# Ulrich bundles on Del Pezzo threefolds

Ciro Ciliberto  Ciro Ciliberto, Dipartimento di Matematica, Università di Roma
Tor Vergata, Via della Ricerca Scientifica, 00173 Roma, Italy
<EMAIL_ADDRESS>,  Flaminio Flamini  Flaminio Flamini, Dipartimento
di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica,
00173 Roma, Italy<EMAIL_ADDRESS>and  Andreas Leopold Knutsen
Andreas Leopold Knutsen, Department of Mathematics, University of Bergen,
Postboks 7800, 5020 Bergen, Norway<EMAIL_ADDRESS>
###### Abstract.

We prove that for any $r\geqslant 2$ the moduli space of stable Ulrich bundles
of rank $r$ and determinant ${\mathcal{O}}_{X}(r)$ on any smooth Fano
threefold $X$ of index two is smooth of dimension $r^{2}+1$ and that the same
holds true for even $r$ when the index is four, in which case no odd–rank
Ulrich bundles exist. In particular this shows that any such threefold is
Ulrich wild. As a preliminary result, we give necessary and sufficient
conditions for the existence of Ulrich bundles on any smooth projective
threefold in terms of the existence of a curve in the threefold enjoying
special properties.

##  1\. Introduction

A vector bundle ${\mathcal{E}}$ on a smooth (complex) projective variety
$X\subset{\mathbb{P}}^{m}$ is said to be an _Ulrich bundle_ if
$h^{i}({\mathcal{E}}(-p))=0$ for all $i\geqslant 0$ and all $1\leqslant
p\leqslant\dim(X)$. The study of such bundles originates in the paper [Ul]
from 1984, where they were considered in the framework of commutative algebra
because they enjoy suitable extremal cohomological properties. The attention
of algebraic geometers was drawn by the paper [ESW], where, among other
things, the Chow form of a projective variety $X$ is computed using Ulrich
bundles on $X$, if they exist.

In recent years there has been a lot of work on Ulrich bundles, mainly
investigating the following problems: given a variety
$X\subset{\mathbb{P}}^{m}$, does there always exist an Ulrich bundle on $X$?
What are the possible ranks for Ulrich bundles? If Ulrich bundles exist, are
they stable, and what are their moduli? (Recall that Ulrich bundles are always
semistable and the notions of stability and slope–stability coincide, cf. [CH,
Thm. 2.9]).

Although a lot is known about these problems for specific classes of varieties
(e.g., curves, Segre and Grassmann varieties, rational scrolls, complete
intersections, some classes of surfaces and threefolds, etc.) the above
questions are still open in their full generality (for nice surveys on the
subject, see for instance [Be1, CH, Co, CMP]).

In this paper we consider smooth Fano threefolds of even index, also called
Del Pezzo threefolds (cf. the beginning of § 4 for the definition and
classification of such varieties). In the index–two case our main result is:

######  Theorem 1.1.

Let $X\subset{\mathbb{P}}^{m}$ be a smooth Fano threefold of index two with
$\omega_{X}\simeq{\mathcal{O}}_{X}(-2)$. For every integer $r\geqslant 2$, the
moduli space of stable Ulrich bundles of rank $r$ and determinant
${\mathcal{O}}_{X}(r)$ on $X$ is smooth of dimension $r^{2}+1$.

In the remaining case of even index we prove:

######  Theorem 1.2.

Let $X\subset{\mathbb{P}}^{9}$ be the $2$–Veronese embedding of
${\mathbb{P}}^{3}$. Then:

  * (i)

$X$ carries no Ulrich bundles of odd rank;

  * (ii)

for every even integer $r\geqslant 2$, the moduli space of stable Ulrich
bundles of rank $r$ and determinant ${\mathcal{O}}_{X}(r)$ on $X$ is smooth of
dimension $r^{2}+1$.

Our results in particular show that any such threefold $X$ is _Ulrich wild_
(recall that, as suggested by an analogous definition in [DG], a variety $X$
is said to be Ulrich wild if it possesses families of dimension $p$ of
pairwise non–isomorphic, indecomposable, Ulrich bundles for arbitrarily large
$p$). Note that there are only very few cases of varieties known to carry
stable Ulrich bundles of infinitely many ranks, and even fewer of _any_ rank,
namely curves, Del Pezzo surfaces and more recently general blow-ups of the
plane (cf. [CFK2]).

Some special cases of the theorems were known already: the rank–two cases were
proved by Beauville in [Be1, Prop. 6.1 and 6.4] and, in the case of the
$2$–Veronese, already in [ESW, Prop. 5.10–5.11], without the statement about
stability. The case of arbitrary rank on the cubic threefold in
${\mathbb{P}}^{4}$ was proved in [CH, Prop. 5.4 and Thm. 5.7] under the
additional hypothesis that the threefold is general, and for any cubic
threefold in [LMS, Thm. B]; the case of intersections of two quadrics ($d=4$)
was proved in [CKL, Thm. 1.1], and the case $d=5$ was proved in [LP, Thm.
1.1]. Our proof of Theorem 1.1 is different from the ones in these papers, and
provides a uniform treatment of all Del Pezzo threefolds, whereas our proof of
Theorem 1.2(ii) is similar to the one in [CH]. Theorem 1.1 proves the
conjecture stated in [LP, p. 276].

Also note that the cases of rank–one Ulrich bundles on Del Pezzo threefolds
are well–known (cf., e.g., [Be1, CFaM1, CFaM2, CFiM] and see the proof of
Proposition 4.4). For other works regarding Ulrich bundles, or more generally
$ACM$ bundles, of rank two on Fano threefolds, we refer to [ACo, Be2, Be4,
Be3, BF, CFaM1, CFaM2, CFiM, MT].

We do not claim the irreducibility of the moduli spaces in Theorems 1.1 and
1.2. Something is known in the rank–two case, cf. Remark 4.5, and
irreducibility for all ranks in the cases $d=4$ and $5$ follows from the
explicit description of the moduli spaces given in [CKL, Thm. 1.1] and [LP,
Thm. 1.1], whereas it was very recently proved in the case of the cubic
threefold in [FP, Thm. 1.4]. Irreducibility in the remaining cases is an
interesting open question.

Theorems 1.1 and 1.2(ii) will be proved in § 4. Before that, in Theorem 3.1,
we give necessary and sufficient conditions for the existence of a rank
$r\geqslant 2$ Ulrich bundle on any smooth projective threefold
$X\subset{\mathbb{P}}^{m}$ in terms of the existence of a quite peculiar curve
$C$ on $X$, and the peculiarity of this curve may explain why Ulrich bundles
often seem quite hard to find. Despite the complexity of the conditions that
the curve has to verify, in certain cases they simplify a bit, as in the case
of Fano threefolds: see Theorem 3.5 and Corollaries 3.7 and 3.9, the latter
proving Theorem 1.2(i). We finish the paper with some remarks and speculations
concerning interesting maps to moduli spaces of curves arising from this
correspondence between Ulrich bundles and curves (cf. § 5).

Throughout the paper we work over the field of complex numbers.

Acknowledgements: C. Ciliberto and F. Flamini are members of GNSAGA of the
Istituto Nazionale di Alta Matematica “F. Severi” and acknowledge support from
the MIUR Excellence Department Project awarded to the Department of
Mathematics, University of Rome Tor Vergata, CUP E83C18000100006. A. L.
Knutsen acknowledges support from the Trond Mohn Foundation Project “Pure
Mathematics in Norway” and the Meltzer Research Fund grant 261756 of the
Research Council of Norway. The authors thank G. Casnati for interesting
comments concerning Remark 4.5, Kyoung-Seog Lee for interesting correspondence
and the Referee for useful feedback.

##  2\.  A preliminary lemma

This short section is devoted to the proof of the following:

######  Lemma 2.1.

Let $X\subset{\mathbb{P}}^{m}$ be a smooth, irreducible, projective variety
with $\deg(X)>1$ and let ${\mathcal{E}}$ be an Ulrich bundle on $X$. Then
$h^{0}({\mathcal{E}}^{*})=0$.

###### Proof.

Assume by contradiction that $h^{0}({\mathcal{E}}^{*})>0$ and pick a non-zero
section $s^{*}\in H^{0}({\mathcal{E}}^{*})$. Since
$h^{0}({\mathcal{E}})=\deg(X)\operatorname{rk}({\mathcal{E}})\geqslant\deg(X)>1$
by [Be1, (3.1)], we have ${\mathcal{E}}\not\simeq{\mathcal{O}}_{X}$. It
follows that $\operatorname{coker}(s^{*})\neq 0$, and by dualizing we obtain

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{F}}:=\operatorname{ker}(s)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{X},}$

where ${\mathcal{F}}\neq 0$. In particular,
$\operatorname{rk}({\mathcal{F}})=\operatorname{rk}({\mathcal{E}})-1>0$ and
$\operatorname{im}(s)$ is an ideal sheaf, whence

$c_{1}({\mathcal{F}})=c_{1}({\mathcal{E}})-c_{1}(\operatorname{im}(s))=c_{1}({\mathcal{E}})+D,$

for an effective divisor $D$ on $X$. Thus, denoting as usual by $\mu$ the
_slope_ of a sheaf, we have

$\mu({\mathcal{F}})=\frac{c_{1}({\mathcal{F}})\cdot{\mathcal{O}}_{X}(1)^{\dim(X)-1}}{\operatorname{rk}({\mathcal{F}})}>\frac{c_{1}({\mathcal{E}})\cdot{\mathcal{O}}_{X}(1)^{\dim(X)-1}}{\operatorname{rk}({\mathcal{E}})}=\mu({\mathcal{E}}),$

contradicting that ${\mathcal{E}}$, being Ulrich, is slope–semistable ([CH,
Thm. 2.9(a)]). ∎

######  Remark 2.2.

The assumption $\deg(X)>1$ in Lemma 2.1 is essential, as on ${\mathbb{P}}^{n}$
trivial bundles are Ulrich, and in fact the only Ulrich bundles (cf., e.g.,
[Be1, Thm. 2.3]).

##  3\. Curves associated to Ulrich bundles on threefolds

In this section we will prove that the existence of Ulrich bundles on a smooth
projective threefold is connected to the existence of smooth curves on the
threefold with particular properties. Before giving the result, we fix some
notation. Let $X$ be a smooth projective threefold, $C\subset X$ a curve and
$D$ a divisor on $X$. The short exact sequence

(1)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{J}}_{C/X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

tensored by ${\mathcal{O}}_{X}(D+K_{X})$ determines a coboundary map

$d:H^{1}(C,{\mathcal{O}}_{C}(D+K_{X}))\longrightarrow
H^{2}(X,{\mathcal{J}}_{C/X}(D+K_{X})),$

whose dual, by Serre duality, is

(2)
$d^{*}:\operatorname{Ext}_{X}^{1}({\mathcal{J}}_{C/X}(D),{\mathcal{O}}_{X})\longrightarrow
H^{0}(C,\omega_{C}(-D-K_{X})).$

Moreover, for any subspace

$W\subseteq\operatorname{Ext}_{X}^{1}({\mathcal{J}}_{C/X}(D),{\mathcal{O}}_{X})\simeq\left(H^{2}(X,{\mathcal{J}}_{C/X}(D+K_{X}))\right)^{*}$

we obtain a surjection

(3) $\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
48.55186pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{H^{2}(X,{\mathcal{J}}_{C/X}(D+K_{X}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
72.62965pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern-3.0pt\lower
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
72.62965pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{W^{*}}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

Since, by Serre duality,

$\displaystyle\operatorname{Ext}_{X}^{1}({\mathcal{J}}_{C/X}(D),W^{*}\otimes{\mathcal{O}}_{X})\;\;\;$
$\displaystyle\simeq$
$\displaystyle\operatorname{Ext}_{X}^{1}({\mathcal{J}}_{C/X}(D+K_{X}),\omega_{X})\otimes
W^{*}\simeq$ $\displaystyle\;\;\;\;\;\;\;\simeq
H^{2}({\mathcal{J}}_{C/X}(D+K_{X}))^{*}\otimes W^{*}$ $\displaystyle\simeq$
$\displaystyle\operatorname{Hom}_{X}(H^{2}({\mathcal{J}}_{C/X}(D+K_{X}),W^{*}),$

we obtain an extension

(4)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W^{*}\otimes{\mathcal{O}}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{J}}_{C/X}(D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

corresponding to the cocycle (3), that is, so that the coboundary map

$\textstyle{H^{2}({\mathcal{J}}_{C/X}(D+K_{X}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{3}(W^{*}\otimes\omega_{X})\simeq
W^{*}\otimes H^{3}(\omega_{X})\simeq W^{*}}$

of (4) tensored by $\omega_{X}$ is the map (3). For any
${\mathcal{L}}\in\operatorname{Pic}(X)$, we may twist (4) by ${\mathcal{L}}$
and obtain a coboundary map

(5) $\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
43.42218pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{H^{2}(X,{\mathcal{L}}(D)\otimes{\mathcal{J}}_{C/X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
67.49997pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
67.49997pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{W^{*}\otimes
H^{3}(X,{\mathcal{L}})}$}}}}}}}\ignorespaces}}}}\ignorespaces,$

which we will call the coboundary map induced by $(W,{\mathcal{L}})$.

######  Theorem 3.1.

Let $X\subset{\mathbb{P}}^{m}$ be a smooth projective threefold with
$\deg(X)>1$ and $D$ be a divisor on $X$. Let $r\geqslant 2$ be an integer. Set
$H:={\mathcal{O}}_{X}(1)$.

Then there is a bijection between the set of Ulrich bundles ${\mathcal{E}}$ on
$X$ satisfying

(6)
$\operatorname{rk}({\mathcal{E}})=r\;\;\mbox{and}\;\;\operatorname{det}({\mathcal{E}})={\mathcal{O}}_{X}(D)$

and the set of pairs $(C,W)$, where $C\subset X$ is a smooth curve and
$W\subseteq\operatorname{Ext}_{X}^{1}({\mathcal{J}}_{C/X}(D),{\mathcal{O}}_{X})$
is an $(r-1)$–dimensional subspace, such that

(7) $\displaystyle d^{*}(W)\;\;\mbox{generates $\omega_{C}(-K_{X}-D)$ \;\;(cf.
\eqref{eq:dstar}),}$ (8) $\displaystyle H^{2}\cdot
D=\frac{r}{2}\left(H^{2}\cdot K_{X}+4H^{3}\right),$ (9) $\displaystyle H\cdot
C=\frac{r}{12}\left(K_{X}^{2}\cdot H+c_{2}(X)\cdot
H-22H^{3}\right)-\frac{1}{2}\left(H\cdot D\cdot K_{X}-H\cdot D^{2}\right),$
(10) $\displaystyle
g(C)=rH^{3}-r\chi(X,{\mathcal{O}}_{X})-\frac{1}{6}D^{3}+\frac{1}{4}K_{X}\cdot
D^{2}$ $\displaystyle\hskip 56.9055pt-\frac{1}{12}\left(K_{X}^{2}\cdot
D+c_{2}(X)\cdot D\right)+C\cdot D+1,$ (11) $\displaystyle
h^{0}(K_{X}+3H-D)=0,$ (12) $\displaystyle h^{0}({\mathcal{J}}_{C/X}(D-H))=0,$
(13) $\displaystyle
h^{1}({\mathcal{J}}_{C/X}(D-pH))=0\;\;\mbox{for}\;\;p\in\\{1,2,3\\},$ (14)
$\displaystyle\mbox{the coboundary
map}\;\delta:H^{2}({\mathcal{J}}_{C/X}(D-3H))\longrightarrow W^{*}\otimes
H^{3}({\mathcal{O}}_{X}(-3H))$ $\displaystyle\mbox{induced by
$(W,{\mathcal{O}}_{X}(-3H))$ is either injective or surjective}.$

Via the above bijection, ${\mathcal{E}}$ sits in an exact sequence

(15)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W^{*}\otimes{\mathcal{O}}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{J}}_{C/X}(D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

######  Remark 3.2.

As $C$ is the $(r-2)$–degeneracy locus of $u_{r-1}$, it follows that its class
of cohomology is $c_{2}(E)$, see for instance [Ot, (2.11)].

We point out that the curve $C$ could be the 0–curve, i.e., $C$ could be
empty. In this case $H\cdot C=0$, $g(C)=1$,
$\mathcal{I}_{C/X}=\mathcal{O}_{X}$ and (7) is vacuous. This case never
happens if $D$ is big and nef, in particular if ${\rm
Pic}(X)\simeq\mathbb{Z}$, since in this case
$\operatorname{Ext}_{X}^{1}({\mathcal{O}}_{X}(D),{\mathcal{O}}_{X})=0$.

For examples where $C$ is the 0–curve, see [LM].

###### Proof.

Let ${\mathcal{E}}$ be an Ulrich bundle of rank $r$ on $X$ with
$\operatorname{det}({\mathcal{E}})={\mathcal{O}}_{X}(D)$. Then ${\mathcal{E}}$
is globally generated by, e.g., [Be1, Thm. 2.3(i)]. We can pick a general
$(r-1)$–dimensional subspace $V_{r-1}\subset H^{0}({\mathcal{E}})$ and let
$V_{r-1}\otimes{\mathcal{O}}_{X}\stackrel{{\scriptstyle
u_{r-1}}}{{\longrightarrow}}{\mathcal{E}}$ be the evaluation morphism. It is
well–known, see, e.g., [Ot, Thm. 2.8] or [Ban, Thm. 1], that
$\operatorname{coker}(u_{r-1})\simeq{\mathcal{J}}_{C/X}(D)$ where $C$, the
degeneracy locus of $u_{r-1}$, is a smooth curve, since $\dim(X)=3$. Setting
$W:=V_{r-1}^{*}$, we thus have an exact sequence like (15). We have
$h^{3}({\mathcal{E}}(K_{X}))=h^{0}({\mathcal{E}}^{*})=0$ by Serre duality and
Lemma 2.1. Therefore (15), tensored by ${\mathcal{O}}_{X}(K_{X})$, yields a
surjection $H^{2}({\mathcal{J}}_{C/X}(D+K_{X}))\to\\!\\!\\!\to W^{*}$, whence,
by duality

$W\subset\left(H^{2}({\mathcal{J}}_{C/X}(D+K_{X}))\right)^{*}\simeq\operatorname{Ext}_{X}^{1}({\mathcal{J}}_{C/X}(D),{\mathcal{O}}_{X}).$

The required correspondence associates to ${\mathcal{E}}$ the pair $(C,W)$.

Note that, dualizing (15), we find

(16)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{X}(-D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{W\otimes{\mathcal{O}}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\omega_{C}(-K_{X}-D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

hence we have the surjection
$W\otimes{\mathcal{O}}_{X}\longrightarrow\omega_{C}(-K_{X}-D)$ that proves
(7).

Conversely, assume we have a smooth curve $C\subset X$ and an
$(r-1)$–dimensional subspace
$W\subseteq\operatorname{Ext}_{X}^{1}({\mathcal{J}}_{C/X}(D),{\mathcal{O}}_{X})$
such that (7) is satisfied. As explained in the beginning of the section, this
defines a sheaf ${\mathcal{E}}$ on $X$ fitting in an exact sequence like (15).
Tensoring (1) by ${\mathcal{O}}_{X}(D)$ gives

$\operatorname{\mathfrak{e}\mathfrak{x}\mathfrak{t}}_{{\mathcal{O}}_{X}}^{i}({\mathcal{J}}_{C/X}(D),{\mathcal{O}}_{X})=0\;\;\;\mbox{for}\;\;i\geqslant
2$

and

$\operatorname{\mathfrak{e}\mathfrak{x}\mathfrak{t}}_{{\mathcal{O}}_{X}}^{1}({\mathcal{J}}_{C/X}(D),{\mathcal{O}}_{X})\simeq\operatorname{\mathfrak{e}\mathfrak{x}\mathfrak{t}}_{{\mathcal{O}}_{X}}^{2}({\mathcal{O}}_{C}(D),{\mathcal{O}}_{X})\simeq\omega_{C}(-K_{X}-D).$

From (15) we therefore find

$\operatorname{\mathfrak{e}\mathfrak{x}\mathfrak{t}}_{{\mathcal{O}}_{X}}^{i}({\mathcal{E}},{\mathcal{O}}_{X})=0,\;\mbox{for}\;\;i\geqslant
2,$

and

$\operatorname{\mathfrak{e}\mathfrak{x}\mathfrak{t}}_{{\mathcal{O}}_{X}}^{1}({\mathcal{E}},{\mathcal{O}}_{X})=\operatorname{coker}\Big{[}W\otimes{\mathcal{O}}_{X}\longrightarrow\omega_{C}(-K_{X}-D)\Big{]}=0,$

by (7). Therefore, ${\mathcal{E}}$ is locally free on $X$.

Thus, to finish the proof of the theorem, we have left to show that if
${\mathcal{E}}$ is a vector bundle verifying (6) and fitting into an exact
sequence like (15), with $C\subset X$ a smooth curve, $D$ a divisor and $W$ an
$(r-1)$–dimensional subspace of
$\operatorname{Ext}_{X}^{1}({\mathcal{J}}_{C/X}(D),{\mathcal{O}}_{X})$
satisfying (7), then ${\mathcal{E}}$ is Ulrich if and only if (8)–(14) hold.

Tensoring (15) by ${\mathcal{O}}_{X}(-pH)$, one sees that ${\mathcal{E}}$ is
Ulrich if and only if

(17)
$h^{i}({\mathcal{J}}_{C/X}(D-pH))=0,\;\;\mbox{for}\;i\in\\{0,1,3\\},\;p\in\\{1,2,3\\}$

and

(18) $\displaystyle\mbox{the coboundary
maps}\;\delta_{p}:H^{2}({\mathcal{J}}_{C/X}(D-pH))\longrightarrow W^{*}\otimes
H^{3}({\mathcal{O}}_{X}(-pH))$ $\displaystyle\mbox{are isomorphisms
for}\;\;p\in\\{1,2,3\\}.$

Condition (17) with $i=3$ is equivalent to $h^{3}({\mathcal{O}}_{X}(D-pH))=0$
for $p\in\\{1,2,3\\}$, which is equivalent to condition (11) by Serre duality.
Condition (17) with $i=0$ is equivalent to condition (12), whereas (17) with
$i=1$ is condition (13).

Consider now (18). Conditions (11)–(13), which as we just saw are equivalent
to (17), yield that the domain of $\delta_{p}$ has dimension
$\chi({\mathcal{J}}_{C/X}(D-pH))$. On the other hand, the target of
$\delta_{p}$ has dimension
$(r-1)h^{3}({\mathcal{O}}_{X}(-pH))=(1-r)\chi({\mathcal{O}}_{X}(-pH))$ by
Kodaira vanishing. Thus, (18) is equivalent to the conditions

(19)
$\displaystyle\chi({\mathcal{J}}_{C/X}(D-pH))=(1-r)\chi({\mathcal{O}}_{X}(-pH))\;\;\mbox{for}\;\;p\in\\{1,2,3\\},$
and (20) $\displaystyle\delta_{p}\;\;\mbox{is either injective or
surjective}\;\;\mbox{for}\;\;p\in\\{1,2,3\\}.$

Suppose for a moment that condition (19) is verified. For each $1\leqslant
p\leqslant 3$ we have a commutative diagram

$\textstyle{H^{2}({\mathcal{J}}_{C/X}(D-3H))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{3}}$$\textstyle{V_{r}\otimes
H^{3}({\mathcal{O}}_{X}(-3H))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H^{2}({\mathcal{J}}_{C/X}(D-pH))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta_{p}}$$\textstyle{V_{r}\otimes
H^{3}({\mathcal{O}}_{X}(-pH)),}$

which shows that $\delta_{1},\delta_{2},\delta_{3}$ are surjective (or,
equivalently, injective) if and only if $\delta_{3}$ is surjective (or,
equivalently, injective). Thus, condition (20) is equivalent to (14).

Finally, consider condition (19). It can be rewritten as

(21)
$0=-\chi({\mathcal{O}}_{X}(D-pH))+\chi({\mathcal{O}}_{C}(D-pH))+(1-r)\chi({\mathcal{O}}_{X}(-pH)),\;\;{\rm
for}\;\;p\in\\{1,2,3\\}.$

Using Riemann–Roch on $X$ and $C$, the right hand side can be rewritten as

$\displaystyle\frac{r\,H^{3}}{6}\,p^{3}+\left(\frac{r\,K_{X}\cdot
H^{2}}{4}-\frac{D\cdot H^{2}}{2}\right)\,p^{2}+$
$\displaystyle+\left(\frac{r}{12}(K_{X}^{2}+c_{2}(X))\cdot H-\frac{H\cdot
D\cdot K_{X}}{2}+\frac{H\cdot D^{2}}{2}-C\cdot H\right)\,p+$
$\displaystyle\frac{1}{2}K_{X}\cdot D^{2}+C\cdot
D+1-g(C)-r\chi({\mathcal{O}}_{X})-\frac{1}{6}D^{3}-\frac{K_{X}\cdot
D^{2}}{4}-\frac{1}{12}(K_{X}^{2}+c_{2}(X))\cdot D.$

Dividing by the leading coefficient $\frac{r\,H^{3}}{6}$ gives a monic, cubic
polynomial in $p$, which, by condition (19), must coincide with
$(p-1)(p-2)(p-3)=p^{3}-6p^{2}+11p-6$.

Equating the coefficients of the terms of degrees $2$, $1$ and $0$ gives,
respectively, (8), (9) and (10). ∎

Despite the complexity of conditions (7)–(14) in Theorem 3.1, in certain cases
some of them may be replaced by slightly easier ones, as shown in the
following:

######  Lemma 3.3.

In the same setting as in Theorem 3.1, we have:

  * (i)

If $h^{i}({\mathcal{O}}_{X}(-D))=0$ for $i\in\\{1,2\\}$, then $W$ can be
identified with a subspace of $H^{0}(\omega_{C}(-K_{X}-D))$, and condition (7)
says that $W$ generates $\omega_{C}(-K_{X}-D)$.

  * (ii)

If $h^{i}({\mathcal{O}}_{X}(D-3H))=0$ for $i\in\\{1,2\\}$, then condition (14)
is equivalent to

(22) $\displaystyle\mbox{the multiplication map}\;\nu:W\otimes
H^{0}({\mathcal{O}}_{X}(K_{X}+3H))\longrightarrow H^{0}(\omega_{C}(3H-D))$
$\displaystyle\mbox{is either injective or surjective}.$

###### Proof.

(i) The assumptions yield that $d^{*}$ is an isomorphism, proving the
assertion.

(ii) The assumptions yield that the coboundary map of (1) tensored by
${\mathcal{O}}_{X}(D-3H)$,

$\gamma:H^{1}({\mathcal{O}}_{C}(D-3H)\longrightarrow
H^{2}({\mathcal{J}}_{C/X}(D-3H)),$

is an isomorphism. The composed map
$\delta\circ\gamma:H^{1}({\mathcal{O}}_{C}(D-3H)\to V_{r-1}\otimes
H^{3}({\mathcal{O}}_{X}(-3H))$ is the dual of the multiplication map $\nu$, by
Serre duality. Hence we see that condition (14) is equivalent to (22). ∎

######  Remark 3.4.

Let ${\mathcal{E}}$ be an Ulrich bundle of rank $r$ sitting in an exact
sequence like (15), and set $V_{r-1}:=W^{*}$. Since ${\mathcal{E}}$ is
generated by global sections ([Be1, Thm. 2.3(i)]) and is nontrivial (Lemma
2.1), the divisor $D$ in Theorem 3.1 must be effective and nonzero (by
Porteous). Pick a general $r$-dimensional subspace $V_{r}\subseteq
H^{0}({\mathcal{E}})$ containing $V_{r-1}$ and consider the evaluation
morphism $V_{r}\otimes{\mathcal{O}}_{X}\stackrel{{\scriptstyle
u_{r}}}{{\longrightarrow}}{\mathcal{E}}$. It is well–known (see again, e.g.,
[Ot, Thm. 2.8] or [Ban, Thm. 1]) that
${\mathcal{L}}:=\operatorname{coker}(u_{r})$ is supported on a member $S$ of
$|\operatorname{det}\,({\mathcal{E}})|=|D|$ (the degeneracy locus of $u_{r}$),
which is smooth as $\dim(X)=3$, and that ${\mathcal{L}}$ is locally free of
rank one on $S$. More precisely, by the Snake Lemma, we have a commutative
diagram

(23)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V_{r-1}\otimes{\mathcal{O}}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hskip
17.07182ptu_{r-1}}$$\textstyle{{\mathcal{E}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{J}}_{C/X}(D)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V_{r}\otimes{\mathcal{O}}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hskip
5.69046ptu_{r}}$$\textstyle{{\mathcal{E}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{L}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{{\mathcal{O}}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$

from which we see that $C\subset S$ (corresponding to the non–zero section of
${\mathcal{J}}_{C/X}(D)$ appearing in the rightmost column) and
${\mathcal{L}}\simeq{\mathcal{O}}_{S}(D-C)$. Dualizing the lower horisontal
sequence in (23) we find

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V_{r}^{*}\otimes{\mathcal{O}}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{\mathfrak{e}\mathfrak{x}\mathfrak{t}}_{{\mathcal{O}}_{X}}^{1}({\mathcal{L}},{\mathcal{O}}_{X})\simeq{\mathcal{O}}_{S}(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

which shows that ${\mathcal{O}}_{S}(C)$ is globally generated by $r$ sections.

In the case of smooth embedded Fano threefolds, we get:

######  Theorem 3.5.

Let $X\subset{\mathbb{P}}^{m}$ be a smooth (Fano) threefold of degree $d>1$
with $\omega_{X}\simeq{\mathcal{O}}_{X}(-\alpha)$, for $\alpha\in\\{1,2,3\\}$.
Then $X$ carries an Ulrich bundle ${\mathcal{E}}$ satisfying

$\operatorname{rk}({\mathcal{E}})=r\geqslant
2\;\;\mbox{and}\;\;\operatorname{det}({\mathcal{E}})={\mathcal{O}}_{X}(m),\;\;\mbox{with}\;\;m\in{\mathbb{Z}}^{+},$

if and only if

(24) $\displaystyle m=\frac{r}{2}(4-\alpha)$

and there exists a smooth curve $C\subset X$ such that

(25)
$\displaystyle\deg(C)=\frac{rd}{12}(\alpha^{2}-22)+\frac{2r}{\alpha}+\frac{md}{2}(\alpha+m)$
(26) $\displaystyle g(C)=r(d-1)+\frac{1}{3}m^{3}d+\frac{1}{4}\alpha
m^{2}d+\frac{1}{12}\left[r(\alpha^{2}-22)-\alpha^{2}\right]md+\frac{2(r-1)}{\alpha}m+1,$
(27) $\displaystyle\omega_{C}(\alpha-m)\;\;\mbox{is generated by an
$(r-1)$-dimensional space}\;\;W,$ (28) the multiplication map
$\displaystyle\nu:W\otimes H^{0}({\mathcal{O}}_{X}(3-\alpha))\longrightarrow
H^{0}(\omega_{C}(3-m))$ $\displaystyle\mbox{ is either injective or
surjective},$ (29) $\displaystyle\mbox{the restriction
map}\;\;H^{0}(X,{\mathcal{O}}_{X}(m-p))\longrightarrow
H^{0}(C,{\mathcal{O}}_{C}(m-p))$ $\displaystyle\mbox{is an isomorphism
for}\;\;p\in\\{1,2,3\\}.$

###### Proof.

We apply Theorem 3.1. Using Lemma 3.3(i), we see that condition (27) is (7).
Conditions (8), (9) and (10) in Theorem 3.1 are, respectively, (24), (25) and
(26). (To compute $c_{2}(X)\cdot H$, we use
$1=\chi(X,\mathcal{O}_{X})=\frac{\alpha}{24}c_{2}\cdot H$ by, e.g., [Fu, Exc.
15.2.5].)

Now consider condition (11) in Theorem 3.1. It reads
$h^{0}({\mathcal{O}}_{X}(3-\alpha-m))=0$, which by (24) is equivalent to
$3-\alpha-\frac{r}{2}\left(4-\alpha\right)<0$, which is clearly satisfied.

Next consider conditions (12)–(13) in Theorem 3.1. Together they are
equivalent to

(30)
$h^{i}(X,{\mathcal{J}}_{C/X}(m-p))=0\;\;\mbox{for}\;\;i\in\\{0,1\\},\;p\in\\{1,2,3\\}.$

From the short exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{J}}_{C/X}(m-p)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{X}(m-p)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{C}(m-p)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

and the fact that $h^{1}({\mathcal{O}}_{X}(m-p))=0$, we see that (30) is
equivalent to (29).

Finally, by Lemma 3.3(ii), condition (14) in Theorem 3.1 can be restated as
(28). ∎

######  Remark 3.6.

Suppose ${\mathcal{E}}$ is an Ulrich bundle on a smooth Fano threefold $X$ as
in Theorem 3.5. Then
$h^{i}({\mathcal{E}}^{*})=h^{3-i}({\mathcal{E}}(K_{X}))=h^{3-i}({\mathcal{E}}(-\alpha))=0$
for all $0\leqslant i\leqslant 3$, so that the cohomology of (16) yields
$W=H^{0}(\omega_{C}(\alpha-m))$.

We will make use of Theorem 3.5 in the proof of our main results in the next
section. In the rest of this section we give some consequences and examples.

######  Corollary 3.7.

Let $X\subset{\mathbb{P}}^{g+1}$ be a smooth Fano threefold of degree $2g-2$
such that $\omega_{X}\simeq{\mathcal{O}}_{X}(-1)$ and
$\operatorname{Pic}(X)\simeq{\mathbb{Z}}[{\mathcal{O}}_{X}(1)]$. Then $X$
carries no Ulrich bundles of odd rank.

###### Proof.

It is well–known that varieties of degree $d>1$ with Picard group generated by
the hyperplane class cannot carry Ulrich line bundles (see, e.g., [Be1, § 4]).
As for rank $r\geqslant 2$, Theorem 3.5 (with $\alpha=1$) yields that
$m=\frac{3r}{2}$ must be an integer, cf. (24). ∎

Recall that Fano threefolds satisfying the hypotheses of the corollary (called
prime Fano threefolds of index one) are classified and that
$g\in\\{3,\ldots,10,12\\}$ (see [IP]). In [CFK1] we prove that all such Fano
threefolds carry Ulrich bundles of all even ranks.

######  Example 3.8.

Let $X\subset{\mathbb{P}}^{g+1}$ be a smooth Fano threefold of degree $2g-2$
such that $\omega_{X}\simeq{\mathcal{O}}_{X}(-1)$. Theorem 3.5 yields that $X$
carries a rank–two Ulrich bundle ${\mathcal{E}}$ such that
$\operatorname{det}({\mathcal{E}})={\mathcal{O}}_{X}(m)$ if and only if $m=3$
and there exists a smooth curve $C\subset X$ of degree $5g-1$ such that
$\omega_{C}\simeq{\mathcal{O}}_{C}(2)$ (in particular $g(C)=5g$) and
$C\subset{\mathbb{P}}^{g+1}$ is non–degenerate and linearly and quadratically
normal.

Since in [CFK1] we prove the existence of such rank–two Ulrich bundles in the
cases where $\operatorname{Pic}(X)\simeq{\mathbb{Z}}[{\mathcal{O}}_{X}(1)]$,
the existence of such a curve will follow as a consequence.

The following proves Theorem 1.2(i):

######  Corollary 3.9.

Let $X\subset{\mathbb{P}}^{9}$ be the $2$–Veronese embedding of
${\mathbb{P}}^{3}$. Then $X$ carries no Ulrich bundles of odd rank.

###### Proof.

We have $\omega_{X}\simeq{\mathcal{O}}_{X}(-2)$. If there exists an Ulrich
bundle of rank $r$ on $X$, then Theorem 3.5 (with $\alpha=2$) yields the
existence of a curve of degree $r(4r-3)$ (since $m=r$ by (24)). As
${\mathcal{O}}_{X}(1)$ is $2$–divisible, the degree must be even, whence $r$
must be even. ∎

######  Example 3.10.

Let $X\subset{\mathbb{P}}^{d+1}$ be a smooth Fano threefold of degree $d$ such
that $\omega_{X}\simeq{\mathcal{O}}_{X}(-2)$.

(i) Theorem 3.5 yields that $X$ carries a rank-two Ulrich bundle
${\mathcal{E}}$ such that
$\operatorname{det}({\mathcal{E}})\simeq{\mathcal{O}}_{X}(m)$ if and only if
$m=2$ and there exists a smooth elliptic normal curve $C\subset X$ of degree
$d+2$. The curve $C$ is contained in a smooth surface
$S\in|{\mathcal{O}}_{X}(2)|$ (cf. Remark 3.4), which is a $K3$ surface.

The existence of rank–two Ulrich bundles on $X$ was proved in [Be1, Prop. 6.1]
by proving the existence of such an elliptic curve (cf. Proposition 4.3
below). (The existence in the case of cubic threefolds in ${\mathbb{P}}^{4}$
had previously been proved in [CH, Prop. 5.1].)

(ii) Theorem 3.5 yields that $X$ carries a rank–three Ulrich bundle
${\mathcal{E}}$ such that
$\operatorname{det}({\mathcal{E}})\simeq{\mathcal{O}}_{X}(m)$ if and only if
$m=3$ and there exists a smooth curve $C\subset X$ of degree $3d+3$ and genus
$2d+4$ such that $\omega_{C}(-1)$ is globally generated (by two sections),
$C\subset{\mathbb{P}}^{d+1}$ is linearly and quadratically normal and the
Petri map

(31) $\mu_{0,{\mathcal{O}}_{X}(1)}:H^{0}({\mathcal{O}}_{C}(1))\otimes
H^{0}(\omega_{C}(-1))\longrightarrow H^{0}(\omega_{C})$

is injective. (Indeed, by (29) and Remark 3.6 the Petri map equals the map
$\nu$ in (28).) The curve $C$ is contained in a smooth surface
$S\in|{\mathcal{O}}_{X}(3)|$ (cf. Remark 3.4), which is a canonical surface,
i.e., $\omega_{S}\simeq{\mathcal{O}}_{S}(1)$.

In the case where $X$ is a general smooth cubic threefold in
${\mathbb{P}}^{4}$, i.e., $d=3$, the existence of rank–three Ulrich bundles on
$X$ was proved in [CH, Prop. 5.4] by constructing a curve $C$ satisfying the
above conditions using Macaulay2 (cf. [CH, App. A.2]). By contrast, we can
deduce the existence of such a curve $C$ as a consequence of Theorem 1.1.

##  4\. Ulrich bundles on Del Pezzo threefolds

This section is devoted to the proofs of Theorems 1.1 and 1.2(ii), recalling
that part (i) of the latter has been proved in Corollary 3.9.

We recall that a smooth Fano threefold $X$ has even index if there exists an
ample divisor $H$ on $X$ such that $K_{X}=-2H$. If $H$ is very ample, it is
well–known that it embeds $X$ as a subvariety of degree $d=H^{3}$ in
${\mathbb{P}}^{d+1}$. The general curve section of $X$ is an elliptic curve
and the general hyperplane section is a smooth Del Pezzo surface. For this
reason, such threefolds are also called Del Pezzo threefolds. The index of $X$
is the order of divisibility of $K_{X}$ in $\operatorname{Pic}(X)$. By the
classification of such varieties (see, e.g., [IP]), there is one case of index
four, namely the $2$–Veronese embedding of ${\mathbb{P}}^{3}$ in
${\mathbb{P}}^{9}$ (where $d=8$), and the following cases of index two:

  * $d=3$:

$X$ is a smooth cubic hypersurface in $\mathbb{P}^{4}$, with
$\operatorname{Pic}(X)=\mathbb{Z}[H]$;

  * $d=4$:

$X$ is a smooth complete intersection of type $(2,2)$ in $\mathbb{P}^{5}$,
with $\operatorname{Pic}(X)=\mathbb{Z}[H]$;

  * $d=5$:

$X$ is a smooth section with a linear space $\Lambda\simeq\mathbb{P}^{6}$ of
the Grassmannian $\mathbb{G}(1,4)$ sitting in $\mathbb{P}^{9}$ via its Plücker
embedding; one has ${\rm Pic}(X)=\mathbb{Z}[H]$.

  * $d=6$:

two cases occur: either $X$ is the Segre embedding of
${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$ in
${\mathbb{P}}^{7}$, in which case $\operatorname{Pic}(X)\simeq\mathbb{Z}^{3}$,
or $X$ is a smooth hyperplane section of the Segre embedding of
$\mathbb{P}^{2}\times\mathbb{P}^{2}$ in $\mathbb{P}^{8}$, and
$\operatorname{Pic}(X)\simeq\mathbb{Z}^{2}$;

  * $d=7$:

$X$ is isomorphic to the blow–up of $\mathbb{P}^{3}$ at a point $p$, embedded
in $\mathbb{P}^{8}$ via the proper transform on $X$ of the linear system of
quadrics in $\mathbb{P}^{3}$ passing through the point $p$. In this case one
has $\operatorname{Pic}(X)\simeq\mathbb{Z}^{2}$.

We will need the following lemma.

######  Lemma 4.1.

Let $X\subset{\mathbb{P}}^{d+1}$ be a smooth Del Pezzo threefold of degree
$d$.

  * (i)

If ${\mathcal{U}}$ is an Ulrich bundle on $X$ and ${\mathcal{E}}$ is any
vector bundle on $X$ satisfying
$h^{0}({\mathcal{E}}(-1))=h^{1}({\mathcal{E}}(-2))=0$, then
$h^{2}({\mathcal{U}}\otimes{\mathcal{E}}^{*})=h^{3}({\mathcal{U}}\otimes{\mathcal{E}}^{*})=0$.
In particular, for any Ulrich bundle ${\mathcal{U}}$ on $X$ one has
$h^{2}({\mathcal{U}}\otimes{\mathcal{U}}^{*})=h^{3}({\mathcal{U}}\otimes{\mathcal{U}}^{*})=0$.

  * (ii)

If ${\mathcal{E}}_{i}$, $i\in\\{1,2\\}$, are vector bundles on $X$ satisfying

(32)
$\operatorname{rk}({\mathcal{E}}_{i})=r_{i},\;\;c_{1}({\mathcal{E}}_{i})=[{\mathcal{O}}_{X}(r_{i})]\;\;\mbox{and}\;\;c_{2}({\mathcal{E}}_{i})\cdot{\mathcal{O}}_{X}(1)=\frac{r_{i}}{2}\left(r_{i}d-d+2\right),$

then $\chi({\mathcal{E}}_{1}\otimes{\mathcal{E}}_{2}^{*})=-r_{1}r_{2}$.

###### Proof.

(i) Since ${\mathcal{U}}$ is Ulrich, there exists a resolution of the form

$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{{\mathbb{P}}^{d+1}}(-2)^{\oplus
a_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{{\mathbb{P}}^{d+1}}(-1)^{\oplus
a_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{{\mathbb{P}}^{d+1}}^{\oplus
a_{0}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{U}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

where $a_{i}\in{\mathbb{Z}}^{+}$ (cf. e.g. [Be1, Thm. 1]). Tensoring by
${\mathcal{E}}^{*}$, we obtain

$\textstyle{\cdots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}^{*}(-2)^{\oplus
a_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}^{*}(-1)^{\oplus
a_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{{\mathcal{E}}^{*}}^{\oplus
a_{0}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{U}}\otimes{\mathcal{E}}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

This implies, by Serre duality and the given assumptions, that

$\displaystyle h^{3}({\mathcal{U}}\otimes{\mathcal{E}}^{*})\leqslant
h^{3}({{\mathcal{E}}^{*}})^{\oplus
a_{0}}=a_{0}h^{3}({\mathcal{E}}^{*})=a_{0}h^{0}({\mathcal{E}}(-2))\leqslant
a_{0}h^{0}({\mathcal{E}}(-1))=0,$ $\displaystyle
h^{2}({\mathcal{U}}\otimes{\mathcal{E}}^{*})\leqslant
a_{0}h^{2}({\mathcal{E}}^{*})+a_{1}h^{3}({\mathcal{E}}^{*}(-1))=a_{0}h^{1}({\mathcal{E}}(-2))+a_{1}h^{0}({\mathcal{E}}(-1))=0.$

(ii) We borrow an argument from [CH, Proof of Prop. 5.6]. Set
${\mathcal{E}}^{\prime}_{i}:={\mathcal{E}}_{i}^{*}(2)$. Then one may check
that properties (32) hold with ${\mathcal{E}}_{i}$ replaced by
${\mathcal{E}}_{i}^{\prime}$, whence by Riemann–Roch

$\chi({\mathcal{E}}_{1}\otimes{\mathcal{E}}_{2}^{*}(-1))=\chi({\mathcal{E}}^{\prime}_{1}\otimes{{\mathcal{E}}^{\prime}_{2}}^{*}(-1)).$

At the same time, Serre duality gives

$\chi({\mathcal{E}}_{1}\otimes{\mathcal{E}}_{2}^{*}(-1))=-\chi({\mathcal{E}}_{1}^{*}\otimes{\mathcal{E}}_{2}(-1))=-\chi({\mathcal{E}}^{\prime}_{1}\otimes{{\mathcal{E}}^{\prime}_{2}}^{*}(-1)),$

so that we must have
$\chi({\mathcal{E}}_{1}\otimes{\mathcal{E}}_{2}^{*}(-1))=0$. Taking a general
$S\in|{\mathcal{O}}_{X}(1)|$, which is a Del Pezzo surface of degree $d$, the
restriction sequence thus yields
$\chi({\mathcal{E}}_{1}\otimes{\mathcal{E}}_{2}^{*})=\chi({\mathcal{E}}_{1}|_{S}\otimes{\mathcal{E}}_{2}^{*}|_{S})$.
For vector bundles ${\mathcal{U}}$ and ${\mathcal{V}}$ on a surface $S$, the
Riemann-Roch theorem gives

$\displaystyle\chi({\mathcal{U}}\otimes{\mathcal{V}}^{*})=\frac{1}{2}\left(\operatorname{rk}({\mathcal{V}})c_{1}({\mathcal{U}})^{2}+\operatorname{rk}({\mathcal{U}})c_{1}({\mathcal{V}})^{2}\right)-\operatorname{rk}({\mathcal{V}})c_{2}({\mathcal{U}})-\operatorname{rk}({\mathcal{U}})c_{2}({\mathcal{V}})$
$\displaystyle-
c_{1}({\mathcal{U}})c_{1}({\mathcal{V}})-\frac{1}{2}\left(\operatorname{rk}({\mathcal{V}})c_{1}({\mathcal{U}})-\operatorname{rk}({\mathcal{U}})c_{1}({\mathcal{V}})\right)\cdot
K_{S}+\operatorname{rk}({\mathcal{U}})\operatorname{rk}({\mathcal{V}})\chi({\mathcal{O}}_{S}).$

Inserting ${\mathcal{U}}={\mathcal{E}}_{1}|_{S}$ and
${\mathcal{V}}={\mathcal{E}}_{2}|_{S}$, and using that
$\operatorname{rk}({\mathcal{E}}_{i}|_{S})=r_{i}$,
$\operatorname{det}({\mathcal{E}}_{i}|_{S})=\operatorname{det}({\mathcal{E}}_{i})|_{S}={\mathcal{O}}_{S}(r_{i})$
and $c_{2}({\mathcal{E}}_{i}|_{S})=c_{2}({\mathcal{E}}_{i})\cdot
S=\frac{r_{i}}{2}\left(r_{i}d-d+2\right)$ by (32), the result follows. ∎

The following settles all statements but the nonemptiness claims in Theorems
1.1 and 1.2(ii), recalling that stable bundles are simple:

######  Proposition 4.2.

Let $X\subset{\mathbb{P}}^{d+1}$ be a smooth Del Pezzo threefold of degree
$d$. If nonempty, the moduli space of simple rank–$r$ Ulrich bundles with
determinant ${\mathcal{O}}_{X}(r)$ on $X$ is smooth of dimension $r^{2}+1$.

###### Proof.

Let ${\mathcal{E}}$ be a member of the moduli space in question. By simpleness
$h^{0}({\mathcal{E}}\otimes{\mathcal{E}}^{*})=1$; moreover
$h^{2}({\mathcal{E}}\otimes{\mathcal{E}}^{*})=h^{3}({\mathcal{E}}\otimes{\mathcal{E}}^{*})=0$
by Lemma 4.1(i). It is then well-known that the moduli space is smooth at
$[{\mathcal{E}}]$ (see, e.g., [CH, Prop. 2.10]), of dimension

$h^{1}({\mathcal{E}}\otimes{\mathcal{E}}^{*})=-\chi({\mathcal{E}}\otimes{\mathcal{E}}^{*})+h^{0}({\mathcal{E}}\otimes{\mathcal{E}}^{*})=r^{2}+h^{0}({\mathcal{E}}\otimes{\mathcal{E}}^{*})=r^{2}+1,$

using Lemma 4.1(ii). ∎

We have left to prove the existence parts of Theorems 1.1 and 1.2(ii). Since
both Ulrichness and stability are open conditions, it suffices to prove the
existence of one stable Ulrich bundle on $X$ (this observation had been done
already in [CFaM1, Lemma 2.3]). This will be proved by induction on the rank
$r$, starting with the case $r=2$.

###  4.1.  The cases of rank $r=2$

The next result is due to Beauville [Be1]; we will briefly recall part of the
proof, because we will need it later.

######  Proposition 4.3 (Beauville).

Let $X\subset{\mathbb{P}}^{d+1}$ be a smooth Del Pezzo threefold of degree
$d$. There exists a rank-two Ulrich bundle on $X$ with determinant
${\mathcal{O}}_{X}(2)$. Moreover, the moduli space parametrizing such bundles
is $5$–dimensional.

###### Proof.

By Example 3.10(i) it suffices to prove that $X$ contains a smooth elliptic
normal curve of degree $d+2$. This is done in [Be1, Lemma 6.2]. We recall the
proof when $d\leqslant 7$: take a general $S\in|{\mathcal{O}}_{X}(1)|$. This
is a smooth Del Pezzo surface, obtained by blowing up $9-d$ points on
${\mathbb{P}}^{2}$ in general position. Consider the linear system on $S$
corresponding to quartic curves in ${\mathbb{P}}^{2}$ passing doubly through
two of the blown-up points and simply through the rest. This is a
$(d+1)$-dimensional base point free linear system, whose general member is a
smooth elliptic curve $E_{0}$ of degree $d+1$. For any point $p\in E_{0}$,
consider any line $\ell_{0}$ in $X$ through $p$ not lying on $S$. As
$\ell_{0}\cdot S=1$, the curves $E_{0}$ and $\ell_{0}$ intersect transversely
at one point. In [Be1, Pf. of Lemma 6.2] it is proved that $E_{0}\cup\ell_{0}$
deforms to a smooth elliptic curve $E$ in $X$, which is the desired curve.

The statement about dimension of the moduli space is proved in [Be1, Prop.
6.4]. ∎

The existence of a stable Ulrich bundle is proved in the following result,
recalling that slope–stability implies stability.

######  Proposition 4.4.

Let $X\subset{\mathbb{P}}^{d+1}$ be a smooth Del Pezzo threefold of degree
$d$. There exists a rank-two slope–stable Ulrich bundle on $X$ with
determinant ${\mathcal{O}}_{X}(2)$.

###### Proof.

Let ${\mathcal{E}}$ be a rank–two Ulrich bundle on $X$ with determinant
${\mathcal{O}}_{X}(2)$. If ${\mathcal{E}}$ is not slope–stable, we have a
destabilizing sequence

(33)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{F}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{G}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

with ${\mathcal{F}}$ and ${\mathcal{G}}$ of rank one. By saturating the
sequence, we may assume that ${\mathcal{G}}$ is torsion-free. It follows from
[CH, Thm. 2.9] that both ${\mathcal{F}}$ and ${\mathcal{G}}$ are Ulrich line
bundles of slope $\mu({\mathcal{E}})=d$ (see also [Be1, (3.2)]).

In the cases where
$\operatorname{Pic}(X)\simeq{\mathbb{Z}}[{\mathcal{O}}_{X}(1)]$, that is, when
$d\in\\{3,4,5\\}$, there exist no Ulrich line bundles on $X$ (see again [Be1,
§4]), so we have a contradiction. There are again no Ulrich line bundles on
$X$ when $d=7$ (cf. [CFiM, Cor. 2.5 and Prop. 2.7]) and $d=8$ (cf. Corollary
3.9), and we have the same contradiction. We have left to treat the two cases
with $d=6$.

If $X$ is a smooth hyperplane section of
${\mathbb{P}}^{2}\times{\mathbb{P}}^{2}$ in its Segre embedding in
${\mathbb{P}}^{8}$, the only Ulrich line bundles on $X$ are
$\pi_{i}^{*}{\mathcal{O}}_{{\mathbb{P}}^{2}}(2)$, $i=1,2$, where
$\pi_{i}:X\to{\mathbb{P}}^{2}$ are the projections (cf. [CFaM2, Cor. 2.7]).
Thus, in (33) we have
${\mathcal{F}}\simeq\pi_{1}^{*}{\mathcal{O}}_{{\mathbb{P}}^{2}}(2)$ and
${\mathcal{G}}\simeq\pi_{2}^{*}{\mathcal{O}}_{{\mathbb{P}}^{2}}(2)$, or
vice–versa. Since

$\dim(\operatorname{Ext}^{1}_{X}(\pi_{1}^{*}{\mathcal{O}}_{{\mathbb{P}}^{2}}(2),\pi_{2}^{*}{\mathcal{O}}_{{\mathbb{P}}^{2}}(2)))=3,$

and similarly with $\pi_{1}$ and $\pi_{2}$ interchanged (cf. [CFaM2, Example
5.4]), bundles which are not slope–stable fill up a locus of dimension at most
$2$ in the moduli space, which is $5$–dimensional by the last statement in
Proposition 4.3. Hence there exist slope–stable rank–two Ulrich bundles on $X$
with determinant ${\mathcal{O}}_{X}(2)$.

If $X\simeq{\mathbb{P}}^{1}\times{\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$, one
may similarly check that the only possibilities for ${\mathcal{F}}$ and
${\mathcal{G}}$ in (33) are
$\pi_{1}^{*}{\mathcal{O}}_{{\mathbb{P}}^{1}}(2)\otimes\pi_{3}^{*}{\mathcal{O}}_{{\mathbb{P}}^{1}}(1)$
and
$\pi_{2}^{*}{\mathcal{O}}_{{\mathbb{P}}^{1}}(2)\otimes\pi_{3}^{*}{\mathcal{O}}_{{\mathbb{P}}^{1}}(1)$,
where $\pi_{i}:X\to{\mathbb{P}}^{1}$ are the three projections, up to
permutations (cf. [CFaM1, Prop. 3.2]). Since

$\dim(\operatorname{Ext}^{1}_{X}(\pi_{2}^{*}{\mathcal{O}}_{{\mathbb{P}}^{1}}(2)\otimes\pi_{3}^{*}{\mathcal{O}}_{{\mathbb{P}}^{1}}(1),\pi_{1}^{*}{\mathcal{O}}_{{\mathbb{P}}^{1}}(2)\otimes\pi_{3}^{*}{\mathcal{O}}_{{\mathbb{P}}^{1}}(1)))=3$

(cf. [CFaM1, Proof of Prop. 6.5]), once again bundles which are not
slope–stable fill up a locus of dimension at most $2$, and we are done as
above. ∎

This concludes the proofs of Theorems 1.1 and 1.2 in the case $r=2$.

######  Remark 4.5.

In the cases $d\in\\{6,7\\}$ the moduli space of rank–two Ulrich bundles with
determinant ${\mathcal{O}}_{X}(2)$ is reducible. For
$X\simeq{\mathbb{P}}^{1}\times{\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$, by
[CFaM1, Thm.B(3)] and taking into account possible permutations of the
generators $h_{1},h_{2},h_{3}$ in the expressions of $c_{2}({\mathcal{E}})$,
the moduli space consists of six irreducible components of dimension $5$,
three of which are generically smooth containing just one point representing
the $S$–equivalence class of strictly slope–semistable Ulrich bundles whereas
the general point corresponds to a slope–stable Ulrich bundle, and three
components are smooth contaning only slope–stable Ulrich bundles. Reducibility
also occurs for a smooth hyperplane section of the Segre variety
$\mathbb{P}^{2}\times\mathbb{P}^{2}$, cf. [CFaM2, Thm. 5.6], and for the case
$d=7$, cf. [CFiM, Thm.B(1)], where in both cases the components are
characterized by different $c_{2}({\mathcal{E}})$. By contrast, when $d=3$,
the case when $X\subset{\mathbb{P}}^{4}$ is a cubic threefold, the moduli
space is birational to the intermediate Jacobian of $X$, and is thus
irreducible (cf. [Be4]).

###  4.2. Conclusion of the proof of Theorem 1.1

Let $X\subset{\mathbb{P}}^{d+1}$ be a smooth Fano threefold of degree $d$ and
index two with $\omega_{X}\simeq{\mathcal{O}}_{X}(-2)$. Denote by
${\mathcal{H}}_{1}(X)$ the Hilbert scheme of lines contained in $X$, which by
[IP, Prop. 3.3.5(i)] is smooth, projective, of pure dimension two. (It is
irreducible for $3\leqslant d\leqslant 5$ (cf. [KPS, Thm. 1.1.1]) and
reducible for $d=6,7$ (cf. [IP, Prop. 3.5.6], but we will not need this.)

######  Lemma 4.6.

Let $X\subset{\mathbb{P}}^{d+1}$ be a smooth Fano threefold of degree $d$ and
index two with $\omega_{X}\simeq{\mathcal{O}}_{X}(-2)$. Let $E\subset
S_{2}\in|{\mathcal{O}}_{X}(2)|$ be a general curve and a general surface
associated to a rank-two Ulrich bundle on $X$ as in Theorem 3.5 and Remark
3.4, and let $\ell$ be a general line in (a component of)
${\mathcal{H}}_{1}(X)$. Then the restriction map induces an isomorphism

$\textstyle{H^{0}({\mathcal{O}}_{S_{2}}(E))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{H^{0}({\mathcal{O}}_{\ell\cap
S_{2}}(E)).}$

###### Proof.

Consider the curve $E_{0}$ contained in the smooth hyperplane section $S$ of
$X$ and the line $\ell_{0}$ as in the proof of Proposition 4.3. Then
$\ell_{0}$ is contained in a hyperplane section $S^{\prime}\neq S$ of $X$. For
$\ell\subset X$ general line in (a component of) ${\mathcal{H}}_{1}(X)$, we
have $\ell\cap\ell_{0}=\emptyset$ and $\ell\cap S$ is one point. Hence $\ell$
intersects $E_{0}\cup\ell_{0}$ in at most one point, and the same is true
replacing $E_{0}$ with any curve linearly equivalent to it on $S$. Therefore,
as one deforms the pair $(E_{0}\cup\ell_{0},S\cup S^{\prime})$ to a pair
$(E,S_{2})$, where $S_{2}\in|{\mathcal{O}}_{X}(2)|$ is a smooth, irreducible
$K3$ surface and $E\subset S_{2}$ is a smooth elliptic normal curve of degree
$d+2$, the line $\ell$ intersects any member of the pencil $|E|$ on $S_{2}$ in
at most one point. Since $\ell\cap S_{2}$ consists of two points, we have
$H^{0}({\mathcal{J}}_{(\ell\cap S_{2})/S_{2}}(E))=0$. Thus, the restriction
map $H^{0}({\mathcal{O}}_{S_{2}}(E))\to H^{0}({\mathcal{O}}_{\ell\cap
S_{2}}(E))$ is injective. As both spaces are two-dimensional, the map is an
isomorphism. ∎

Using induction on $r$ we will prove the following statement, which concludes
the proof of (the nonemptiness part of) Theorem 1.1.

######  Proposition 4.7.

Let $X\subset{\mathbb{P}}^{d+1}$ be a smooth Fano threefold of degree $d$ and
index two such that $\omega_{X}\simeq{\mathcal{O}}_{X}(-2)$. For every integer
$r\geqslant 2$ there exists an irreducible component $\mathfrak{U}_{r}$ of the
moduli space of slope–semistable vector bundles of rank $r$ and determinant
${\mathcal{O}}_{X}(r)$ on $X$ such that its general member is a slope–stable
Ulrich bundle. Moreover, for $r\geqslant 3$, $\mathfrak{U}_{r}$ contains a
closed subscheme $\mathfrak{U}_{r}^{\rm{ext}}$ of dimension at most
$r^{2}-r+2$ consisting of extensions ${\mathcal{E}}$ of the form

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}_{r-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{J}}_{\ell/X}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

where ${\mathcal{E}}_{r-1}$ is a general member of $\mathfrak{U}_{r-1}$ and
$\ell$ is a general line in (a component of) ${\mathcal{H}}_{1}(X)$.

The case $r=2$ has been proved in Propositions 4.3 and 4.4. Assume now we have
proved Proposition 4.7 for some $r\geqslant 2$. We will prove that it also
holds for $r+1$.

We denote by ${\mathcal{E}}_{r}$ a general member of $\mathfrak{U}_{r}$. Since
${\mathcal{E}}_{r}$ is Ulrich, by Theorem 3.5 and (15) it sits in a short
exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{C}}^{r-1}\otimes{\mathcal{O}}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}_{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{J}}_{C_{r}/X}(r)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

where $C_{r}$ is a smooth irreducible curve in $X$ numerically equivalent to
$c_{2}({\mathcal{E}}_{r})$, cf. Remark 3.2, with

(34) $\deg(C_{r})=\frac{r}{2}\left(rd-d+2\right).$

Pick a general line in (a component of) ${\mathcal{H}}_{1}(X)$.

######  Lemma 4.8.

We have

  * (i)

${\mathcal{E}}_{r}|_{\ell}\simeq{\mathcal{O}}_{{\mathbb{P}}^{1}}(1)^{\oplus
r}$,

  * (ii)

$h^{2}({\mathcal{E}}_{r}^{*}(-1)\otimes{\mathcal{J}}_{\ell/X})=r$.

###### Proof.

(i) Recall from Theorem 3.1 and Remark 3.4 that there is a smooth surface
$S_{r}\in|{\mathcal{O}}_{X}(r)|$ containing $C_{r}$ and an exact sequence

(35)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}_{r}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{X}^{\oplus
r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{S_{r}}(C_{r})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Restricting to $\ell$, and setting $Z_{r}:=\ell\cap S_{r}$ (a scheme of length
$r$), we obtain

(36)
---
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}_{r}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{X}^{\oplus
r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{S_{r}}(C_{r})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}_{r}^{*}|_{\ell}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{\ell}^{\oplus
r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{Z_{r}}(C_{r})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$

(The map
${\mathcal{E}}_{r}^{*}|_{\ell}\longrightarrow{\mathcal{O}}_{\ell}^{\oplus r}$
is injective because ${\mathcal{E}}_{r}^{*}|_{\ell}$ is locally free on $\ell$
and therefore it does not contain torsion subsheaves.) Since
${\mathcal{E}}_{r}$ is Ulrich, we have
$H^{i}({\mathcal{E}}_{r}^{*})=H^{3-i}({\mathcal{E}}_{r}(-2))=0$ for all $i$,
whence we have a commutative diagram

$\textstyle{H^{0}({\mathcal{O}}_{X}^{\oplus
r})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{H^{0}({\mathcal{O}}_{S_{r}}(C_{r}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\textstyle{H^{0}({\mathcal{O}}_{\ell}^{\oplus
r})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{H^{0}({\mathcal{O}}_{Z_{r}}(C_{r})),}$

where all four spaces are isomorphic to ${\mathbb{C}}^{r}$. Grant for the
moment the following:

(37) $f$ is injective.

Then $f$ is an isomorphism, and it follows that the map $g$ must be
surjective, whence an isomorphism as well. Going back to (36), we see that
$h^{i}({\mathcal{E}}_{r}^{*}|_{\ell})=0$ for $i\in\\{0,1\\}$. Since
$\deg({\mathcal{E}}_{r}^{*}|_{\ell})=-c_{1}({\mathcal{E}}_{r})\cdot\ell=-{\mathcal{O}}_{X}(r)\cdot\ell=-r$,
then
${\mathcal{E}}_{r}^{*}|_{\ell}\simeq{\mathcal{O}}_{{\mathbb{P}}^{1}}(-1)^{\oplus
r}$, proving (i).

We now prove (37), which is equivalent to

(38) $H^{0}({\mathcal{J}}_{Z_{r}/S_{r}}(C_{r}))=0.$

The case $r=2$ follows from Lemma 4.6, with $C_{2}=E$.

If $r\geqslant 3$, we may use Proposition 4.7 (recalling that by the induction
hypothesis we are assuming that it holds for $r$) and specialize
${\mathcal{E}}_{r}$ in a one-parameter flat family over the disc
${\mathbb{D}}$ to a vector bundle ${\mathcal{E}}^{\prime}_{r}$ sitting in an
extension

(39)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}_{r-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}^{\prime}_{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{J}}_{\ell^{\prime}/X}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

with ${\mathcal{E}}_{r-1}$ general in $\mathfrak{U}_{r-1}$ and $\ell^{\prime}$
a general line in (a component of) ${\mathcal{H}}_{1}(X)$. Then the surface
$S_{r}$ in (35) specializes to $S_{r-1}\cup S_{1}$, where
$S_{r-1}\in|{\mathcal{O}}_{X}(r-1)|$ and $S_{1}\in|{\mathcal{O}}_{X}(1)|$ are
smooth surfaces. In this process $Z_{r}$, the intersection scheme of $\ell$
with $S_{r}$, specializes to the disjoint union $Z_{r-1}\cup x$, where
$Z_{r-1}$ is the intersection scheme of $\ell$ with $S_{r-1}$ and $x:=\ell\cap
S_{1}$. By generality of $\ell$, we have $x\not\in S_{r-1}$, so that $x\not\in
Z_{r-1}$. By (39), we have

(40)
$[C_{r}]=c_{2}({\mathcal{E}}_{r})=c_{2}({\mathcal{E}}^{\prime}_{r})=\left[C_{r-1}\cup\ell^{\prime}\cup
I\right],\;\;\mbox{with}\;\;I:=S_{1}\cap S_{r-1},$

where $C_{r-1}$ sits in

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{C}}^{r-2}\otimes{\mathcal{O}}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}_{r-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{J}}_{C_{r-1}/X}(r-1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

We want to prove that none of the curves in the limit linear systems of
$|{\mathcal{O}}_{S_{r}}(C_{r})|$ contains the scheme $Z_{r-1}\cup x$. To do
so, consider the family $\pi:\mathcal{S}\to{\mathbb{D}}$, whose general fiber
is a smooth surface $S_{r}\in|{\mathcal{O}}_{X}(r)|$ and whose central fiber
is $S_{r-1}\cup S_{1}$. Then, by [Se2, Chp. 2] or [Fr, §2], the singular locus
$\mathfrak{Z}$ of $\mathcal{S}$ is supported at a divisor in the linear system
$|{\mathcal{N}}_{I/S_{1}}\otimes{\mathcal{N}}_{I/S_{r-1}}|$ on $I$, where
${\mathcal{N}}_{I/S_{j}}$ denotes the normal bundle of $I$ in $S_{j}$ for
$j=1,r-1$, which has degree $(r-1)^{2}d+(r-1)d=r(r-1)d$. One can make
$\mathcal{S}$ smooth by making a small resolution at each of the singular
points, cf., e.g., [CLM, p. 647], in such a way that the general fiber is
unaltered and the central fibre is replaced by $S_{r-1}\cup\widetilde{S}_{1}$,
where $\widetilde{S}_{1}\to S_{1}$ is the blow–up along the points of
$\mathfrak{Z}$ on $I$. By abuse of notation, we still denote by
$\ell^{\prime}$ the strict transform of $\ell^{\prime}$ on
$\widetilde{S}_{1}$. We also still denote $S_{r-1}\cap\widetilde{S}_{1}$ by
$I$.

A limit linear system of $|{\mathcal{O}}_{S_{r}}(C_{r})|$ consists of the
union of suitable linear systems on $S_{r-1}$ and $\widetilde{S}_{1}$ matching
along $I$. Precisely, by (40), a limit linear system is of the form

$\Lambda_{k}:=\Big{\\{}(D_{1},D_{2})\in|{\mathcal{O}}_{S_{r-1}}(C_{r-1}+I-kI)|\times|{\mathcal{O}}_{\widetilde{S}_{1}}(\ell^{\prime}+kI)|\;:\;D_{1}\cap
I=D_{2}\cap I\Big{\\}},$

and we have $k\geqslant 0$ since
$|{\mathcal{O}}_{\widetilde{S}_{1}}(\ell^{\prime}-I)|$ is empty. By induction,
no member of $|{\mathcal{O}}_{S_{r-1}}(C_{r-1})|$ contains $Z_{r-1}$, whence
also no member of $|{\mathcal{O}}_{S_{r-1}}(C_{r-1}+I-kI)|$, for any $k>0$,
contains $Z_{r-1}$. Hence, for $k>0$ no curve in $\Lambda_{k}$ contains
$Z_{r-1}\cup x$. Let us see that also no curve in $\Lambda_{0}$ contains
$Z_{r-1}\cup x$. In fact, as $\ell$ and $\ell^{\prime}$ are general lines on
$X$, they do not intersect, whence $x\not\in\ell^{\prime}$. Since for $k=0$ we
have
$|{\mathcal{O}}_{\widetilde{S}_{1}}(\ell^{\prime}+kI)|=|{\mathcal{O}}_{\widetilde{S}_{1}}(\ell^{\prime})|=\\{\ell^{\prime}\\}$,
we see that no curve in $\Lambda_{0}$ contains $Z_{r-1}\cup x$. Thus, (38)
follows, proving (37).

(ii) Consider the short exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}_{r}^{*}(-1)\otimes{\mathcal{J}}_{\ell/X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}_{r}^{*}(-1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}_{r}^{*}(-1)|_{\ell}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

Since ${\mathcal{E}}_{r}$ is Ulrich, we have
$h^{i}({\mathcal{E}}_{r}^{*}(-1))=h^{3-i}({\mathcal{E}}_{r}(-1))=0$ for
$i\in\\{0,1,2,3\\}$. Moreover, by (i) we have
$h^{1}({\mathcal{E}}_{r}^{*}(-1)|_{\ell})=h^{1}({\mathcal{O}}_{\ell}(-2)^{\oplus
r})=r$, and the result follows. ∎

Since

(41)
$\operatorname{Ext}_{X}^{1}({\mathcal{J}}_{\ell/X}(1),{\mathcal{E}}_{r})\simeq\operatorname{Ext}_{X}^{1}({\mathcal{J}}_{\ell/X}(-1)\otimes{\mathcal{E}}_{r}^{*},\omega_{X})\simeq
H^{2}({\mathcal{E}}_{r}^{*}(-1)\otimes{\mathcal{J}}_{\ell/X})^{*}\simeq{\mathbb{C}}^{r}$

by Lemma 4.8(ii), there is an $(r-1)$-dimensional family of nonsplit
extensions

(42)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}_{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{F}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{J}}_{\ell/X}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

In the next three lemmas (4.9–4.11) we list a number of properties of any
sheaf ${\mathcal{F}}$ obtained in this way.

######  Lemma 4.9.

Any sheaf ${\mathcal{F}}$ as in (42) is locally free.

###### Proof.

From the sequence

(43)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{J}}_{\ell/X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{\ell}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

twisted by ${\mathcal{O}}_{X}(1)$ we find that

$\operatorname{\mathfrak{e}\mathfrak{x}\mathfrak{t}}^{i}_{{\mathcal{O}}_{X}}({\mathcal{J}}_{\ell/X}(1),{\mathcal{O}}_{X})\simeq\operatorname{\mathfrak{e}\mathfrak{x}\mathfrak{t}}^{i+1}_{{\mathcal{O}}_{X}}({\mathcal{O}}_{\ell}(1),{\mathcal{O}}_{X})\simeq
0\;\;\mbox{for}\;\;i\geqslant 2,$

(as $\operatorname{codim}(\ell,X)=2$) and

$\operatorname{\mathfrak{e}\mathfrak{x}\mathfrak{t}}^{1}_{{\mathcal{O}}_{X}}({\mathcal{J}}_{\ell/X}(1),{\mathcal{O}}_{X})\simeq\operatorname{\mathfrak{e}\mathfrak{x}\mathfrak{t}}^{2}_{{\mathcal{O}}_{X}}({\mathcal{O}}_{\ell}(1),{\mathcal{O}}_{X})\simeq\operatorname{\mathfrak{e}\mathfrak{x}\mathfrak{t}}^{2}_{{\mathcal{O}}_{X}}({\mathcal{O}}_{\ell},\omega_{X})\otimes\mathcal{O}_{X}(1)\simeq\omega_{\ell}(1)\simeq{\mathcal{O}}_{{\mathbb{P}}^{1}}(-1).$

By (42) and the fact that ${\mathcal{E}}_{r}$ is locally free, we therefore
get

$\operatorname{\mathfrak{e}\mathfrak{x}\mathfrak{t}}^{i}_{{\mathcal{O}}_{X}}({\mathcal{F}},{\mathcal{O}}_{X})\simeq\operatorname{\mathfrak{e}\mathfrak{x}\mathfrak{t}}^{i}_{{\mathcal{O}}_{X}}({\mathcal{J}}_{\ell/X}(1),{\mathcal{O}}_{X})=0,\;\;\mbox{for}\;\;i\geqslant
2$

and

(44)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{X}(-1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{F}}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}_{r}^{*}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{{\mathcal{O}}_{{\mathbb{P}}^{1}}(-1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{\mathfrak{e}\mathfrak{x}\mathfrak{t}}^{1}_{{\mathcal{O}}_{X}}({\mathcal{F}},{\mathcal{O}}_{X})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

To prove that ${\mathcal{F}}$ is locally free, we need to show that $\alpha$
is surjective. We argue by contradiction.

If $\alpha=0$, then dualizing the left part of (44) we obtain

(45)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}_{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{F}}^{**}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{X}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Ulrichness of ${\mathcal{E}}_{r}$ yields

$\operatorname{Ext}_{X}^{1}({\mathcal{O}}_{X}(1),{\mathcal{E}}_{r})\simeq\operatorname{Ext}_{X}^{1}({\mathcal{O}}_{X},{\mathcal{E}}_{r}(-1))\simeq
H^{1}({\mathcal{E}}_{r}(-1))=0,$

whence (45) splits. Combining (42) and (45) with the natural map
$h:{\mathcal{F}}\to{\mathcal{F}}^{**}$, we get

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}_{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{{\mathcal{F}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{{\mathcal{J}}_{\ell/X}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}_{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\textstyle{{\mathcal{F}}^{**}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{{\mathcal{O}}_{X}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

where $p:{\mathcal{F}}^{**}\to{\mathcal{E}}_{r}$ is the projection map induced
by the splitting of (45). Since $p\circ
j=\operatorname{id}_{{\mathcal{E}}_{r}}$, we see that $p\circ
h:{\mathcal{F}}\to{\mathcal{E}}_{r}$ induces a splitting of (42) as well, a
contradiction.

If $\alpha\neq 0$ and $\alpha$ is not surjective, then
$\operatorname{im}(\alpha)\simeq{\mathcal{O}}_{{\mathbb{P}}^{1}}(-1-a)$ for
some $a>0$. By Lemma 4.8(i) we would obtain a nonzero morphism
${\mathcal{O}}_{{\mathbb{P}}^{1}}(-1)\to{\mathcal{O}}_{{\mathbb{P}}^{1}}(-1-a)$
for $a>0$, which is impossible. ∎

######  Lemma 4.10.

Let ${\mathcal{F}}$ be as in (42). Then:

  * (i)

$h^{i}({\mathcal{F}}(-j))=0$ for all $i\geqslant 0$ and $j\in\\{1,2\\}$,

  * (ii)

$h^{i}({\mathcal{F}}(-3))=\begin{cases}0,&i\in\\{0,1\\},\\\
1,&i\in\\{2,3\\},\end{cases}$

  * (iii)

$h^{3}({\mathcal{F}}\otimes{\mathcal{F}}^{*})=0$,

  * (iv)

$\chi({\mathcal{F}}\otimes{\mathcal{F}}^{*})=-(r+1)^{2}$.

###### Proof.

Items (i)-(ii) follow by straightforward computations using the sequences (42)
and (43), suitably twisted.

By tensoring (42) by ${\mathcal{F}}^{*}$ and taking cohomology we find

$h^{3}({\mathcal{F}}\otimes{\mathcal{F}}^{*})\leqslant
h^{3}({\mathcal{E}}_{r}\otimes{\mathcal{F}}^{*})+h^{3}({\mathcal{F}}^{*}\otimes{\mathcal{J}}_{\ell/X}(1)).$

By (i) and Lemma 4.1(i) we get
$h^{3}({\mathcal{E}}_{r}\otimes{\mathcal{F}}^{*})=0$. Moreover,

$h^{3}({\mathcal{F}}^{*}\otimes{\mathcal{J}}_{\ell/X}(1))\leqslant
h^{2}({\mathcal{F}}^{*}(1)|_{\ell})+h^{3}({\mathcal{F}}^{*}(1))=h^{0}({\mathcal{F}}(-3))=0,$

again by (i). This proves (iii).

Finally, by (42) we have:

$\displaystyle c_{1}({\mathcal{F}})$ $\displaystyle=$ $\displaystyle
c_{1}({\mathcal{E}}_{r})+c_{1}({\mathcal{O}}_{X}(1))=c_{1}({\mathcal{O}}_{X}(r+1)),$
$\displaystyle c_{2}({\mathcal{F}})\cdot{\mathcal{O}}_{X}(1)$ $\displaystyle=$
$\displaystyle\Big{(}c_{2}({\mathcal{E}}_{r})+c_{2}({\mathcal{J}}_{\ell/X}(1))+c_{1}({\mathcal{E}}_{r})\cdot
c_{1}({\mathcal{J}}_{\ell/X}(1))\Big{)}\cdot{\mathcal{O}}_{X}(1)$
$\displaystyle=$
$\displaystyle\Big{(}[C_{r}]+[\ell]+[{\mathcal{O}}_{X}(r)\cdot{\mathcal{O}}_{X}(1)]\Big{)}\cdot{\mathcal{O}}_{X}(1)=\deg(C_{r})+\deg(\ell)+rd$
$\displaystyle=$
$\displaystyle\frac{r}{2}\left(rd-d+2\right)+1+rd=\frac{r+1}{2}\left((r+1)d-d+2\right)$

(where we have used (34)); thus (iv) follows from Lemma 4.1(ii). ∎

######  Lemma 4.11.

Let ${\mathcal{F}}$ be as in (42) and let ${\mathcal{G}}$ be a destabilizing
subsheaf of ${\mathcal{F}}$. Then
${\mathcal{G}}^{*}\simeq{\mathcal{E}}_{r}^{*}$. In particular,
$\mu({\mathcal{G}})=\mu({\mathcal{F}})$ whence ${\mathcal{F}}$ is
slope–semistable.

###### Proof.

We note that (42) yields
$\mu({\mathcal{F}})=\frac{{\mathcal{O}}_{X}(r+1)\cdot{\mathcal{O}}_{X}(1)^{2}}{r+1}=d$.
Assume that ${\mathcal{G}}$ is a destabilizing subsheaf of ${\mathcal{F}}$,
that is
$0<\operatorname{rk}({\mathcal{G}})\leqslant\operatorname{rk}({\mathcal{F}})-1=r$
and $\mu({\mathcal{G}})\geqslant d$. Define

${\mathcal{Q}}:=\operatorname{im}\\{{\mathcal{G}}\subset{\mathcal{F}}\longrightarrow{\mathcal{J}}_{\ell/X}(1)\\}\;\;\mbox{and}\;\;{\mathcal{K}}:=\operatorname{ker}\\{{\mathcal{G}}\longrightarrow{\mathcal{Q}}\\}.$

Then we may put (42) into a commutative diagram with exact rows and columns:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{K}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{G}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{Q}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}_{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{F}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{J}}_{\ell/X}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{K}}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{F}}/{\mathcal{G}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{Q}}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$

defining ${\mathcal{K}}^{\prime}$ and ${\mathcal{Q}}^{\prime}$. We have
$\operatorname{rk}({\mathcal{Q}})\leqslant 1$.

Assume that $\operatorname{rk}({\mathcal{Q}})=0$. Since
${\mathcal{J}}_{\ell/X}(1)$ is torsion-free, we must have ${\mathcal{Q}}=0$,
whence ${\mathcal{K}}\simeq{\mathcal{G}}$. Since
$\mu({\mathcal{K}})=\mu({\mathcal{G}})\geqslant d=\mu({\mathcal{E}}_{r})$ and
${\mathcal{E}}_{r}$ is slope–stable (being a general member of
$\mathfrak{U}_{r}$), we must have
$\operatorname{rk}({\mathcal{K}})=\operatorname{rk}({\mathcal{E}}_{r})=r$. It
follows that $\operatorname{rk}({\mathcal{K}}^{\prime})=0$. Since

$c_{1}({\mathcal{K}})=c_{1}({\mathcal{E}}_{r})-c_{1}({\mathcal{K}}^{\prime})=[{\mathcal{O}}_{X}(r)]-D^{\prime},$

where $D^{\prime}$ is an effective divisor supported on the codimension one
locus of the support of ${\mathcal{K}}^{\prime}$, we have

$d\leqslant\mu({\mathcal{K}})=\frac{\left({\mathcal{O}}_{X}(r)-D^{\prime}\right)\cdot{\mathcal{O}}_{X}(1)^{2}}{r}=d-\frac{D^{\prime}\cdot{\mathcal{O}}_{X}(1)^{2}}{r}.$

Hence $D^{\prime}=0$ (and $\mu({\mathcal{K}})=d$), which means that
${\mathcal{K}}^{\prime}$ is supported in codimension at least two. Thus,
$\operatorname{\mathfrak{e}\mathfrak{x}\mathfrak{t}}_{{\mathcal{O}}_{X}}^{i}({\mathcal{K}}^{\prime},{\mathcal{O}}_{X})=0$
for $i\leqslant 1$, and it follows that
${\mathcal{K}}^{*}\simeq{\mathcal{E}}_{r}^{*}$ and
$\mu({\mathcal{G}})=\mu({\mathcal{F}})$, as desired.

Assume that $\operatorname{rk}({\mathcal{Q}})=1$. Then
$\operatorname{rk}({\mathcal{K}})=\operatorname{rk}({\mathcal{G}})-1\leqslant
r-1<r=\operatorname{rk}({\mathcal{E}}_{r})$ and
$\operatorname{rk}({\mathcal{Q}}^{\prime})=0$; in particular
${\mathcal{Q}}^{\prime}$ is a torsion sheaf. Since

$c_{1}({\mathcal{K}})=c_{1}({\mathcal{G}})-c_{1}({\mathcal{Q}})=c_{1}({\mathcal{G}})-c_{1}({\mathcal{J}}_{\ell/X}(1))+c_{1}({\mathcal{Q}}^{\prime})=c_{1}({\mathcal{G}})-c_{1}({\mathcal{O}}_{X}(1))+D,$

where $D$ is an effective divisor supported on the codimension one locus of
the support of ${\mathcal{Q}}^{\prime}$, we have

$\displaystyle\mu({\mathcal{K}})$ $\displaystyle=$
$\displaystyle\frac{\Big{(}c_{1}({\mathcal{G}})-c_{1}({\mathcal{O}}_{X}(1))+D\Big{)}\cdot{\mathcal{O}}_{X}(1)^{2}}{\operatorname{rk}({\mathcal{K}})}\geqslant\frac{c_{1}({\mathcal{G}})\cdot{\mathcal{O}}_{X}(1)^{2}-{\mathcal{O}}_{X}(1)^{3}}{\operatorname{rk}({\mathcal{K}})}$
$\displaystyle=$
$\displaystyle\frac{\mu({\mathcal{G}})\operatorname{rk}({\mathcal{G}})-d}{\operatorname{rk}({\mathcal{K}})}\geqslant\frac{d\operatorname{rk}({\mathcal{G}})-d}{\operatorname{rk}({\mathcal{K}})}=\frac{d(\operatorname{rk}({\mathcal{K}})+1)-d}{\operatorname{rk}({\mathcal{K}})}=d.$

This contradicts the slope-stability of ${\mathcal{E}}_{r}$. ∎

We denote by $\mathfrak{U}_{r+1}^{\rm{ext}}$ the family of vector bundles on
$X$ consisting of all extensions ${\mathcal{F}}$ of the form (42), for a
general (stable) ${\mathcal{E}}_{r}$ in $\mathfrak{U}_{r}$ and $\ell$ a
general line in (a component of) ${\mathcal{H}}_{1}(X)$. Note that all the
elements are semistable by the previous lemma. We denote by
$\mathfrak{U}_{r+1}$ the component of the moduli space of semistable bundles
on $X$ containing $\mathfrak{U}_{r+1}^{\rm{ext}}$. To finish the proof of
Proposition 4.7, we need to prove that the general member in
$\mathfrak{U}_{r+1}$ is Ulrich and slope–stable. This will be done in Lemma
4.13 below, after an intermediate result:

######  Lemma 4.12.

One has $\dim(\mathfrak{U}_{r+1}^{\rm{ext}})<\dim(\mathfrak{U}_{r+1})$.

###### Proof.

By Lemma 4.10(iii)-(iv) and standard deformation theory we have

$\displaystyle\dim(\mathfrak{U}_{r+1})$ $\displaystyle\geqslant$
$\displaystyle
h^{1}({\mathcal{F}}\otimes{\mathcal{F}}^{*})-h^{2}({\mathcal{F}}\otimes{\mathcal{F}}^{*})=-\chi({\mathcal{F}}\otimes{\mathcal{F}}^{*})+h^{0}({\mathcal{F}}\otimes{\mathcal{F}}^{*})$
$\displaystyle\geqslant$ $\displaystyle(r+1)^{2}+1=r^{2}+2r+2.$

On the other hand, using (41), we have

$\displaystyle\dim(\mathfrak{U}_{r+1}^{\rm{ext}})$ $\displaystyle\leqslant$
$\displaystyle\dim(\mathfrak{U}_{r})+\dim{\mathbb{P}}(\operatorname{Ext}_{X}^{1}({\mathcal{J}}_{\ell/X}(1),{\mathcal{E}}_{r}))+\dim({\mathcal{H}}_{1}(X))$
$\displaystyle=$ $\displaystyle(r^{2}+1)+(r-1)+2=r^{2}+r+2.$

The result follows. ∎

######  Lemma 4.13.

The general member in $\mathfrak{U}_{r+1}$ is Ulrich and slope–stable.

###### Proof.

Let ${\mathcal{E}}$ be a general member of $\mathfrak{U}_{r+1}$. By Lemma
4.10(i)-(ii) and semicontinuity we have $h^{i}({\mathcal{E}}(-j))=0$ for all
$i\geqslant 0$ and $j\in\\{1,2\\}$, $h^{i}({\mathcal{E}}(-3))=0$ for
$i\in\\{0,1\\}$ and $\chi({\mathcal{E}}(-3))=0$. To prove that ${\mathcal{E}}$
is Ulrich, we therefore have left to prove that $h^{3}({\mathcal{E}}(-3))=0$,
equivalently $h^{0}({\mathcal{E}}^{*}(1))=0$.

Assume therefore, to get a contradiction, that $h^{0}({\mathcal{E}}^{*}(1))>0$
for a general ${\mathcal{E}}$ in $\mathfrak{U}_{r+1}$, and consider a nonzero
section $s\in H^{0}({\mathcal{E}}^{*}(1))$. Setting
${\mathcal{Q}}:=\operatorname{coker}(s)$, we have

(46)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\textstyle{{\mathcal{E}}^{*}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{Q}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

We may assume that this specializes to a nonzero section $s_{0}\in
H^{0}({\mathcal{F}}^{*}(1))$ for all ${\mathcal{F}}$ in
$\mathfrak{U}_{r+1}^{\rm{ext}}$ and we have an exact sequence

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{O}}_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{0}}$$\textstyle{{\mathcal{F}}^{*}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{Q}}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

where ${\mathcal{Q}}_{0}:=\operatorname{coker}(s_{0})$. Dualizing and twisting
by ${\mathcal{O}}_{X}(1)$, we obtain

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{Q}}_{0}^{*}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{F}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{0}^{*}}$$\textstyle{{\mathcal{O}}_{X}(1),}$

and one has $\operatorname{im}(s_{0}^{*})={\mathcal{J}}_{Z/X}(1)$ for some
(possibly empty) subscheme $Z\subset X$. Since

$\displaystyle c_{1}({\mathcal{Q}}_{0}^{*}(1))$ $\displaystyle=$
$\displaystyle
c_{1}({\mathcal{F}})-c_{1}({\mathcal{J}}_{Z/X}(1))=c_{1}({\mathcal{O}}_{X}(r+1))-c_{1}({\mathcal{O}}_{X}(1))+c_{1}({\mathcal{O}}_{Z}(1))$
$\displaystyle=$ $\displaystyle c_{1}({\mathcal{O}}_{X}(r))+Z^{\prime},$

where $Z^{\prime}$ is an effective divisor supported on the codimension-one
locus of $Z$, we have

$\displaystyle\mu({\mathcal{Q}}_{0}^{*}(1))$ $\displaystyle=$
$\displaystyle\frac{\Big{(}c_{1}({\mathcal{O}}_{X}(r))+Z^{\prime}\Big{)}\cdot{\mathcal{O}}_{X}(1)^{2}}{\operatorname{rk}({\mathcal{Q}}_{0}^{*}(1))}\geqslant\frac{c_{1}({\mathcal{O}}_{X}(r))\cdot{\mathcal{O}}_{X}(1)^{2}}{r}=\frac{rd}{r}=d.$

Lemma 4.11 therefore yields
$\left({\mathcal{Q}}_{0}^{*}(1)\right)^{*}\simeq{\mathcal{E}}_{r}^{*}$, whence
${\mathcal{Q}}_{0}^{*}(1)\simeq{\mathcal{E}}_{r}$, as
${\mathcal{Q}}_{0}^{*}(1)$ is reflexive (by [Ha, Prop. 1.1]). It follows that
${\mathcal{Q}}^{*}(1)$ is a deformation of ${\mathcal{E}}_{r}$, that is,
$[{\mathcal{Q}}^{*}(1)]\in\mathfrak{U}_{r}$. The dual of (46) twisted by
${\mathcal{O}}_{X}(1)$ gives

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{Q}}^{*}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s^{*}}$$\textstyle{{\mathcal{O}}_{X}(1),}$

and for reasons of Chern classes we must have
$\operatorname{im}(s^{*})\simeq{\mathcal{J}}_{\ell/X}(1)$ for a line
$\ell\subset X$. Therefore $[{\mathcal{E}}]\in\mathfrak{U}_{r+1}^{\rm{ext}}$,
contradicting Lemma 4.12. We have thus proved that ${\mathcal{E}}$ is Ulrich.

If the general member of $\mathfrak{U}_{r+1}$ were not slope–stable, we could
find a one-parameter flat family of bundles $\\{{\mathcal{E}}^{(t)}\\}$ over
the disc ${\mathbb{D}}$ such that ${\mathcal{E}}^{(t)}$ is a general member of
$\mathfrak{U}_{r+1}$ for $t\neq 0$ and ${\mathcal{E}}^{(0)}$ lies in
$\mathfrak{U}_{r+1}^{\rm{ext}}$, and such that we have, for $t\neq 0$, a
destabilizing sequence

(47)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{G}}^{(t)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{E}}^{(t)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathcal{F}}^{(t)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0,}$

which we can take to be saturated, that is, such that ${\mathcal{F}}^{(t)}$ is
torsion free, whence so that ${\mathcal{F}}^{(t)}$ and ${\mathcal{G}}^{(t)}$
are (Ulrich) vector bundles (see [CH, Thm. 2.9] or [Be1, (3.2)]).

The limit of
${\mathbb{P}}({\mathcal{F}}^{(t)})\subset{\mathbb{P}}({\mathcal{E}}^{(t)})$
defines a subvariety of ${\mathbb{P}}({\mathcal{E}}^{(0)})$ of the same
dimension as ${\mathbb{P}}({\mathcal{F}}^{(t)})$, whence a coherent sheaf
${\mathcal{F}}^{(0)}$ of rank $\operatorname{rk}({\mathcal{F}}^{(t)})$ with a
surjection ${\mathcal{E}}^{(0)}\twoheadrightarrow{\mathcal{F}}^{(0)}$.
Denoting by ${\mathcal{G}}^{(0)}$ its kernel, we have
$\operatorname{rk}({\mathcal{G}}^{(0)})=\operatorname{rk}({\mathcal{G}}^{(t)})$
and $c_{1}({\mathcal{G}}^{(0)})=c_{1}({\mathcal{G}}^{(t)})$. Hence, (47)
specializes to a destabilizing sequence for $t=0$. Lemma 4.11 yields that
${{\mathcal{G}}^{(0)}}^{*}$ is the dual of a member of $\mathfrak{U}_{r}$. It
follows that ${{\mathcal{G}}^{(t)}}^{*}$ is a deformation of the dual of a
member of $\mathfrak{U}_{r}$, whence ${\mathcal{G}}^{(t)}$ is a deformation of
a member of $\mathfrak{U}_{r}$, as both are locally free. Therefore,
$[{\mathcal{G}}^{(t)}]\in\mathfrak{U}_{r}$ and we obtain the same
contradiction to Lemma 4.12 as above. ∎

The proof of Proposition 4.7 is now complete, whence also the proof of Theorem
1.1.

###  4.3. Conclusion of the proof of Theorem 1.2(ii)

Let $X\subset{\mathbb{P}}^{9}$ be the $2$–Veronese embedding of
${\mathbb{P}}^{3}$. We imitate the proof of the existence of Ulrich bundles of
rank $r\geqslant 2$ on a general cubic threefold in [CH, Thm. 5.7]. The
strategy is as follows. We define $\mathfrak{U}_{2}$ to be any component of
the moduli space of slope–stable rank–$2$ Ulrich bundles of determinant
${\mathcal{O}}_{X}(2)$, which is nonempty by Propositions 4.3 and 4.4. Then we
proceed to construct inductively (irreducible components of) moduli spaces
$\mathfrak{U}_{2h}$ of rank–$2h$ Ulrich bundles, for all integers $h\geqslant
2$, in the following way: assume that we have constructed, for an integer
$h\geqslant 1$, an irreducible moduli space $\mathfrak{U}_{2h}$ parametrizing
rank–2h Ulrich bundles of determinant ${\mathcal{O}}_{X}(2h)$ whose general
member is slope–stable. In particular, for any
$[{\mathcal{E}}_{2h}]\in\mathfrak{U}_{2h}$, we have that

(48) $c_{2}({\mathcal{E}}_{2h})\cdot{\mathcal{O}}_{X}(1)=2h(8h-3)$

by (25) in Theorem 3.5 with $\alpha=2$ (cf. Remark 3.2).

######  Lemma 4.14.

For general $[{\mathcal{E}}_{2}]\in\mathfrak{U}_{2}$ and
$[{\mathcal{E}}^{\prime}_{2h}]\in\mathfrak{U}_{2h}$, such that
${\mathcal{E}}_{2}\not\simeq{\mathcal{E}}^{\prime}_{2}$ when $h=1$, one has
$\dim(\operatorname{Ext}^{1}({\mathcal{E}}^{\prime}_{2h},{\mathcal{E}}_{2}))=4h$.

###### Proof.

By Lemma 4.1(i) we have
$h^{2}({\mathcal{E}}_{2}\otimes({\mathcal{E}}^{\prime}_{2h})^{*})=h^{3}({\mathcal{E}}_{2}\otimes({\mathcal{E}}^{\prime}_{2h})^{*})=0$.
Moreover $h^{0}({\mathcal{E}}_{2}\otimes({\mathcal{E}}^{\prime}_{2h})^{*})=0$,
because ${\mathcal{E}}_{2}$ and ${\mathcal{E}}^{\prime}_{2h}$ are slope–stable
bundles of the same slope, namely $8$, which are not isomorphic. Hence

$\dim(\operatorname{Ext}^{1}({\mathcal{E}}^{\prime}_{2h},{\mathcal{E}}_{2}))=h^{1}({\mathcal{E}}_{2}\otimes({\mathcal{E}}^{\prime}_{2h})^{*})=-\chi({\mathcal{E}}_{2}\otimes({\mathcal{E}}^{\prime}_{2h})^{*})=4h,$

by Lemma 4.1(ii), since conditions (32) hold true by (48). ∎

By the previous lemma, there exist nonsplit extensions of the form

(49)
$0\longrightarrow{\mathcal{E}}_{2}\longrightarrow{\mathcal{F}}_{2h+2}\longrightarrow{\mathcal{E}}^{\prime}_{2h}\longrightarrow
0,$

for general $[{\mathcal{E}}_{2}]\in\mathfrak{U}_{2}$ and
$[{\mathcal{E}}^{\prime}_{2h}]\in\mathfrak{U}_{2h}$. We have that
$\operatorname{rk}({\mathcal{F}}_{2h+2})=2h+2$,
$c_{1}({\mathcal{F}}_{2h+2})=[{\mathcal{O}}_{X}(2h+2)]$ and
${\mathcal{F}}_{2h+2}$ is easily seen to be Ulrich. We denote by
$\mathfrak{U}_{2h+2}^{\rm ext}$ the family of such nonsplit extentions and we
define $\mathfrak{U}_{2h+2}$ to be the component of the moduli space of Ulrich
bundles containing $\mathfrak{U}_{2h+2}^{\rm ext}$.

To finish the proof of Theorem 1.2(ii), we have left to prove that the general
element of $\mathfrak{U}_{2h+2}$ is slope–stable. To this end, we need the
following two lemmas.

######  Lemma 4.15.

One has $\dim(\mathfrak{U}_{2h+2}^{\rm ext})<\dim(\mathfrak{U}_{2h+2})$.

###### Proof.

One checks that in (49) one has
$\mu({\mathcal{E}}_{2})=\mu({\mathcal{E}}^{\prime}_{2h})=8$. Thus, by [CH,
Lemma 4.2], the bundle ${\mathcal{F}}_{2h+2}$ is simple, whence
$\dim(\mathfrak{U}_{2h+2})=(2h+2)^{2}+1=4h^{2}+8h+5$ by Proposition 4.2. On
the other hand, by Lemma 4.14, we have

$\displaystyle\dim(\mathfrak{U}_{2h+2}^{\rm ext})$ $\displaystyle\leqslant$
$\displaystyle\dim(\mathfrak{U}_{2})+\dim(\mathfrak{U}_{2h})+\dim({\mathbb{P}}({\rm
Ext}^{1}({\mathcal{E}}^{\prime}_{2h},{\mathcal{E}}_{2})))$ $\displaystyle=$
$\displaystyle 5+(4h^{2}+1)+(4h-1)=4h^{2}+4h+5.$

∎

######  Lemma 4.16.

Let ${\mathcal{F}}_{2h+2}$ be as in (49) and let ${\mathcal{G}}$ be a
destabilizing subsheaf of ${\mathcal{F}}_{2h+2}$. Then
${\mathcal{G}}^{*}\simeq{\mathcal{E}}_{2}^{*}$.

###### Proof.

This is identical to the proof of Lemma 4.11 and left to the reader. ∎

To prove that the general element of $\mathfrak{U}_{2h+2}$ is slope–stable one
applies almost verbatim the argument of the last part of the proof of Lemma
4.13: indeed, arguing by contradiction, if it were not slope–stable then one
applies Lemma 4.16 to prove that it lies in $\mathfrak{U}_{2h+2}^{\rm ext}$,
which contradicts Lemma 4.15. This completes the proof of Theorem 1.2.

##  5\. Final remarks and speculations

Let $X\subset{\mathbb{P}}^{d+1}$ be a smooth Fano threefold of degree $d$ and
index two with $\omega_{X}\simeq{\mathcal{O}}_{X}(-2)$. Theorem 1.1 proves
that for every $r\geqslant 2$, there exists a smooth,
$\left(r^{2}+1\right)$–dimensional moduli space $\mathfrak{U}_{X,r}$,
parametrizing a family of Ulrich bundles of rank $r$ and determinant
${\mathcal{O}}_{X}(r)$ on $X$. For any ${\mathcal{E}}\in\mathfrak{U}_{X,r}$,
one has $h^{0}({\mathcal{E}})=rd$ (cf., e.g., [Be1, (3.1)]). By Theorems 3.1
and 3.5, given a general subspace $V$ of dimension $r-1$ in
$H^{0}({\mathcal{E}})$, the inclusion
$V\otimes{\mathcal{O}}_{X}\longrightarrow{\mathcal{E}}$ drops rank along a
smooth curve $C$ of genus

$g_{r,d}=\frac{1}{3}rd(r^{2}-3r+2)+(r-1)^{2}$

(cf. (24) and (26)).

Consider the moduli space $\mathfrak{F}_{2,d}$ of isomorphisms classes of
smooth Fano threefolds of index two and degree $d$ and the universal moduli
space $\mathfrak{U}_{r,d}$ of Ulrich bundles of rank $r$ over
$\mathfrak{F}_{2,d}$: a point in $\mathfrak{U}_{r,d}$ is a pair
$([X],[{\mathcal{E}}])$, where $[X]\in\mathfrak{F}_{2,d}$ and
$[{\mathcal{E}}]\in\mathfrak{U}_{X,r}$. Let $\mathbb{G}$ be the Grassmannian
bundle over $\mathfrak{U}_{r,d}$, whose fibre over $([X],[{\mathcal{E}}])$ is
$\operatorname{Grass}(r-1,H^{0}(X,{\mathcal{E}}))$. By the above
considerations, we have a rational map

$\varphi_{r,d}:\mathbb{G}\dasharrow{\mathcal{M}}_{g_{r,d}},$

where ${\mathcal{M}}_{g}$ is the moduli space of smooth projective curves of
genus $g$. Its image is clearly uniruled, and, as such, it is an interesting
sublocus of ${\mathcal{M}}_{g_{r,d}}$. In particular, it is a natural question
to understand for which $d$ and $r$ the map $\varphi_{r,d}$ is dominant.

Let us focus on low values of $r$. For $r=2$ one has $g_{2,d}=1$ and it is
easy to deduce from the proof of Proposition 4.3 that $\varphi_{2,d}$ is
dominant for all $d$.

The next step is $r=3$. In this case $g_{3,d}=2d+4$, and, as we have seen in
Example 3.10, through the embedding in $X$ the curve $C$ admits a linearly and
quadratically normal embedding as a curve of degree $3d+3$ in
${\mathbb{P}}^{d+1}$ with injective Petri map $\mu_{0,{\mathcal{O}}_{C}(1)}$
(cf. (31)). (We also note that the linear system $|{\mathcal{O}}_{C}(1)|$ has
Brill–Noether number $\rho(2d+4,d+1,3d+3)=0$.) This suggests that the map
$\varphi_{3,d}$ might also be dominant.

A result in this direction is the following:

######  Proposition 5.1.

The map $\varphi_{3,d}$ is dominant for $3\leqslant d\leqslant 4$.

###### Idea of the proof.

We give an idea in the case $d=3$, where the target space is
${\mathcal{M}}_{10}$. The proof in the case $d=4$, where the target space is
${\mathcal{M}}_{12}$, is similar.

Since $\rho(10,4,12)=0$, there are linearly normal curves with general moduli
in $\mathbb{P}^{4}$ of degree 12 and genus 10. It suffices to prove that such
a general curve lies on a smooth cubic hypersurface; indeed the remaining
properties as in Example 3.10(ii) are automatically satisfied since the curve
has general moduli. Actually one proves that the general such curve lies on a
general cubic hypersurface.

To do so, let $\mathcal{H}$ be the component of the Hilbert scheme containing
the above curves with general moduli. An easy count of parameters shows that
$\dim(\mathcal{H})=51$. Consider the incidence variety

$I=\\{(C,F)\in\mathcal{H}\times|\mathcal{O}_{\mathbb{P}^{4}}(3)|:C\subset
F\\}$

with the two projections $p_{1}:I\to\mathcal{H}$ and
$p_{2}:I\to|\mathcal{O}_{\mathbb{P}^{4}}(3)|$. The map $p_{1}$ is dominant and
the general fibre has dimension 7 (because the general curve $C\in\mathcal{H}$
is of maximal rank). So $\dim(I)=58$. One proves that $p_{2}$ is dominant by
proving that the general fibre of $p_{2}$ has dimension 24 (recall that
$\dim(|\mathcal{O}_{\mathbb{P}^{4}}(3)|)=34$). It suffices to show that there
is one $F\in|\mathcal{O}_{\mathbb{P}^{4}}(3)|$ such that $p_{2}^{-1}(F)$ has
dimension 24. One proves this by degeneration, showing that this is the case
if $F$ is the general union of a quadric and a hyperplane, which is the
technical and lengthy part of the proof that we skip. ∎

As a consequence, one has that ${\mathcal{M}}_{10}$ and ${\mathcal{M}}_{12}$
are uniruled, which is no big news, because we know since a long time that
${\mathcal{M}}_{10}$ and ${\mathcal{M}}_{12}$ are unirational (cf. [AS, Se1]).

The proof of Proposition 5.1, based on delicate counts of parameters, is
rather lenghty and we do not dwell on it here.

The question remains, how does $\varphi_{3,d}$ behave for $5\leqslant
d\leqslant 7$? We suspect that in these cases $\varphi_{3,d}$ is dominant onto
a codimension 3 closed subset of ${\mathcal{M}}_{2d+4}$. Let us briefly
explain the reason for this expectation. First of all note that the moduli
spaces $\mathfrak{F}_{2,d}$ consist of one isolated point for $5\leqslant
d\leqslant 7$ (see [IP]). Hence

$\dim(\mathbb{G})=10+2(3d-2)=6d+6.$

Since $\dim({\mathcal{M}}_{2d+4})=6d+9$, the map $\varphi_{3,d}$ cannot be
dominant. We conjecture that it is generically finite onto its image, which
would prove that $\varphi_{3,d}$ dominates a codimension 3 closed subset of
${\mathcal{M}}_{2d+4}$.

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