# The CM representation type of projective varieties

Daniele Faenzi<EMAIL_ADDRESS>Institut de Mathématiques de
Bourgogne, UMR CNRS 5584, Université de Bourgogne, 9 Avenue Alain Savary, BP
47870, 21078 Dijon Cedex, France  and  Joan Pons-Llopis<EMAIL_ADDRESS>Université de Pau et des Pays de l’Adour
Avenue de l’Université - BP 576 - 64012 PAU Cedex - France

###### Abstract.

We show that all projective integral varieties which are not cones and have a
Cohen-Macaulay coordinate ring are of wild CM type, except for a few cases
which we completely classify.

###### Key words and phrases:

ACM vector sheaves and bundles, Ulrich sheaves, MCM modules, Graded Cohen-
Macaulay rings, Representation type

###### 2010 Mathematics Subject Classification:

14F05; 13C14; 14J60; 16G60

Both authors partially supported by GEOLMI ANR-11-BS03-0011

## Introduction

A classical result in representation theory of quivers is Gabriel’s theorem,
stating that a connected quiver supports only finitely many irreducible
representations (i.e., of indecomposable modules over the associated path
algebra) if and only if it is of type $A$, $D$, $E$. The classification of
tame quivers as Euclidean graphs, or extended Dynkin diagrams, of type
$\tilde{A}$, $\tilde{D}$, $\tilde{E}$ came shortly afterwards. Remarkably, any
other quiver supports arbitrarily large families of indecomposable
representations, i.e., to be of wild representation type.

In algebraic geometry, the relevant problem in representation theory of
algebras concerns the complexity of the category of maximal Cohen-Macaulay
modules over the coordinate ring $\boldsymbol{k}[X]$ of a closed subvariety
$X\subset\mathds{P}^{n}$ defined over an algebraically closed field
$\boldsymbol{k}$. For $\dim(X)>0$, assuming $\boldsymbol{k}[X]$ to be Cohen-
Macaulay (so $X$ is said to be arithmetically Cohen-Macaulay, briefly ACM),
these in turn correspond to ACM sheaves, namely coherent sheaves $\mathscr{E}$
on $X$ without intermediate cohomology i.e. satisfying
$\operatorname{H}^{i}(X,\mathscr{E}(t))=0$ for all $t$ and $0<i<\dim(X)$. For
hypersurfaces, in view of [Eis80] these modules also correspond to matrix
factorizations, which in turn are related to mirror symmetry, cf. [Orl04].

In this sense, reduced projective ACM varieties of finite CM type have been
classified in [EH88], cf. also [BGS87, AR87, Knö87, Her78]. Their list (for
positive dimension) consists of rational normal curves, projective spaces,
smooth quadrics, the Veronese surface in $\mathds{P}^{5}$ and the cubic scroll
in $\mathds{P}^{4}$. This ties in with Grothendieck’s classical result on the
splitting of vector bundles over $\mathds{P}^{1}$, [Gro57], in turn rooted in
ideas of Segre, [Seg84].

Our goal here is to tell what happens to other varieties. Smooth cubic
surfaces are of wild CM type according to the main point of [CH11]. Some more
varieties have been proved to fall in this class, for instance all Segre
varieties besides the CM finite ones (cf. [CMRPL12]), del Pezzo surfaces (cf.
[MRPL12, CKM13]), positive-dimensional hypersurfaces of degree at least $4$
and some complete intersections (cf. [DT14]) some Fano varieties (cf.
[MRPL14]), the triple Veronese embedding of any variety (cf. [MR15]).

As for CM tame projective varieties, besides smooth elliptic curves (by
seminal work of Atiyah, [Ati57], also related to classical work of Segre, cf
[Seg86]) and rational chains of type $\tilde{A}$ (cf. [DG01, DG93], cf. also
[BD04]), the only known examples were given in [FM14] and consist of smooth
quartic surface scrolls in $\mathds{P}^{5}$. Here we prove the following
result.

######  Theorem.

Let $X\subset\mathds{P}^{n}$ be a closed integral ACM subvariety of positive
dimension which is not a cone. Then $X$ is of wild CM type unless $X$ is one
of the following:

  1. i)

a linear subspace;

  2. ii)

a smooth quadric;

  3. iii)

a rational normal curve;

  4. iv)

a smooth or nodal curve of arithmetic genus $1$;

  5. v)

a smooth rational surface scroll of degree $d\leq 4$;

  6. vi)

the Veronese surface in $\mathds{P}^{5}$.

Our result implies a strong version of the finite-tame-wild trichotomy for the
class of varieties appearing in the previous theorem, namely that any such
variety $X$ falls in exactly one of the following classes:

  * Finite:

there are only finitely many indecomposable ACM sheaves over $X$ up to
isomorphism. This happens in cases (i), (ii), (iii), (v) for $d=3$ and (vi).

  * Tame:

for any given rank $r$, the parameter space of indecomposable non-isomorphic
ACM sheaves of rank at most $r$ consists of the union of finitely many
isolated points and connected curves of genus $0$ (in case (v) and $d=4$) or
$1$ (in case (iv)).

  * Wild:

$X$ is of wild CM type in the algebraic sense and hence in the geometric
sense, i.e. $X$ admits families of arbitrarily large dimension of
indecomposable non-isomorphic ACM sheaves.

It should be noted that, although our assumptions on the variety $X$ are
rather mild, a different behaviour might be expected when relaxing them. For
instance a quadric cone of corank $1$ is CM countable, cf. [BGS87]; several
other examples of this kind exist over local rings, cf. for instance [LW12]
for a detailed picture, see also [Sto13]. More varieties of tame type appear
from germs of elliptic singularities, cf. [Kah89] or from non-isolated affine
surface singularities, see also [BD10].

Let us indicate the strategy of our proof. Let $Y$ be a variety of the class
under examination: an integral closed ACM subscheme of $\mathds{P}^{n}$ which
is not a cone. The first step is to isolate some essential datum in order to
build large families of indecomposable non-isomorphic MCM modules on
$\boldsymbol{k}[Y]$. In Theorem 1 we establish that this should be a pair of
ACM sheaves $\mathscr{A}$ and $\mathscr{B}$ on $Y$, whose only endomorphisms
are homotheties (i.e. $\mathscr{A}$ and $\mathscr{B}$ are simple), having no
non-trivial maps in either direction $\mathscr{A}\to\mathscr{B}$ or
$\mathscr{B}\to\mathscr{A}$ and such that the extension space
$\operatorname{Ext}^{1}_{Y}(\mathscr{B},\mathscr{A})$ is sufficiently large,
namely of dimension $w\geq 3$.

Indeed, from this datum we construct a fully faithful exact functor from the
category $\mathbf{Rep}_{\Upsilon}$ of finite-dimensional representations of
the Kronecker quiver $\Upsilon=\Upsilon_{w}$ with $w$ arrows to the category
$\mathbf{MCM}_{\boldsymbol{k}[Y]}$ of MCM modules over $\boldsymbol{k}[Y]$.
This quiver should be seen as parametrizing ACM sheaves appearing as extension
of copies of $\mathscr{A}$ and $\mathscr{B}$. In turn, by a standard argument,
the existence of such functor suffices to prove CM wildness of $Y$.

However, constructing the sheaves $\mathscr{A}$ and $\mathscr{B}$ turns out to
be quite complicated in general. For this we need our next result, Theorem 2,
which shows how to deduce CM wildness of $X$ from CM wildness of $Y$ when $Y$
is a linear section of $X$ of codimension $c>0$. For this, we need to further
assume that $\mathscr{A}$ and $\mathscr{B}$ are Ulrich sheaves, i.e. their
modules of global sections have the maximal number of generators. We see this
as a further prove of the importance of these sheaves, cf. [ESW03, ES09]. We
also need the extra assumption $\operatorname{H}^{0}(Y,\omega_{Y}(m-c))\neq
0$, which turns out to hold everywhere except for varieties of minimal degree,
where in turn we do not need it.

The idea of Theorem 2 is that taking the $c$-th syzygy
$\boldsymbol{\Omega}^{c}_{\boldsymbol{k}[X]}$ of the
$\boldsymbol{k}[X]$-module of an MCM module $L$ over $\boldsymbol{k}[Y]$ one
obtains an MCM module over $\boldsymbol{k}[X]$ and that this entails no
essential loss of information if $L$ is Ulrich. In fact, for
$\boldsymbol{\Omega}_{\boldsymbol{k}[X]}^{c}$ to be a functor we need to pass
to the stable category $\underline{\mathbf{MCM}}_{\boldsymbol{k}[X]}$ where we
quotient out by morphisms factoring through a free module. The point is that
the stable syzygy functor $\underline{\boldsymbol{\Omega}}^{c}$ is fully
faithful on Ulrich modules. The proof of this fact relies on a duality
argument and on cohomology vanishing of Ulrich sheaves.

The next result, Theorem 3, shows how to put these two ingredients together.
Indeed, by resolving over $\boldsymbol{k}[X]$ the module of global sections of
the universal extension of the sheaves $\mathscr{B}$ by $\mathscr{A}$ over $Y$
needed for Theorem 1 and taking its $c$-th syzygy, we get a functor
$\mathbf{Rep}_{\Upsilon}\to\mathbf{MCM}_{X}$ whose stabilization is fully
faithful by Theorem 2. Then the functor itself is not quite fully faithful,
nevertheless is a representation embedding, i.e. it sends non-isomorphic
(resp. irreducible) representations to non-isomorphic (resp. indecomposable)
modules, and this suffices to show CM wildness of $X$.

Summing up, it only remains to construct the Ulrich sheaves $\mathscr{A}$ and
$\mathscr{B}$ as above over a linear section $Y$ of $X$, which we take to be
of dimension $1$ when the sectional genus $p$ of $X$ is at least $2$, or of
dimension $2$ for $p\leq 1$. This is the content of §5, which we now review.

We first take care of the case $p=0$, i.e. of varieties of minimal degree.
Here, our assumptions combined with del Pezzo and Bertini’s classification say
that $X$ is a rational normal scroll or the Veronese surface in
$\mathds{P}^{5}$, and these varieties are easily dealt with. Note that here is
where all CM-finite cases appear, together with the two sporadic cases given
by the two rational quartic surface scrolls.

Next, the case $p\geq 2$ is quite easily seen to provide only CM wild
varieties, as $\mathscr{A}$ and $\mathscr{B}$ can be taken to be sufficiently
general bundles of rank $2$ over $Y$ of slope $\deg(Y)+p-1$.

Finally, in the case $p=1$ our proof of the existence of $\mathscr{A}$ and
$\mathscr{B}$ is based on a detailed study of Ulrich sheaves on surfaces of
almost minimal degree. When $Y$ smooth (i.e. $Y$ is a del Pezzo surface) or
even just normal, this can be carried out via the well-known Hartshorne-Serre
construction associated with general sets of $\deg(Y)+2$ points of $Y$. On the
other hand, when $Y$ is not normal, we take $\mathscr{A}$ and $\mathscr{B}$ to
be Ulrich sheaves of rank $1$ which are not locally free, in fact the main
point is to bound from below the number of independent global sections of the
first local $\mathscr{E}xt$ sheaf between $\mathscr{A}$ and $\mathscr{B}$. To
accomplish this, according to the classification of these surfaces, it
suffices to settle a finite number of cases. In the main one (when $Y$ is
normalized by a scroll) we define $\mathscr{A}$ and $\mathscr{B}$ to be direct
images of certain line bundles via the normalization map $\bar{Y}\to Y$, and
reduce the result to counting the intersection points of the moving part of
the associated curves.

##  1\. CM wild varieties

Let $\boldsymbol{k}$ be an algebraically closed field, and let
$S=\boldsymbol{k}[x_{0},\ldots,x_{n}]$ be the graded polynomial ring in $n+1$
indeterminates. Write $\boldsymbol{k}[X]$ for the homogeneous coordinate ring
of a closed subscheme $X$ of $\mathds{P}^{n}:=\mathbf{Proj}(S)$, and
$\mathscr{O}_{X}(1)$ for the restriction to $X$ of
$\mathscr{O}_{\mathds{P}^{n}}(1)$.

###  1.1. ACM varieties and modules

We first recall some basic terminology for various Cohen-Macaulay properties
of varieties, sheaves and modules.

######  Definition 1.

Let $X\subset\mathds{P}^{n}$ be a closed subscheme of dimension $m$. Then $X$
is arithmetically Cohen-Macaulay (ACM) if the ring $\boldsymbol{k}[X]$ of $X$
is Cohen-Macaulay, i.e. if $\boldsymbol{k}[X]$ has a graded $S$-free
resolution of length $n-m$.

Given a closed subscheme $X\subset\mathds{P}^{n}$, we set
$R=\boldsymbol{k}[X]$ and let $\mathbf{MCM}_{R}$ be the category of finitely
generated maximal Cohen-Macaulay (MCM) modules over $R$. If $X$ is an ACM
variety of positive dimension, then the sheafification functor provides an
equivalence between $\mathbf{MCM}_{R}$ and the full subcategory of
$\mathbf{Coh}_{X}$ consisting of arithmetically Cohen-Macaulay (ACM) sheaves,
i.e. coherent sheaves whose $R$-module of global sections is MCM. Given $M$ in
$\mathbf{MCM}_{R}$ of positive rank, the number of independent minimal
generators of $M$ is at most $\deg(X)\operatorname{rk}(M)$. For the
corresponding sheaf $\mathscr{E}$ over $X$, assuming
$\operatorname{H}^{0}(X,\mathscr{E}(-1)))=0$, we have:

(1.1)
$\dim_{\boldsymbol{k}}\operatorname{H}^{0}(X,\mathscr{E})\leq\deg(X)\operatorname{rk}(\mathscr{E}).$

We say that $\mathscr{E}$ is an Ulrich sheaf on $X$ and that $M$ is an Ulrich
module over $R$ if equality is attained in (1.1). Put $\mathbf{Ulr}_{R}$ for
the full subcategory of $\mathbf{MCM}_{R}$ consisting of Ulrich modules.

We say that a coherent sheaf $\mathscr{E}$ over $X$ is simple if its only
endomorphisms are homotheties. Given two coherent sheaves $\mathscr{E}$ and
$\mathscr{F}$ over $X$, we write $\chi(\mathscr{E},\mathscr{F})$ for the
alternating sum of the dimension of
$\operatorname{Ext}^{i}_{X}(\mathscr{E},\mathscr{F})$. We abbreviate
$\chi(\mathscr{F})=\chi(\mathscr{O}_{X},\mathscr{F})$.

###  1.2. CM wildness

We will consider a couple of related notions of CM wildness of a variety.
Algebraically this means that, for any finitely generated associative
$\boldsymbol{k}$-algebra $\Lambda$, the category of MCM modules over
$R=\boldsymbol{k}[X]$ contains, in some sense, the category
$\mathbf{FMod}_{\Lambda}$ of left $\Lambda$-modules which have finite
dimension over $\boldsymbol{k}$.

######  Definition 2.

Let $X\subset\mathds{P}^{n}$ be an equidimensional closed subscheme and let
$R$. For any finitely generated associative $\boldsymbol{k}$-algebra
$\Lambda$, and any $(R,\Lambda)$-bimodule $\boldsymbol{M}$, flat over
$\Lambda$, define the functor:

$\displaystyle\Phi_{\boldsymbol{M}}:\mathbf{FMod}_{\Lambda}$
$\displaystyle\to\mathbf{Mod}_{R},$ $\displaystyle N$
$\displaystyle\mapsto\boldsymbol{M}\operatorname{\otimes}_{\Lambda}N.$

The variety $X$ is of wild CM type if, for any $\Lambda$ as above, there is
$\boldsymbol{M}$ such that $\Phi_{\boldsymbol{M}}$ takes values in
$\mathbf{MCM}_{R}$ and is a representation embedding, i.e.:

  1. a)

the module $N$ is decomposable whenever $\Phi_{\boldsymbol{M}}(N)$ is;

  2. b)

for any pair $(N,N^{\prime})$ of modules in $\mathbf{FMod}_{\Lambda}$, we
have:

$\Phi_{\boldsymbol{M}}(N)\simeq\Phi_{\boldsymbol{M}}(N^{\prime})\Leftrightarrow
N\simeq N^{\prime}.$

enumerate The variety $X$ is of wild Ulrich type if moreover: enumerate[[a)]]

  3. c)

for any $N$ in $\mathbf{FMod}_{\Lambda}$, $\Phi_{\boldsymbol{M}}(N)$ is
Ulrich.

The variety $X$ is said to be strictly CM wild if for any $\Lambda$ as above
there is $\boldsymbol{M}$ such that $\Phi_{\boldsymbol{M}}$ takes values in
$\mathbf{MCM}_{R}$ and is fully faithful, i.e.:

$\operatorname{Hom}_{\Lambda}(N,N^{\prime})\simeq\operatorname{Hom}_{R}(\Phi(N),\Phi(N^{\prime}))_{0}.$

If moreover $\Phi_{\boldsymbol{M}}(N)$ is Ulrich for all $N$, then $X$ is
strictly Ulrich wild.

######  Remark 1.

The following facts are well-known.

  1. i)

If the subscheme $X$ is strictly CM wild, then it is of wild CM type, and if
$X$ is strictly Ulrich wild then it is of wild Ulrich type.

  2. ii)

If $X$ is of wild CM type, then for any $r\in\mathds{N}$ there are families of
dimension at least $r$ consisting of indecomposable ACM sheaves on $X$, all
non-isomorphic to one another. In other words, $X$ is of wild CM type in the
geometric sense. If $X$ is of wild Ulrich type, these families can be taken to
consist of Ulrich sheaves.

  3. iii)

Any fully faithful exact functor
$\Phi:\mathbf{FMod}_{\Lambda}\to\mathbf{Mod}_{R}$ is of the form
$\Phi_{\boldsymbol{M}}$ for some $\Lambda$-flat $(R,\Lambda)$-bimodule
$\boldsymbol{M}$. enumerate Let $w\geq 1$ be an integer and consider the
Kronecker quiver $\Upsilon=\Upsilon_{w}$ with two vertices and $w$ arrows from
the first vertex to the second. Write $\mathbf{Rep}_{\Upsilon}$ for the
Abelian category of finite-dimensional $\boldsymbol{k}$-representations of
$\Upsilon$. enumerate[[i)]]

  4. iv)

To check that $X$ is strictly CM wild (resp., of wild CM type), it suffices to
construct a fully faithful exact functor (resp., a representation embedding):

$\Phi:\mathbf{Rep}_{\Upsilon}\to\mathbf{MCM}_{R}$

where $\Upsilon=\Upsilon_{w}$ is the Kronecker quiver with $w\geq 3$. If
moreover $\Phi(\mathscr{R})$ is Ulrich for any $\mathscr{R}$ in
$\mathbf{Rep}_{\Upsilon}$, then $X$ is strictly Ulrich wild (resp., of wild
Ulrich type).

##  2\. Strict CM wildness from extensions

One of the main tools that we use to build large families of MCM modules (or
ACM sheaves) is to start with two such sheaves and build extensions in a
systematic way. This goes as follows. Let $X\subset\mathds{P}^{n}$ be a closed
subscheme, and let $\mathscr{A}$ and $\mathscr{B}$ be two coherent sheaves on
$X$ such that:

$\operatorname{Ext}^{1}_{X}(\mathscr{B},\mathscr{A})\neq 0.$

Set $W=\operatorname{Ext}^{1}_{X}(\mathscr{B},\mathscr{A})$ and consider the
projective space $\mathds{P}W$ of lines through the origin in $W$. Then, over
$X\times\mathds{P}W$, there is a universal extension:

$0\to\mathscr{A}\boxtimes\mathscr{O}_{\mathds{P}W}\to\mathscr{U}\to\mathscr{B}\boxtimes\mathscr{O}_{\mathds{P}W}(-1)\to
0,$

where we write $p$ and $q$ for the projections from $X\times\mathds{P}W$ to
$X$ and $\mathds{P}W$, and for $\mathscr{E}\in\mathbf{Coh}_{X}$ and
$\mathscr{F}\in\mathbf{Coh}_{\mathds{P}W}$, we set
$\mathscr{E}\boxtimes\mathscr{F}=p^{*}(\mathscr{E})\operatorname{\otimes}q^{*}(\mathscr{F})$.
Then we consider:

$\Phi_{\mathscr{U}}=\mathbf{R}p_{*}(q^{*}(-)\operatorname{\otimes}\mathscr{U}):\mathbf{D}^{b}(\mathbf{Coh}_{\mathds{P}W})\to\mathbf{D}^{b}(\mathbf{Coh}_{X}).$

It is clear that:

$\Phi_{\mathscr{U}}(\mathscr{O}_{\mathds{P}W})\simeq\mathscr{A},\qquad\Phi_{\mathscr{U}}(\Omega_{\mathds{P}W}(1))\simeq\mathscr{B}[-1].$

Set
$w=\dim_{\boldsymbol{k}}\operatorname{Ext}^{1}_{X}(\mathscr{B},\mathscr{A})$
and consider the Kronecker quiver $\Upsilon=\Upsilon_{w}$. Then, the natural
isomorphism
$W\simeq\operatorname{Hom}_{\mathds{P}W}(\Omega_{\mathds{P}W}(1),\mathscr{O}_{\mathds{P}W})$
provides an equivalence:

(2.1)
$\Xi:\mathbf{D}^{b}(\mathbf{Rep}_{\Upsilon})\simeq\langle\Omega_{\mathds{P}W}(1),\mathscr{O}_{\mathds{P}W}\rangle\subset\mathbf{D}^{b}(\mathbf{Coh}_{\mathds{P}W}).$

Explicitly, this is described as follows. Choose a basis
$(e_{1},\ldots,e_{w})$ of
$W=\operatorname{Ext}^{1}_{X}(\mathscr{B},\mathscr{A})$. Let $\mathscr{R}$ be
a representation of $\Upsilon$ having dimension vector $(a,b)$. Then
$\mathscr{R}$ corresponds to the choice of $w$ linear maps
$m_{1},\ldots,m_{w}:\boldsymbol{k}^{a}\to\boldsymbol{k}^{b}$. Take the
element:

(2.2) $\xi=\sum m_{i}\otimes
e_{i}\in\operatorname{Hom}_{\boldsymbol{k}}(\boldsymbol{k}^{b},\boldsymbol{k}^{a})\operatorname{\otimes}W.$

Then, under
$W\simeq\operatorname{Hom}_{\mathds{P}W}(\Omega_{\mathds{P}W}(1),\mathscr{O}_{\mathds{P}W})$,
we obtain from $\xi$ a morphism:

(2.3) $M:\Omega_{\mathds{P}W}(1)^{b}\to\mathscr{O}_{\mathds{P}W}^{a}.$

The cone of $M$ is the element of $\mathbf{D}^{b}(\mathbf{Coh}_{\mathds{P}W})$
associated with $\mathscr{R}$ via $\Xi$. This is directly extended to
morphisms.

######  Theorem 1.

Let $X\subset\mathds{P}^{n}$ be a closed subscheme and let $\mathscr{A}$ and
$\mathscr{B}$ be simple coherent sheaves on $X$ satisfying:

$\displaystyle\operatorname{Hom}_{X}(\mathscr{A},\mathscr{B})=\operatorname{Hom}_{X}(\mathscr{B},\mathscr{A})=0.$

Set
$w=\dim_{\boldsymbol{k}}\operatorname{Ext}^{1}_{X}(\mathscr{B},\mathscr{A})$,
assume $w\neq 0$, and put $\Upsilon=\Upsilon_{w}$. Then the restriction $\Phi$
of $\Phi_{\mathscr{U}}\circ\Xi$ to $\mathbf{Rep}_{\Upsilon}$ is a fully
faithful exact functor:

$\Phi:\mathbf{Rep}_{\Upsilon}\to\mathbf{Coh}_{X}.$

###### Proof.

Let us first give a more explicit description of $\Phi$. At the level of
objects, given a representation $\mathscr{R}$ of the quiver
$\mathbf{Rep}_{\Upsilon}$ with dimension vector $(a,b)$ let
$(m_{1},\ldots,m_{w})$ be the $w$ linear maps associated with $\mathscr{R}$
and let $\xi$ be as in (2.2). Then $\Phi(\mathscr{R})$ fits as middle term of
a representative of the extension class corresponding to $\xi$:

$0\to\mathscr{A}^{a}\to\Phi(\mathscr{R})\to\mathscr{B}^{b}\to 0.$

Let us now check that this is well-defined on morphisms. Let $\mathscr{S}$ be
another representation of $\Upsilon$, of dimension vector $(c,d)$,
corresponding to the linear maps $(n_{1},\ldots,n_{w})$. A morphism
$\lambda:\mathscr{R}\to\mathscr{S}$ of representations is given by linear maps
$\alpha:\boldsymbol{k}^{a}\to\boldsymbol{k}^{c}$ and
$\beta:\boldsymbol{k}^{b}\to\boldsymbol{k}^{d}$ such that:

(2.4) $n_{i}\alpha=\beta m_{i},\qquad\mbox{for all $i=1,\ldots,w$.}$

Recall that
$\operatorname{End}_{X}(\mathscr{A})=\langle\operatorname{\mathrm{id}}_{\mathscr{A}}\rangle_{\boldsymbol{k}}$,
and consider the map of coherent sheaves
$\beta_{\mathscr{A}}=\beta\operatorname{\otimes}\operatorname{\mathrm{id}}_{\mathscr{A}}:\mathscr{A}^{a}\to\mathscr{A}^{c}$.
Then, it is well-known that this map defines a morphism of extensions:

(2.5)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{A}^{a}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{\mathscr{A}}}$$\textstyle{\Phi(\mathscr{R})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{\mathscr{B}^{b}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{A}^{c}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{\mathscr{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{\mathscr{B}^{b}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

for a certain sheaf $\mathscr{D}$ representing the element:

$\sum\beta m_{i}\otimes
e_{i}\in\operatorname{Ext}^{1}_{X}(\mathscr{B}^{b},\mathscr{A}^{c}).$

Analogously
$\alpha_{\mathscr{B}}=\alpha\operatorname{\otimes}\operatorname{\mathrm{id}}_{\mathscr{B}}:\mathscr{B}^{b}\to\mathscr{B}^{d}$
defines:

(2.6)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{A}^{c}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{\prime}}$$\textstyle{\mathscr{D}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi^{\prime}}$$\scriptstyle{p^{\prime}}$$\textstyle{\mathscr{B}^{b}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{\mathscr{B}}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{A}^{c}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Phi(\mathscr{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{B}^{d}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

with upper row representing an extension class in:

$\sum n_{i}\alpha\otimes
e_{i}\in\operatorname{Ext}^{1}_{X}(\mathscr{B}^{b},\mathscr{A}^{c}).$

Now we can use (2.4) to see that lower extension of (2.5) is the same as the
upper one (2.6). Then the morphisms $\phi^{\prime}$ and $\phi$ of extensions
compose to map from $\Phi(\mathscr{R})$ to $\Phi(\mathscr{S})$.

It is clear that this construction agrees with the definition of $\Phi$.
Indeed, the representation $\mathscr{R}$ is mapped by $\Xi$ to the cone of the
matrix $M$ of (2.3), which in turn is sent by $\Phi_{\mathscr{U}}$ to the cone
of:

$\Phi_{\mathscr{U}}(M):\mathscr{B}^{b}[-1]\to\mathscr{A}^{a}.$

By construction this cone is represented by the extension class
$\Phi(\mathscr{R})$.

Now it only remains to use this setup to show that the functor $\Phi$ is fully
faithful. To check this, consider a commutative diagram:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{0}$$\scriptstyle{i}$$\textstyle{\mathscr{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda}$$\scriptstyle{p}$$\textstyle{\mathscr{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{0}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{\prime}}$$\textstyle{\mathscr{D}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{\prime}}$$\textstyle{\mathscr{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

Since $p^{\prime}\circ\lambda=0$, we have
$\operatorname{Im}\lambda\subseteq\mathscr{A}$. But $\lambda i=0$ implies that
$\lambda$ factors as:

$\textstyle{\mathscr{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\lambda}$$\textstyle{\mathscr{A}}$$\textstyle{\mathscr{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{D}/\mathscr{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\bar{\lambda}}$

If $\lambda\neq 0$, this would give a nonzero map
$\bar{\lambda}:\mathscr{B}\to\mathscr{A}$, contradicting
$\operatorname{Hom}_{X}(\mathscr{B},\mathscr{A})=0$. With this in mind, it is
easy to deduce injectiveness of the natural map:

(2.7)
$\operatorname{Hom}_{\Upsilon}(\mathscr{R},\mathscr{S})\to\operatorname{Hom}_{X}(\Phi(\mathscr{R}),\Phi(\mathscr{S})).$

As for surjectiveness, given a morphism
$\mu:\Phi(\mathscr{R})\to\Phi(\mathscr{S})$, we compose $\mu$ on one side with
the projection $\Phi(\mathscr{S})\to\mathscr{B}^{d}$, and with the injection
$\mathscr{A}^{a}\to\Phi(\mathscr{R})$ on the other side. We obtain thus a map
$\mathscr{A}^{a}\to\mathscr{B}^{d}$, which must vanish since
$\operatorname{Hom}_{X}(\mathscr{A},\mathscr{B})=0$. We deduce that $\mu$
defines maps $\mathscr{A}^{a}\to\mathscr{A}^{b}$ and
$\mathscr{B}^{c}\to\mathscr{B}^{d}$, which must be of the form
$\beta\operatorname{\otimes}\operatorname{\mathrm{id}}_{\mathscr{A}}$ and
$\alpha\operatorname{\otimes}\operatorname{\mathrm{id}}_{\mathscr{B}}$ by the
assumption that $\mathscr{A}$ and $\mathscr{B}$ are simple. It is now clear
that the pair $(\beta,\alpha)$ defines a morphism $\mathscr{R}\to\mathscr{S}$
whose image under (2.7) is $\mu$. ∎

We deduce a criterion for an ACM variety being strictly CM wild.

######  Corollary 1.

In the hypothesis of Theorem 1, we have:

  1. i)

if $w\geq 3$ and $\mathscr{A}$ and $\mathscr{B}$ are ACM, then $X$ is strictly
CM wild;

  2. ii)

if moreover $\mathscr{A}$ and $\mathscr{B}$ are Ulrich, then $X$ is strictly
Ulrich wild.

###### Proof.

By construction of the functor $\Phi$ of Theorem 1, the sheaf
$\Phi(\mathscr{R})$ associated with a representation $\mathscr{R}$ of
$\Upsilon$ is ACM (respectively, Ulrich) if $\mathscr{A}$ and $\mathscr{B}$
are ACM (respectively, Ulrich). We conclude using Remark 1. ∎

######  Remark 2.

The hypothesis $\operatorname{Hom}_{X}(\mathscr{B},\mathscr{A})=0$ in Theorem
1 is necessary. Indeed, if
$\operatorname{Hom}_{X}(\mathscr{B},\mathscr{A})\neq 0$ we could consider the
map:

$\phi:\mathscr{D}\xrightarrow{p}\mathscr{B}\xrightarrow{}\mathscr{A}\xrightarrow{i}\mathscr{D}$

This map $\phi$ is not zero, and makes the following diagram commute:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{0}$$\scriptstyle{i}$$\textstyle{\mathscr{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\scriptstyle{p}$$\textstyle{\mathscr{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{0}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{\prime}}$$\textstyle{\mathscr{D}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{\prime}}$$\textstyle{\mathscr{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

Notice, however that any $\phi$ fitting in such a diagram will be nilpotent.

As an explicit example, let $X\subset\mathds{P}^{m+1}$ be an hypersurface of
degree $d$ and $Z\subset X$ an AG subscheme of codimension two and index
$i_{Z}$, where:

$i_{Z}=\max\\{s\mid\operatorname{H}^{m-1}(X,\mathscr{I}_{Z/X}(s))\neq 0\\}.$

Define $e=i_{Z}+m+2-d$ and assume that $e<0$ so that
$\operatorname{Hom}_{X}(\mathscr{I}_{Z/X}(e),\mathscr{O}_{X})\neq 0$. Let
$\mathscr{D}$ be the non-trivial extension of $\mathscr{I}_{Z/X}(e)$ by
$\mathscr{O}_{X}$ (one can show that this extension exists and is unique up to
a nonzero scalar, by the definition of $e$ and by Serre duality).

When $Z$ is not a complete intersection inside $X$, $\mathscr{D}$ is
indecomposable. Anyway $\mathscr{D}$ is an ACM sheaf of rank $2$ over $X$
which is never simple, as it always admits a nilpotent endomorphism. The
conclusion of Theorem 1 clearly fails in this case.

##  3\. Stable syzygies of Ulrich modules

Let $X\subset\mathds{P}^{n}$ be a closed ACM subscheme of dimension $m\geq 1$,
let $Y$ be a linear section of $X$ of codimension $c<m$. Set
$T=\boldsymbol{k}[Y]$, $R=\boldsymbol{k}[X]$ and write $\omega_{Y}$ for the
dualizing sheaf of $Y$. Looking at a finitely generated graded module $L$ over
$T$ as a graded module over $R$, we take its minimal graded $R$-free
resolution:

(3.1) $0\leftarrow L\leftarrow
F_{0}\xleftarrow{d_{1}}F_{1}\leftarrow\cdots\leftarrow
F_{\ell-1}\xleftarrow{d_{\ell}}F_{\ell}\leftarrow\cdots$

Write $\boldsymbol{\Omega}^{\ell}_{R}(L)$ for the $\ell$-th syzygy of $L$ over
$R$, i.e. $\boldsymbol{\Omega}^{\ell}_{R}(L)=\operatorname{Im}(d_{\ell})$. It
is well-known that, if $L$ is MCM over $T$, then
$\boldsymbol{\Omega}_{R}^{\ell}(L)$ is MCM over $R$ for $\ell\geq c$.

Let $\underline{\mathbf{MCM}}_{R}$ the stable category of maximal Cohen-
Macaulay modules over $R$. Given $M,M^{\prime}$ in $\mathbf{MCM}_{R}$, we
write $\underline{\mathrm{Hom}}_{R}(M,M^{\prime})$ for the morphisms in the
this category, namely the morphisms of degree $0$ from $M$ to $M^{\prime}$,
modulo the ideal of morphisms (of degree zero) that factor through a free
$R$-module. We write $\Pi$ for the stabilization functor:

$\Pi:\mathbf{MCM}_{R}\to\underline{\mathbf{MCM}}_{R}.$

For $\ell\geq c$, we have also the $\ell$-th syzygy stable functor:

$\displaystyle\underline{\boldsymbol{\Omega}}^{\ell}:\mathbf{MCM}_{T}$
$\displaystyle\to\underline{\mathbf{MCM}}_{R},$ $\displaystyle L$
$\displaystyle\mapsto\boldsymbol{\Omega}_{R}^{\ell}(L).$

######  Theorem 2.

Assume $\operatorname{H}^{0}(Y,\omega_{Y}(m-c-1))\neq 0$. Then the restriction
to $\mathbf{Ulr}_{T}$ of the $c$-th stable syzygy functor provides a fully
faithful embedding:

$\displaystyle\underline{\boldsymbol{\Omega}}^{c}:\mathbf{Ulr}_{T}$
$\displaystyle\to\underline{\mathbf{MCM}}_{R}.$

The following lemma will be one of the keys of our analysis. Given a finitely
generated graded $R$-module $M$, we write $\langle M_{d}\rangle$ for the
graded submodule of $M$ generated by the elements of degree at most $d$ of
$M$. We also write $M^{*}=\operatorname{Hom}_{R}(M,R)$.

######  Lemma 1.

Fix the hypothesis as in Theorem 2, let $L$ be an Ulrich module over $T$, and
set $M=\boldsymbol{\Omega}_{R}^{c}(L)$. Then we have a functorial exact
sequence:

$0\to\langle M^{*}_{\leq{1-c}}\rangle\to
M^{*}\to\operatorname{Hom}_{T}(L,T(c))\to 0.$

###### Proof.

Recall that $L$ is generated in a single degree, and that the number of
minimal generators of $L$ equals $\alpha_{0}=\deg(X)\operatorname{rk}(L)$. In
other words, we can assume $F_{0}\simeq R^{\alpha_{0}}$. By the minimality of
the resolution, we have, for $i\geq 1$:

$F_{\ell}\simeq\bigoplus_{j\geq\ell}R(-j)^{\alpha_{\ell,j}},\qquad\mbox{for
some integers $\alpha_{\ell,j}$}.$

Also, $X$ and $Y$ are not linear subspaces, so that $L$ has no free summands.

The coherent sheaf over the variety $X$ associated with $L$ is of the form
$i_{*}(\mathscr{L})$, where $\mathscr{L}$ is an Ulrich sheaf over $Y$ and
$i:Y\hookrightarrow X$ is the obvious inclusion. Then sheafifying (3.1) we get
an exact complex:

(3.2) $0\leftarrow
i_{*}\mathscr{L}\leftarrow\mathscr{F}_{0}\xleftarrow{d_{1}}\mathscr{F}_{1}\leftarrow\cdots\leftarrow\mathscr{F}_{\ell-1}\xleftarrow{d_{\ell}}\mathscr{F}_{\ell}\leftarrow\cdots$

where
$\mathscr{F}_{\ell}\simeq\bigoplus_{j\geq\ell}\mathscr{O}_{X}(-j)^{\alpha_{\ell,j}}$.
Set $\mathscr{M}$ for the sheafification of $M$.

Now apply $\mathscr{H}om_{X}(-,\omega_{X})$ to (3.2) and use
$\mathscr{E}xt^{k}_{X}(i_{*}\mathscr{L},\omega_{X})=0$ for $k<c$. Note also
that Grothendieck duality gives:

$\mathscr{E}xt^{c}_{X}(i_{*}\mathscr{L},\omega_{X})\simeq
i_{*}\mathscr{H}om_{Y}(\mathscr{L},\omega_{Y}).$

We obtain a long exact sequence:

$0\to\mathscr{F}_{0}^{\vee}\operatorname{\otimes}\omega_{X}\to\cdots\to\mathscr{F}_{c-2}^{\vee}\operatorname{\otimes}\omega_{X}\xrightarrow{d_{c-1}^{*}}\mathscr{F}_{c-1}^{\vee}\operatorname{\otimes}\omega_{X}\xrightarrow{\pi}\mathscr{M}^{\vee}\to
i_{*}\mathscr{H}om_{Y}(\mathscr{L},\omega_{Y})\to 0.$

Next, we apply $\mathscr{H}om_{X}(\omega_{X},-)$ to this sequence and use that
$X$ is Cohen-Macaulay so that
$\mathscr{H}om_{X}(\omega_{X},\omega_{X})\simeq\mathscr{O}_{X}$. Also, we
easily see that:

$\mathscr{H}om_{X}(\omega_{X},i_{*}\mathscr{H}om_{Y}(\mathscr{L},\omega_{Y}))\simeq\mathscr{H}om_{Y}(i^{*}\omega_{X},\mathscr{H}om_{Y}(\mathscr{L},\omega_{Y}))\simeq\mathscr{H}om_{Y}(\mathscr{L},\mathscr{H}om_{Y}(i^{*}\omega_{X},\omega_{Y})),$

as the first isomorphism is obvious, while the second one follows from a
standard exchange argument of homomorphism groups. Moreover, since $Y$ is cut
in $X$ by $c$ hyperplanes forming a regular sequence, adjunction theory says
that $\omega_{Y}\simeq i^{*}\omega_{Y}(c)$. Therefore, using
$\mathscr{H}om_{Y}(\omega_{Y},\omega_{Y})\simeq\mathscr{O}_{Y}$ we get:

$\mathscr{H}om_{Y}(\mathscr{L},\mathscr{H}om_{Y}(i^{*}\omega_{X},\omega_{Y}))\simeq\mathscr{H}om_{Y}(\mathscr{L},\mathscr{O}_{Y}(c)).$

Putting together, we get the long exact sequence:

$0\to\mathscr{F}_{0}^{\vee}\to\cdots\to\mathscr{F}_{c-2}^{\vee}\xrightarrow{d_{c-1}^{*}}\mathscr{F}_{c-1}^{\vee}\xrightarrow{\pi}\mathscr{M}^{\vee}\to
i_{*}\mathscr{H}om_{Y}(\mathscr{L},\mathscr{O}_{Y}(c))\to 0.$

Taking global sections of this sequence, we just get the result of applying
$\operatorname{Hom}_{R}(-,T)$ to (3.1), i.e. the long exact sequence:

(3.3) $0\to F_{0}^{*}\to\cdots\to
F_{c-2}^{*}\xrightarrow{d_{c-1}^{*}}F_{c-1}^{*}\xrightarrow{\pi}M^{*}\to\operatorname{Hom}_{T}(L,T(c))\to
0.$

Now comes the main point, namely that $\operatorname{Hom}_{T}(L,T)_{-1}=0$. To
see this, recall that
$\operatorname{Hom}_{T}(L,T)_{-1}\simeq\operatorname{Hom}_{Y}(\mathscr{L},\mathscr{O}_{Y}(1))$
and that $\mathscr{L}$ is Ulrich on $Y$ as well as
$\mathscr{H}om_{Y}(\mathscr{L},\omega_{Y})$. Then, by [ESW03, Proposition 2.1]
we get:

$\operatorname{Hom}_{Y}(\mathscr{L},\omega_{Y}(m-c))\simeq\operatorname{H}^{m-c}(Y,\mathscr{L}(c-m))^{*}=0.$

Note that, by the assumption $\operatorname{H}^{0}(Y,\omega_{Y}(m-c-1))\neq 0$
we get an embedding:

$\mathscr{O}_{Y}(1)\hookrightarrow\omega_{Y}(m-c).$

Therefore, $\operatorname{Hom}_{Y}(\mathscr{L},\omega_{Y}(m-c))=0$ implies
$\operatorname{Hom}_{Y}(\mathscr{L},\mathscr{O}_{Y}(1))=0$.

We have thus established that $\operatorname{Hom}_{T}(L,T(c))$ contains no
element of degree $\leq 1-c$. Also, we may write:

$F_{c-1}^{*}=R(c-1)^{\alpha_{c-1,c-1}}\oplus
R(c)^{\alpha_{c-1,c}}\oplus\cdots$

Then, (3.3) says that $F_{c-1}^{*}$ generates all the elements of $M^{*}$ of
degree at most $1-c$, i.e. the image of $\pi$ is the submodule $\langle
M^{*}_{\leq{1-c}}\rangle$ of $M^{*}$. This is clearly functorial, and the
lemma is now proved. ∎

###### Proof of Theorem 2.

Let $L$ and $N$ be two Ulrich modules over $T$. Our goal will be to describe
two mutually inverse maps:

$\operatorname{Hom}_{T}(L,N)_{0}\leftrightarrows\underline{\mathrm{Hom}}_{R}(\boldsymbol{\Omega}^{c}_{R}(L),\boldsymbol{\Omega}^{c}_{R}(N))_{0}.$

Set $M=\boldsymbol{\Omega}^{c}_{R}(L)$ and $P=\boldsymbol{\Omega}^{c}_{R}(N)$.
First, let $\varphi:L\to N$ belong to $\operatorname{Hom}_{T}(L,N)_{0}$.
Consider the minimal graded free resolutions of $L$ and $N$ over $R$ and
choose a lifting of $\varphi$ to these resolutions:

(3.4)
$\textstyle{0}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
L\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
F_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{0}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\cdots}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
F_{c-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{c-1}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\varphi}}$$\textstyle{0}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
N}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
G_{0}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\cdots}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
G_{c-1}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces P}$

The morphism $\tilde{\varphi}$ induced on the $c$-th syzygy modules gives the
class $\bar{\varphi}$ in $\underline{\mathrm{Hom}}_{R}(M,P)_{0}$. This does
not depend on the choice of the lifting $\varphi_{i}$, as any other choice
would provide a map $\tilde{\varphi}^{\prime}$ such that
$\tilde{\varphi}-\tilde{\varphi}^{\prime}$ factors through a free module.

Conversely, given $\bar{\psi}\in\underline{\mathrm{Hom}}_{R}(M,P)_{0}$, we
choose a representative $\psi:M\to P$ and take its dual $\psi^{*}:P^{*}\to
M^{*}$. Since $\psi^{*}$ is homogeneous of degree $0$, it maps elements of
degree at most $1-c$ in $P^{*}$ to elements of degree at most $1-c$ in
$M^{*}$. By Lemma 1 we obtain a diagram:

(3.5)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\langle
M^{*}_{\leq{1-c}}\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Hom}_{T}(L,T(c))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\langle
P^{*}_{\leq{1-c}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi^{*}}$$\textstyle{P^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi^{*}}$$\textstyle{\operatorname{Hom}_{T}(N,T(c))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{\psi}}$$\textstyle{0}$

We wish to associate with $\bar{\psi}$ the morphism $\hat{\psi}^{*}:L\to N$.
To do this, we have to check that $\hat{\psi}$ does not depend on the choice
of the representative $\psi$ of $\bar{\psi}$. By definition any other
representative differs from $\psi$ by a map $\zeta:M\to P$ that factors
through a free module, which we call $F$, i.e. $\zeta=\zeta_{2}\zeta_{1}$ with
$\zeta_{1}:M\to F$ and $\zeta_{2}:F\to G$. Therefore $\zeta^{*}$ factors
through $F^{*}$ and again by Lemma 1 we get the commutative diagram:

|
---|---
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\langle
M^{*}_{\leq{1-c}}\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Hom}_{T}(L,T(c))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\langle
F^{*}_{\leq{1-c}}\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\zeta_{1}^{*}}$$\textstyle{F^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\zeta_{1}^{*}}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\langle
P^{*}_{\leq{1-c}}\rangle\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\zeta_{2}^{*}}$$\textstyle{P^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\zeta_{2}^{*}}$$\textstyle{\operatorname{Hom}_{T}(N,T(c))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hat{\zeta}}$$\textstyle{0}$

Call $G$ the quotient $F^{*}/\langle F^{*}_{\leq{1-c}}\rangle$. The diagram
says that $\hat{\zeta}$ factors through $G$.

Now observe that $G$ is a free module. Indeed, any direct summand of $F$ takes
the form $R(a)$ for some $a\in\mathds{Z}$, and:

(3.6) $\langle
R(a)_{\leq{1-c}}\rangle=\left\\{\begin{array}[]{ll}R(a),&\mbox{if $a\geq
c-1$,}\\\ 0,&\mbox{if $a<c-1$}.\end{array}\right.$

Therefore $G$ is the direct sum of all summands $R(a)$ of $F^{*}$ with
$a<c-1$, hence $G$ is free. But $\operatorname{Hom}_{T}(N,T(c))$ is a torsion
module, So it admits no non-trivial morphism with target in $G$, and therefore
$\hat{\zeta}=0$.

Let us now check that these maps are mutually inverse. Given
$\varphi\in\operatorname{Hom}_{T}(L,N)_{0}$, we consider a representative
$\psi=\tilde{\varphi}$ of the class $\bar{\varphi}$. Dualizing (3.4) we obtain
a commutative diagram:

$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F_{c-1}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{M^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Hom}_{T}(L,T(c))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G_{c-1}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi_{c-1}^{*}}$$\textstyle{P^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi^{*}}$$\textstyle{\operatorname{Hom}_{T}(N,T(c))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi^{*}}$$\textstyle{0}$

This diagram is the extension of (3.5) to a minimal resolution of $\langle
P^{*}_{\leq{1-c}}\rangle$ and $\langle M^{*}_{\leq{1-c}}\rangle$. This says
that $\hat{\psi}=\varphi^{*}$, so $\hat{\psi}^{*}=\varphi$.

Conversely, let $\psi$ be a representative of
$\bar{\psi}\in\underline{\mathrm{Hom}}_{R}(M,P)_{0}$ and set
$\varphi=\hat{\psi}^{*}$. Lift the map $\langle
P^{*}_{\leq{1-c}}\rangle\to\langle M^{*}_{\leq{1-c}}\rangle$ induced by $\psi$
to the minimal graded free resolutions of these modules and dualize to obtain:

$\textstyle{0}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
L\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
F_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{0}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\cdots}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
F_{c-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{c-1}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
0}$$\textstyle{0}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
N}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
G_{0}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\cdots}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
G_{c-1}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
P}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces 0}$

Then $\psi$ is induced by a lifting of $\varphi:L\to N$ to the minimal
resolutions of $L$ and $N$. Our proof is thus complete. ∎

We isolate the following easy consequence of Lemma 1.

######  Lemma 2.

If $L$ is an Ulrich module $L$ over $T$, then $\boldsymbol{\Omega}^{c}_{R}(L)$
has no free summands.

###### Proof.

Suppose $\boldsymbol{\Omega}^{c}_{R}(L)=M\oplus F$, with $F$ a non-zero free
direct summand. Then by Lemma 1 we have:

$0\to\langle M_{\leq 1-c}\rangle\oplus\langle F_{\leq
1-c}\rangle\xrightarrow{\pi}M\oplus F\to\operatorname{Hom}_{T}(L,T(c))\to 0,$

with $\pi$ diagonal. Then, (3.6) says that the restriction of $\pi$ is an
isomorphism between $\langle F_{\leq 1-c}\rangle$ and $F$, as obviously $L$
has no free summand. Therefore, looking at (3.3) we see that $d_{c}$ is
surjective onto $F$, which contradicts minimality of the resolution (3.1). ∎

######  Example 1.

Let $X\subset\mathds{P}^{m+1}$ be a hypersurface of degree $d$ and $Y$ be a
linear section of codimension $c$ of $X$. Then, by matrix factorization the
minimal graded free resolution of an Ulrich module $L$ of rank $r$ on
$T=\boldsymbol{k}[Y]$ reads:

(3.7) $0\leftarrow L\leftarrow T^{rd}\leftarrow T(-1)^{rd}\leftarrow
T(-d)^{rd}\leftarrow\cdots$

Since $L(-d)\simeq\ker(T(-1)^{rd}\to T^{rd})$, this yields a resolution:

(3.8) $0\leftarrow\operatorname{Hom}_{T}(L,T(d-1))\leftarrow T^{rd}\leftarrow
T(-1)^{rd}\leftarrow T(-d)^{rd}\leftarrow\cdots$

Combining (3.7) with the Koszul resolution of $Y$ in $X$ we get a resolution
over $R=\boldsymbol{k}[X]$:

(3.9) $0\leftarrow L\leftarrow R^{rd}\leftarrow R(-1)^{rd(c+1)}\leftarrow
R(-2)^{rd(c+{c\choose 2})}\oplus R(-d)^{rd}\leftarrow\cdots$

The $k$-th term $F_{k}$ of this resolution looks as follows (here
$\varepsilon\in\\{0,1\\}$):

$F_{k}=\bigoplus_{2h+\varepsilon+j=k}R(-(j+hd+\varepsilon))^{{c\choose j}rd}.$

Let $M=\boldsymbol{\Omega}^{c}_{R}(L)$. The resolution of the dualized syzygy
$M^{*}$ starts with:

$\cdots\to R(c-d)^{rd(c+1)}\oplus F_{c-2}^{*}\to R(c-d+1)^{rd}\oplus
F_{c-1}^{*}\to M^{*}\to 0.$

We may now remove from this the dual of the truncation at
$M=\boldsymbol{\Omega}^{c}_{R}(L)$ of (3.9), which is to say, by Lemma 1, the
resolution of $\langle M^{*}_{1-c}\rangle$. The residual strand recovers
precisely (3.8), twisted by $R(c-d+1)$. The two strands of the resolution do
not mix if $d>2$.

######  Remark 3.

The assumption $\operatorname{H}^{0}(Y,\omega_{Y}(m-c-1))\neq 0$ is crucial,
as shown by the following example. Take
$X=\mathds{P}^{1}\times\mathds{P}^{2}\subset\mathds{P}^{5}$ and let $Y$ be a
cubic scroll obtained as hyperplane section of $X$. Note that in this case
$\operatorname{H}^{0}(Y,\omega_{Y}(1))=0$. If $F$ is a line in $Y$ and $H$ is
the class of a hyperplane in $Y$, then $\mathscr{L}=\mathscr{O}_{Y}(H-F)$ is
an Ulrich line bundle on $Y$, whose sheafified minimal
$\mathscr{O}_{X}$-resolution reads:

$0\leftarrow
i_{*}\mathscr{L}\leftarrow\mathscr{O}_{X}^{4}\leftarrow\mathscr{O}_{X}^{9}(-1)\leftarrow\mathscr{O}_{X}^{18}(-2)\leftarrow\cdots$

The sheafified first syzygy $\mathscr{M}$ of $\mathscr{L}$ has a resolution:

$0\leftarrow\mathscr{M}^{*}\leftarrow\mathscr{O}_{X}^{5}\leftarrow\mathscr{O}_{X}^{5}(-1)\leftarrow\mathscr{O}_{X}^{9}(-2)\leftarrow\mathscr{O}_{X}^{18}(-3)\leftarrow\cdots$

In this case,
$\mathscr{H}om_{Y}(\mathscr{L},\mathscr{O}_{Y}(1))\simeq\mathscr{O}_{Y}(F)$
appears as cokernel of a map $\mathscr{O}_{X}^{4}\to\mathscr{M}$, but this
cannot be singled out by degree reasons only from the first map of the above
resolution.

##  4\. CM wildness from syzygies of Ulrich extensions

Let us fix the setup for this section. Let $X\subset\mathds{P}^{n}$ be an ACM
subscheme, put $R=\boldsymbol{k}[X]$ let and $Y$ be a linear section of $X$ of
codimension $c$ and dimension $m\geq 1$. Let $\mathscr{A}$ and $\mathscr{B}$
be two simple Ulrich sheaves on $Y$. Set
$W=\operatorname{Ext}^{1}_{Y}(\mathscr{B},\mathscr{A})$,
$w=\dim_{\boldsymbol{k}}W$, and assume $w\neq 0$. Put $\Upsilon=\Upsilon_{w}$.
Over $Y\times\mathds{P}W$, there is a universal extension:

$0\to\mathscr{A}\boxtimes\mathscr{O}_{\mathds{P}W}\to\mathscr{U}\to\mathscr{B}\boxtimes\mathscr{O}_{\mathds{P}W}(-1)\to
0.$

Take the sheafified minimal graded free resolutions of $\mathscr{A}$ and
$\mathscr{B}$ as $\mathscr{O}_{X}$-modules, pull-back via $p$ to
$X\times\mathds{P}W$, and use the mapping cone construction to build a minimal
graded free resolution of $\mathscr{U}$ over $X\times\mathds{P}W$:

---
$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{A}\boxtimes\mathscr{O}_{\mathds{P}W}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{U}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{B}\boxtimes\mathscr{O}_{\mathds{P}W}(-1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{F}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\boxtimes\mathscr{O}_{\mathds{P}W}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{H}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{0}}$$\textstyle{\mathscr{G}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\boxtimes\mathscr{O}_{\mathds{P}W}(-1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{\vdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\vdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\vdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{F}_{c-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\boxtimes\mathscr{O}_{\mathds{P}W}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{H}_{c-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{G}_{c-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\boxtimes\mathscr{O}_{\mathds{P}W}(-1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{F}_{c}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\boxtimes\mathscr{O}_{\mathds{P}W}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{H}_{c}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{c}}$$\textstyle{\mathscr{G}_{c}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\boxtimes\mathscr{O}_{\mathds{P}W}(-1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{\vdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\vdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\vdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

Here,
$\mathscr{H}_{i}=\mathscr{F}_{i}\boxtimes\mathscr{O}_{\mathds{P}W}\oplus\mathscr{G}_{i}\boxtimes\mathscr{O}_{\mathds{P}W}(-1)$.
Set:

$\mathscr{V}=\operatorname{Im}(d_{c}).$

Then we consider:

$\Phi_{\mathscr{V}}=\mathbf{R}p_{*}(q^{*}(-)\operatorname{\otimes}\mathscr{V}):\mathbf{D}^{b}(\mathbf{Coh}_{\mathds{P}W})\to\mathbf{D}^{b}(\mathbf{Coh}_{X}).$

The following lemma is clear.

######  Lemma 3.

Let $T=\boldsymbol{k}[Y]$, and set $L$ and $N$ for the modules of global
sections of $\mathscr{A}$ and $\mathscr{B}$. Let $\mathscr{M}$ and
$\mathscr{N}$ be the sheafifications of $\boldsymbol{\Omega}_{R}^{c}(L)$ and
$\boldsymbol{\Omega}_{R}^{c}(N)$. Then:

$\Phi_{\mathscr{V}}(\mathscr{O}_{\mathds{P}W})\simeq\mathscr{M},\qquad\Phi_{\mathscr{V}}(\Omega_{\mathds{P}W}(1))\simeq\mathscr{N}[-1].$

Now consider the equivalence $\Xi$ of (2.1). Then the restriction of
$\Phi_{\mathscr{V}}\circ\Xi$ to $\mathbf{Rep}_{\Upsilon}$, composed with the
global sections functor, gives an exact functor:

$\Psi:\mathbf{Rep}_{\Upsilon}\to\mathbf{MCM}_{R}.$

######  Theorem 3.

Assume
$\operatorname{Hom}_{Y}(\mathscr{A},\mathscr{B})=\operatorname{Hom}_{Y}(\mathscr{B},\mathscr{A})=0$
and $\operatorname{H}^{0}(Y,\omega_{Y}(m-c-1))\neq 0$. Then $\Psi$ is a
representation embedding. In particular if $w\geq 3$ then $X$ is of wild CM
type.

###### Proof.

By construction we have the commutative diagram of functors:

$\textstyle{\mathbf{Rep}_{\Upsilon}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi}$$\scriptstyle{\Phi}$$\textstyle{\mathbf{MCM}_{R}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Pi}$$\textstyle{\mathbf{Ulr}_{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\underline{\boldsymbol{\Omega}}_{R}^{c}}$$\textstyle{\underline{\mathbf{MCM}}_{R}}$

We proved in Theorem 2 that $\underline{\boldsymbol{\Omega}}^{c}_{R}$ is fully
faithful, and in Theorem 1 that $\Phi$ is also fully faithful. So the same
happens to $\underline{\boldsymbol{\Omega}}^{c}_{R}\circ\Phi$ and therefore to
$\Pi\circ\Psi$.

Therefore, if $\mathscr{R}$ and $\mathscr{S}$ are two representations of
$\Upsilon$ such that $\Psi(\mathscr{R})\simeq\Psi(\mathscr{S})$, we see that
$\Pi(\Psi(\mathscr{R}))\simeq\Pi(\Psi(\mathscr{S}))$ and hence
$\mathscr{R}\simeq\mathscr{S}$ by full faithfulness.

Moreover, if $\Psi(\mathscr{R})$ is decomposable, then
$\operatorname{Hom}_{R}(\Psi(\mathscr{R}),\Psi(\mathscr{R}))_{0}$ contains a
non-trivial idempotent $\psi$. The class $\bar{\psi}$ is also an idempotent,
which is trivial if and only if the direct summand of $\Psi(\mathscr{R})$
associated with $\psi$ is free. But this cannot happen by Lemma 2, again by
full faithfulness of $\Pi\circ\Psi$, $\bar{\psi}$ corresponds to a non-trivial
idempotent of $\mathscr{R}$, so $\mathscr{R}$ is also decomposable. This
finishes the proof that $\Psi$ is a representation embedding. The consequence
that $X$ is of wild CM type is clear. ∎

######  Example 2.

The first class of varieties where it is a priori unknown how to construct
large families of ACM bundles is given by general cubic hypersurfaces of
dimension $m\geq 4$. We can do this with Theorem 3. Indeed, start with a cubic
hypersurface $X$ in $\mathds{P}^{m+1}$, sufficiently general to admit a smooth
surface section $Y$. Then we may take $\mathscr{A}=\mathscr{O}_{Y}(A)$ and
$\mathscr{B}=\mathscr{O}_{Y}(B)$, where $A$ and $B$ are twisted cubics in $Y$
meeting at $5$ points, cf. [Fae08] : these will satisfy the assumptions of
Theorem 3. In the next section we will see how to deal in a similar fashion
with any variety besides the non-wild varieties listed in the main result.

##  5\. Proof of the main result

We prove here the main theorem of the introduction. Let
$X\subset\mathds{P}^{n}$ be an integral closed ACM subvariety of dimension
$m\geq 1$ and degree $d$, and let $Y$ be an integral one-dimensional linear
section of $X$. We argue on the arithmetic genus $p$ of $Y$, i.e. the
sectional genus of $X$. We also introduce the $\Delta$-genus of $X$, defined
as $\Delta(X)=d-m+n-1$; it is well-known that $\Delta(X)\geq 0$.

###  5.1. Minimal degree

In case $p=0$ we have $\Delta(X)=0$, which means that $X$ has minimal degree.
According to the classification of del Pezzo and Bertini of these varieties,
cf. [EH87], we have that $X$ is smooth, for otherwise $X$ would be a cone. In
this case $X$ is of finite CM type (cf. [AR87, EH88]) if it is a linear space
(cf. [Hor64]), or a smooth quadric (see [Knö87]), or the Veronese surface in
$\mathds{P}^{5}$, or a smooth cubic scroll in $\mathds{P}^{4}$ (see [AR89]),
or a rational normal curve. Also, $X$ is of tame CM type if it is a rational
normal surface scroll of degree $4$, cf. [FM14]. Besides these cases, $X$ is
strictly Ulrich wild, as we see by applying Theorem 1 to the Ulrich line
bundles constructed in [MR13, FM14].

###  5.2. Sectional genus at least 2

Let us look at the case of (arithmetic) genus $p\geq 2$. Consider the
normalization $\bar{Y}$ of $Y$, which is a smooth irreducible curve, let $g$
be the genus of $\bar{Y}$ and $\sigma:\bar{Y}\to Y$ be the normalization.

We first consider an Ulrich sheaf of rank one on $Y$. According to [ESW03], to
construct $\mathscr{L}_{0}$ on $Y$ we need to pick a sufficiently general line
bundle $\bar{\mathscr{L}}_{0}$ of degree $d+p-1$ on $\bar{Y}$ and define
$\mathscr{L}_{0}=\sigma_{*}(\bar{\mathscr{L}}_{0})$. Since $Y$ is integral,
the sheaf $\mathscr{L}_{0}$ is simple, being reflexive of rank $1$.

Now we consider two general flat deformations $\mathscr{L}_{1}$ and
$\mathscr{L}_{2}$ of $\mathscr{L}_{0}$ in the compactified Jacobian of $Y$.
These are locally free sheaves of rank $1$ on $Y$, both having the same degree
as $\mathscr{L}_{0}$, namely again $d+p-1$. The condition that
$\mathscr{L}_{k}$ is Ulrich amounts to
$\operatorname{H}^{i}(Y,\mathscr{L}_{k}(-1))=0$ for $i=0,1$. By assumption
this vanishing holds for $k=0$, hence by semicontinuity also for $k=1,2$, so
that $\mathscr{L}_{1}$ and $\mathscr{L}_{2}$ are Ulrich line bundles on $Y$.
Since the dimension of the compactified Jacobian of $Y$ is $p\geq 2$, we may
assume that $\mathscr{L}_{1}$ and $\mathscr{L}_{2}$ are not isomorphic.
Therefore, since the $\mathscr{L}_{i}$ are invertible, are not isomorphic and
have the same Hilbert polynomial, by irreducibility of $Y$ we deduce
$\operatorname{Hom}_{Y}(\mathscr{L}_{i},\mathscr{L}_{j})=0$ for $i\neq j$.

Next, since $\mathscr{L}_{1}$ and $\mathscr{L}_{2}$ are locally free of rank
$1$, we have $\chi(\mathscr{L}_{i},\mathscr{L}_{i})=1-p$ for $i=1,2$.
Therefore, we may choose $\mathscr{A}$ and $\mathscr{B}$ as two sheaves
corresponding to non-trivial elements of
$\operatorname{Ext}^{1}_{Y}(\mathscr{L}_{i},\mathscr{L}_{i})$, and write:

$\displaystyle 0\to\mathscr{L}_{1}\to\mathscr{A}\to\mathscr{L}_{1}\to 0,$
$\displaystyle 0\to\mathscr{L}_{2}\to\mathscr{B}\to\mathscr{L}_{2}\to 0.$

It is clear that $\mathscr{A}$ and $\mathscr{B}$ are locally free Ulrich
sheaves of rank $2$. Also it is easy to check that $\mathscr{A}$ and
$\mathscr{B}$ are simple and satisfy:

$\displaystyle\operatorname{Hom}_{Y}(\mathscr{A},\mathscr{B})=\operatorname{Hom}_{Y}(\mathscr{B},\mathscr{A})=0.$
$\displaystyle\chi(\mathscr{A},\mathscr{B})=\chi(\mathscr{B},\mathscr{A})=4(1-p).$

We obtain the following:

$\dim_{\boldsymbol{k}}\operatorname{Ext}^{1}_{Y}(\mathscr{A},\mathscr{B})=\dim_{\boldsymbol{k}}\operatorname{Ext}^{1}_{Y}(\mathscr{B},\mathscr{A})=4(p-1).$

Note that the non vanishing condition of Theorem 2 reduces to
$\operatorname{H}^{0}(Y,\omega_{Y})\neq 0$, which is obviously true. Now
Theorem 3 implies that $X$ is of wild representation type.

###  5.3. Sectional genus 1, normal del Pezzo surfaces

Finally, let us consider the case of sectional genus $p=1$. Since $X$ is ACM,
it turns out that the $\Delta$-genus of $X$ is also one, cf. [Fuj90, page 45],
namely $X$ is of almost minimal degree. According to Fujita’s terminology, $X$
is a del Pezzo variety.

####  5.3.1. General discussion and the case of elliptic curves

This time, $X$ is of tame CM type if it is a smooth elliptic curve (see
[Ati57]) or a rational curve of arithmetic genus $1$ with only an ordinary
double point (see [DG01]), which settles the case $m=1$.

For higher dimension, we have to work with a surface section $Y$ of $X$
instead of a curve section. So we let $Y$ be a general linear section of $X$
of dimension $2$, in particular $Y$ is an integral projective ACM surface
which is not a cone, having degree $d$ in $\mathds{P}^{d}$. These surfaces
constitute a generalization of classical del Pezzo surfaces to allow certain
singularities. They are arithmetically Gorenstein, as if fact
$\omega_{Y}\simeq\mathscr{O}_{Y}(-1)$. In particular $Y$ satisfies the non
vanishing condition of Theorem 2, namely
$\operatorname{H}^{0}(Y,\omega_{Y}(1))\neq 0$.

Surfaces of almost minimal degree and actually even higher dimensional
varieties of almost minimal degree, are completely classified cf. [Fuj82] for
the case of normal varieties, and [Rei94, BS07] for the non-normal case. We
will actually rely on the complete classification of (singular) del Pezzo
surfaces of degree $3$ and $4$. We refer to [LPS11, LPS12] for this theory,
rooted in work of Schläfli and Cayley, see for instance [Abh60, BW79].

For our purpose, we have to study in two different ways the case of normal
surfaces and that of non-normal ones. Indeed, in the normal case we can treat
Ulrich bundles of rank $2$ in view of the good knowledge of sets of points in
$Y$ and Hartshorne-Serre correspondence, while we may not rely on rank-$1$
Ulrich bundles (one reason is that they do not even exist for cubic surfaces
with an $E_{6}$ singularity, cf. [Dol12, Theorem 9.3.6]). On the other hand
for the non-normal case we do use rank-$1$ Ulrich sheaves coming from the
normalization $\bar{Y}$, which are under control since $\bar{Y}$ is a surface
of minimal degree.

####  5.3.2. Hartshorne-Serre correspondence for normal del Pezzo surfaces

We assume from now on in this subsection that $Y$ is (locally) normal. This
case is treated via the Hartshorne-Serre construction in the spirit for
instance of [CH11], where we only need to modify the argument slightly so that
it applies to singular surfaces too. So let us consider a set $Z\subset
Y\subset\mathds{P}^{d}$ of $d+2$ points in general linear position. We have:

$\operatorname{Ext}^{1}_{Y}(\mathscr{I}_{Z|Y}(2),\mathscr{O}_{Y})^{*}\simeq\operatorname{Ext}^{1}_{Y}(\mathscr{O}_{Y},\mathscr{I}_{Z|Y}(1))\simeq\operatorname{H}^{1}(Y,\mathscr{I}_{Z|Y}(1))\simeq\boldsymbol{k},$

where we used that $\omega_{Y}\simeq\mathscr{O}_{Y}(-1)$ and that $Z$ spans
the whole $\mathds{P}^{d}$. Through Serre correspondence, a non-zero element
$\lambda\in\boldsymbol{k}\simeq\operatorname{H}^{1}(Y,\mathscr{I}_{Z|Y}(1))$
provides a coherent sheaf $\mathscr{F}$ of rank $2$ that fits into the short
exact sequence:

(5.1) $0\to\mathscr{O}_{Y}\to\mathscr{F}_{Z}\to\mathscr{I}_{Z|Y}(2)\to 0.$

Given a set of points $Z$ of $d+2$ points of $Y$, write $[Z]$ for the
corresponding element of the Hilbert scheme $\operatorname{Hilb}^{d+2}(Y)$ of
subschemes of length $d+2$ of $Y$. In the next lines, stability is always with
respect to $\mathscr{O}_{Y}(1)$.

######  Lemma 4.

Let $Y\subset\mathds{P}^{d}$ be an integral normal ACM surface of degree $d$.

  1. i)

If $Z$ is a reduced set of $d+2$ points in general linear position then
$\mathscr{F}_{Z}$ is Ulrich.

  2. ii)

If moreover $Z$ is disjoint from the singular locus of $Y$ then
$\mathscr{F}_{Z}$ is locally free.

  3. iii)

If $[Z]$ is general enough in $\operatorname{Hilb}^{d+2}(Y)$, then the sheaf
$\mathscr{F}_{Z}$ is stable.

###### Proof.

The fact that $\mathscr{F}_{Z}$ is Ulrich follows from the form of the minimal
graded free resolution of the ideal of $Z$, which can be extracted from
[MRPL12], so (i) is clear. The fact that $\mathscr{F}_{Z}$ is locally free is
immediate when $Y$ is smooth, by the Auslander-Buchsbaum formula. If $Y$ is
singular and $Z$ does not meet the singular locus of $Y$, then $Z$ is locally
a complete intersection. On the other hand, we have:

$\lambda\in\operatorname{Ext}^{1}(\mathscr{I}_{Z|Y}(2),\mathscr{O}_{Y})\simeq\operatorname{H}^{0}(Y,\mathscr{E}xt^{1}_{\mathscr{O}_{Y}}(\mathscr{I}_{Z|Y}(2),\mathscr{O}_{Y})),$

so for any $x\in Y$, the element
$\lambda_{x}\in\mathscr{E}xt^{1}_{\mathscr{O}_{Y}}(\mathscr{I}_{Z|Y}(2),\mathscr{O}_{Y}))_{x}$
represents a non-trivial extension of $(\mathscr{F}_{Z})_{x}$. Moreover,
taking dual of the short exact sequence:

$0\to\mathscr{I}_{Z|Y}(2)\to\mathscr{O}_{Y}(2)\to\mathscr{O}_{Z}(2)\to 0$

we deduce:

$\mathscr{E}xt^{1}_{\mathscr{O}_{Y}}(\mathscr{I}_{Z|Y}(2),\mathscr{O}_{Y})\simeq\mathscr{E}xt^{2}_{\mathscr{O}_{Y}}(\mathscr{O}_{Z},\mathscr{O}_{Y})\simeq\omega_{Z}.$

and therefore
$\mathscr{E}xt^{1}_{\mathscr{O}_{Y}}(\mathscr{I}_{Z|Y}(2),\mathscr{O}_{Y})_{x}$
is generated by $\lambda_{x}$. Then we can apply [OSS80, Lemma 5.1.2] to
conclude that $(\mathscr{F}_{Z})_{x}$ is free. We have proved (ii).

The proof of (iii) will be done by a dimension count. If the Ulrich bundle
$\mathscr{F}_{Z}$ is not stable, a straightforward modification of [CHGS12],
relying on the fact that $\omega_{Y}\simeq\mathscr{O}_{Y}(-1)$, shows that
$\mathscr{F}_{Z}$ is an extension

$0\to\mathscr{A}\to\mathscr{F}_{Z}\to\mathscr{B}\to 0,$

where $\mathscr{A}$ and $\mathscr{B}$ are Ulrich sheaves of rank $1$. By
construction, the global section $s\in\operatorname{H}^{0}(Y,\mathscr{F}_{Z})$
associated with $Z$ vanishes in codimension two. Therefore $s$ cannot lie in
$\operatorname{H}^{0}(Y,\mathscr{A})\subset\operatorname{H}^{0}(Y,\mathscr{F}_{Z})$,
as $s$ would then then vanish in codimension $1$. Therefore, we construct the
following commutative diagram:

$\textstyle{\mathscr{O}_{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\textstyle{\mathscr{O}_{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{E}_{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{I}_{Z|Y}(2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where $\mathscr{T}$ is a torsion sheaf defined by the diagram. This says that:

$\
\operatorname{H}^{0}(Y,\mathscr{I}_{Z/Y}\operatorname{\otimes}\mathscr{A}^{\vee}(2))\neq
0,$

namely $Z$ lies on a divisor $D$ from the linear system
$|\mathscr{A}^{\vee}(2)|$. Since $\mathscr{A}^{\vee}(2)$ is also an Ulrich
sheaf (again because $\omega_{Y}\simeq\mathscr{O}_{Y}(-1)$), the dimension of
the linear system $|\mathscr{A}^{\vee}(2)|$ is $d-1$. Therefore, the family of
subschemes of $Y$ consisting of $(d+2)$-tuples of points lying on a divisor of
class $\mathscr{A}^{\vee}(2)$ is at most:

$\operatorname{dim}\operatorname{Hilb}^{d+2}(D)+d-1=2d+1<2d+4=\operatorname{dim}\operatorname{Hilb}^{d+2}(Y).$

Then, $\mathscr{F}_{Z}$ is stable as soon as $[Z]$ lies away from the union of
proper closed algebraic subsets of $\operatorname{Hilb}^{d+2}(Y)$, one for
each Ulrich sheaf of rank $1$ over $Y$. Now the statement is clear since all
of these sheaves have discrete parameter space. Indeed, any such sheaf
corresponds to the linear equivalence class of a Weil divisor of $Y$, and
since $Y$ is a normal regular surface, the divisor class group of $Y$ is
discrete. ∎

This construction can be performed in quite a general setup, for instance for
cubic surfaces one even knows, if $\boldsymbol{k}$ is not algebraically
closed, the degree of the field extension needed to construct
$\mathscr{E}_{Z}$, cf. [Tan14]. However what we need is the following result.

######  Proposition 1.

Let $\mathscr{E}$, $\mathscr{F}$ be non-isomorphic stable Ulrich bundles
coming from 5.1. Then:

$\displaystyle\operatorname{Hom}_{Y}(\mathscr{E},\mathscr{F})=\operatorname{Hom}_{Y}(\mathscr{F},\mathscr{E})=0,$
$\displaystyle\dim_{\boldsymbol{k}}\operatorname{Ext}^{1}_{Y}(\mathscr{E},\mathscr{F})=4.$

###### Proof.

The first two statements concerning morphisms are clear since we can suppose
that $\mathscr{E}$ and $\mathscr{F}$ are stable with the same slope. For the
first statement, since $\mathscr{E}$ is locally free, we have
$\operatorname{Ext}^{1}_{Y}(\mathscr{E},\mathscr{F})\simeq\operatorname{H}^{1}(Y,\mathscr{E}^{\vee}\otimes\mathscr{F})$.
Tensoring the short exact sequence 5.1 by $\mathscr{E}^{\vee}$ and considering
the associated long exact sequence of global sections, since
$\operatorname{H}^{1}(Y,\mathscr{E}^{\vee})=\operatorname{H}^{2}(Y,\mathscr{E}^{\vee})=0$,
we get an isomorphism

(5.2)
$\operatorname{Ext}^{1}_{Y}(\mathscr{E},\mathscr{F})\simeq\operatorname{H}^{1}(Y,\mathscr{E}^{\vee}\otimes\mathscr{I}_{Z|Y}(2)).$

On the other hand, the exact sequence defining $Z\subset Y$ twisted by
$\mathscr{O}_{Y}(2)$ reads:

(5.3) $0\to\mathscr{I}_{Z|Y}(2)\to\mathscr{O}_{Y}(2)\to\mathscr{O}_{Z}(2)\to
0.$

Taking into account that $\mathscr{E}^{\vee}\simeq\mathscr{E}(-2)$ we obtain
$\operatorname{H}^{0}(Y,\mathscr{E}^{\vee}\otimes\mathscr{I}_{Z|Y}(2))=0$.
Then, tensoring (5.3) by $\mathscr{E}^{\vee}$ taking global sections and
combining with (5.2) we get:

$0\to\operatorname{H}^{0}(Y,\mathscr{E}^{\vee}(2))\to\operatorname{H}^{0}(Y,\mathscr{E}^{\vee}\otimes\mathscr{O}_{Z}(2))\to\operatorname{Ext}^{1}_{Y}(\mathscr{E},\mathscr{F})\to
0.$

Now we know that $\mathscr{E}^{\vee}(2)\simeq\mathscr{E}$ has $2d$ independent
global sections, as $\mathscr{E}$ is Ulrich. On the other hand, since $Z$ has
length $d+2$, so that $\mathscr{E}^{\vee}\otimes\mathscr{O}_{Z}(2)$ is just a
vector space of rank $2$ concentrated at $d+2$ points, hence
$\dim_{\boldsymbol{k}}\operatorname{H}^{0}(Y,\mathscr{E}^{\vee}\otimes\mathscr{O}_{Z}(2))=2d+4$.
We conclude that
$\dim_{\boldsymbol{k}}\operatorname{Ext}^{1}_{Y}(\mathscr{E},\mathscr{F})=4$.
∎

We may argue that this proposition says that any surface $Y$ of almost minimal
degree supports infinitely many Ulrich bundles of rank $2$ with determinant
$\mathscr{O}_{Y}(2)$, in particular any cubic surface admits (over an
algebraically closed field) infinitely many pfaffian representations. The
normal case is now complete by virtue of Theorem 3.

###  5.4. Del Pezzo surfaces normalized by a rational normal scroll

This time $Y$ is a non-normal ACM surface of degree $d$ in $\mathds{P}^{d}$,
we let ${\bar{Y}}$ be the normalization of $Y$ and $\sigma:{\bar{Y}}\to Y$ the
normalization map. It turns out that these surfaces are completely classified
cf. [Rei94], and that $\bar{Y}$ is a rational normal scroll or a Veronese
surface. Also, the locus where $\sigma$ is not an isomorphism is a line
$L\subset Y$ and ${\bar{L}}=\sigma^{*}L$ is a conic in ${\bar{Y}}$. In this
subsection we are going to treat the case when ${\bar{Y}}$ is a rational
normal scroll leaving the case of the Veronese surface for the next
subsection.

So let us consider the two non-isomorphic rank-1 Ulrich bundles over
${\bar{Y}}$, normalized so that they have $d$ independent global section,
where $d=d_{Y}$ is the degree of ${\bar{Y}}$ as a rational normal scroll,
which is the same as the degree of $Y$. These line bundles are
$\mathscr{O}_{\bar{Y}}((d-1)F)$, where $F$ is a fibre of the scroll map
${\bar{Y}}\to\mathds{P}^{1}$, i.e. $F$ is just a line of the scroll, and
$\mathscr{O}_{\bar{Y}}(H-F)$, where $H$ is the hyperplane section of
${\bar{Y}}$. Let us write $A=(d-1)F$ and $B=H-F$. We have:

$\dim_{\boldsymbol{k}}\operatorname{Ext}^{1}_{\bar{Y}}(\mathscr{O}_{\bar{Y}}(A),\mathscr{O}_{\bar{Y}}(B))=d-2.$

Set $\mathscr{A}=\sigma_{*}\mathscr{O}_{\bar{Y}}(A)$ and
$\mathscr{B}=\sigma_{*}\mathscr{O}_{\bar{Y}}(B)$. These are Ulrich sheaves of
rank $1$ on $Y$. We also have
$\operatorname{Ext}^{2}_{\bar{Y}}(\mathscr{O}_{\bar{Y}}(A),\mathscr{O}_{\bar{Y}}(B))=0$,
so the local-to-global spectral sequence degenerates to:

$\operatorname{Ext}^{1}_{Y}(\mathscr{A},\mathscr{B})\simeq\operatorname{Ext}^{1}_{\bar{Y}}(\mathscr{O}_{\bar{Y}}(A),\mathscr{O}_{\bar{Y}}(B))\oplus\operatorname{H}^{0}(Y,\mathscr{E}xt^{1}_{Y}(\mathscr{A},\mathscr{B})),$

and similarly:

$\operatorname{Ext}^{1}_{Y}(\mathscr{B},\mathscr{A})\simeq\operatorname{H}^{0}(Y,\mathscr{E}xt^{1}_{Y}(\mathscr{B},\mathscr{A})).$

Therefore, in order to conclude by Theorem 3, we only need to prove that in
the cases $d=3,4$ either
$\operatorname{H}^{0}(Y,\mathscr{E}xt^{1}_{Y}(\mathscr{B},\mathscr{A}))$ has
dimension at least $3$ or that
$\operatorname{H}^{0}(Y,\mathscr{E}xt^{1}_{Y}(\mathscr{A},\mathscr{B}))$ has
dimension at least $5-d$. Luckily, these two cases are completely classified
up to projective equivalence, cf. [LPS11, LPS12].

####  5.4.1. The non-normal locus

We said that $Y$ is not normal along a line $L$. We have:

(5.4)
$0\to\mathscr{O}_{Y}\to\sigma_{*}\mathscr{O}_{\bar{Y}}\to\mathscr{O}_{L}(-1)\to
0.$

In order to see this notice that the restriction of $\sigma$ is $2:1$ as a map
${\bar{L}}\to L$. Therefore $\sigma_{*}\mathscr{O}_{\bar{Y}}$ is locally free
of rank $2$ over $L$, and
$(\sigma_{*}\mathscr{O}_{\bar{Y}})|_{L}\simeq\sigma_{*}\mathscr{O}_{\bar{L}}$,
so we get an exact sequence:

$0\to\mathscr{O}_{L}\to\sigma_{*}\mathscr{O}_{\bar{L}}\to Q\to 0,$

where $Q$ is a line bundle on $L$. From the previous exact sequence we have
$\chi(Q)=0$, namely $Q\simeq\mathscr{O}_{L}(-1)$.

This gives with no difficulty the isomorphism:

(5.5)
$\mathscr{E}xt^{1}_{Y}(\sigma_{*}\mathscr{O}_{\bar{Y}},\sigma_{*}\mathscr{O}_{\bar{Y}})\simeq\mathscr{E}xt^{1}_{Y}(\mathscr{O}_{L},\mathscr{O}_{L}).$

Next we may compute a locally free resolution of $\mathscr{O}_{L}$ (in fact
the sheafified minimal graded free resolution of the module of global sections
of $\mathscr{O}_{L}$) over $Y$, relying on the classification appearing in
[LPS11, LPS12]. This reads, when $d=4$:

$0\leftarrow\mathscr{O}_{L}\leftarrow\mathscr{O}_{Y}\leftarrow\mathscr{O}_{Y}(-1)^{3}\leftarrow\mathscr{O}_{Y}(-2)^{5}\leftarrow\mathscr{O}_{Y}(-3)^{7}\leftarrow\cdots,$

and for $d=3$:

(5.6) $\displaystyle
0\leftarrow\mathscr{O}_{L}\leftarrow\mathscr{O}_{Y}\xleftarrow{d_{1}}\mathscr{O}_{Y}(-1)^{2}\xleftarrow{d_{2}}\mathscr{O}_{Y}(-2)\oplus\mathscr{O}_{Y}(-3)\leftarrow\mathscr{O}_{Y}(-4)^{2}\leftarrow\cdots$

This gives:

(5.7)
$\displaystyle\mathscr{E}xt^{1}_{Y}(\mathscr{O}_{L},\mathscr{O}_{L})\simeq\mathscr{O}_{L}\oplus\mathscr{O}_{L}(1),$
$\displaystyle\mbox{for $d=4$},$ (5.8)
$\displaystyle\mathscr{E}xt^{1}_{Y}(\mathscr{O}_{L},\mathscr{O}_{L})\simeq\mathscr{O}_{L}(1)^{2},$
$\displaystyle\mbox{for $d=3$}.$

We may also note that $\mathscr{E}xt^{i}_{Y}(\mathscr{O}_{L},\mathscr{O}_{L})$
is locally free of rank $2$ on $L$ for all $i>0$.

####  5.4.2. Curves associated with Ulrich bundles

A general global section of $\mathscr{O}_{\bar{Y}}(B)$ vanishes along a
rational normal curve $B$ of degree $d-1$ in ${\bar{Y}}$, while any non-zero
global section of $\mathscr{O}_{\bar{Y}}(A)$ vanishes along the union $A$ of
$d-1$ fibres, which are pairwise disjoint lines contained in ${\bar{Y}}$. All
this is clear from the description of divisors on ${\bar{Y}}$, independently
on the characteristic of $\boldsymbol{k}$. It turns out that the zero locus of
a general global section of $\mathscr{A}$ is a curve ${\hat{A}}$ which is the
union of $\sigma A$ and $L$, and the same happens to $\mathscr{B}$ with
obvious notation. We write $A^{\prime}=\sigma A$ and $B^{\prime}=\sigma B$, so
${\hat{A}}=L\cup A^{\prime}$ and ${\hat{B}}=L\cup B^{\prime}$. Actually
$B^{\prime}$ cuts $L$ at a single point $p_{0}$, while $A^{\prime}$ cuts $L$
at $d-1$ points $p_{1},\ldots,p_{d-1}$, one for each component of
$A^{\prime}$. Both ${\hat{A}}$ and ${\hat{B}}$ are ACM curves in $Y$.

$A^{\prime}$$L$$B^{\prime}$$p_{0}$$p_{1}$$p_{2}$$p_{3}$ Figure 1. Curves
associated with Ulrich bundles for a quartic surface normalized by a rational
scroll.

When $B$ is smooth (which happens if the defining section of
$\mathscr{O}_{\bar{Y}}(B)$ is general enough), setting $p$ (resp.
$p^{\prime}$) for a point of $B$ (resp. of $B^{\prime}$), we get:

$\mathscr{O}_{\bar{Y}}(B)|_{B}\simeq\mathscr{O}_{B}((d-2)p),\qquad\mathscr{B}|_{B^{\prime}}\simeq\mathscr{O}_{B^{\prime}}((d-2)p^{\prime})$

We also have:

$\mathscr{O}_{\bar{Y}}(A)|_{A}\simeq\mathscr{O}_{A},\qquad\mathscr{A}|_{A^{\prime}}\simeq\mathscr{O}_{A^{\prime}}.$

There is an exact commutative diagram:

(5.9)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{O}_{Y}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{A}|_{{\hat{A}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\sigma_{*}\mathscr{O}_{\bar{Y}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathscr{A}|_{A^{\prime}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

This induces:

(5.10)
$0\to\mathscr{O}_{L}(-1)\to\mathscr{A}|_{{\hat{A}}}\to\mathscr{O}_{A^{\prime}}\to
0.$

Similarly we get:

(5.11)
$0\to\mathscr{O}_{L}(-1)\to\mathscr{B}|_{{\hat{B}}}\to\mathscr{O}_{B^{\prime}}((d-2)p^{\prime})\to
0.$

We also have that the curves $A^{\prime}$ and $B^{\prime}$ intersect
transversely at $d-1$ points, none of which lies in $L$, when $A$ and $B$ are
general enough.

####  5.4.3. Computation of Ext sheaves

Our goal now is to give a lower bound for the number of independent global
sections of the coherent sheaf
$\mathscr{E}xt^{1}_{Y}(\mathscr{A},\mathscr{B})$. We know already that this
sheaf is supported on $L$.

First, we replace each instance of $A$ in the diagram (5.9) by $B$ and we
apply $\mathscr{H}om_{Y}(\mathscr{A},-)$ to the first row of the resulting
exact commutative diagram. Since $\mathscr{A}$ is locally CM we get an
isomorphism:

(5.12)
$\mathscr{E}xt^{1}_{Y}(\mathscr{A},\mathscr{B})\simeq\mathscr{E}xt^{1}_{Y}(\mathscr{A},\mathscr{B}|_{{\hat{B}}}),$

Next, applying $\mathscr{H}om_{Y}(-,\mathscr{B}|_{{\hat{B}}})$ to the second
row of (5.9) we get a long exact sequence:

$\cdots\to\mathscr{E}xt^{i}_{Y}(\mathscr{O}_{A^{\prime}},\mathscr{B}|_{{\hat{B}}})\xrightarrow{\varsigma_{i}}\mathscr{E}xt^{i}_{Y}(\mathscr{A},\mathscr{B}|_{{\hat{B}}})\to\mathscr{E}xt^{i}_{Y}(\sigma_{*}\mathscr{O}_{{\bar{Y}}},\mathscr{B}|_{{\hat{B}}})\to\cdots$

Note that, for all $i$, the sheaf
$\mathscr{E}xt^{i}_{Y}(\mathscr{O}_{A^{\prime}},\mathscr{B}|_{{\hat{B}}})$ is
supported at $A^{\prime}\cap{\hat{B}}$ and so has finite length. For $i=2$ we
need something more precise, namely the next lemma.

######  Lemma 5.

The sheaf
$\mathscr{E}xt^{2}_{Y}(\mathscr{O}_{A^{\prime}},\mathscr{B}|_{{\hat{B}}})$ has
length $d-1$.

###### Proof.

We first note that (5.11) easily implies
$\mathscr{E}xt^{2}_{Y}(\mathscr{O}_{A^{\prime}},\mathscr{B}|_{{\hat{B}}})\simeq\mathscr{E}xt^{2}_{Y}(\mathscr{O}_{A^{\prime}},\mathscr{O}_{L}(-1))$
since $A^{\prime}$ and $B$ meet transversely along $B^{\prime}$, i.e. away
from the non-normal locus $L$. In turn, this sheaf is obviously isomorphic to
$\mathscr{E}xt^{2}_{Y}(\mathscr{O}_{A^{\prime}},\mathscr{O}_{L})$. So we would
like to check that
$\mathscr{E}xt^{2}_{Y}(\mathscr{O}_{A^{\prime}},\mathscr{O}_{L})$ has length
$d-1$ by showing that it has rank $1$ over each of the points
$p_{1},\ldots,p_{d-1}$ of $L\cap A^{\prime}$.

Looking at each point $p_{i}$ we have to show that the line $A_{i}\subset Y$
defined as the irreducible component of $A^{\prime}$ passing through $p_{i}$
satisfies
$\mathscr{E}xt^{2}_{Y}(\mathscr{O}_{A_{i}},\mathscr{O}_{L})\simeq\mathscr{O}_{p_{i}}$.
The situation is local, and the cases $d=3$ and $d=4$ do not differ in this
sense, so we just look at $d=3$. We consider the sheafified minimal graded
free resolution of the module of global sections of $\mathscr{O}_{A_{i}}$,
which in turn looks precisely like (5.6) with $L$ replaced with $A_{i}$. Of
course,
$\boldsymbol{\Omega}^{1}(\mathscr{O}_{A_{i}})\simeq\mathscr{I}_{A_{i}/Y}$, and
we see that
$\boldsymbol{\Omega}^{2}(\mathscr{O}_{A_{i}})\simeq\mathscr{I}_{C_{i}/Y}(-1)$,
where $C_{i}$ is the residual conic of $A_{i}$ with respect to a plane section
of $Y$ containing $A_{i}$. Write $\iota_{j}$ for the restriction of $d_{j}$ to
$\boldsymbol{\Omega}^{j}(\mathscr{O}_{A_{i}})$.

$A_{i}$$L$$C_{i}$$p_{i}$ Figure 2. Lines $L$ and $A_{i}$ with residual conic
$C_{i}$.

Next we observe that, since $A_{i}$ meets $L$ transversely at $p_{i}$, the
sheaf $\mathscr{H}om_{Y}(\mathscr{I}_{A_{i}/Y},\mathscr{O}_{L})$ is just the
dual over $L$ of $\mathscr{I}_{p_{i}/L}$, that is $\mathscr{O}_{L}(1)$. The
curve $C_{i}$ also meets $L$ transversely at $p_{i}$, so:

(5.13)
$\mathscr{H}om_{Y}(\mathscr{I}_{A_{i}/Y},\mathscr{O}_{L})\simeq\mathscr{O}_{L}(1)\simeq\mathscr{H}om_{Y}(\mathscr{I}_{C_{i}/Y},\mathscr{O}_{L}),$

Now we apply then $\mathscr{H}om_{Y}(-,\mathscr{O}_{L})$ to our resolution of
$\mathscr{O}_{A_{i}}$. We first note that:

$\mathscr{E}xt^{2}_{Y}(\mathscr{O}_{A_{i}},\mathscr{O}_{L})\simeq\mathscr{E}xt^{1}_{Y}(\mathscr{I}_{A_{i}/Y},\mathscr{O}_{L}).$

Then, using (5.13) we get an exact sequence:

$0\to\mathscr{O}_{L}(1)\to\mathscr{O}_{L}(1)^{2}\xrightarrow{\mathscr{H}om_{Y}(\iota_{2},\mathscr{O}_{L})}\mathscr{O}_{L}(2)\to\mathscr{E}xt^{2}_{Y}(\mathscr{O}_{A_{i}},\mathscr{O}_{L})\to
0$

Now it clearly turns out that
$\mathscr{E}xt^{2}_{Y}(\mathscr{O}_{A_{i}},\mathscr{O}_{L})\simeq\mathscr{O}_{p_{i}}$,
which finishes the proof. ∎

To continue our computation, we observe that the sheaf
$\mathscr{E}xt^{1}_{Y}(\mathscr{A},\mathscr{B})$ is locally free of rank $2$
over $L$. This follows from (5.5), (5.7) and (5.8) since
$\mathscr{O}_{{\bar{Y}}}(B)$ and $\mathscr{O}_{{\bar{Y}}}(A)$ locally behave
like $\mathscr{O}_{{\bar{Y}}}$, so $\mathscr{B}$ and $\mathscr{A}$ locally
behave like $\sigma_{*}\mathscr{O}_{{\bar{Y}}}$. Then $\varsigma_{i}=0$ for
$i=1,2$, because we have just seen that the middle sheaf is locally free over
$L$ in view of (5.12) and the leftmost sheaf has finite length.

Set
$\epsilon=\chi(\mathscr{E}xt^{1}_{Y}(\sigma_{*}\mathscr{O}_{{\bar{Y}}},\mathscr{B}|_{{\hat{B}}}))$.
We have shown via Lemma 5:

$\dim_{\boldsymbol{k}}\operatorname{Ext}^{1}_{Y}(\mathscr{A},\mathscr{B})\geq\chi(\mathscr{E}xt^{1}_{Y}(\mathscr{A},\mathscr{B}))\geq\epsilon-d+1.$

It remains to compute $\epsilon$. Observe that in order to finish we have to
show $\epsilon\geq 4$. To get this, we apply
$\mathscr{H}om_{Y}(\sigma_{*}\mathscr{O}_{{\bar{Y}}},-)$ to (5.10) after
replacing $A$ with $B$. Using (5.4), it is easy to observe that:

$\mathscr{E}xt^{1}_{Y}(\sigma_{*}\mathscr{O}_{{\bar{Y}}},\mathscr{O}_{L}(-1))\simeq\mathscr{E}xt^{1}_{Y}(\mathscr{O}_{L},\mathscr{O}_{L}).$

Note that
$\mathscr{E}xt^{i}_{Y}(\sigma_{*}\mathscr{O}_{{\bar{Y}}},\mathscr{B}|_{B^{\prime}})$
is supported at $B^{\prime}$ for all $i$ and
$\mathscr{E}xt^{i}_{Y}(\sigma_{*}\mathscr{O}_{{\bar{Y}}},\mathscr{O}_{L})$ is
locally free over $L$, so we see get an exact sequence:

$0\to\mathscr{E}xt^{1}_{Y}(\mathscr{O}_{L},\mathscr{O}_{L})\to\mathscr{E}xt^{1}_{Y}(\sigma_{*}\mathscr{O}_{{\bar{Y}}},\mathscr{B}|_{{\hat{B}}})\to\mathscr{E}xt^{1}_{Y}(\sigma_{*}\mathscr{O}_{{\bar{Y}}},\mathscr{O}_{B^{\prime}})\to
0.$

Now, in view of (5.8) and (5.7), it suffices to compute
$\chi(\mathscr{E}xt^{1}_{Y}(\sigma_{*}\mathscr{O}_{{\bar{Y}}},\mathscr{O}_{B^{\prime}}))\geq
1$. But using (5.4) we see that:

$\mathscr{E}xt^{1}_{Y}(\sigma_{*}\mathscr{O}_{{\bar{Y}}},\mathscr{O}_{B^{\prime}})\simeq\mathscr{E}xt^{1}_{Y}(\mathscr{O}_{L}(-1),\mathscr{O}_{B^{\prime}}),$

and this is just a skyscraper sheaf over the point $p_{0}=L\cap B^{\prime}$.
The proof is now finished.

######  Remark 4.

According to [LPS11, LPS12] there are $2$ inequivalent integral non-normal
cubic surfaces which are not cones in characteristic different from $2$, and
one more in characteristic $2$; the normalization of all of them is a smooth
cubic scroll in $\mathds{P}^{4}$. Also, there are three inequivalent integral
non-normal surfaces of degree $4$ in $\mathds{P}^{4}$ whose normalization is a
quartic scroll and which are not cones, one of them appearing in
characteristic $2$. All these cases can be treated with the method we
outlined. The computations sketched here can also be carried out explicitly
with Macaulay2, [GS], cf. the webpage
http://dfaenzi.perso.math.cnrs.fr/code/cm-wild

###  5.5. Quartic del Pezzo surfaces normalized by the Veronese surface

We still have to study the case when ${\bar{Y}}$ is the Veronese surface.
Again there is a line $L\subset Y$ which is the non-normal locus of $Y$. In
this case, the two quadrics threefolds $Q_{1},Q_{2}$ whose equations generate
$I_{Y}$ can be chosen so that $Q_{1}$ is singular along a line
$M\subset\mathds{P}^{4}$. The intersection $M\cap Y$ consists a double point
$\vec{q}$ whose reduced structure $q$ lies in $L$. Also, $Q_{2}$ is a cone
over a point $p\in L$. Then, $Q_{1}$ admits a unique linear symmetric
determinantal representation by a $2\times 2$ matrix whose cokernel
$\mathscr{L}_{1}$ is a rank-1 Ulrich sheaf on $Q_{1}$ with $2$ independent
global sections, corresponding to a pencil of planes $\Lambda_{1}$ in $Q_{1}$.
On the other hand $Q_{2}$ supports two non-isomorphic rank-1 Ulrich sheaves
$\mathscr{L}_{2}$ and $\mathscr{L}_{2}^{\prime}$, one for each of the two
distinct pencils of planes $\Lambda_{2}$ and $\Lambda_{2}^{\prime}$ in
$Q_{2}$. We define:

$\mathscr{A}=\mathscr{L}_{1}\otimes\mathscr{L}_{2},\qquad\mathscr{B}=\mathscr{L}_{1}\otimes\mathscr{L}_{2}^{\prime}.$

One can check now that $\mathscr{A}$ and $\mathscr{B}$ are rank-1 Ulrich
sheaves on $Y$, having each $4$ independent global sections, as an easy
application of [ESW03, Proposition 2.1].

Moreover, $\mathscr{A}$ and $\mathscr{B}$ are not isomorphic, as one can see
by comparing the curves associated with them. Indeed, the curves associated
with $\mathscr{A}$ corresponding to decomposable tensors in
$\operatorname{H}^{0}(Y,\mathscr{A})\simeq\operatorname{H}^{0}(Q_{1},\mathscr{L}_{1})\otimes\operatorname{H}^{0}(Q_{2},\mathscr{L}_{2})$
are reducible of the form $\Lambda_{1}\cap Q_{2}\cup\Lambda_{2}\cap Q_{1}$,
and these curves cannot appear as zero locus of global sections in
$\operatorname{H}^{0}(Y,\mathscr{B})$, as $\Lambda_{2}^{\prime}$ would lie in
the same pencil as $\Lambda_{2}$. It follows that:

$\operatorname{Hom}_{Y}(\mathscr{A},\mathscr{B})=0=\operatorname{Hom}_{Y}(\mathscr{B},\mathscr{A}),$

as any morphism $\mathscr{A}\to\mathscr{B}$ would be an isomorphism, since the
source and target are torsionfree of rank $1$ with the same Hilbert
polynomial, and the variety $Y$ is integral.

It remains to compute $\operatorname{Ext}^{1}_{Y}(\mathscr{A},\mathscr{B})$.
We would like to show that this vector space has dimension $3$, which suffices
to conclude by Theorem 3. Note that $\mathscr{L}_{1}$ is locally free away
from $M$, while $\mathscr{L}_{2}$ and $\mathscr{L}_{2}^{\prime}$ are locally
free except at $p$, so that $\mathscr{A}$ is locally free away from $p$ and
$q$. The same happens to $\mathscr{B}$, in particular
$\mathscr{E}xt^{1}_{Y}(\mathscr{A},\mathscr{B})$ is supported (set-
theoretically) at $p$ and $q$ so it has finite length.

By our analysis of reducible curves appearing as zero loci of sections of
decomposable tensors in $\mathscr{A}$, we see that the only singularity of
such a curve is, when $\Lambda_{1}$ and $\Lambda_{2}$ are general enough, the
point $\Lambda_{1}\cap\Lambda_{2}$. Therefore a general global section of
$\mathscr{A}$ will vanish on a smooth irreducible curve, actually a rational
normal quartic $A$ in $Y$, and one has:

(5.14) $0\to\mathscr{O}_{Y}\to\mathscr{A}\to\omega_{A}\to 0,\qquad
0\to\mathscr{O}_{Y}\to\mathscr{B}\to\omega_{B}\to 0,$

where $\omega_{A}$ and $\omega_{B}$ are invertible on $A$ and $B$.

Any such curve $A$ or $B$ contains $\vec{q}$ and $p$. Also, two curves $A$ and
$B$ corresponding to sufficiently general global sections of $\mathscr{A}$ and
$\mathscr{B}$ intersect at the length-$3$ subscheme consisting of the fixed
points $\vec{q}$ and $p$, and moreover at $3$ more simple points
$p_{1},p_{2},p_{3}$ which move along with the choice of $A$ and $B$. This says
that $\mathscr{E}xt^{1}_{Y}(\omega_{A},\omega_{B})$ has length $6$ and is
supported at $\vec{q}$, $p$, $p_{1}$, $p_{2}$, $p_{3}$.

$\vec{q}$$A$$B$$L$$p$ Figure 3. Curves associated with Ulrich bundles for a
quartic surface normalized by the Veronese surface.

Actually this can already be seen on the reducible curves by direct
computation. So:

(5.15)
$\dim_{\boldsymbol{k}}\operatorname{Ext}^{1}_{Y}(\omega_{A},\omega_{B})=6.$

Now applying $\operatorname{Hom}_{Y}(-,\mathscr{B})$ to the first of (5.14) we
get:

$0\to\operatorname{H}^{0}(Y,\mathscr{B})\to\operatorname{Ext}^{1}_{Y}(\omega_{A},B)\to\operatorname{Ext}^{1}_{Y}(\mathscr{A},\mathscr{B})\to
0.$

Recall that $\mathscr{B}$ is Ulrich and has thus $4$ independent global
sections. On the other hand, one immediately computes
$\operatorname{Ext}^{1}_{Y}(\omega_{Y},\mathscr{O}_{Y})\simeq\boldsymbol{k}$
and $\operatorname{Ext}^{2}_{Y}(\omega_{Y},\mathscr{O}_{Y})=0$, while of
course $\operatorname{Hom}_{Y}(\omega_{A},\omega_{B})=0$. So applying
$\operatorname{Hom}_{Y}(\omega_{A},-)$ to the second sequence of (5.14) one
gets:

$0\to\boldsymbol{k}\to\operatorname{Ext}^{1}_{Y}(\omega_{A},\mathscr{B})\to\operatorname{Ext}^{1}_{Y}(\omega_{A},\omega_{B})\to
0.$

Using these sequences and (5.15) we finally obtain
$\dim_{\boldsymbol{k}}\operatorname{Ext}^{1}_{Y}(\mathscr{A},\mathscr{B})=3$.

In characteristic $2$, there is another surface $Y$ of degree $4$, not
projectively equivalent to $Y$, whose normalization is the Veronese surface,
cf. case 8 of [LPS12]. However this case can be treated in the same way as the
present one, so we omit the details.

###  5.6. Further remarks

The hypothesis that $X$ is integral and not a cone can be removed for
sectional genus at least $3$ thanks to the following remark. In order to state
it, we will use the notion of $h$-vector, cf. [Mig98] for a thorough
exposition.

Given an ACM closed subscheme $X\subset\mathds{P}^{n}$ of dimension $m\geq 1$,
the Hilbert function of a linear Artinian reduction of $X$ is eventually zero
and therefore it is encoded on a finite sequence of positive integers
$(1,h_{1},\ldots,h_{s})$. This sequence is known as the _h-vector_ of $X$. Its
length is $s+1$. The sectional genus $p$ of $X$ can be recovered by the
following formula:

$p=\sum_{i=2}^{s}(i-1)h_{i}.$

######  Lemma 6.

Let $X\subset\mathds{P}^{n}$ be an ACM projective scheme of sectional genus at
least $3$. Then $X$ is of wild representation type.

###### Proof.

Set $R:=\boldsymbol{k}[X]$, so that $R$ is a graded CM ring. We have $p\geq 3$
so either $s\geq 3$ or $h_{2}\geq 3$; assume $s\geq 3$. Let $0\neq f\in R$ be
a homogeneous element of degree $a$ and $Y=X\cap\mathds{V}(f)$. Then the
Hilbert series $H_{\boldsymbol{k}[X]}$ and $H_{\boldsymbol{k}[Y]}$ of
$\boldsymbol{k}[X]$ and $\boldsymbol{k}[Y]$ are related as follows:

$H_{\boldsymbol{k}[Y]}(t)=(1-t^{a})H_{\boldsymbol{k}[X]}(t).$

Suppose now $m=1$. Let $f_{1},f_{2}\in R$ (respectively, $g_{1},g_{2}\in R$)
be a $R$-regular sequences with $\deg f_{i}=1$ (respectively $\deg g_{i}=2$).
Consider then $V=\boldsymbol{k}[X]/(f_{1},f_{2})$ and
$W=\boldsymbol{k}[X]/(g_{1},g_{2})$. The Hilbert series of $V$ and $W$
correspond, respectively, to the standard $h$-vector of $X$ and to the weighed
one with weight $(2,2)$. Now, the Hilbert functions $H_{V}$ and $H_{W}$ of $V$
and of $W$ are related by:

$H_{W}(t)=(1+t)^{2}H_{V}(t).$

In particular, the weighted $h$-vector $(1,k_{1},\ldots,k_{t})$ of weight
$(2,2)$ of $X$ and the $h$-vector $(1,h_{1},\ldots,h_{s})$ of $X$ are related
by:

$k_{i}=h_{i}+2h_{i-1}+h_{i-2}.$

Therefore our claim follows from [DT14, Theorem 2.1], as well as the cases
$m>1$ (since the $h$-vector does not change by linear section) and $h_{2}\geq
3$. ∎

In fact, we have a more detailed information: if $g_{i}:=f_{i}^{2}$, then the
resolution of $V=\boldsymbol{k}[X]/(f_{1},f_{2})$ over
$W=\boldsymbol{k}[Y]/(g_{1},g_{2})$ reads:

$0\leftarrow V\leftarrow W\leftarrow W(-1)^{2}\leftarrow
W(-2)^{3}\leftarrow\cdots$

Then one easily gets the already mentioned relation between the Hilbert
functions of $V$ and $W$. We thank Aldo Conca for pointing out to us this
fact.

######  Acknowledgments.

The second named author would like to thank to the University of Bourgogne for
the hospitality during the stay when part of this work was done.

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