM bundles in higher rank on cubic surfaces and threefolds. A
general existence result for Ulrich bundles on hypersurfaces is given in
[BHU91]. But in general, the existence of Ulrich sheaves on a projective
scheme $X$ is not known.

######  Remark 3.5.

Liaison has become an established technique in algebraic geometry. The
greatest activity, however, has been in the last quarter-century beginning
with the work of Peskine and Szpiró in 1974(see [PeS74]). Liaison is a
powerful tool for constructing examples.

######  Definition 3.6.

We say $Y_{2}$ is obtained by an elementary biliaison of height $m$ from
$Y_{1}$, if there exists an ACM scheme $Z\subset\mathbb{P}^{n}$, of dimension
$r+1$ containing $Y_{1}$ and $Y_{2}$, such that $Y_{2}\thicksim Y_{1}+mH$ on
$Z$ . (here $\thicksim$ means linear equivalence of divisors on $Z$ in the
sense of [Har94].) The equivalence relation generated by elementary biliaisons
is called simply biliaison. If $X$ is a complete intersection scheme in
$\mathbb{P}^{n}$, we speak of CI-biliaison. If $Y_{1}$, $Y_{2}$, $Z$ are
contained in some projective scheme $X\subseteq\mathbb{P}^{n}$, we speak of
biliaison (CI-biliaison) on $X$.

Casanellas and Hartshorne gave a criterion for when two codimension $2$
subschemes $Y_{1}$, $Y_{2}$ of a normal ACM projective scheme $X$ are in the
same biliaison class (see [CaH04, Theorem 3.1]). As application, we have the
following corollary.

######  Corollary 3.7.

Let $\mathcal{F}$ be an Ulrich bundle on a cubic foufold $X$. Then surfaces
constructed from $\mathcal{F}$ as in Theorem 3.2 are ACM and belong to the
same $CI$-biliaison equivalence class on $X$.

###### Proof.

It follows immediately from Theorem 3.2, Theorem 3.1 in [CaH04]) and
Proposition 3.3 ∎

Now we will prove a converse to the Proposition 3.3.

######  Theorem 3.8.

Assume that $Y$ is an ACM smooth surface with the minimal free resolution 3
for $r\geq 3$. Let $X$ be a general cubic fourfold containing $Y$. Then $Y$
has an $\mathcal{N}$-type resolution with $\mathcal{F}$ being an Ulrich bundle
of rank $r$

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{X}^{r-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathcal{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{I}_{Y/X}(r)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

###### Proof.

Consider $\Gamma_{\bullet}(\mathcal{O}_{Y})$ as a $R_{X}$-module, the periodic
part of its minimal free resolution yields, up to twist, matrix factorization
of the form

$\textstyle{R^{3r}(-3)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{R^{3r}(-1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad\varphi}$$\textstyle{R^{3r}.}$

Let $F^{\bullet}$ and $\overline{G^{\bullet}}$ be minimal free resolutions of
the section ring $\Gamma_{\bullet}(\mathcal{O}_{Y})$ as an $R$-module, and
$R_{X}$-module, respectively. Let $\varphi$ be the map
$\overline{G_{4}}\to\overline{G_{3}}$, and $\mathcal{F}=\widetilde{{\rm
coker}\varphi}(-3)$. Since $\deg\det\varphi=1\cdot(3r)=3\cdot r$,
$\mathcal{F}$ is an ACM sheaf of rank $r$, and so the Shamash resolution is

$\to\mathcal{O}_{X}^{3r}(-r-4)\to\mathcal{O}_{X}^{3r}(-r-3)\to\mathcal{O}_{X}^{3r}(-r-1)\oplus\mathcal{O}_{X}(-3)\to\mathcal{O}^{2r+1}_{X}(-r)\oplus\mathcal{O}_{X}(-3)\to\mathcal{I}_{Y/X}\to
0.$

Thus, the Shamash resolution is always non-minimal, and in a minimal
resolution a cancellation occurs, causing $\beta_{1,3}=8$, if $r=3$, and
$\beta_{1,3}=1$, otherwise to decrease by one. Such cancellation corresponds
to the equation $f$ of $X$ in $R$ becoming superfluous in $R_{X}=R/(f)$. Since
$\mathcal{F}=\widetilde{{\rm coker}\varphi}(-3)$, the sheaf $\mathcal{F}$ is
the sheafification of a MCM module over $R_{X}$. Thus, the sheaf $\mathcal{F}$
is an Ulrich bundle of rank $r$. By the definition of the Shamash resolution,
the map $\overline{G_{3}}\to\overline{G_{2}}$ factorizes through
$\mathcal{F}$. Thus, we obtain a map
$\alpha:\mathcal{O}_{X}^{r-1}(-r)\to\mathcal{F}$.

Now, we apply the functor ${\rm Hom}(\bullet,\omega_{X})$ to $\alpha$, and
obtain $\alpha^{*}(-r+1):\mathcal{F}^{*}(-r+1)\to\mathcal{O}_{X}^{r-1}$. The
cokernel of this map coincides by construction with the cokernel of the dual
of the sheafification of the last map of $F^{\bullet}$

$\mathcal{O}_{X}^{3r}(-r+1)\to\mathcal{O}_{X}^{r-1},$

which is a presentation of $\omega_{Y}$ by duality on $\mathbb{P}^{5}$. Since
${\rm rank}\mathcal{F}=r=(r-1)+1$, both $\alpha^{*}(-r+1)$, and $\alpha$ drop
rank in expected codimension $2$. Applying again ${\rm
Hom}(\bullet,\omega_{X})$ to $\alpha^{*}(-r+1)$ we get that $\alpha$ is
injective. It follows from the Hilbert-Burch Theorem that we have an exact
sequence

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{O}_{X}^{r-1}(-r)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
67.1275pt\raise 4.50694pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{\alpha}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 90.47934pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
90.47934pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathcal{F}}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
127.00714pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
127.00714pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\mathcal{O}_{X}(\ell)}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
181.80832pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
181.80832pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{O}_{Y}(\ell)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
236.02615pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
236.02615pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{0}$}}}}}}}\ignorespaces}}}}\ignorespaces,$

for some $\ell$. By applying again ${\rm Hom}(\bullet,\omega_{X})$ to this
last exact sequence one gets that $\alpha^{*}(-r)$ is a presentation of
$\omega_{Y}(-\ell)$. Hence, we have $\ell=0$, as required. ∎

######  Theorem 3.9.

Let $\mathcal{F}$ be a vector bundle of rank $r\geq 3$ on a cubic fourfold $X$
in $\mathbb{P}^{5}$. Then the following statements are equivalent.

  1. $1)$

$\mathcal{F}$ is an Ulrich bundle of rank $r$.

  2. $2)$

$\mathcal{F}$ is isomorphic to a vector bundle obtained from a surface
$Y\subset X$ such that $\mathcal{I}_{Y}$ has the minimal free resolution 3.

  3. $3)$

There exists a $3r\times 3r$ matrix $M$ of linear forms on $\mathbb{P}^{5}$
such that the sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{\mathbb{P}^{5}}^{3r}(-1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{M}$$\textstyle{\mathcal{O}_{\mathbb{P}^{5}}^{3r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

is exact.

###### Proof.

$1)\Rightarrow 2)$: This follows from Proposition 3.3.
$2)\Rightarrow 3)$: This follows from Theorem 3.8.
$3)\Rightarrow 1)$: This is trivial. ∎

##  4\.  Ulrich bundles of small rank

First, let us describe a little more details on the existence of Ulrich
bundles on smooth cubic fourfolds. In [KiS20, Proposition 2.5], Kim and
Schreyer showed that if $\mathcal{F}$ is be an Ulrich bundle of rank $r$ on a
very general cubic fourfold then $r$ is divisible by 3 and $r\geq 6$. Using
invariant theory, Manivel [Man19, Propostion 2.2] found a family of very
general cubic fourfolds having an Ulrich sheaf of rank $9$. In [Bea00],
Beauville showed that there exists a smooth cubic fourfold having an Ulrich
bundle of rank $2$. The set of such cubic fourfolds contains an open (but it
is not open) subset of the divisor $\mathcal{C}_{14}$([BRS19]). Hence, it is
natural to ask the existence question for rank $3$ Ulrich bundles on smooth
cubic fourfolds. The goal of this section is to prove Theorem 4.7, which
answers this question.

Now, let $\mathcal{H}_{9,4}$ be the component of the Hilbert scheme of curves
of degree $d=9$ and genus $g=4$ in $\mathbb{P}^{4}$ which dominate the moduli
spaces $\mathcal{M}_{4}$ of curves of genus $4$.

######  Proposition 4.1.

Let $C\in\mathcal{H}_{9,4}$ be a general point. Then $C$ is of maximal rank
and homogeneous coordinate ring $R_{C}=R/I_{C}$ and the section ring
$\Gamma_{\bullet}(\mathcal{O}_{C})$ have minimal free resolutions with the
following Betti tables:

|  | $0$ | $1$ | $2$ | $3$ | $4$
---|---|---|---|---|---|---
0 |  | $1$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$
1 |  | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$
2 |  | $\cdot$ | $11$ | $18$ | $9$ | $1$
| $0$ | $1$ | $2$ | $3$
---|---|---|---|---
0 | $1$ | $\cdot$ | $\cdot$ | $\cdot$
1 | $1$ | $5$ | $\cdot$ | $\cdot$
2 | $\cdot$ | $1$ | $8$ | $4$

###### Proof.

Assuming that the restriction map $H^{0}(\mathcal{O}_{\mathbb{P}^{4}}(m))\to
H^{0}(\mathcal{O}_{C}(m))$ has maximal rank. Then we have the following
statements.

  * •

The Hilbert series of the homogeneous coordinate ring of $C$ is

$H_{C}(t)=1+5t+15t^{2}+24t^{3}+33t^{4}+42t^{5}+11t^{6}+60t^{7}+\ldots.$

  * •

The Hartshorne–Rao module

$H^{1}_{\bullet}(\mathcal{I}_{C})\cong k(-1)$

is a vector space of dimension $1$ concentrated in degree $1$.

  * •

The ideal sheaf $\mathcal{I}_{C}$ is $3$-regular.

  * •

The homogeneous ideal $I_{C}=H_{\bullet}^{0}(\mathcal{I}_{C})$ has a $11$
generator in degree $3$.

  * •

The Hilbert numerator has shape

$(1-t)^{4}H_{C}(t)=1-11t^{3}+18t^{4}-9t^{6}+1.$

Hence, smooth maximal rank curves in $\mathcal{H}_{9,4}$ have a Betti table as
required. To establish that a general point in $C\in\mathcal{H}_{9,4}$ is a
maximal rank curve, it suffices to produce a single maximal rank example. We
will explicitely construct such examples. Moreover, by inspection we find that
the general $C$ lies on a smooth cubic hypersurface $X$.

Now let $Z=\mathbb{P}^{2}(p_{1},\ldots,p_{10})$ is a blow-up of
$\mathbb{P}^{2}$ in $10$ points in general position by the complete linear
system

$H_{Z}=(5;2^{2},1^{8})=5L-E_{1}-E_{2}-\sum\limits_{i=3}^{10}E_{i}.$

Then $Z$ is a smooth surface of degree $9$ and sectional genus $4$ in
$\mathbb{P}^{6}$. Let $q\in\mathbb{P}^{6}$ be general point and
$\phi_{q}:\mathbb{P}^{6}\dashrightarrow\mathbb{P}^{5}$ the projection from $q$
onto a hyperplane and let $Y_{1}=\phi_{q}(Z)\subseteq\mathbb{P}^{5}$.
Therefore, $Y_{1}$ is a surface of degree $9$ and sectional genus $4$ and
$K_{Y_{1}}^{2}=-1$. Now we take a general hyperplane of $Y_{1}$, we get that
curve as requried.

The Betti table of the resolution of the section ring
$\Gamma^{\bullet}(\mathcal{O}_{C})$ as an $R$-module can be deduced with the
same method, since $H_{\Gamma^{\bullet}(\mathcal{O}_{C})}(t)=H_{C}(t)+t$.

∎

######  Remark 4.2.

In the computation above, we need exhibit explicit examples of modules, curves
or surfaces defined over the rationals $\mathbb{Q}$ or complex numbers
$\mathbb{C}$ satisfying some open conditions on their Betti numbers. Since
such examples defined over an open part ${\rm Spec}\mathbb{Z}$, the same
result holds for algebraically closed fields of positive characteristic except
for possibly finitely many primes $p$. Thus, in principle, one could try to
verify the result one by one for the remaining primes, and then by
semicontinuity, we get the same result holds for the rationals $\mathbb{Q}$ or
complex numbers $\mathbb{C}$. In this paper, we provide explicit
constructions/equations of examples with special geometric features. In an
ancillary file (see [TY20]) we have implemented our constructions in the
computer algebra software Macaulay2 [GSM2].

Now let $\mathcal{W}_{12,10}$ be the irreducible component of the Hilbert
scheme parametrizing projective Cohen-Macaulay smooth curves
$C\subseteq\mathbb{P}^{4}$ of degree $12$, and genus $10$ which dominates the
moduli space $\mathcal{M}_{10}$.

######  Corollary 4.3.

Let $C\in\mathcal{W}_{12,10}$ be a general point. Then $C$ is of maximal rank
and the homogeneous coordinate ring $R_{C}=R/I_{C}$ and the section ring
$\Gamma_{\bullet}(\mathcal{O}_{C})$ have minimal free resolutions with the
following Betti tables:

(4) | $0$ | $1$ | $2$ | $3$
---|---|---|---|---
0 | $1$ | $\cdot$ | $\cdot$ | $\cdot$
1 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$
2 | $\cdot$ | $8$ | $9$ | $\cdot$
3 | $\cdot$ | $\cdot$ | $\cdot$ | $2$

###### Proof.

As in Proposition 4.1, we can compute the expected Betti tables of the
$R$-resolutions of $R/I_{C}$ and $\Gamma_{\bullet}(\mathcal{O}_{C})$. ∎

######  Corollary 4.4.

A general cubic hypersurface in $\mathbb{P}^{4}$ contains a family of
dimension $24$ of curves of genus $10$ and degree $12$.

###### Proof.

Let $C\in\mathcal{W}_{12,10}$ be a general point and let $X$ be a general
cubic threefold containing it. Then $\chi(\mathcal{N}_{C/X})=12(5-3)=24$ by
[ScT18, Lemma 5.3.]. We claim that $h^{1}(\mathcal{N}_{C/X})=0$. It is
sufficient to check this vanishing on one example, as can be done with the
Macaulay2. Hence, $h^{0}(\mathcal{N}_{C/X})=24$. Let $\mathcal{C}$ be the
space of cubic threefolds containing a general curve $C$ of genus $10$ and
degree $12$, up to projective equivalences. Since
$h^{0}(\mathbb{P}4,\mathcal{I}_{C}(3))-1=8-1=7$. We have

$\dim\mathcal{C}=\dim\mathcal{M}_{10}+\rho(10,4,12)+7-24=10,$

where the Bril-Noether number $\rho(g,r,d):=g-(r+1)(g+r-d)$.

∎

######  Proposition 4.5.

There exists a projective surface in $\mathbb{P}^{5}$ of degree $9$ sectional
genus $4$ such that the homogeneous coordinate ring $R_{Y}=R/I_{Y}$ and the
section ring $\Gamma_{\bullet}(\mathcal{O}_{Y})$ have minimal free resolutions
with the Betti tables as in Proposition 4.1. In particular, the minimal
resolution of $\Gamma_{\bullet}(\mathcal{O}_{Y})$ as a module over the
homogeneous coordinate ring of a cubic fourfold $X\supset Y$ is eventually
$2$-periodic with Betti numbers

| $0$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$
---|---|---|---|---|---|---|---
0 | $1$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$
1 | $1$ | $5$ | $1$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$
2 | $\cdot$ | $1$ | $9$ | $9$ | $\cdot$ | $\cdot$ | $\cdot$
3 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $9$ | $9$ | $\cdot$

###### Proof.

Let $C$ be a smooth curve as in Proposition 4.1. Then it has minimal free
resolution

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.42361pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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29.42361pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 131.77719pt\raise 0.0pt\hbox{\hbox{\kern
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131.77719pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{{R^{\prime}}^{18}(-4)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
177.63107pt\raise 5.1875pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8264pt\hbox{$\scriptstyle{\varphi}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 194.92065pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
194.92065pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{{R^{\prime}}^{11}(-3)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
258.06412pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
258.06412pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{R^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
297.27426pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
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297.27426pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{R_{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
339.34738pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
339.34738pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{0}$}}}}}}}\ignorespaces}}}}\ignorespaces,$

where ${R^{\prime}}=k[x_{0},\ldots,x_{4}]$. Recall that we say that a sequence
of $R^{\prime}$-modules

$\textstyle{\ldots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M_{i+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{i+1}}$$\textstyle{M_{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{i}}$$\textstyle{M_{i-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ldots}$

is exact at $M_{i}$ if and only if $\mathrm{Im}\ \varphi_{i+1}=\mathrm{Ker}\
\varphi_{i}$. The sequence is exact if it is exact at every $M_{i}$ in the
sequence. Now we will construct surface $Y$ as follows.

  * •

Let $A$ be an $11\times 18$ matrix of indeterminates and $B$ be an $18\times
9$ matrix of indeterminates. Let $T=k[A,B]$ be the polynomial ring generated
by all the indeterminates $A_{ij}$ and $B_{ij}$ and
$T^{\prime}=R^{\prime}\otimes T$. Consider a sequence $F_{\bullet}$ of
$T^{\prime}$-modules

$F_{\bullet}:\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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19.8205pt\raise 6.1111pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75pt\hbox{$\scriptstyle{\psi+B}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 41.95297pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
41.95297pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{{T^{\prime}}^{18}(-4)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
82.71657pt\raise 6.07222pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.71112pt\hbox{$\scriptstyle{\varphi+A}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 104.65894pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
104.65894pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{{T^{\prime}}^{11}(-3).}$}}}}}}}\ignorespaces}}}}\ignorespaces$

A sequence $F_{\bullet}$ of $T^{\prime}$-modules is exact if and only if

$A\circ\psi+\varphi\circ B\equiv 0\text{ and }AB\equiv 0$

  * •

Since $\psi$ and $\varphi$ is linear matrixes over polynomial ring
$R^{\prime}$, the condition $A\circ\psi+\varphi\circ B\equiv 0$ gives a linear
relationship on the indeterminates $A_{ij}$ and $B_{ij}$. Let $m+1$ is numbers
of independent variables on such linear relationship. Let
$S=k[y_{0},\ldots,y_{m}]$. Then the condition $AB\equiv 0$ defined a varieties
$V$ such that a point of $V$ is corespondent to a smooth projective surface
$Y(p)$ of degree $9$ sectional genus $4$. In the current case, they are $m=14$
and $V$ is a cone in $\mathbb{P}^{14}$ of dimensional $8$ generated by $23$
quadric elements.

Hence, we get that surface $Y$ has minimal free resolutions with the Betti
tables as in Proposition 4.1. Moreover, the Hartshorne–Rao module:

$H^{1}_{\bullet}(\mathcal{I}_{Y})\cong k(-1)$

is a vector space of dimension $1$ concentrated in degree $1$. Since
$H_{\Gamma^{\bullet}(\mathcal{O}_{Y})}(t)=H_{Y}(t)+t$, the Betti table of the
resolution of the section ring as an $R$-module is as in Proposition 4.1. ∎

Recall that a smooth cubic fourfold $X$ in $\mathbb{P}^{5}$ is special, if
there is an embedding of a saturated rank-$2$ lattice

$L_{\delta}:=\langle h^{2},Y\rangle\hookrightarrow A(X),$

where $A(X)$ is the lattice of middle Hodge classes, $h\in{\rm Pic}(X)$ is the
hyperplane class, $Y$ is an algebraic surface not homologous to a complete
intersection, and $\delta$ is the determinant of the intersection matrix of
$\langle h^{2},Y\rangle$. In 2000, Hassett showed in [Has00] that the locus
$\mathcal{C}_{\delta}$ of special cubic fourfolds $X$ of a discriminant
$\delta$ is an irreducible divisor which is nonempty if and only if $\delta>6$
and $\delta\equiv 0,2\pmod{6}$. Now let $\mathcal{H}_{12,10}$ be the
irreducible component of the Hilbert scheme parametrizing projective Cohen-
Macaulay smooth surfaces $Y\subseteq\mathbb{P}^{5}$ of degree $12$, and
sectional genus $10$ which dominates the moduli space $\mathcal{W}_{12,10}$.
With the above notation, we have the following proposition.

######  Proposition 4.6.

Let $Y\in\mathcal{H}_{12,10}$ be a general point. Then $Y$ is of maximal rank
and the homogeneous coordinate ring $R_{Y}=R/I_{Y}$ and the section ring
$\Gamma_{\bullet}(\mathcal{O}_{Y})$ have minimal free resolutions with the
Betti tables 4. Moreover, if $X$ is a general cubic fourfold containing $Y$
then $X\in\mathcal{C}_{18}$.

###### Proof.

Assuming that the restriction map $H^{0}(\mathcal{O}_{\mathbb{P}^{5}}(m))\to
H^{0}(\mathcal{O}_{Y}(m))$ has maximal rank. Then we have the following
statements.

  * •

The Hilbert series of the homogeneous coordinate ring of $Y$ is

$H_{Y}(t)=1+6t+21t^{2}+48t^{3}+87t^{4}+138t^{5}+201t^{6}+276t^{7}+\ldots.$

  * •

The Hartshorne–Rao module

$H^{1}_{\bullet}(\mathcal{I}_{Y})=H^{2}_{\bullet}(\mathcal{I}_{Y})=0.$

  * •

The Hilbert numerator has shape

$(1-t)^{5}H_{Y}(t)=1-8t^{3}+9t^{4}-2t^{6}.$

Since $H_{Y}(t)=H_{\Gamma_{\bullet}(\mathcal{O}_{Y})}(t)$, the above Betti
table of the resolution $F_{\bullet}$ of the section ring
$\Gamma_{\bullet}(\mathcal{O}_{Y})$ as an $R$-module. Finally, we compute the
Betti number of $\Gamma_{\bullet}(\mathcal{O}_{Y})$ as an $R_{X}$-module. The
(possibly non-minimal) resolution has Betti table as above.

To show that the Betti tables are indeed the expected ones and that, a
posteriori, a general surface $Y$ is of maximal rank, we only need to exhibit
a concrete example, which we construct from Theorem 3.9 and Proposition 4.5.

Now let $X$ be a general cubic fourfold containing $Y$. By computing the self-
intersection of $Y\subseteq X$ using the formula from [Has00],

$Y^{2}=c_{2}(\mathcal{N}_{Y/X})=6H^{2}+3H\cdot K+K^{2}-\chi_{top}=54.$

Hence, we have $\delta=3Y^{2}-d^{2}=3\cdot 54-12^{2}=18$. Hence
$[X]\in\mathcal{C}_{18}$.

∎

######  Theorem 4.7.

Let $X$ be a general special cubic fourfold in $\mathcal{C}_{18}$. Then the
moduli space of stable rank $3$ Ulrich bundles on a special cubic fourfold $X$
with first Chern class $c_{1}=3H$, where $H$ is the hyperplane class, and
$c_{2}=12$ is nonempty and smooth of dimension $10$.

###### Proof.

By Proposition 4.6, moduli space of stable rank $3$ Ulrich bundles on a
special cubic fourfold $X$ is nonempty. Now we will show that a general
$[X]\in\mathcal{C}_{18}$ contains a surface $[Y]\in\mathcal{H}_{12,10}$. In
fact, $\dim\mathcal{H}_{12,10}=63$ and $\mathcal{H}_{12,10}$ is generically
smooth. Indeed, one can verify that $h^{1}(\mathcal{N}_{Y/\mathbb{P}^{5}})=0$
for $[Y]\in\mathcal{H}_{12,10}$. Let
$\mathcal{C}\subseteq|H^{0}(\mathbb{P}^{5}(3))|\cong\mathbb{P}^{55}$ be the
open set corresponding to smooth cubic hypersurfaces. Since
$h^{0}(\mathcal{I}_{Y}(3))=5$ so that the locus
$\mathcal{D}_{18}=\\{([Y],[X]):Y\subseteq
X\\}\subseteq\mathcal{H}\times\mathcal{C}$ has dimension $63+8-1=70$. The
image of $\pi_{2}:\mathcal{D}_{18}\to\mathcal{C}$ has dimension at most $54$
because the general cubic does not contain any $Y$ belonging to $\mathcal{H}$.
For every $[X]\in\pi_{2}(\mathcal{D}_{18})$ we have

$\dim(\pi^{-1}_{2}([X]))\geq\dim(\mathcal{D}_{18})-\dim(\pi_{2}(\mathcal{D}_{18}))\geq
70-54=16.$

Since $h^{0}(\mathcal{N}_{Y/X})\geq\dim_{[Y\subset X]}(\pi^{-1}_{2}([X]))$ for
every $[Y\subset X]\in\pi^{-1}_{2}([X])$, to show that a general
$X\in\mathcal{C}_{18}$ contains a surface $Y$, it is sufficient to verify that
$h^{0}(\mathcal{N}_{Y/X})=16$ for a general $Y$ and for a smooth
$X\in|H^{0}(\mathcal{I}_{Y}(3))|$. We verified this via Maucaulay2 and we can
conclude that $h^{0}(\mathcal{N}_{Y/X})=16$.

To show that the moduli space is smooth of dimension $10$, we just compute
$h^{1}(\mathcal{F}\otimes\mathcal{F}^{\vee})=10$ and
$h^{2}(\mathcal{F}\otimes\mathcal{F}^{\vee})=h^{3}(\mathcal{F}\otimes\mathcal{F}^{\vee})=0$.
This is elementary (see the proof of Theorem 1.3).

∎

######  Remark 4.8.

$1)$ As an application of the result above and [KuM09, Theorem 4.3], one gets
that the smooth locus of any moduli space of stable rank $3$ Ulrich bundles on
a special cubic fourfold $X\in\mathcal{C}_{18}$ carries a closed symplectic
form.
$2)$ The Hilbert scheme parametrizing projective Cohen-Macaulay smooth
surfaces of degree $12$, and sectional genus $10$ has least two components. In
fact, let $X$ be a cubic fourfold containing an elliptic ruled surface $T$ as
in Theorem 2 in [AHTV19]. Consider $\Gamma_{\bullet}(\mathcal{O}_{T})$ as a
$R_{X}$-module, the periodic part of its minimal free resolution yields, up to
twist, matrix factorization of the form

$\textstyle{R^{9}(-4)\oplus
R(-3)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{R^{9}(-2)\oplus
R(-3)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\quad\varphi}$$\textstyle{R^{9}(-1)\oplus
R.}$

Thus there exists an ACM surface $Y$ such that homogeneous coordinate ring
$R_{Y}=R/I_{Y}$ and the section ring $\Gamma_{\bullet}(\mathcal{O}_{Y})$ have
minimal free resolutions with the Betti tables

| $0$ | $1$ | $2$ | $3$
---|---|---|---|---
0 | $1$ | $\cdot$ | $\cdot$ | $\cdot$
1 | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$
2 | $\cdot$ | $8$ | $9$ | $1$
3 | $\cdot$ | $\cdot$ | $1$ | $2$

Now, for the proof of Theorem 1.3 we need the following result.

######  Lemma 4.9.

There exists a smooth cubic fourfold having both Ulrich bundles of rank $2$
and $3$.

###### Proof.

Let $p_{1},\ldots,p_{10}$ be points in general position in $\mathbb{P}^{2}$
and $C_{0}$ a quadric curve passing four points $p_{3},\ldots,p_{6}$. Let
$Z=\mathbb{P}^{2}(p_{1},\ldots,p_{10})$ is a blow-up of $\mathbb{P}^{2}$ in
$10$ points in general position by the complete linear system
$H_{Z}=(5;2^{2},1^{8})=5L-E_{1}-E_{2}-\sum\limits_{i=3}^{10}E_{i}.$ Then $Z$
is a smooth surface of degree $9$ and sectional genus $4$ in $\mathbb{P}^{6}$.
Let $C_{1}$ be the strict transform of $C_{0}$ on $Z$. Then $C_{1}$ is a
smooth rational curve of degree $6$.

Let $q$ be a point belong a trisecant line to the surface $Z$ and
$\phi_{q}:\mathbb{P}^{6}\dashrightarrow\mathbb{P}^{5}$ the projection from $q$
onto a hyperplane and let $Y=\phi_{q}(Z)\subseteq\mathbb{P}^{5}$. Therefore,
$Y$ is a surface of degree $9$ and sectional genus $4$ and $K_{Y}^{2}=-1$.
Moreover, $Y$ has the minimal free resolution with the Betti tables as in
Proposition 4.1.

On the other hand, $C_{2}=\phi_{q}(C_{1})\subseteq\mathbb{P}^{5}$ is a smooth
curve on $Y$ with the minimal free resolution

| $0$ | $1$ | $2$ | $3$ | $4$ | $5$
---|---|---|---|---|---|---
0 | $1$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$
1 | $\cdot$ | $8$ | $11$ | $3$ | $\cdot$ | $\cdot$
2 | $\cdot$ | $\cdot$ | $4$ | $10$ | $6$ | $1$

Then $C_{2}$ is contained in a smooth rational normal scroll surface $S_{1}$
of degree $4$. Moreover $S$ is contained in a smooth rational normal scroll
cubic threefold $\Sigma$. Let $X$ be a smooth cubic fourfold which contains
both surfaces $Y$ and $S_{1}$. Then there exists a quintic del Pezzo surface
$S$ such that

$X\cap\Sigma=S\cup S_{1}.$

Then by Theorem 2.5 and Proposition 4.5, $X$ has Ulrich bundles of rank $2$
and $3$. Notice that, in an ancillary file (see [TY20]), we provide the
explicit homogeneous ideal of surfaces $Y$, $S$ in the cubic fourfold $X$. ∎

######  Remark 4.10.

In the proof of Lemma 4.9, we give an explicit example of a triple $S$,
$Y\subset X$ over a finite field with the help of Macaulay2 and then
establishing the lemma in characteristic $0$ with semi-continuity. By Theorem
2.5 and 4.7, $[X]\in\mathcal{C}_{14}\cap\mathcal{C}_{18}$. Since $S$ and $Y$
intersect transversally in $16$ points, $X$ is a rational cubic fourfold in
$\mathcal{C}_{18}$ associated with a good sextic del Pezzo fibration(see
[AHTV19, Definition 11]) and with one nontrivial Brauer class. Notice that
intersection $\mathcal{C}_{14}\cap\mathcal{C}_{18}$ has $9$ irreducible
components and H. Awada also gave an other example of a rational cubic
fourfold in $\mathcal{C}_{18}$ associated with a good sextic del Pezzo
fibration and with one nontrivial Brauer class(see [Awa19]).

###### Proof of Theorem 1.3.

Let $X$ be as in Lemma 4.9. Then $[X]\in\mathcal{C}_{14}\cap\mathcal{C}_{18}$.
Since $\mathcal{F}$ is an Ulrich bundle of rank $r\geq 2$, it has the
following minimal free resolution in $\mathbb{P}^{5}$

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{O}_{\mathbb{P}^{5}}^{3r}(-1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{\mathcal{O}_{\mathbb{P}^{5}}^{3r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

by [CaH11, Proposition 3.7]. Tensoring with $F^{\vee}$, we get a right exact
sequence on $X$,

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}^{\vee}(-1)^{3r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{\mathcal{F^{\vee}}^{3r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}\otimes\mathcal{F}^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Now $\mathcal{F}^{\vee}(2)$ is an Ulrich bundle, so $\mathcal{F}^{\vee}$ and
$\mathcal{F}^{\vee}(-1)$ have no cohomology. It follows from cohomology
sequences on $X$, that
$H^{2}(\mathcal{F}\otimes\mathcal{F}^{\vee})=H^{3}(\mathcal{F}\otimes\mathcal{F}^{\vee})=0$.
Therefore, the moduli space is smooth. Furthermore,
$\chi(\mathcal{F}\otimes\mathcal{F}^{\vee})=\chi(\mathcal{F}_{H}\otimes\mathcal{F}^{\vee}_{H})=-r^{2}$
by [CaH11, Corollary 2.13], where $\mathcal{F}_{H}$ is the general hyperplane
section of $\mathcal{F}$, since $c_{1}(\mathcal{F})=rH$. For $\mathcal{F}$
stable or simple we have $h^{0}(\mathcal{F}\otimes\mathcal{F}^{\vee})=1$, and
$h^{2}(\mathcal{F}\otimes\mathcal{F}^{\vee})=h^{3}(\mathcal{F}\otimes\mathcal{F}^{\vee})=0$,
so $h^{1}(\mathcal{F}\otimes\mathcal{F}^{\vee})=r^{2}+1$ is the dimension of
the moduli space.

It remains to show the existence. We proceed by induction on $r$, the cases
$r=2$, $3$ by Lemma 4.9. So let $r\geq 4$, and choose $\mathcal{E}$ stable of
rank $2$, and $\mathcal{F}$ stable of rank $r-2$, different from
$\mathcal{E}$. Then $h^{i}(\mathcal{E}\otimes\mathcal{F}^{\vee})=0$ for $i=0$,
$2$, $3$, so
$h^{1}(\mathcal{E}\otimes\mathcal{F}^{\vee})=-\chi(\mathcal{E}\otimes\mathcal{F}^{\vee})=-\chi(\mathcal{E}_{H}\otimes\mathcal{F}^{\vee}_{H})=2(r-2)$
by [CaH11, Corollary 2.13]. In particular, this number is positive, so there
exist non-split extensions

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{G}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

and the new bundle $G$ will be a simple Ulrich bundle of rank $r$. We consider
the modular family of these simple bundles, which will be smooth of dimension
$r^{2}+1$ by the above observations. If the general simple bundle in this
family is not stable, it must have the same splitting type as the ones just
constructed. However, the dimension of the family of extensions above is

$\dim{\mathcal{E}}+\dim\mathcal{F}+\dim({\rm
Ext}^{1}(\mathcal{F},\mathcal{E}))-1=r^{2}-2r+5.$

Since $r\geq 4$, this number is strictly less than $r^{2}+1$. We conclude that
the general simple bundle of rank $r$ is stable, so stable bundles exist. ∎

For a very general hypersurface $X\subseteq\mathbb{P}^{3}$ of degree at least
$4$, the Noether-Lefschetz theorem says that every curve $C\subset X$ is a
complete intersection of $X$ with a surface in $\mathbb{P}^{3}$, i.e. $C=X\cap
S$ where $S\subset\mathbb{P}^{3}$ is a surface. As a consequence of this
theorem, any ACM line bundle on such an $X$ is the restriction of a line
bundle on $\mathbb{P}^{3}$. One might ask whether this generalizes in some way
to higher codimension and higher dimensional hypersurfaces. Motivated by this,
Griffiths and Harris conjectured whether subvarieties $Y$ of codimension two
of a hypersurface $X\subset\mathbb{P}^{n}$ can be obtained by intersecting
with a subvariety of codimension two of the ambient space $\mathbb{P}^{n}$. We
shall call subvarieties $Y\subset X\subset\mathbb{P}^{n}$ which are not
intersections of $X$ with any subvariety of codimension two of
$\mathbb{P}^{n}$ as distinguished. C. Voisin very soon proved that a general
threefold $X\subset\mathbb{P}^{4}$ always contains distinguished curves
$C\subset X$, thus proving that this conjecture is false. In [KRR09], it is
shown that there exists a large class of distinguished ACM subvarieties in
smooth hypersurfaces of dimension at least three and degree at least two. But
such ACM subvarieties may not give smooth ones. The next goal in this section
is to provide large families of distinguished ACM subvarieties on special
cubic fourfolds.

######  Theorem 4.11.

Let $\mathcal{F}$ be an Ulrich bundle on a cubic foufold $X$ and $Y$ a surface
constructed from $\mathcal{F}$ as in Theorem 3.2. Then $Y$ is a distinguished
ACM subvarieties on $X$.

###### Proof.

It follows immediately that Proposition 4 in [KRR09] and Theorem 3.9.

∎

######  Theorem 4.12.

Let $X$ be a very general cubic fourfold in $\mathbb{P}^{5}$ containing a
plane. Then the moduli space of stable rank $4$ Ulrich bundles on a cubic
fourfold $X$ is nonempty and smooth of dimension $17$.

###### Proof.

The argument can be divided in a few parts.
Step 1: There exists stable rank $4$ Ulrich bundles $\mathcal{F}$ on a cubic
fourfold $X$. Let $Z=\mathbb{P}^{2}(p_{1},\ldots,p_{10})$ is a blow-up of
$\mathbb{P}^{2}$ in $10$ points in general position by the complete linear
system

$H_{Z}=(6;2^{5},1^{5})=6L-\sum\limits_{i=1}^{5}2E_{i}-\sum\limits_{i=6}^{10}E_{i}.$

Then $Z$ is smooth surface of degree $11$ and sectional genus $5$ in
$\mathbb{P}^{7}$. Let $\ell\subset\mathbb{P}^{7}$ be general line, and
$\phi_{\ell}:\mathbb{P}^{7}\dashrightarrow\mathbb{P}^{5}$ be the projection
from $\ell$ onto a hyperplane. Put
${Y_{1}}=\phi_{\ell}(Z)\subseteq\mathbb{P}^{5}$. Therefore, ${Y_{1}}$ is a
surface of degree $11$ and sectional genus $5$ and $K_{Y_{1}}^{2}=-1$.
Moreover, homogeneous coordinate ring $R_{Y_{1}}=R/I_{Y_{1}}$ have the
following Betti table

|  | $0$ | $1$ | $2$ | $3$ | $4$ | $5$
---|---|---|---|---|---|---|---
0 |  | $1$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$
1 |  | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$
2 |  | $\cdot$ | $1$ | $\cdot$ | $\cdot$ | $\cdot$ | $\cdot$
3 |  | $\cdot$ | $25$ | $65$ | $63$ | $28$ | $5$
|  | $0$ | $1$ | $2$ | $3$
---|---|---|---|---|---
0 |  | $1$ | $\cdot$ | $\cdot$ | $\cdot$
1 |  | $2$ | $7$ | $\cdot$ | $\cdot$
2 |  | $\cdot$ | $1$ | $10$ | $5$

Let $X$ be a unique smooth cubic fourfold containing the surface ${Y_{1}}$.
Consider $\Gamma_{*}(\mathcal{O}_{Y_{1}})$ as a $R_{X}$-module, the periodic
part of its minimal free resolution yields, up to twist, matrix factorization
of the form

$\textstyle{R^{12}_{X}}$$\textstyle{R^{12}_{X}(-1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
R^{12}_{X}(-3).}$$\scriptstyle{\varphi}$

Let $F^{\bullet}$ and $\overline{G^{\bullet}}$ be minimal free resolutions of
the section ring $\Gamma_{*}(\mathcal{O}_{Y_{1}})$ as an $R$-module and an
$R_{X}$-module, respectively. We denote by $\varphi$ the syzygy map
$\textstyle{\overline{G_{3}}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\overline{G_{4}}}$
and $\mathcal{F}=\widetilde{{\rm coker}\varphi}(-3)$. Then $\mathcal{F}$ is
rank $4$ Ulrich bundles $\mathcal{F}$ on the cubic fourfold $X$.

Step 2: If $X$ is very general cubic fourfold in $\mathbb{P}^{5}$ containing a
plane, then $X$ contains a surface $Y$ of degree $22$ as in Theorem 3.9. Since
$\mathcal{F}$ is rank $4$ Ulrich bundles $\mathcal{F}$ on the cubic fourfold
$X$, by Theorem 3.9, there exists an ACM surface $Y$ of degree $22$ as in
Theorem 3.9. Moreover, the irreducible component of the Hilbert scheme
parametrizing the surfaces $Y\subset\mathbb{P}^{5}$ has dimension $91$ and it
is generically smooth. Indeed, one can verify that
$h^{1}(\mathcal{N}_{Y/\mathbb{P}^{5}})=0$. Let
$\mathcal{C}\subseteq|H^{0}(\mathbb{P}^{5}(3))|\cong\mathbb{P}^{55}$ be the
open set corresponding to smooth cubic hypersurfaces. Since
$h^{0}(\mathcal{I}_{Y}(3))=1$ so that the locus
$\mathcal{D}_{8}=\\{([Y],[X]):Y\subseteq
X\\}\subseteq\mathcal{H}\times\mathcal{C}$ has dimension $91+1-1=91$. The
image of $\pi_{2}:\mathcal{D}_{8}\to\mathcal{C}$ has dimension at most $54$
because the general cubic does not contain any $Y$ belonging to $\mathcal{H}$.
For every $[X]\in\pi_{2}(\mathcal{D}_{8})$ we have

$\dim(\pi^{-1}_{2}([X]))\geq\dim(\mathcal{D}_{8})-\dim(\pi_{2}(\mathcal{D}_{8}))\geq
91-54=37.$

Since $h^{0}(\mathcal{N}_{Y/X})\geq\dim_{[Y\subset X]}(\pi^{-1}_{2}([X]))$ for
every $[Y\subset X]\in\pi^{-1}_{2}([X])$, to show that a general
$X\in\mathcal{C}_{8}$ contains a surface $Y$ it is sufficient to verify that
$h^{0}(\mathcal{N}_{Y/X})=37$ for a general $Y$ and for a smooth
$X\in|H^{0}(\mathcal{I}_{Y}(3))|$, see also [Nue17] for a similar argument. We
verified this via Maucaulay2 and we can conclude that
$h^{0}(\mathcal{N}_{Y/X})=37$. Since $[X]\in\mathcal{C}_{8}$ if and only if
$X$ contains a plane, we get statement as required.

Step 3: Dimension and stableness. To complete the diminsioness statement of
Theorem, we should prove that for stable rank $4$ Ulrich bundles $\mathcal{F}$
on $X$, we have $h^{1}(\mathcal{F}\otimes\mathcal{F}^{\vee})=17$ and
$h^{2}(\mathcal{F}\otimes\mathcal{F}^{\vee})=h^{3}(\mathcal{F}\otimes\mathcal{F}^{\vee})=0$.
For this we can use the same deformation argument as in the proof of Theorem
1.3.

By Theorem 4.7, there are no rank $2$ or $3$ Ulrich bundles on a very general
cubic fourfold $X$ containing a plane $P$. Therefore, as observed in [CaH12,
Section 5], $\mathcal{F}$ is stable.

∎

######  Corollary 4.13.

Let $X$ be a very general cubic fourfold in $\mathbb{P}^{5}$ containing a
plane or a general special cubic fourfold in $\mathcal{C}_{14}$ or
$\mathcal{C}_{18}$. Then $X$ contains distinguished smooth surfaces $Y$.

###### Proof.

These facts follow immediately from Theorems 3.9, 4.7, 1.3, 4.11 and 4.12 ∎

## References


# Performance of Large Aperture UCCA Arrays in a 5G User Dense Network

Md Imrul Hasan  Electrical Engineering
The University of Texas at Dallas
Texas, USA
<EMAIL_ADDRESS>Mohammad Saquib  Electrical Engineering
The University of Texas at Dallas
Texas, USA
<EMAIL_ADDRESS>
###### Abstract

The transmitted signals in the fifth generation (5G) wireless networks suffer
from significant path loss due to the use of higher frequencies in Sub-6 GHz
and millimeter-wave (mmWave) bands. Inter-user interference in an ultra-dense
network offers additional challenges to provide a high data rate. Therefore,
it is desirable to generate narrow beams to extend the coverage of a 5G
network by increasing antenna gain and improve its capacity by reducing the
inter-user interference. This fact leads us to address the use of large
aperture uniform concentric circular antenna (UCCA) arrays for 5G beamforming
in massive multiple-input-multiple-output (MIMO) technology. Our analysis
demonstrates that a UCCA with a larger antenna element spacing is capable of
generating a significantly narrower beam with a moderate side-lobe level than
a rectangular planar antenna (RPA) array while operating with the same number
of antenna elements. This capability of the UCCA is analyzed to discover the
performance gain of a 5G network.

###### Index Terms:

5G beamforming, large aperture concentric circular array, spectral efficiency,
spatial separation distance.

##  I Introduction

In recent years, with the advent of the internet of things (IoT), and the
increasing data rate of media-rich applications, the demand for wireless
bandwidth has been expanding very rapidly. The fifth-generation (5G) network
is introduced to support this high demand for spectrum [1]. This network uses
higher frequencies in Sub-6 GHz (S-band, C-Band, etc.) and millimeter-wave
(mmWave) bands, both of which are prone to high path loss. Moreover, in a case
when two user equipments (UEs) are spatially close to each other, their data
rate decreases, and in the worst scenario, either of the two UEs can be
dropped by the scheduler. A possible solution to these limitations is the use
of massive multiple-input multiple-output (mMIMO) technology which offers
higher gain and interference suppression capability by allowing its antenna
array to focus narrow beams towards a user. These narrow beams also allow
spatial multiplexing while increasing the number of UEs for the same
time/frequency resources. Therefore, an antenna array capable of offering
narrower beams is always desirable.

There exist many types of geometries for 2-D mMIMO antenna arrays. Most of the
works in the literature focus on the use of rectangular planar antenna (RPA)
arrays [2, 3]. However, uniform concentric circular antenna (UCCA) arrays also
have multiple advantages; the flexibility in array pattern synthesis and
design [4]. The large aperture antenna array is introduced in order to
decrease the correlation among the closely spaced UEs [5]. Our analysis
reveals that for a given number of antenna elements, an RPA array with a large
aperture (element spacing is greater than or equal to half of the wavelength)
results in many higher side lobes, whereas, a UCCA offers a significantly
narrower main beam with moderate side lobe level (SLL). This observation leads
us to investigate the performance of a large aperture UCCA for different
element spacing in the context of 5G beamforming.

##  II Antenna Array Model

Figure 1: UCCA array geometry. Figure 2: Inter-UE spatial separation in 2D.

Let’s provide the mathematical model of a UCCA array assuming the desired
signal of wavelength $\lambda$ impinges upon the array from the elevation
angle $\theta_{0}$ and the azimuthal angle $\phi_{0}$. The geometry of an UCCA
array is depicted in Fig. 1, where the ${i}^{\underline{\text{th}}}$ ring in
the array with a radius $r_{i}$, contains $N_{i}$ omni-directional elements;
$i=1,2,...,k$. If $d_{i}$ denotes the inter-element distance in the
${i}^{\underline{\text{th}}}$ ring, then $d_{i}=\Gamma\left({2\pi
r_{i}/N_{i}}\right)$, where the function $\Gamma$ rounds up the fraction to
the upper integer value. The array factor of the UCCA directed towards
$(\theta_{0},\phi_{0})$ is given by [6]

$F_{\rm{UCCA}}(\theta,\phi)=1+\sum_{i=1}^{k}\sum_{j=1}^{N_{i}}B_{ij}e^{i(2\pi
r_{i}/\lambda)f_{ij}(\theta,\,\phi)},$ (1)

where $B_{ij}$ the beamformer weight for the ${j}^{\underline{\text{th}}}$
element in the ${i}^{\underline{\text{th}}}$ ring,

$f_{ij}(\theta,\,\phi)=\sin\theta\cos(\phi-\phi_{ij})-\sin\theta_{0}\cos(\phi_{0}-\phi_{ij})\,,$

and $\phi_{ij}=2\pi(j-1)/N_{i}$.

##  III Beamforming Performance Criteria

In 5G networks, the narrow beam formed by an mMIMO facilitates more UEs within
the same spatial dimension. To evaluate how densely we can allow beams to
support neighboring UEs in 3D, the idea of beam packing is explored
considering an arbitrary sphere. This number is calculated by the maximum
number of supported half-power beamwidths (HPBWs) in the elevation
($\theta_{\rm{3dB}}$), and the azimuthal ($\phi_{\rm{3dB}}$) planes. The
overall beam packing gain can be defined as

$G_{\rm{BP}}=\left({\frac{90^{\circ}}{\theta_{\rm{3dB}}}}\right)\left({\frac{360^{\circ}}{\phi_{\rm{3dB}}}}\right).$
(2)

Next, we relate the angular HPBW separation to UE separation distances. As
shown in Fig. 2, these distances can be approximated as [7]

$S_{\phi_{\rm{3dB}}}=2R\,\tan\\{\phi_{\rm{3dB}}/2\\}\,,$ (3)

and

$S_{\theta_{\rm{3dB}}}=R-h\,\tan\\{\arctan(R/h)-\theta_{\rm{3dB}}\\},$ (4)

Figure 3: Beam pattern for element spacing = $\lambda$.

Figure 4: Beam pattern for element spacing = $3\lambda$.

where $R$ is the range and $h$ is the antenna height. In (3),
$S_{\phi_{\rm{3dB}}}$ is the minimum inter UE distance at the boundary on the
azimuthal plane. Similarly, in (4) $S_{\theta_{\rm{3dB}}}$ denotes the minimum
radial separation distance of two UEs on the elevation plane. Note that all
these separation schemes prohibit half-power beam overlapping while
suppressing the interference from any undesired direction.

Now, in order to analyze the quality of signals in 5G networks, we evaluate
the SINR which is given by [8]

${\mathrm{SINR}}={\frac{P}{I+N}}\,,$ (5)

where $P$ is the received power of the signal of interest, $I$ is the
interference power of the other (interfering) signals in the network, and $N$
is the average power of the background noise. In general, the desired signal,
interference, and noise signals are mutually statistically independent. The
above SINR is directly related to the spectral efficiency, which can be
written as [5]

$\mathrm{SE}=\log_{2}\left({1+\rm{SINR}}\right)\\\ \mbox{bps/Hz}.$ (6)

##  IV Numerical Study

In this section, our objectives are a) to perform a comparative study of large
aperture UCCA arrays with different element spacings on the system level using
(2)-(4), and b) to provide insights into the link level performance of the
arrays using (5) and (6). In this study, we use a uniform RPA array as the
baseline performer. The beam pattern of the RPA array towards the desired user
is well known and obtained by [9]

Figure 5: HPBW for variable element spacing in a UCCA array.

Figure 6: Area size of a UCCA array with variable element spacing.

$\centering
F_{\rm{RPA}}(\theta,\phi)=\sum_{m=0}^{N_{x}-1}\sum_{n=0}^{N_{y}-1}A_{mn}e^{i(2\pi/\lambda)\\{m\times
h(\theta,\,\phi)+n\times g(\theta,\,\phi)\\}},\@add@centering$ (7)

where the number of elements along the $x$ and $y$ axis are $N_{x}$, and
$N_{y}$ with element spacing $d_{x}$, and $d_{y}$. Here $A_{mn}$ denotes the
beamformer weight, and

$h(\theta,\,\phi)=d_{x}\left({\sin\theta\cos\phi-\sin\theta_{0}\,cos\,\phi_{0}}\right)\,,$
$g(\theta,\,\phi)=d_{y}\left({\sin\theta\sin\phi-\sin\theta_{0}\sin\phi_{0}}\right)\,.$

Assume, the frequency of interest is 28 GHz band, and the signal of interest
is at $(\theta_{0},\phi_{0})=(30^{\circ},60^{\circ})$. For the UCCA arrays, we
use the total number of rings, $k=5$, and equal inter ring and inter element
spacings. These yield the number of antenna elements in the UCCA array,
$N_{\rm{UCCA}}=98$. As for the RPA array beamforming, we use $N_{x}=N_{y}=10$,
hence, the total number of elements, $N_{\rm{RPA}}=100$. The normalized beam
pattern of the above arrays can be defined by

$N_{\rm{array}}(\theta,\phi)=\left({|F_{\rm{array}}(\theta,\phi)|^{2}/|F_{\rm{array}}(\theta_{0},\phi_{0})|^{2}}\right).$
(8)

The beam patterns are depicted in Fig. 3 and 4 for RPA and UCCA arrays
considering element spacing $>\lambda/2$ , Here, we have assumed uniform
illumination ($A_{mn}=B_{ij}=1$) for both the array structures [9].

Figure 7: Spatial separation distance on the elevation plane.

Figure 8: Spatial separation distance on the azimuthal plane.

Notice that although both the arrays have the same maximum gain towards the
desired UE, the RPA array with element spacing $>\lambda/2$ is highly
vulnerable to interference from other directions. Therefore, a larger element
spacing is not recommendable for an RPA array. On the other hand, the UCCA
array is capable of producing significantly narrower beams without noticeably
increasing the SLL as the element spacing increases. The HPBWs for those beams
are depicted in Fig. 6 for variable element spacing in UCCA arrays on both the
azimuthal and the elevation plane. Here, it can be seen that as the spacing
increases, the HPBW decreases in a UCCA array. Of course, these narrower beams
come at the cost of larger antenna area sizes; see Fig. 6. Here, please notice
that due to the use of mmWave frequency, the array size does not grow
significantly, although it increases with the square of the antenna element
spacing.

In the following numerical examples, we consider UCCA array with two different
element spacings, 1) $d_{i}=2\lambda$, and 2) $d_{i}=3\lambda$, namely
$\rm{UCCA}_{2\lambda}$, and $\rm{UCCA}_{3\lambda}$, respectively. Now we find
the beam packing gain $G_{\rm{BP}}$ of the above two arrays w.r.t. the
conventional RPA array ($d_{x}=d_{y}=\lambda/2$). The HPBW of
$\rm{UCCA}_{2\lambda}$:
$(\theta_{\rm{3dB}},\phi_{\rm{3dB}})=(3.1^{\circ},5.4^{\circ})$, and that of
$\rm{UCCA}_{3\lambda}$:
$(\theta_{\rm{3dB}},\phi_{\rm{3dB}})=(2^{\circ},3.6^{\circ})$. As for the RPA
array, these beamwidths are
$(\theta_{\rm{3dB}},\phi_{\rm{3dB}})=(12^{\circ},20.7^{\circ})$. Using (2), we
obtain the beam packing gain of $\rm{UCCA_{2\lambda}}$, and
$\rm{UCCA_{3\lambda}}$ arrays over the RPA array as 9 and 30 times,
respectively.

In the next numerical example, our goal is to observe the minimum spatial
separation distance as a function of the base station (BS) transmitter range
using (3), and (4); see Fig. 8 and 8, where we use the practical BS range of
5G networks. Here, it can be noticed that as expected the
$\rm{UCCA_{3\lambda}}$, performs better than the $\rm{UCCA_{2\lambda}}$ array
in placing the UEs more closely in a ultra dense network. For instance, at a
distance 50 m from the BS the $\rm{UCCA}_{2\lambda}$, and
$\rm{UCCA}_{3\lambda}$ are capable of offering 3.87 and 5.81 times more
closely spaced UEs w.r.t. the RPA array while considering
$S_{\phi_{3\rm{dB}}}$, and for $S_{\theta_{3\rm{dB}}}$ these numbers are 2.42
and 3.47 times, respectively.

To get a better insight into the leakage power, we list the maximum SLL of the
above arrays in Table. I for performance comparison.

TABLE I: Side Lobe Level SLL (dB) | RPA | $\rm{UCCA}_{2\lambda}$ | $\rm{UCCA}_{3\lambda}$
---|---|---|---
Maximum $\rm{SLL}_{\theta}$ | -24.11 | -18.06 | -15.64
Maximum $\rm{SLL}_{\phi}$ | -14.64 | -14.07 | -11.55

It can be seen that as expected the maximum SLL is higher in $\rm{UCCA}$ array
than RPA array and among two UCCA arrays the $\rm{UCCA_{3\lambda}}$ is
expected to suffer most in terms of inter-user interference. Notably, for all
three arrays, this interference is more significant on the azimuthal plane
than on the elevation plane.

TABLE II: Performance on the elevation plane

Array |  SINR (dB) |  $\rm{\mathbf{SE}}$ (bps/Hz)
---|---|---
RPA | 6.03 | 2.32
$\rm{UCCA}_{2\lambda}$ | 12.09 | 4.10
$\rm{UCCA}_{3\lambda}$ | 12.52 | 4.24

TABLE III: Performance on the azimuthal plane

Array |  SINR (dB) |  $\rm{\mathbf{SE}}$ (bps/Hz)
---|---|---
RPA | -1.37 | 0.79
$\rm{UCCA}_{2\lambda}$ | 10.88 | 3.73
$\rm{UCCA}_{3\lambda}$ | 12.15 | 4.12

Finally, we want to compare the SINR, and maximum spectral efficiency offered
by the arrays of interests. We consider all the UEs are distributed within
$0^{\circ}$ to $90^{\circ}$ in both the planes (azimuthal and elevation) and
the signal of interest is at $(\theta_{0},\phi_{0})=(30^{\circ},60^{\circ})$.
We perform 10000 realizations with arbitrary angles for $U=10$ interfering UEs
on the boundary. We assume that the antenna elements are isotropic, there is
no inter-element coupling, and the inner product of channel vectors is very
much related to the antenna array pattern in far-field approximation [5]. For
user equal power allocation, and SNR = $10$ dB, the SINR, and maximum spectral
efficiency for the arrays are calculated using (6), and presented in Table
III, and III for comparison. Notice that the UCCA achieves significant
improvement on SINR, and SE than the RPA array. $\rm{UCCA}_{2\lambda}$, and
$\rm{UCCA}_{3\lambda}$ arrays are capable of providing 4.72 times and 5.22
times more spectral efficiency than the RPA array on the azimuthal plane. As
expected, the $\rm{UCCA}_{3\lambda}$ shows slightly better performance than
the $\rm{UCCA}_{2\lambda}$ due to its narrower HPBW. On the elevation plane,
the $\rm{UCCA}_{2\lambda}$, and $\rm{UCCA}_{3\lambda}$ provide more than 6 dB
gain in SINR while obtaining 1.77 and 1.83 times spectral efficiency gain,
respectively over the RPA array.

##  V Conclusions

In this paper, a comprehensive performance analysis of large aperture UCCA
arrays was presented considering the conventional RPA array as the baseline
performer. Our numerical study demonstrated that in a user dense 5G network, a
larger aperture UCCA array performs better than the RPA array due to its
capability of generating narrower beams while operating with a lesser number
of antenna elements. For instance, a UCCA array with 98 antenna elements and
$2\lambda$ antenna element spacing offered approximately 9 times beam packing
gain, 4.72, and 1.77 times spectral efficiency gain on the azimuthal and the
elevation plane, respectively over a $10\times 10$ RPA array. Those gains were
enhanced to 30, 5.22, and 1.83, respectively when the antenna element spacing
was increased to $3\lambda$.

## References

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