Assumptions 1.8 Let $\varphi:X_{0}\to\mathbb{F}_{0}$ be the scroll map and $F$
be the $\varphi$-fibre. Let $r\geqslant 2$ be any integer. Then the moduli
space of rank-$r$ vector bundles $\mathcal{U}_{r}$ on $X_{0}$ which are Ulrich
w.r.t. $\xi$ and with first Chern class

$\displaystyle
c_{1}(\mathcal{U}_{r})=\begin{cases}r\xi+\varphi^{*}\mathcal{O}_{\mathbb{F}_{0}}(3,b_{0}-3)+\varphi^{*}\mathcal{O}_{\mathbb{F}_{0}}\left(\frac{r-3}{2},\frac{(r-3)}{2}(b_{0}-2)\right),&\mbox{if
$r$ is odd},\\\
r\xi+\varphi^{*}\mathcal{O}_{\mathbb{F}_{0}}\left(\frac{r}{2},\frac{r}{2}(b_{0}-2)\right),&\mbox{if
$r$ is even}.\end{cases}$

is not empty and it contains a generically smooth component $\mathcal{M}(r)$
of dimension

$\displaystyle\dim(\mathcal{M}(r))=\begin{cases}\frac{(r^{2}-1)}{4}(6b_{0}-4),&\mbox{if
$r$ is odd},\\\ \frac{r^{2}}{4}(6b_{0}-4)+1,&\mbox{if $r$ is
even},\end{cases}$

with $b_{0}\geqslant 2$ by (1.6). Moreover the general point
$[\mathcal{U}_{r}]\in\mathcal{M}(r)$ corresponds to a slope-stable vector
bundle, of slope w.r.t. $\xi$ given by $\mu(\mathcal{U}_{r})=8b_{0}-k_{0}-3$.

###### Proof.

It directly follows from Theorem 3.1, (4.6), (4.7) and from Lemmas 4.5, 4.7
and Proposition 4.8, where by abuse of notation we have used the same symbol
$\mathcal{M}(r)$ for the smooth modular family which gives rise to the
generically smooth irreducible component of the corresponding moduli space. ∎

###  4.2. Higher rank Ulrich vector bundles on $3$-fold scrolls over
$\mathbb{F}_{e}$, $e>0$

Here we focus on the case $e>0$. The strategy we will use is slightly
different from that used in § 4.1.More precisely, as before we will
inductively define irreducible families of vector bundles on $(X_{e},\xi)$
whose general members will be slope–stable Ulrich bundles obtained, by
induction, as deformations of extensions of lower ranks Ulrich bundles. The
main difference with respect to the case $e=0$ is that there are no Ulrich
line bundles w.r.t. $\xi$ on $(X_{e},\xi)$ when $e>0$, as it follows from
Theorem 2.1 (cf. Main Theorem-(a)).

Therefore, in the even rank case, our starting point for the inductive process
will be given by the use of rank-$2$ Ulrich vector bundles as in Theorem 3.4.
Extensions, recursive procedures, deformations and moduli theory, will then
allow us to construct slope-stable Ulrich vector bundles on $(X_{e},\xi)$ of
even ranks $r=2h$, for any $h\geqslant 2$, and to study their modular
components.

For odd ranks, instead, we will use some results in [2] concerning rank-$3$
vector bundles over $\mathbb{F}_{e}$ which turn out to be Ulrich w.r.t.
$c_{1}(\mathcal{E}_{e})=3C_{e}+b_{e}f=\mathcal{O}_{\mathbb{F}_{e}}(3,b_{e})$,
for any $e>0$, and then we will apply Theorem 1.7 to obtain rank-$3$ vector
bundles on $(X_{e},\xi)$, $e>0$, which are Ulrich w.r.t. $\xi$ (cf. Theorem
4.14). Then such rank-$3$ bundles along with $[\mathcal{U}_{2}]\in\mathcal{M}$
general as in Theorem 3.4, will allow us to apply an inductive process on
$X_{e}$ also for odd ranks $r=2h+1$, for any $h\geqslant 1$. In order to avoid
confusion, we will separately treat even ranks $r=2h$ and odd ranks $r=2h+1$,
for any $h\geqslant 1$.

Even ranks: in the even rank cases, set $r=2h$, for $h\geqslant 1$ an integer.
We start by defining the irreducible scheme $\mathcal{M}(2)$ to be the
component $\mathcal{M}$ as in Theorem 3.4. Recall that
$\mathcal{M}(2)=\mathcal{M}$ is generically smooth, of dimension
$\dim(\mathcal{M}(2))=6b_{e}-9e-3$ and that the general member
$[\mathcal{U}_{2}]\in\mathcal{M}(2)$ is a rank-$2$ vector bundle on $X_{e}$
which is Ulrich and slope-stable w.r.t. $\xi$, of slope
$\mu(\mathcal{U}_{2})=8b_{e}-k_{e}-12e-3$, whose first Chern class is
$c_{1}(\mathcal{U}_{2})=2\xi+\varphi^{*}\mathcal{O}_{\mathbb{F}_{e}}(1,b_{e}-e-2)$.

Assume by induction we have constructed an irreducible scheme
$\mathcal{M}(2h-2)$, for some $h\geqslant 2$; similarly as in [15] we define
$\mathcal{M}(2h)$ to be the (possibly empty a priori) component of the moduli
space of Ulrich bundles on $(X_{e},\xi)$ containing bundles $\mathcal{F}_{2h}$
that are non–trivial extensions of the form

(4.29)
$0\to\mathcal{U}^{\prime}_{2}\to\mathcal{F}_{2h}\to\mathcal{U}_{2h-2}\to 0,$

with $[\mathcal{U}^{\prime}_{2}]\in\mathcal{M}(2)$,
$[\mathcal{U}_{2h-2}]\in\mathcal{M}(2h-2)$ and such that
$\mathcal{U}^{\prime}_{2}\not\cong\mathcal{U}_{2h-2}$ when $h=2$. Similarly as
in the the case $e=0$ in § 4.1, we let $\mathcal{M}(2h)^{\rm ext}$ denote the
locus in $\mathcal{M}(2h)$ of bundles that are non-trivial extensions of the
form (4.29).

In the next results we will prove that non-trivial extensions as in (4.29)
always exist and that $\mathcal{M}(2h)^{\rm ext}\neq\emptyset$, so in
particular $\mathcal{M}(2h)\neq\emptyset$, for any $h\geqslant 2$. In
statements and proofs below we will use the following notation:
$\mathcal{U}^{\prime}_{2}$ will correspond to a general member of
$\mathcal{M}(2)$ and $\mathcal{U}_{2h-2}$ to a general member of
$\mathcal{M}(2h-2)$, with
$\mathcal{U}^{\prime}_{2}\not\cong\mathcal{U}_{2h-2}$ when $h=2$. We will
denote by $\mathcal{F}_{2h}$ a general member of $\mathcal{M}(2h)^{\rm ext}$
and, in bounding cohomologies, we will use the fact that $\mathcal{U}_{2h}$
specializes to $\mathcal{F}_{2h}$ in an irreducible flat family.

All vector bundles $\mathcal{F}_{2h}$, $h\geqslant 2$, recursively defined as
in (4.29) are of rank $2h$ and Ulrich w.r.t. $\xi$, since extensions of
bundles which are Ulrich w.r.t. $\xi$ are again Ulrich w.r.t. $\xi$. Their
first Chern class is given by

(4.30)
$c_{1}(\mathcal{F}_{2h}):=2h\xi+\varphi^{*}\mathcal{O}_{\mathbb{F}_{e}}(h,h(b_{e}-e-2))=h\;c_{1}(\mathcal{U}_{2}),$

where $h=\frac{r}{2}$, whose slope w.r.t. $\xi$ is

(4.31) $\mu(\mathcal{F}_{2h})=8b_{e}-k_{e}-12e-3.$

From Theorem 1.4-(a), any such bundle $\mathcal{F}_{2h}$ is strictly
semistable and slope-semistable, being extensions of Ulrich bundles of the
same slope.

######  Lemma 4.10.

Let $h\geqslant 1$ be an integer and assume $\mathcal{M}(2k)\neq\emptyset$ for
all $1\leqslant k\leqslant h$. Then

  * (i)

$h^{j}(\mathcal{U}_{2h}\otimes{\mathcal{U}^{\prime}_{2}}^{\vee})=h^{j}(\mathcal{U}^{\prime}_{2}\otimes\mathcal{U}_{2h}^{\vee})=0$
for $j=2,3$,

  * (ii)

$\chi(\mathcal{U}_{2h}\otimes{\mathcal{U}^{\prime}_{2}}^{\vee})=\chi(\mathcal{U}^{\prime}_{2}\otimes\mathcal{U}_{2h}^{\vee})=h(4+9e-6b_{e})$,

  * (iii)

$h^{j}(\mathcal{U}_{2h}\otimes\mathcal{U}_{2h}^{\vee})=0$ for $j=2,3$,

  * (iv)

$\chi(\mathcal{U}_{2h}\otimes\mathcal{U}_{2h}^{\vee})=h^{2}(4+9e-6b_{e})$.

###### Proof.

For $h=1$, (iii) and (iv) follow from the proof of Theorem 3.4. As for (i),
the vanishings hold when $\mathcal{U}_{2}^{\prime}=\mathcal{U}_{2}$ once again
by the proof of Theorem 3.4, and thus, by semi-continuity, they also hold for
a general pair
$([\mathcal{U}_{2}^{\prime}],[\mathcal{U}_{2}])\in\mathcal{M}(2)\times\mathcal{M}(2)$.
Similarly, (ii) follows from the proof of Theorem 3.4, since the given $\chi$
is constant as $\mathcal{U}_{2}$ and $\mathcal{U}^{\prime}_{2}$ vary in
$\mathcal{M}(2)$.

We now prove the statements for any integer $h\geqslant 2$ by induction.
Assume therefore that they are satisfied for all positive integers $k$ less
than $h$.

(i) Let $j\in\\{2,3\\}$. By specialization and (4.29) we have

$h^{j}(\mathcal{U}_{2h}\otimes{\mathcal{U}^{\prime}_{2}}^{\vee})\leqslant
h^{j}(\mathcal{F}_{2h}\otimes{\mathcal{U}^{\prime}_{2}}^{\vee})\leqslant
h^{j}(\mathcal{U}^{\prime}_{2}\otimes{\mathcal{U}^{\prime}_{2}}^{\vee})+h^{j}(\mathcal{U}_{2h-2}\otimes{\mathcal{U}^{\prime}_{2}}^{\vee}),$

and the latter are $0$ by induction. Similarly, by specialization and the dual
of (4.29) we have

$h^{j}(\mathcal{U}^{\prime}_{2}\otimes{\mathcal{U}_{2h}}^{\vee})\leqslant
h^{j}(\mathcal{U}^{\prime}_{2}\otimes{\mathcal{F}_{2h}}^{\vee})\leqslant
h^{j}(\mathcal{U}^{\prime}_{2}\otimes{\mathcal{U}^{\prime}_{2}}^{\vee})+h^{j}(\mathcal{U}^{\prime}_{2}\otimes{\mathcal{U}_{2h-2}}^{\vee}),$

which are again $0$ by induction.

(ii) By specialization, (4.29) and induction we have

$\displaystyle\chi(\mathcal{U}_{2h}\otimes{\mathcal{U}^{\prime}_{2}}^{\vee})$
$\displaystyle=$
$\displaystyle\chi(\mathcal{F}_{2h}\otimes{\mathcal{U}^{\prime}_{2}}^{\vee})=\chi(\mathcal{U}^{\prime}_{2}\otimes{\mathcal{U}^{\prime}_{2}}^{\vee})+\chi(\mathcal{U}_{2h-2}\otimes{\mathcal{U}^{\prime}_{2}}^{\vee})$
$\displaystyle=$
$\displaystyle(4+9e-6b_{e})+(h-1)(4+9e-6b_{e})=h(4+9e-6b_{e}).$

Likewise, by specialization, the dual of (4.29) and induction, the same holds
for $\chi(\mathcal{U}^{\prime}_{2}\otimes{\mathcal{U}_{2h}}^{\vee})$.

(iii) Let $j=2,3$; by specialization, (4.29) and its dual we have

$\displaystyle h^{j}(\mathcal{U}_{2h}\otimes\mathcal{U}_{2h}^{\vee})$
$\displaystyle\leqslant$ $\displaystyle
h^{j}(\mathcal{F}_{2h}\otimes\mathcal{F}_{2h}^{\vee})\leqslant
h^{j}(\mathcal{U}^{\prime}_{2}\otimes\mathcal{F}_{2h}^{\vee})+h^{j}(\mathcal{U}_{2h-2}\otimes\mathcal{F}_{2h}^{\vee})$
$\displaystyle\leqslant$ $\displaystyle
h^{j}(\mathcal{U}^{\prime}_{2}\otimes{\mathcal{U}^{\prime}_{2}}^{\vee})+h^{j}(\mathcal{U}^{\prime}_{2}\otimes\mathcal{U}_{2h-2}^{\vee})+h^{j}(\mathcal{U}_{2h-2}\otimes{\mathcal{U}^{\prime}_{2}}^{\vee})+h^{j}(\mathcal{U}_{2h-2}\otimes\mathcal{U}_{2h-2}^{\vee}),$

which are all $0$ by induction.

(iv) By specialization, (4.29) and its dual we have

$\displaystyle\chi(\mathcal{U}_{2h}\otimes\mathcal{U}_{2h}^{\vee})$
$\displaystyle=$
$\displaystyle\chi(\mathcal{F}_{2h}\otimes\mathcal{F}_{2h}^{\vee})=\chi(\mathcal{U}^{\prime}_{2}\otimes\mathcal{F}_{2h}^{\vee})+\chi(\mathcal{U}_{2h-2}\otimes\mathcal{F}_{2h}^{\vee})$
$\displaystyle=$
$\displaystyle\chi(\mathcal{U}^{\prime}_{2}\otimes{\mathcal{U}^{\prime}_{2}}^{\vee})+\chi(\mathcal{U}^{\prime}_{2}\otimes\mathcal{U}_{2h-2}^{\vee})+\chi(\mathcal{U}_{2h-2}\otimes{\mathcal{U}^{\prime}_{2}}^{\vee})+\chi(\mathcal{U}_{2h-2}\otimes\mathcal{U}_{2h-2}^{\vee}).$

By induction, this equals
$(4+9e-6b_{e})+2(h-1)(4+9e-6b_{e})+(h-1)^{2}(4+9e-6b_{e})=h^{2}(4+9e-6b_{e})$.
∎

######  Proposition 4.11.

For all integers $h\geqslant 1$ the scheme $\mathcal{M}(2h)$ is not empty and
its general member $[\mathcal{U}_{2h}]\in\mathcal{M}(2h)$ corresponds to a
rank-$2h$ vector bundle $\mathcal{U}_{2h}$ which is Ulrich w.r.t. $\xi$and
which satisfies

$c_{1}(\mathcal{U}_{2h})=2h\xi+\varphi^{*}\mathcal{O}_{\mathbb{F}_{e}}(h,h(b_{e}-e-2))\;{\rm
and}\;h^{j}(\mathcal{U}_{2h}\otimes\mathcal{U}_{2h}^{\vee})=0,\;j=2,3.$

###### Proof.

We prove this by induction on $h$, the case $h=1$ being satisfied by the
choice of $\mathcal{M}(2)$. Therefore, let $h\geqslant 2$; for general
$[\mathcal{U}_{2h-2}]\in\mathcal{M}(2h-2)$ and
$[\mathcal{U}^{\prime}_{2}]\in\mathcal{M}(2)$, one has

$\displaystyle\dim({\rm
Ext}^{1}(\mathcal{U}_{2h-2},\mathcal{U}^{\prime}_{2}))=h^{1}(\mathcal{U}^{\prime}_{2}\otimes\mathcal{U}_{2h-2}^{\vee}).$

By Lemma 4.10-(i) we have that
$h^{j}(\mathcal{U}^{\prime}_{2}\otimes\mathcal{U}_{2h-2}^{\vee})=0$, for
$j=2,3$. Therefore

$\displaystyle\chi(\mathcal{U}^{\prime}_{2}\otimes\mathcal{U}_{2h-2}^{\vee})=h^{0}(\mathcal{U}^{\prime}_{2}\otimes\mathcal{U}_{2h-2}^{\vee})-h^{1}(\mathcal{U}^{\prime}_{2}\otimes\mathcal{U}_{2h-2}^{\vee})$

so, by specialization and invariance of $\chi$ in irreducible families, we
have

$\displaystyle\dim({\rm
Ext}^{1}(\mathcal{U}_{2h-2},\mathcal{U}^{\prime}_{2}))$ $\displaystyle=$
$\displaystyle
h^{1}(\mathcal{U}^{\prime}_{2}\otimes\mathcal{U}_{2h-2}^{\vee})$
$\displaystyle=$
$\displaystyle-\chi(\mathcal{U}^{\prime}_{2}\otimes\mathcal{U}_{2h-2}^{\vee})+h^{0}(\mathcal{U}^{\prime}_{2}\otimes\mathcal{U}_{2h-2}^{\vee})$
$\displaystyle\geqslant$
$\displaystyle-\chi(\mathcal{U}^{\prime}_{2}\otimes\mathcal{U}_{2h-2}^{\vee})=-\chi(\mathcal{U}^{\prime}_{2}\otimes\mathcal{F}_{2h-2}^{\vee})=$
$\displaystyle=$ $\displaystyle(h-1)(6b_{e}-9e-4)>0$

the latter equality following from Lemma 4.10-(ii) (with $h$ replaced by
$h-1$) whereas the last strict inequality following from $h\geqslant 2$.
Hence, by its very definition, one has that $\mathcal{M}(2h)^{\rm ext}$, and
so also $\mathcal{M}(2h)$, is not empty.

The members of $\mathcal{M}(2h)$ have rank $2h$ and first Chern class
$2h\xi+\varphi^{*}\mathcal{O}_{\mathbb{F}_{e}}(h,h(b_{e}-e-2))$ as in (4.30),
since $c_{1}(\mathcal{U}_{2h})=c_{1}(\mathcal{F}_{2h})$ being constant in
$\mathcal{M}(2h)$. It is immediate that extensions of Ulrich bundles are still
Ulrich, so the general member $\mathcal{U}_{2h}$ of $\mathcal{M}(2h)$ is an
Ulrich bundle. It also satisfies
$h^{j}(\mathcal{U}_{2h}\otimes\mathcal{U}_{2h}^{\vee})=0$ for $j=2,3$ by Lemma
4.10-(iii). ∎

We need to prove that the general member of $\mathcal{M}(2h)$ corresponds to a
slope–stable vector bundle, that $\mathcal{M}(2h)$ is generically smooth and
we need to compute the dimension at its general point $[\mathcal{U}_{2h}]$. We
will again prove all these facts by induction on $h$. Similarly as in the case
$e=0$, we need the following auxiliary result.

######  Lemma 4.12.

Let $\mathcal{F}_{2h}$ correspond to a general member of $\mathcal{M}(2h)^{\rm
ext}$, sitting in an extension like (4.29). Assume furthermore that
$\mathcal{U}^{\prime}_{2}$ and $\mathcal{U}_{2h-2}$ are slope–stable. Let
$\mathcal{D}$ be a destabilizing subsheaf of $\mathcal{F}_{2h}$. Then
$\mathcal{D}^{\vee}\cong{\mathcal{U}_{2}^{\prime}}^{\vee}$ and
$\left(\mathcal{F}_{2h}/\mathcal{D}\right)^{\vee}\cong\mathcal{U}_{2h-2}^{\vee}$.

###### Proof.

The proof is almost identical to that of Lemma 4.6, so the reader is referred
therein. ∎

######  Proposition 4.13.

For all integers $h\geqslant 1$ the scheme $\mathcal{M}(2h)$ is not empty,
generically smooth of dimension

$\displaystyle\dim(\mathcal{M}(2h))=h^{2}(6b_{e}-9e\color[rgb]{0,0,0}-4)+1.$

Its general member corresponds to a slope-stable bundle $\mathcal{U}_{2h}$
whose slope w.r.t. $\xi$ is $\mu(\mathcal{U}_{2h})=8b_{e}-k_{e}-12e-3$.
Furthermore, $\mathcal{M}(2h)$ properly contains the locally closed subscheme
$\mathcal{M}(2h)^{\rm ext}$, namely $\dim(\mathcal{M}(2h)^{\rm
ext})<\dim(\mathcal{M}(2h))$.

###### Proof.

We prove this by induction on $h$, the case $h=1$ being satisfied by
$\mathcal{M}(2)$ as in Theorem 3.4.

Let therefore $h\geqslant 2$ and assume that we have proved the lemma for all
positive integers $k\leqslant h-1$; we will prove it for $h$.

The slope of the members of $\mathcal{M}(2)$ and $\mathcal{M}(2h-2)$ are both
equal to $8b_{e}-k_{e}-12e-3$ as in (4.31). Thus, by [11, Lemma 4.2], the
general member $[\mathcal{F}_{2h}]\in\mathcal{M}(2h)^{\rm ext}$ corresponds to
a simple bundle. Hence, by semi-continuity, also the general member
$[\mathcal{U}_{2h}]\in\mathcal{M}(2h)$ corresponds to a simple bundle which
also satisfies $h^{j}(\mathcal{U}_{2h}\otimes\mathcal{U}_{2h}^{\vee})=0$,
$j=2,3$, by Lemma 4.10-(iii).

Therefore $\mathcal{M}(2h)$ is smooth at $[\mathcal{U}_{2h}]$ (see, e.g., [11,
Prop. 2.10]) with

$\displaystyle\dim(\mathcal{M}(2h))$ $\displaystyle=$ $\displaystyle
h^{1}(\mathcal{U}_{2h}\otimes\mathcal{U}_{2h}^{\vee})=-\chi(\mathcal{U}_{2h}\otimes\mathcal{U}_{2h}^{\vee})+h^{0}(\mathcal{U}_{2h}\otimes\mathcal{U}_{2h}^{\vee})$
$\displaystyle=$ $\displaystyle h^{2}(6b_{e}-9e-4)+1,$

using the facts that $h^{0}(\mathcal{U}_{2h}\otimes\mathcal{U}_{2h}^{\vee})=1$
as $\mathcal{U}_{2h}$ is simple, and that
$\chi(\mathcal{U}_{2h}\otimes\mathcal{U}_{2h}^{\vee})=h^{2}(4+9e-6b_{e})$ by
Lemma 4.10-(iv). This proves that $\mathcal{M}(2h)$ is generically smooth of
the stated dimension.

Finally, we prove that $\mathcal{U}_{2h}$ general is slope–stable and that
$\dim(\mathcal{M}(2h)^{\rm ext})<\dim(\mathcal{M}(2h))$. If $\mathcal{U}_{2h}$
general were not slope-stable then, as in the proof of Proposition 4.8, we
could find a one-parameter family of bundles $\\{\mathcal{U}_{2h}^{(t)}\\}$
over the disc $\Delta$ such that $\mathcal{U}_{2h}^{(t)}$ is a general member
of $\mathcal{M}(2h)$ for $t\neq 0$ and $\mathcal{U}_{2h}^{(0)}$ lies in
$\mathcal{M}(2h)^{\rm ext}$, and such that we have a destabilizing sequence

(4.34) $0\to\mathcal{D}^{(t)}\to\mathcal{U}_{2h}^{(t)}\to\mathcal{G}^{(t)}\to
0$

for $t\neq 0$, which we can take to be saturated, that is, such that
$\mathcal{G}^{(t)}$ is torsion free, whence so that $\mathcal{D}^{(t)}$ and
$\mathcal{G}^{(t)}$ are (Ulrich) vector bundles (see [11, Thm. 2.9] or [5,
(3.2)]). The limit of
$\mathbb{P}(\mathcal{G}^{(t)})\subset\mathbb{P}(\mathcal{U}_{2h}^{(t)})$
defines a subvariety of $\mathbb{P}(\mathcal{U}_{2h}^{(0)})$ of the same
dimension as $\mathbb{P}(\mathcal{G}^{(t)})$, whence a coherent sheaf
$\mathcal{G}^{(0)}$ of rank ${\rm rk}(\mathcal{G}^{(t)})$ with a surjection
$\mathcal{U}_{2h}^{(0)}\twoheadrightarrow\mathcal{G}^{(0)}$. Denoting by
$\mathcal{D}^{(0)}$ its kernel, we have ${\rm rk}(\mathcal{D}^{(0)})={\rm
rk}(\mathcal{D}^{(t)})$ and
$c_{1}(\mathcal{D}^{(0)})=c_{1}(\mathcal{D}^{(t)})$. Hence, (4.34) specializes
to a destabilizing sequence for $t=0$.

Lemma 4.12 yields that ${\mathcal{D}^{(0)}}^{\vee}$ (resp.,
${\mathcal{G}^{(0)}}^{\vee}$) is the dual of a member of $\mathcal{M}(2)$
(resp., of $\mathcal{M}(2h)$). It follows that ${\mathcal{D}^{(t)}}^{\vee}$
(resp., ${\mathcal{G}^{(t)}}^{\vee}$) is a deformation of the dual of a member
of $\mathcal{M}(2))$ (resp., of $\mathcal{M}(2h)$), whence that
${\mathcal{D}^{(t)}}$ (resp., ${\mathcal{G}^{(t)}}$) is a deformation of a
member of $\mathcal{M}(2)$ (resp., $\mathcal{M}(2h)$), as both are locally
free. It follows that $[\mathcal{U}_{2h}^{(t)}]\in\mathcal{M}(2h)^{\rm ext}$
for $t\neq 0$. Thus,

(4.35) $\mathcal{M}(2h)^{\rm ext}=\mathcal{M}(2h).$

On the other hand we have

(4.36) $\dim(\mathcal{M}(2h)^{\rm ext})\leqslant\dim(\mathbb{P}({\rm
Ext}^{1}(\mathcal{U}_{2h-2},\mathcal{U}^{\prime}_{2})))+\dim(\mathcal{M}(2h-2))+\dim(\mathcal{M}(2)),$

for general $[\mathcal{U}_{2h-2}]\in\mathcal{M}(2h-2)$ and
$[\mathcal{U}^{\prime}_{2}]\in\mathcal{M}(2)$. As $\mathcal{U}_{2h-2}$ and
$\mathcal{U}^{\prime}_{2}$ are slope–stable by induction, of the same slope,
we have $h^{0}(\mathcal{U}^{\prime}_{2}\otimes\mathcal{U}_{2h-2}^{\vee})=0$.
Lemma 4.10-(i), (ii) and (iii) thus yield

$h^{1}(\mathcal{U}^{\prime}_{2}\otimes\mathcal{U}_{2h-2}^{\vee})=-\chi(\mathcal{U}^{\prime}_{2}\otimes\mathcal{U}_{2h-2}^{\vee})=(h-1)(6b_{e}-9e-4).$

Hence, by (4.36) and (4.2) we have

$\displaystyle\dim(\mathcal{M}(2h)^{\rm ext})$ $\displaystyle\leqslant$
$\displaystyle(h-1)(6b_{e}-9e-4)-1+\left[(h-1)^{2}(6b_{e}-9e-4)+1\right]+(6b_{e}-9e-3)$
$\displaystyle=$
$\displaystyle(h^{2}-h+1)(6b_{e}-9e-4)+1<h^{2}(6b_{e}-9e-4)+1=\dim(\mathcal{M}(2h)),$

as it easily follows from the fact that $h\geqslant 2$. The previous
inequality shows that $\dim(\mathcal{M}(2h)^{\rm ext})<\dim(\mathcal{M}(2h))$,
as stated; in particular (4.35) is a contradiction, which forces also
$\mathcal{U}_{2h}$ general to be slope-stable. ∎

Odd ranks: in odd ranks, set $r=2h+1$, for $h\geqslant 1$ an integer. The
first step is given by the following result.

######  Theorem 4.14.

Let
$(X_{e},\xi)\cong(\mathbb{P}({\mathcal{E}_{e}}),{\mathcal{O}}_{\mathbb{P}({\mathcal{E}_{e}})}(1))$
be a $3$-fold scroll over $\mathbb{F}_{e}$, with $e>0$, and $\mathcal{E}_{e}$
be as in Assumptions 1.8. Let $\varphi:X_{e}\to\mathbb{F}_{e}$ be the scroll
map and $F$ be the $\varphi$\- fibre. Then the moduli space of rank-$3$ vector
bundles on $X_{e}$, which are Ulrich w.r.t. $\xi$, with first Chern class

(4.37) $c_{1}=3\xi+\varphi^{*}\mathcal{O}_{\mathbb{F}_{e}}(3,b_{e}-3)$

is not empty and it contains a generically smooth component $\mathcal{M}(3)$
of dimension

$\dim(\mathcal{M}(3))=2(6b_{e}-9e-4),$

whose general point $[\mathcal{U}_{3}]$ corresponds to a slope-stable vector
bundle $\mathcal{U}_{3}$ of slope w.r.t. $\xi$

(4.38) $\displaystyle\mu(\mathcal{U}_{3})=8b_{e}-k_{e}-12e-3.$

Recalling the expression of the first Chern class in Main Theorem-(c) for
$r=3$, this coincides with that in (4.37) since $\frac{r-3}{2}=0$ in this
case.

###### Proof of Theorem 4.14.

Similarly as in the proof of Theorem 3.4, we consider a rank-3 vector bundle
$\mathcal{A}_{3}$ on $\mathbb{F}_{e}$ which is Ulrich w.r.t.
$c_{1}(\mathcal{E}_{e})=3C_{e}+b_{e}f=\mathcal{O}_{\mathbb{F}_{e}}(3,b_{e})$,
for any $e>0$.

Such a bundle certainly exists, as it follows from [2, Prop. 5.3, Thm. 6.2];
indeed, using same notation therein, we let $r=3$,
$D\in|\mathcal{O}_{\mathbb{F}_{e}}(\alpha,\beta)|$ be any divisor on
$\mathbb{F}_{e}$, polarization
$c_{1}(\mathcal{E}_{e})=3C_{e}+b_{e}f=\mathcal{O}_{\mathbb{F}_{e}}(3,b_{e})$
(in [2] the polarization is denoted by $h$), it follows that any pair $(3,D)$,
satisfying

$T=24e+\frac{b_{e}}{3}(\alpha+3)\in\mathbb{Z}\;\;\mbox{and}\;\;6+\frac{9e}{b_{e}}\leqslant\alpha\leqslant
15-\frac{9e}{b_{e}}$

and such that

$\displaystyle
D(3C_{e}+b_{e}f)=\frac{3}{2}\big{(}2(3C_{e}+b_{e}f)^{2}+(3C_{e}+b_{e}f)K_{\mathbb{F}_{e}}\big{)},$

is an admissable Ulrich pair on $\mathbb{F}_{e}$ w.r.t. the polarization
$\mathcal{O}_{\mathbb{F}_{e}}(3,b_{e})$ (cf. [2, Def. 5.1, Prop. 5.3]).

Because $b_{e}\geqslant 3e+2$ from (1.6), then one has
$7\leqslant\alpha\leqslant 14$. If we therefore take e.g. $\alpha=12$ and
$\beta=4b_{e}-3$, an easy check shows that the previous relations hold true.
Hence the pair
$(3,\mathcal{O}_{\mathbb{F}_{e}}(12,4b_{e}-3))=(3,\;12C_{e}+(4b_{e}-3)f)$ is
an admissable Ulrich pair, thus $\mathbb{F}_{e}$ certainly supports rank-$3$
vector bundles, say $\mathcal{A}_{3}$, which are Ulrich w.r.t.
$c_{1}(\mathcal{E}_{e})=3C_{e}+b_{e}f$, such that
$c_{1}(\mathcal{A}_{3})=12C_{e}+(4b_{e}-3)f=\mathcal{O}_{\mathbb{F}_{e}}(12,4b_{e}-3)$
and which are given as cokernels of appropriate injective vector bundle maps,
(see [2, Thm. 4.1, (4.1) and Thm. 6.2]). They moreover satisfies
$Ext^{2}(\mathcal{A}_{3},\mathcal{A}_{3})=h^{2}(\mathcal{A}_{3}\otimes\mathcal{A}_{3}^{\vee})=0$
(cf. [2, Lemma 6.3]). From [2, Prop. 6.4], any such bundle $\mathcal{A}_{3}$
is slope-stable, hence simple i.e.
$h^{0}(\mathcal{A}_{3}\otimes\mathcal{A}_{3}^{\vee})=1$. These bundles belong
to the moduli space $\mathcal{M}_{c_{1}(\mathcal{E}_{e})}^{U}(3,c_{1},c_{2})$
of rank-3 vector bundles of given Chern classes $c_{1}$, $c_{2}$, which are
Ulrich w.r.t. $c_{1}(\mathcal{E}_{e})$, which is smooth, irreducible, of
dimension

$\dim(\mathcal{M}_{c_{1}(\mathcal{E}_{e})}^{U}(3,c_{1},c_{2}))=h^{1}(\mathcal{A}_{3}\otimes\mathcal{A}_{3}^{\vee})=1-\chi(\mathcal{A}_{3}\otimes\mathcal{A}_{3}^{\vee})=2(6b_{e}-9e-4)$

(cf. [2, Prop. 6.4]). All points of
$\mathcal{M}_{c_{1}(\mathcal{E}_{e})}^{U}(3,c_{1},c_{2})$ correspond to
slope–stable Ulrich bundles, since on $\mathbb{F}_{e}$ there are no Ulrich
line bundles w.r.t.
$c_{1}(\mathcal{E}_{e})=3C_{e}+b_{e}f=\mathcal{O}_{\mathbb{F}_{e}}(3,b_{e})$
from [4, Thm. 2.1] (cf. Theorem 1.4-(b) or [5, Sect.3, (3.2)]).

Using Theorem 1.7, we therefore first consider on $\mathbb{F}_{e}$ bundles
$\mathcal{H}_{3}:=\mathcal{A}_{3}(-c_{1}({\mathcal{E}}_{e}))$, which are of
rank 3, with first Chen class

$c_{1}(\mathcal{H}_{3})=c_{1}(\mathcal{A}_{3})-3c_{1}(\mathcal{E}_{e})=3C_{e}+(b_{e}-3)f=\mathcal{O}_{\mathbb{F}_{e}}(3,b_{e}-3),$

and then, on $X_{e}$, we take

(4.39) $\mathcal{V}_{3}=\xi\otimes\varphi^{*}(\mathcal{H}_{3}),$

which are rank-3 vector bundles on $X_{e}$ which are Ulrich w.r.t. $\xi$, as
it follows from Theorem 1.7, whose first Chern class
$c_{1}:=c_{1}(\mathcal{V}_{3})$ is as in (4.37). From Main Theorem–(a), when
$e>0$, there are no Ulrich line bundles w.r.t. $\xi$ thus, once again by
Theorem 1.4-(b) or [5, Sect.3, (3.2)], $\mathcal{V}_{3}$ is also slope-stable
w.r.t. $\xi$, whose slope is as in (4.38); in particular it is also simple.
Moreover, by Leray’s isomorphism, projection formula and independence on the
twists, one has

(4.40)
$h^{j}(X_{e},\mathcal{V}_{3}\otimes\mathcal{V}_{3}^{\vee})=h^{j}(\mathbb{F}_{e},\mathcal{H}_{3}\otimes\mathcal{H}_{3}^{\vee})=h^{j}(\mathbb{F}_{e},\mathcal{A}_{3}\otimes\mathcal{A}_{3}^{\vee}),\;0\leqslant
j\leqslant 3,$

which implies that bundles $\mathcal{V}_{3}$ fill–up a smooth modular
component $\mathcal{M}(3)$ of the moduli space of rank-3 vector bundles on
$X_{e}$, which are Ulrich w.r.t. $\xi$, whose first Chern class and slope
w.r.t. $\xi$ are as in (4.37), (4.38), respectively, and whose dimension is

$\dim(\mathcal{M}(3))=\dim(\mathcal{M}_{c_{1}(\mathcal{E}_{e})}^{U}(3,c_{1},c_{2}))=2(6b_{e}-9e-4)$

as in [2, Prop. 6.4]. ∎

Together with $\mathcal{M}(3)$ as in Theorem 4.14, consider also the
irreducible scheme $\mathcal{M}(2)$ to be the component $\mathcal{M}$ as in
Theorem 3.4. Recall that $\mathcal{M}(2)$ is generically smooth, of dimension
$\dim(\mathcal{M}(2))=6b_{e}-9e-3$ and that the general member
$[\mathcal{U}_{2}]\in\mathcal{M}(2)$ corresponds to a rank-$2$ vector bundle
on $X_{e}$ which is Ulrich and slope-stable w.r.t. $\xi$, of slope
$\mu(\mathcal{U}_{2})=8b_{e}-k_{e}-12e-3$ and whose first Chern class is
$c_{1}(\mathcal{U}_{2})=2\xi+\varphi^{*}\mathcal{O}_{\mathbb{F}_{e}}(1,b_{e}-e-2)$.

As done in the even rank case, if we assume by induction that we have
constructed, for some integer $h\geqslant 2$, an irreducible scheme
$\mathcal{M}(2h-1)$ which is a generically smooth modular component of the
moduli space of vector bundles of rank $r=2h-1$ on $X_{e}$, which are Ulrich
w.r.t. $\xi$, whose first Chern class is as in Main Theorem–(c) and whose
general point $[\mathcal{U}_{2h-1}]\in\mathcal{M}(2h-1)$ is slope-stable, with
slope $\mu(\mathcal{U}_{2h-1})=8b_{e}-k_{e}-12e-3$ w.r.t. $\xi$, we may
therefore inductively define $\mathcal{M}(2h+1)$ to be the (possibly empty a
priori) component of the moduli space of Ulrich bundles on $(X_{e},\xi)$
containing bundles $\mathcal{F}_{2h+1}$ that are non–trivial extensions of the
form

(4.41) $0\to\mathcal{U}_{2}\to\mathcal{F}_{2h+1}\to\mathcal{U}_{2h-1}\to 0,$

with $[\mathcal{U}_{2}]\in\mathcal{M}(2)$,
$[\mathcal{U}_{2h-1}]\in\mathcal{M}(2h-1)$ general, and we let
$\mathcal{M}(2h+1)^{\rm ext}$ denote the locus in $\mathcal{M}(2h+1)$ of
bundles that are non-trivial extensions of the form (4.41).

In the next results we will prove that non-trivial extensions as in (4.41)
always exist (cf. the proof of Proposition 4.16) and that
$\mathcal{M}(2h+1)^{\rm ext}\neq\emptyset$, so in particular
$\mathcal{M}(2h+1)\neq\emptyset$, for any $h\geqslant 2$.

All vector bundles $\mathcal{F}_{2h+1}$, $h\geqslant 2$, recursively defined
as in (4.41) are of rank $2h+1$ and Ulrich w.r.t. $\xi$, since extensions of
bundles which are Ulrich w.r.t. $\xi$ are again Ulrich w.r.t. $\xi$. Their
first Chern class is given by

$c_{1}(\mathcal{F}_{2h+1})=c_{1}(\mathcal{U}_{2})+c_{1}(\mathcal{U}_{2h-1})=c_{1}(\mathcal{U}_{3})+(h-1)c_{1}(\mathcal{U}_{2}),$

i.e.

(4.42)
$c_{1}(\mathcal{F}_{2h+1}):=(2h+1)\xi+\varphi^{*}\mathcal{O}_{\mathbb{F}_{e}}(3,b_{e}-3)+\varphi^{*}\mathcal{O}_{\mathbb{F}_{e}}(h-1,(h-1)(b_{e}-e-2)),$

where $h=\frac{r-1}{2}$, whose slope w.r.t. $\xi$ is

(4.43) $\mu(\mathcal{F}_{2h+1})=8b_{e}-k_{e}-12e-3.$

From Theorem 1.4-(a), any such bundle $\mathcal{F}_{2h+1}$ is strictly
semistable and slope-semistable, being extension of Ulrich bundles of the same
slope w.r.t. $\xi$.

######  Lemma 4.15.

Let $h\geqslant 1$ be an integer and assume $\mathcal{M}(2k+1)\neq\emptyset$
for all $1\leqslant k\leqslant h$. Then

  * (i)

$h^{j}(\mathcal{U}_{2h+1}\otimes{\mathcal{U}_{2}}^{\vee})=h^{j}(\mathcal{U}_{2}\otimes\mathcal{U}_{2h+1}^{\vee})=0$
for $j=2,3$,

  * (ii)

$\chi(\mathcal{U}_{2h+1}\otimes{\mathcal{U}_{2}}^{\vee})=(h-1)(4+9e-6b_{e})+(15e-10b_{e}+3)$
whereas
$\chi(\mathcal{U}_{2}\otimes{\mathcal{U}}^{\vee}_{2h+1})=(h-1)(4+9e-6b_{e})+(12e-8b_{e}-3)$,

  * (iii)

$h^{j}(\mathcal{U}_{2h+1}\otimes\mathcal{U}^{\vee}_{2h+1})=0$ for $j=2,3$,

  * (iv)

$\chi(\mathcal{U}_{2h+1}\otimes\mathcal{U}^{\vee}_{2h+1})=1+9(h-1)(3e-2b_{e})+((h-1)^{2}+2)(4+9e-6b_{e})$.

###### Proof.

We prove it by induction on $h$. We start with $h=1$, namely $r=3$. In this
case, (iii) and (iv) follow from Theorem 4.14. Indeed, take
$\mathcal{V}_{3}=\xi\otimes\varphi^{*}(\mathcal{H}_{3})$ so, by (4.40) and
semi–continuity, one has

$h^{j}(X_{e},\mathcal{U}_{3}\otimes\mathcal{U}_{3}^{\vee})=h^{j}(X_{e},\mathcal{V}_{3}\otimes\mathcal{V}_{3}^{\vee})=h^{j}(\mathbb{F}_{e},\mathcal{A}_{3}\otimes\mathcal{A}_{3}^{\vee})=0,\;2\leqslant
j\leqslant 3,$

which proves (iii) for $h=1$; moreover, by stability and semi–continuity, one
has
$h^{0}(X_{e},\mathcal{U}_{3}\otimes\mathcal{U}_{3}^{\vee})=h^{0}(X_{e},\mathcal{V}_{3}\otimes\mathcal{V}_{3}^{\vee})=1$,
therefore from the vanishings above one has

$\chi(X_{e},\mathcal{U}_{3}\otimes\mathcal{U}_{3}^{\vee})=1-h^{1}(X_{e},\mathcal{U}_{3}\otimes\mathcal{U}_{3}^{\vee})=1-\dim(\mathcal{M}(3))=1-2(6b_{e}-9e-4)=1+2(4+9e-6b_{e}),$

which also proves (iv) for $h=1$.

As for (i), by semi–continuity, one has

$h^{j}(X_{e},\mathcal{U}_{3}\otimes\mathcal{U}^{\vee}_{2})\leqslant
h^{j}(X_{e},\mathcal{V}_{3}\otimes\mathcal{V}^{\vee}_{2})\;\;{\rm
and}\;\;h^{j}(X_{e},\mathcal{U}_{2}\otimes\mathcal{U}^{\vee}_{3})\leqslant
h^{j}(X_{e},\mathcal{V}_{2}\otimes\mathcal{V}^{\vee}_{3}),$

for any $0\leqslant j\leqslant 3$, where $\mathcal{V}_{2}=\mathcal{V}$ as in
the proof of Theorem 3.4, i.e.
$\mathcal{V}_{2}=\xi\otimes\varphi^{*}(\mathcal{H})$, where
$c_{1}(\mathcal{H})=C_{e}+(b_{e}-e-2)f={\mathcal{O}}_{\mathbb{F}_{e}}(1,b_{e}-e-2)$.
From Leray’s isomorphism, projection formula and invariance under twists, it
follows that, for any $0\leqslant j\leqslant 3$, one has

$h^{j}(X_{e},\mathcal{V}_{3}\otimes\mathcal{V}_{2}^{\vee})=h^{j}(\mathbb{F}_{e},\mathcal{H}_{3}\otimes\mathcal{H}^{\vee})=h^{j}(\mathbb{F}_{e},\mathcal{A}_{3}\otimes\mathcal{H}_{1}^{\vee})$

and

$h^{j}(X_{e},\mathcal{V}_{2}\otimes\mathcal{V}_{3}^{\vee})=h^{j}(\mathbb{F}_{e},\mathcal{H}\otimes\mathcal{H}_{3}^{\vee})=h^{j}(\mathbb{F}_{e},\mathcal{H}_{1}\otimes\mathcal{A}_{3}^{\vee}),$

where $\mathcal{A}_{3}=\mathcal{H}_{3}(c_{1}(\mathcal{E}_{e}))$ is the rank-3
vector bundle on $\mathbb{F}_{e}$ as in the proof of Theorem 4.14 whereas
$\mathcal{H}_{1}=\mathcal{H}(c_{1}(\mathcal{E}_{e}))$ is the rank-2 vector
bundle on $\mathbb{F}_{e}$ as in (3.30), which are both Ulrich w.r.t.
$c_{1}(\mathcal{E}_{e})={\mathcal{O}}_{\mathbb{F}_{e}}(3,b_{e})$. Therefore,
for dimension reasons,
$h^{3}(\mathbb{F}_{e},\mathcal{A}_{3}\otimes\mathcal{H}_{1}^{\vee})=h^{3}(\mathbb{F}_{e},\mathcal{H}_{1}\otimes\mathcal{A}_{3}^{\vee})=0$,
which implies therefore
$h^{3}(X_{e},\mathcal{U}_{3}\otimes\mathcal{U}^{\vee}_{2})=h^{3}(X_{e},\mathcal{U}_{2}\otimes\mathcal{U}^{\vee}_{3})=0$,
as desired.

To prove that
$h^{2}(\mathbb{F}_{e},\mathcal{A}_{3}\otimes\mathcal{H}_{1}^{\vee})=0$ recall
that, from [2, Prop. 6.4], $\mathcal{A}_{3}$ arises as the cokernel of a
general injective vector bundle map of the form

(4.44)
$0\to\mathcal{O}_{\mathbb{F}_{e}}(2,b_{e}-e-1)^{\oplus\gamma}\to\mathcal{O}_{\mathbb{F}_{e}}(2,b_{e}-e)^{\oplus\delta}\oplus\mathcal{O}_{\mathbb{F}_{e}}(3,b_{e}-1)^{\oplus\tau}\to\mathcal{A}_{3}\to
0,$

for suitable positive integers $\gamma,\;\delta,\,\tau$ as in [2, Thm. 4.1],
in our situation one can see that $\gamma=b_{e}-3e+3,\;\delta=b-3e,\,\tau=6$.
Tensoring (4.44) by $\mathcal{H}_{1}^{\vee}$ gives that

$h^{2}(\mathbb{F}_{e},\mathcal{A}_{3}\otimes\mathcal{H}_{1}^{\vee})\leqslant\delta\,h^{2}(\mathbb{F}_{e},\mathcal{H}_{1}^{\vee}\otimes\mathcal{O}_{\mathbb{F}_{e}}(2,b_{e}-e))+\tau\,h^{2}(\mathbb{F}_{e},\mathcal{H}_{1}^{\vee}\otimes\mathcal{O}_{\mathbb{F}_{e}}(3,b_{e}-1)),$

therefore to prove that
$h^{2}(\mathbb{F}_{e},\mathcal{A}_{3}\otimes\mathcal{H}_{1}^{\vee})=0$ it is
enough to show that

(4.45)
$h^{2}(\mathbb{F}_{e},\mathcal{H}_{1}^{\vee}\otimes\mathcal{O}_{\mathbb{F}_{e}}(2,b_{e}-e))=h^{2}(\mathbb{F}_{e},\mathcal{H}_{1}^{\vee}\otimes\mathcal{O}_{\mathbb{F}_{e}}(3,b_{e}-1))=0.$

Taking into account that $\mathcal{H}_{1}$ is of rank 2, i.e.
$\mathcal{H}^{\vee}_{1}\cong\mathcal{H}_{1}(-c_{1}(\mathcal{H}_{1}))$, where
$c_{1}(\mathcal{H}_{1})=C_{e}+(b_{e}-e-2)f=\mathcal{O}_{\mathbb{F}_{e}}(1,b_{e}-e-2)$,
from (3.30) one has that $\mathcal{H}_{1}(-c_{1}(\mathcal{H}_{1}))$ fits in

(4.46)
$0\to\mathcal{O}_{\mathbb{F}_{e}}(2,e+1)\to\mathcal{H}_{1}(-c_{1}(\mathcal{H}_{1}))\to
I_{Z}\otimes\mathcal{O}_{\mathbb{F}_{e}}(3,b_{e}+1)\to 0,$

where $Z\subset\mathbb{F}_{e}$ is a general zero-dimensional subscheme of
$\mathbb{F}_{e}$ of length $\ell(Z)=2b_{e}-3e$. Thus, from (4.46), to prove
(4.45) it is enough to prove that

$h^{2}(\mathcal{O}_{\mathbb{F}_{e}}(4,b_{e}+1))=h^{2}(\mathcal{O}_{\mathbb{F}_{e}}(5,b_{e}+e))=h^{2}(I_{Z}\otimes\mathcal{O}_{\mathbb{F}_{e}}(5,2b_{e}-e+1))=h^{2}(I_{Z}\otimes\mathcal{O}_{\mathbb{F}_{e}}(6,2b_{e}))=0,$

which trivially hold true from either Serre duality on $\mathbb{F}_{e}$ or
from the use of the exact sequence $0\to
I_{Z}\to\mathcal{O}_{\mathbb{F}_{e}}\to\mathcal{O}_{Z}\to 0$. This shows that
$h^{2}(\mathbb{F}_{e},\mathcal{A}_{3}\otimes\mathcal{H}_{1}^{\vee})=0$.

To prove instead that
$h^{2}(\mathbb{F}_{e},\mathcal{H}_{1}\otimes\mathcal{A}_{3}^{\vee})=0$, one
considers the dual exact sequence of (4.44), i.e.

$0\to\mathcal{A}_{3}^{\vee}\to\mathcal{O}_{\mathbb{F}_{e}}(-2,-b_{e}+e)^{\oplus\delta}\oplus\mathcal{O}_{\mathbb{F}_{e}}(-3,-b_{e}+1)^{\oplus\tau}\to\mathcal{O}_{\mathbb{F}_{e}}(-2,-b_{e}+e+1)^{\oplus\gamma}\to
0,$

and tensor it by $\mathcal{H}_{1}$, which gives

$\displaystyle
h^{2}(\mathbb{F}_{e},\mathcal{H}_{1}\otimes\mathcal{A}_{3}^{\vee})\leqslant$
$\displaystyle\delta\,h^{2}(\mathcal{H}_{1}\otimes\mathcal{O}_{\mathbb{F}_{e}}(-2,-b_{e}+e))+\tau\,h^{2}(\mathcal{H}_{1}\otimes\mathcal{O}_{\mathbb{F}_{e}}(-3,-b_{e}+1))+$
$\displaystyle\gamma\,h^{1}(\mathcal{H}_{1}\otimes\mathcal{O}_{\mathbb{F}_{e}}(-2,-b_{e}+e+1)).$

From (3.30), it is enough to prove

$h^{2}(\mathcal{O}_{\mathbb{F}_{e}}(1,e-1))=h^{2}(\mathcal{O}_{\mathbb{F}_{e}})=h^{1}(\mathcal{O}_{\mathbb{F}_{e}}(1,e))=0,$

and

$h^{2}(I_{Z}\otimes\mathcal{O}_{\mathbb{F}_{e}}(2,b_{e}-1))=h^{2}(I_{Z}\otimes\mathcal{O}_{\mathbb{F}_{e}}(1,b_{e}-e))=h^{1}(I_{Z}\otimes\mathcal{O}_{\mathbb{F}_{e}}(2,b_{e}))=0.$

The vanishings of the $h^{2}$’s easily follow from the same reasoning as
above; by Leray’s isomorphism and projection formula one gets
$h^{1}(\mathcal{O}_{\mathbb{F}_{e}}(1,e))=h^{1}(\mathbb{P}^{1},\mathcal{O}_{\mathbb{P}^{1}}(e)\oplus\mathcal{O}_{\mathbb{P}^{1}})=0$.
At last, if we take into account that $Z$ is a zero-dimensional subscheme of
length $\ell(Z)=2b_{e}-3e$ of general points on $\mathbb{F}_{e}$, then $Z$
imposes independent conditions on the linear system
$|\mathcal{O}_{\mathbb{F}_{e}}(2,b_{e})|$ on $\mathbb{F}_{e}$, which is of
dimension $3b_{e}-3e+2>\ell(Z)=2b_{e}-3e$ since $b_{e}\geqslant 3e+2$ by
(1.6); this means that
$h^{1}(I_{Z}\otimes\mathcal{O}_{\mathbb{F}_{e}}(2,b_{e}))=h^{1}(\mathcal{O}_{\mathbb{F}_{e}}(2,b_{e}))$
and the latter is zero by standard computations. This shows that
$h^{2}(\mathbb{F}_{e},\mathcal{H}_{1}\otimes\mathcal{A}_{3}^{\vee})=0$, which
concludes the proof of (i) for $h=1$.

Finally, to prove (ii) for $h=1$, from invariance of $\chi$ in irreducible
families and from above one has

$\chi(X_{e},\mathcal{U}_{3}\otimes\mathcal{U}_{2}^{\vee})=\chi(X_{e},\mathcal{V}_{3}\otimes\mathcal{V}_{2}^{\vee})=\chi(\mathbb{F}_{e},\mathcal{H}_{3}\otimes\mathcal{H}^{\vee})=\chi(\mathbb{F}_{e},\mathcal{A}_{3}\otimes\mathcal{H}_{1}^{\vee})$

and

$\chi(X_{e},\mathcal{U}_{2}\otimes\mathcal{U}_{3}^{\vee})=\chi(X_{e},\mathcal{V}_{3}\otimes\mathcal{V}_{2}^{\vee})=\chi(\mathbb{F}_{e},\mathcal{H}\otimes\mathcal{H}_{3}^{\vee})=\chi(\mathbb{F}_{e},\mathcal{H}_{1}\otimes\mathcal{A}_{3}^{\vee}).$

Since $\mathcal{A}_{3}$ and $\mathcal{H}_{1}$ are both Ulrich bundles w.r.t.
$c_{1}(\mathcal{E}_{e})=3C_{e}+b_{e}f=\mathcal{O}_{\mathbb{F}_{e}}(3,b_{e})$
on $\mathbb{F}_{e}$, let us consider the smooth projective model
$(S,\mathcal{O}_{S}(1))\cong(\mathbb{F}_{e},\mathcal{O}_{\mathbb{F}_{e}}(3,b_{e}))$,
which is a surface, in a suitable projective space, of degree
$d:=\deg(S)=(c_{1}(\mathcal{E}_{e}))^{2}=6b_{e}-9e$. Thus, from [11, Prop.
2.12], one has

$\chi(\mathbb{F}_{e},\mathcal{A}_{3}\otimes\mathcal{H}_{1}^{\vee})=3c_{1}(\mathcal{H}_{1})\cdot
K_{S}-c_{1}(\mathcal{A}_{3})\cdot c_{1}(\mathcal{H}_{1})+6(2d-2)$

and

$\chi(\mathbb{F}_{e},\mathcal{H}_{1}\otimes\mathcal{A}_{3}^{\vee})=2c_{1}(\mathcal{A}_{3})\cdot
K_{S}-c_{1}(\mathcal{A}_{3})\cdot c_{1}(\mathcal{H}_{1})+6(2d-2),$

where $p_{a}(S)=1$ and $K_{S}=-2C_{e}-(e+2)f$. Using that
$c_{1}(\mathcal{H}_{1})=7C_{e}+(3b_{e}-e-2)f$ and
$c_{1}(\mathcal{A}_{3})=12C_{e}+(4b_{e}-3)f$, one gets

$c_{1}(\mathcal{A}_{3})\cdot
c_{1}(\mathcal{H}_{1})=64b_{e}-96e-45,\;c_{1}(\mathcal{A}_{3})\cdot
K_{S}=12e-8b_{e}-18,$ $c_{1}(\mathcal{H}_{1})\cdot
K_{S}=9e-6b_{e}-10,\;6(2d-2)=12(d-1)=72b_{e}-108e-12;$

plugging these computations in the previous formulas, one gets

$\chi(\mathbb{F}_{e},\mathcal{A}_{3}\otimes\mathcal{H}_{1}^{\vee})=15e-10b_{e}+3\;\;{\rm
and}\;\;\chi(\mathbb{F}_{e},\mathcal{H}_{1}\otimes\mathcal{A}_{3}^{\vee})=12e-8b_{e}-3,$

which concludes the proof of (ii) for $h=1$.

We now prove by induction that all statements hold true also for any integer
$h\geqslant 2$. Assume therefore that they are satisfied for all positive
integers $k$ such that $1\leqslant k\leqslant h-1$.

(i) Let $j\in\\{2,3\\}$. By specialization and (4.41) tensored with
${\mathcal{U}_{2}}^{\vee}$, we have

$h^{j}(\mathcal{U}_{2h+1}\otimes{\mathcal{U}_{2}}^{\vee})\leqslant
h^{j}(\mathcal{F}_{2h+1}\otimes{\mathcal{U}_{2}}^{\vee})\leqslant
h^{j}(\mathcal{U}_{2}\otimes{\mathcal{U}_{2}}^{\vee})+h^{j}(\mathcal{U}_{2h-1}\otimes{\mathcal{U}_{2}}^{\vee}),$

and the latter are $0$ by induction. Similarly, by specialization and using
the dual of (4.41) tensored with ${\mathcal{U}_{2}}$ we have

$h^{j}(\mathcal{U}_{2}\otimes{\mathcal{U}^{\vee}_{2h+1}})\leqslant
h^{j}(\mathcal{U}_{2}\otimes{\mathcal{F}^{\vee}_{2h+1}})\leqslant
h^{j}(\mathcal{U}_{2}\otimes{\mathcal{U}_{2}}^{\vee})+h^{j}(\mathcal{U}_{2}\otimes{\mathcal{U}^{\vee}_{2h-1}}),$

which are again $0$ by induction.

(ii) By specialization, (4.41) tensored with ${\mathcal{U}_{2}}^{\vee}$ and
induction we have

$\displaystyle\chi(\mathcal{U}_{2h+1}\otimes{\mathcal{U}_{2}}^{\vee})$
$\displaystyle=$
$\displaystyle\chi(\mathcal{F}_{2h+1}\otimes{\mathcal{U}_{2}}^{\vee})=\chi(\mathcal{U}_{2}\otimes{\mathcal{U}_{2}}^{\vee})+\chi(\mathcal{U}_{2h-1}\otimes{\mathcal{U}_{2}}^{\vee})$
$\displaystyle=$
$\displaystyle(4+9e-6b_{e})+(h-2)(4+9e-6b_{e})+(15e-10b_{e}+3).$

Likewise, by specialization, the dual of (4.41) and induction, we have

$\displaystyle\chi(\mathcal{U}_{2}\otimes{\mathcal{U}}^{\vee}_{2h+1})$
$\displaystyle=$
$\displaystyle\chi(\mathcal{U}_{2}\otimes{\mathcal{F}}^{\vee}_{2h+1})=\chi(\mathcal{U}_{2}\otimes{\mathcal{U}_{2}}^{\vee})+\chi(\mathcal{U}_{2}\otimes{\mathcal{U}}^{\vee}_{2h-1})$
$\displaystyle=$
$\displaystyle(4+9e-6b_{e})+(h-2)(4+9e-6b_{e})+(12e-8b_{e}-3).$

(iii) Let $j=2,3$; by specialization, (4.41) and its dual we have

$\displaystyle h^{j}(\mathcal{U}_{2h+1}\otimes\mathcal{U}_{2h+1}^{\vee})$
$\displaystyle\leqslant$ $\displaystyle
h^{j}(\mathcal{F}_{2h+1}\otimes\mathcal{F}_{2h+1}^{\vee})\leqslant$
$\displaystyle\leqslant$ $\displaystyle
h^{j}(\mathcal{U}_{2}\otimes\mathcal{F}_{2h+1}^{\vee})+h^{j}(\mathcal{U}_{2h-1}\otimes\mathcal{F}_{2h+1}^{\vee})$
$\displaystyle\leqslant$ $\displaystyle
h^{j}(\mathcal{U}_{2}\otimes{\mathcal{U}_{2}}^{\vee})+h^{j}(\mathcal{U}_{2}\otimes\mathcal{U}_{2h-1}^{\vee})+$
$\displaystyle+h^{j}(\mathcal{U}_{2h-1}\otimes{\mathcal{U}_{2}}^{\vee})+h^{j}(\mathcal{U}_{2h-1}\otimes\mathcal{U}_{2h-1}^{\vee}),$

which are all $0$ by induction.

(iv) By specialization, (4.41) and its dual we have

$\displaystyle\chi(\mathcal{U}_{2h+1}\otimes\mathcal{U}^{\vee}_{2h+1})$
$\displaystyle=$
$\displaystyle\chi(\mathcal{F}_{2h+1}\otimes\mathcal{F}^{\vee}_{2h+1})=\chi(\mathcal{U}_{2}\otimes\mathcal{F}^{\vee}_{2h+1})+\chi(\mathcal{U}_{2h-1}\otimes\mathcal{F}^{\vee}_{2h+1})$
$\displaystyle=$
$\displaystyle\chi(\mathcal{U}_{2}\otimes{\mathcal{U}_{2}}^{\vee})+\chi(\mathcal{U}_{2}\otimes\mathcal{U}^{\vee}_{2h-1})+\chi(\mathcal{U}_{2h-1}\otimes{\mathcal{U}_{2}}^{\vee})+\chi(\mathcal{U}_{2h-1}\otimes\mathcal{U}^{\vee}_{2h-1}).$

By induction, this equals

$\displaystyle(4+9e-6b_{e})+(h-2)(4+9e-6b_{e})+(h-2)(4+9e-6b_{e})+(27e-18b_{e})+1+$
$\displaystyle+9(h-2)(3e-2b_{e})+((h-2)^{2}+2)(4+9e-6b_{e})=$
$\displaystyle=1+9(h-1)(3e-2b_{e})+(4+9e-6b_{e})(1+h-2+h-2+(h-2)^{2}+2)=$
$\displaystyle=1+9(h-1)(3e-2b_{e})+((h-1)^{2}+2)(4+9e-6b_{e}).$

∎

######  Proposition 4.16.

For all integers $h\geqslant 1$ the scheme $\mathcal{M}(2h+1)$ is not empty
and its general member $[\mathcal{U}_{2h+1}]\in\mathcal{M}(2h+1)$ corresponds
to a rank-$(2h+1)$ vector bundle $\mathcal{U}_{2h+1}$ which is Ulrich w.r.t.
$\xi$ and which satisfies

$c_{1}(\mathcal{U}_{2h+1})=(2h+1)\xi+\varphi^{*}\mathcal{O}_{\mathbb{F}_{e}}(3,b_{e}-3))+\varphi^{*}\mathcal{O}_{\mathbb{F}_{e}}(h-1,(h-1)(b_{e}-e-2))$

and $h^{j}(\mathcal{U}_{2h+1}\otimes\mathcal{U}_{2h+1}^{\vee})=0$, $j=2,3$.

###### Proof.

We prove this by induction on $h$, the case $h=1$ being satisfied by the
choice of $\mathcal{M}(3)$ as in Theorem 4.14. Therefore, let $h\geqslant 2$;
for general $[\mathcal{U}_{2h-1}]\in\mathcal{M}(2h-1)$ and
$[\mathcal{U}_{2}]\in\mathcal{M}(2)$, one has

$\displaystyle\dim({\rm
Ext}^{1}(\mathcal{U}_{2h-1},\mathcal{U}_{2}))=h^{1}(\mathcal{U}_{2}\otimes\mathcal{U}_{2h-1}^{\vee}).$

By Lemma 4.15-(i) we have that
$h^{j}(\mathcal{U}_{2}\otimes\mathcal{U}_{2h-1}^{\vee})=0$, for $j=2,3$.
Therefore

$\displaystyle\chi(\mathcal{U}_{2}\otimes\mathcal{U}_{2h-1}^{\vee})=h^{0}(\mathcal{U}_{2}\otimes\mathcal{U}_{2h-1}^{\vee})-h^{1}(\mathcal{U}_{2}\otimes\mathcal{U}_{2h-1}^{\vee})$

so, by specialization and invariance of $\chi$ in irreducible families, we
have

$\displaystyle\dim({\rm Ext}^{1}(\mathcal{U}_{2h-1},\mathcal{U}_{2}))$
$\displaystyle=$ $\displaystyle
h^{1}(\mathcal{U}_{2}\otimes\mathcal{U}_{2h-1}^{\vee})$ $\displaystyle=$
$\displaystyle-\chi(\mathcal{U}_{2}\otimes\mathcal{U}_{2h-1}^{\vee})+h^{0}(\mathcal{U}_{2}\otimes\mathcal{U}_{2h-1}^{\vee})$
$\displaystyle\geqslant$
$\displaystyle-\chi(\mathcal{U}_{2}\otimes\mathcal{U}_{2h-1}^{\vee})=-\chi(\mathcal{U}_{2}\otimes\mathcal{F}_{2h-1}^{\vee})$
$\displaystyle=$ $\displaystyle(h-2)(6b_{e}-9e-4)+(8b_{e}-12e+3)>0,$

the latter equality following from Lemma 4.15-(ii) (with $h$ replaced by
$h-1$) whereas the last strict inequality following from $h\geqslant 2$ and
$b_{e}\geqslant 3e+2$ by (1.6).

The above computations prove that $\dim({\rm
Ext}^{1}(\mathcal{U}_{2h-1},\mathcal{U}_{2}))>0$, i.e. there exist non-trivial
extensions as in (4.41), and that the scheme $\mathcal{M}(2h+1)^{\rm ext}$,
and so also $\mathcal{M}(2h+1)$, is not empty.

The members of $\mathcal{M}(2h+1)$ have rank $2h+1$ and first Chern class as
in (4.42), since $c_{1}(\mathcal{U}_{2h+1})=c_{1}(\mathcal{F}_{2h+1})$ being
constant in $\mathcal{M}(2h+1)$. It is immediate that extensions of Ulrich
bundles are still Ulrich, so the general member
$[\mathcal{U}_{2h+1}]\in\mathcal{M}(2h+1)$ corresponds to an Ulrich bundle
w.r.t. $\xi$. It also satisfies
$h^{j}(\mathcal{U}_{2h+1}\otimes\mathcal{U}_{2h+1}^{\vee})=0$ for $j=2,3$ by
Lemma 4.15-(iii). ∎

We need to prove that the general member of $\mathcal{M}(2h+1)$ corresponds to
a vector bundle which is slope–stable w.r.t. $\xi$, that $\mathcal{M}(2h+1)$
is generically smooth and we need to compute the dimension at its general
point $[\mathcal{U}_{2h+1}]$. We will again prove all these facts by induction
on $h$. Similarly as in the previous cases, we need the following auxiliary
result.

######  Lemma 4.17.

Let $\mathcal{F}_{2h+1}$ correspond to a general member of
$\mathcal{M}(2h+1)^{\rm ext}$, sitting in an extension like (4.41). Assume
furthermore that $\mathcal{U}_{2}$ and $\mathcal{U}_{2h-1}$ are slope–stable.
Let $\mathcal{D}$ be a destabilizing subsheaf of $\mathcal{F}_{2h+1}$. Then
$\mathcal{D}^{\vee}\cong{\mathcal{U}_{2}}^{\vee}$ and
$\left(\mathcal{F}_{2h+1}/\mathcal{D}\right)^{\vee}\cong\mathcal{U}_{2h-1}^{\vee}$.

###### Proof.

The proof is identical to that of Lemma 4.12, so the reader is referred
therein. ∎

######  Proposition 4.18.

For all integers $h\geqslant 1$ the scheme $\mathcal{M}(2h+1)$ is not empty,
generically smooth of dimension

$\displaystyle\dim(\mathcal{M}(2h+1))=((h-1)^{2}+2)(6b_{e}-9e-4)+9(h-1)(2b_{e}-3e).$

Its general member corresponds to a slope-stable bundle $\mathcal{U}_{2h+1}$
whose slope w.r.t. $\xi$ is

$\mu(\mathcal{U}_{2h+1})=8b_{e}-k_{e}-12e-3.$

Furthermore, $\mathcal{M}(2h+1)$ properly contains the locally closed
subscheme $\mathcal{M}(2h+1)^{\rm ext}$, namely $\dim(\mathcal{M}(2h+1)^{\rm
ext})<\dim(\mathcal{M}(2h+1))$.

###### Proof.

We prove this by induction on $h$, the case $h=1$ being satisfied by
$\mathcal{M}(3)$ as in Theorem 4.14, where in such a case $\mathcal{M}(3)^{\rm
ext}=\emptyset$.

Let therefore $h\geqslant 2$ and assume that we have proved the statement for
all positive integers $k\leqslant h-1$; we will prove it for $h$.

The slope of the members of $\mathcal{M}(2)$ and $\mathcal{M}(2h-1)$ are both
equal to $8b_{e}-k_{e}-12e-3$ as in (4.43). Thus, by [11, Lemma 4.2], the
general member $[\mathcal{F}_{2h+1}]\in\mathcal{M}(2h+1)^{\rm ext}$ (which is
not empty by Proposition 4.16) corresponds to a simple bundle. Hence, by semi-
continuity, also the general member $[\mathcal{U}_{2h+1}]\in\mathcal{M}(2h+1)$
corresponds to a simple bundle which also satisfies
$h^{j}(\mathcal{U}_{2h+1}\otimes\mathcal{U}^{\vee}_{2h+1})=0$, $j=2,3$, by
Lemma 4.15-(iii).

Therefore $\mathcal{M}(2h+1)$ is smooth at $[\mathcal{U}_{2h+1}]$ (see, e.g.,
[11, Prop. 2.10]) with

$\displaystyle\dim(\mathcal{M}(2h+1))$ $\displaystyle=$ $\displaystyle
h^{1}(\mathcal{U}_{2h+1}\otimes\mathcal{U}^{\vee}_{2h+1})$ $\displaystyle=$
$\displaystyle-\chi(\mathcal{U}_{2h+1}\otimes\mathcal{U}^{\vee}_{2h+1})+h^{0}(\mathcal{U}_{2h+1}\otimes\mathcal{U}_{2h+1}^{\vee})$
$\displaystyle=$ $\displaystyle((h-1)^{2}+2)(6b_{e}-9e-4)+9(h-1)(2b_{e}-3e),$

using the facts that
$h^{0}(\mathcal{U}_{2h+1}\otimes\mathcal{U}^{\vee}_{2h})=1$ as
$\mathcal{U}_{2h+1}$ is simple, and the computation of
$\chi(\mathcal{U}_{2h+1}\otimes\mathcal{U}^{\vee}_{2h+1})$ in Lemma 4.15-(iv).
This proves that $\mathcal{M}(2h+1)$ is generically smooth of the stated
dimension.

Finally, we prove that $\mathcal{U}_{2h+1}$ general is slope–stable and that
$\dim(\mathcal{M}(2h+1)^{\rm ext})<\dim(\mathcal{M}(2h+1))$. If
$\mathcal{U}_{2h+1}$ general were not slope-stable then we could find a one-
parameter family of bundles $\\{\mathcal{U}_{2h+1}^{(t)}\\}$ over the disc
$\Delta$ such that $\mathcal{U}_{2h+1}^{(t)}$ is a general member of
$\mathcal{M}(2h+1)$ for $t\neq 0$ and $\mathcal{U}_{2h+1}^{(0)}$ lies in
$\mathcal{M}(2h+1)^{\rm ext}$, and such that we have a destabilizing sequence

(4.49)
$0\to\mathcal{D}^{(t)}\to\mathcal{U}_{2h+1}^{(t)}\to\mathcal{G}^{(t)}\to 0$

for $t\neq 0$, which we can take to be saturated, that is, such that
$\mathcal{G}^{(t)}$ is torsion free, whence so that $\mathcal{D}^{(t)}$ and
$\mathcal{G}^{(t)}$ are (Ulrich) vector bundles (see [11, Thm. 2.9] or [5,
(3.2)]). The limit of
$\mathbb{P}(\mathcal{G}^{(t)})\subset\mathbb{P}(\mathcal{U}_{2h+1}^{(t)})$
defines a subvariety of $\mathbb{P}(\mathcal{U}_{2h+1}^{(0)})$ of the same
dimension as $\mathbb{P}(\mathcal{G}^{(t)})$, whence a coherent sheaf
$\mathcal{G}^{(0)}$ of rank ${\rm rk}(\mathcal{G}^{(t)})$ with a surjection
$\mathcal{U}_{2h+1}^{(0)}\twoheadrightarrow\mathcal{G}^{(0)}$. Denoting by
$\mathcal{D}^{(0)}$ its kernel, we have ${\rm rk}(\mathcal{D}^{(0)})={\rm
rk}(\mathcal{D}^{(t)})$ and
$c_{1}(\mathcal{D}^{(0)})=c_{1}(\mathcal{D}^{(t)})$. Hence, (4.49) specializes
to a destabilizing sequence for $t=0$.

Lemma 4.17 yields that ${\mathcal{D}^{(0)}}^{\vee}$ (resp.,
${\mathcal{G}^{(0)}}^{\vee}$) is the dual of a member of $\mathcal{M}(3)$
(resp., of $\mathcal{M}(2h+1)$). It follows that ${\mathcal{D}^{(t)}}^{\vee}$
(resp., ${\mathcal{G}^{(t)}}^{\vee}$) is a deformation of the dual of a member
of $\mathcal{M}(2))$ (resp., of $\mathcal{M}(2h+1)$), whence that
${\mathcal{D}^{(t)}}$ (resp., ${\mathcal{G}^{(t)}}$) is a deformation of a
member of $\mathcal{M}(2)$ (resp., $\mathcal{M}(2h+1)$), as both are locally
free. It follows that $[\mathcal{U}_{2h+1}^{(t)}]\in\mathcal{M}(2h+1)^{\rm
ext}$ for $t\neq 0$. Thus,

(4.50) $\mathcal{M}(2h+1)^{\rm ext}=\mathcal{M}(2h+1).$

On the other hand we have

(4.51) $\dim(\mathcal{M}(2h+1)^{\rm ext})\leqslant\dim(\mathbb{P}({\rm
Ext}^{1}(\mathcal{U}_{2h-1},\mathcal{U}_{2})))+\dim(\mathcal{M}(2h-1))+\dim(\mathcal{M}(2)),$

for $[\mathcal{U}_{2h-1}]\in\mathcal{M}(2h-1)$ and
$[\mathcal{U}_{2}]\in\mathcal{M}(2)$ general. Because $\mathcal{U}_{2}$ is
slope–stable and also $\mathcal{U}_{2h-1}$ is slope–stable by induction, of
the same slope, we have
$h^{0}(\mathcal{U}_{2}\otimes\mathcal{U}_{2h-1}^{\vee})=0$. Thus, (4.2) gives

$h^{1}(\mathcal{U}_{2}\otimes\mathcal{U}^{\vee}_{2h-1})=-\chi(\mathcal{U}_{2}\otimes\mathcal{U}^{\vee}_{2h-1})=(h-2)(6b_{e}-9e-4)+(8b_{e}-12e+3).$

Hence, from (4.51), using also (4.2) and the fact that
$\dim(\mathcal{M}(2))=(6b_{e}-9e-3)$ from Theorem 3.4, one has

$\displaystyle\dim(\mathcal{M}(2h+1)^{\rm ext})$ $\displaystyle\leqslant$
$\displaystyle(h-2)(6b_{e}-9e-4)+(8b_{e}-12e+3)-1+$
$\displaystyle+((h-2)^{2}+2)(6b_{e}-9e-4)+9(h-2)(2b_{e}-3e)+(6b_{e}-9e-3)$
$\displaystyle<$
$\displaystyle((h-1)^{2}+2)(6b_{e}-9e-4)+9(h-1)(2b_{e}-3e)=\dim(\mathcal{M}(2h+1)),$

as $h\geqslant 2$. The previous inequality shows that
$\dim(\mathcal{M}(2h+1)^{\rm ext})<\dim(\mathcal{M}(2h+1))$, as stated; in
particular (4.50) is a contradiction, which forces also $\mathcal{U}_{2h+1}$
general to be slope-stable. ∎

The collection of the previous results in even and in odd ranks, respectively,
gives the following:

######  Theorem 4.19.

Let
$(X_{e},\xi)\cong(\mathbb{P}({\mathcal{E}_{e}}),{\mathcal{O}}_{\mathbb{P}({\mathcal{E}_{e}})}(1))$
be a $3$-fold scroll over $\mathbb{F}_{e}$, with $e>0$ and $\mathcal{E}_{e}$
as in Assumptions 1.8. Let $\varphi:X_{e}\to\mathbb{F}_{e}$ be the scroll map
and $F$ be the $\varphi$-fibre. Let $r\geqslant 2$ be any integer.

Then the moduli space of rank-$r$ vector bundles $\mathcal{U}_{r}$ on $X_{e}$
which are Ulrich w.r.t. $\xi$ and with first Chern class

$\displaystyle
c_{1}(\mathcal{U}_{r})=\begin{cases}r\xi+\varphi^{*}\mathcal{O}_{\mathbb{F}_{e}}(3,b_{e}-3)+\varphi^{*}\mathcal{O}_{\mathbb{F}_{e}}\left(\frac{r-3}{2},\frac{(r-3)}{2}(b_{e}-e-2\right),&\mbox{if
$r$ is odd},\\\
r\xi+\varphi^{*}\mathcal{O}_{\mathbb{F}_{e}}\left(\frac{r}{2},\frac{r}{2}(b_{e}-e-2)\right),&\mbox{if
$r$ is even}.\end{cases}$

is not empty and it contains a generically smooth component $\mathcal{M}(r)$
of dimension

$\displaystyle\dim(\mathcal{M}(r))=\begin{cases}\left(\frac{(r-3)^{2}}{4}+2\right)(6b_{e}-9e-4)+\frac{9}{2}(r-3)(2b_{e}-3e),&\mbox{if
$r$ is odd},\\\ \frac{r^{2}}{4}(6b_{e}-9e-4)+1,&\mbox{if $r$ is
even}.\end{cases}$

The general member $[\mathcal{U}_{r}]\in\mathcal{M}(r)$ corresponds to a
slope-stable vector bundle, of slope w.r.t. $\xi$ given by
$\mu(\mathcal{U}_{r})=8b_{e}-k_{e}-12e-3$.

###### Proof.

In the even case, the statement directly follows from Theorem 3.4, (4.30),
(4.31) and from Propositions 4.11, 4.13. For odd cases, the statement follows
from Theorem 4.14, (4.42), (4.43) and from Propositions 4.16, 4.18 ∎

##  5\. Final remarks on Ulrichness over $\mathbb{F}_{e}$, $e\geqslant 0$

As a direct consequence of Main Theorem, Main Corollary, Theorem 1.7 and the
one–to–one correspondence in [23, Proposition 6.2], one has the following
result concerning moduli spaces of rank-$r$ vector bundles on Hirzebruch
surfaces $\mathbb{F}_{e}$, for any $r\geqslant 1$ and any $e\geqslant 0$,
which are Ulrich w.r.t. the very ample line bundle
$c_{1}(\mathcal{E}_{e})=3C_{e}+b_{e}f$, with $b_{e}\geqslant 3e+2$ as in (1.6)
(the case $r=1,2,3$ already known by [4, 12, 2]).

######  Theorem 5.1.

For any integer $e\geqslant 0$, consider the Hirzebruch surface
$\mathbb{F}_{e}$ and let $\mathcal{O}_{\mathbb{F}_{e}}(\alpha,\beta)$ denote
the line bundle $\alpha C_{e}+\beta f$ on $\mathbb{F}_{e}$, where $C_{e}$ and
$f$ are the generators of ${\rm Num}(\mathbb{F}_{e})$.

Consider the very ample polarization $c_{1}(\mathcal{E}_{e})=3C_{e}+b_{e}f$,
where $b_{e}\geqslant 3e+2$. Then:

(a) $\mathbb{F}_{e}$ does not support any Ulrich line bundle w.r.t.
$c_{1}(\mathcal{E}_{e})$ unless $e=0$. In this latter case, the unique line
bundles on $\mathbb{F}_{0}$ which are Ulrich w.r.t. $c_{1}(\mathcal{E}_{e})$
are

$\mathcal{L}_{1}:=\mathcal{O}_{\mathbb{F}_{0}}(5,b_{0}-1)\;{\rm
and}\;\mathcal{L}_{2}:=\mathcal{O}_{\mathbb{F}_{0}}(2,2b_{0}-1).$

(b) Set $e=0$ and let $r\geqslant 2$ be any integer. Then the moduli space of
rank-$r$ vector bundles $\mathcal{H}_{r}$ on $\mathbb{F}_{0}$ which are Ulrich
w.r.t. $c_{1}(\mathcal{E}_{0})$ and with first Chern class

$\displaystyle
c_{1}(\mathcal{H}_{r})=\begin{cases}\mathcal{O}_{\mathbb{F}_{0}}(3(r+1),(r+1)b_{0}-3)\otimes\mathcal{O}_{\mathbb{F}_{0}}\left(\frac{r-3}{2},\frac{(r-3)}{2}(b_{0}-2)\right),&\mbox{if
$r$ is odd},\\\
\mathcal{O}_{\mathbb{F}_{0}}(3r,rb_{0})\otimes\mathcal{O}_{\mathbb{F}_{0}}\left(\frac{r}{2},\frac{r}{2}(b_{0}-2)\right),&\mbox{if
$r$ is even}.\end{cases}$

is not empty and it contains a generically smooth component
$\mathcal{M}_{\mathbb{F}_{0}}(r)$ of dimension

$\displaystyle\dim(\mathcal{M}_{\mathbb{F}_{0}}(r))=\begin{cases}\frac{(r^{2}-1)}{4}(6b_{0}-4),&\mbox{if
$r$ is odd},\\\ \frac{r^{2}}{4}(6b_{0}-4)+1,&\mbox{if $r$ is
even}.\end{cases}$

The general point $[\mathcal{H}_{r}]\in\mathcal{M}_{\mathbb{F}_{0}}(r)$
corresponds to a slope-stable vector bundle.

(c) When $e>0$, let $r\geqslant 2$ be any integer. Then the moduli space of
rank-$r$ vector bundles $\mathcal{H}_{r}$ on $X_{e}$ which are Ulrich w.r.t.
$c_{1}(\mathcal{E}_{e})$ and with first Chern class

$\displaystyle
c_{1}(\mathcal{H}_{r})=\begin{cases}\mathcal{O}_{\mathbb{F}_{e}}(3(r+1),(r+1)b_{e}-3)\otimes\mathcal{O}_{\mathbb{F}_{e}}\left(\frac{r-3}{2},\frac{(r-3)}{2}(b_{e}-e-2)\right),&\mbox{if
$r$ is odd},\\\
\mathcal{O}_{\mathbb{F}_{e}}(3r,rb_{e})\otimes\mathcal{O}_{\mathbb{F}_{e}}\left(\frac{r}{2},\frac{r}{2}(b_{e}-e-2)\right),&\mbox{if
$r$ is even}.\end{cases}$

is not empty and it contains a generically smooth component
$\mathcal{M}_{\mathbb{F}_{e}}(r)$ of dimension

$\displaystyle\dim(\mathcal{M}_{\mathbb{F}_{e}}(r))=\begin{cases}\left(\frac{(r-3)^{2}}{4}+2\right)(6b_{e}-9e-4)+\frac{9}{2}(r-3)(2b_{e}-3e),&\mbox{if
$r$ is odd},\\\ \frac{r^{2}}{4}(6b_{e}-9e-4)+1,&\mbox{if $r$ is
even}.\end{cases}$

The general point $[\mathcal{H}_{r}]\in\mathcal{M}_{\mathbb{F}_{e}}(r)$
corresponds to a slope-stable vector bundle.

###### Proof.

(a) When $e>0$, the fact that $\mathbb{F}_{e}$ does not support line bundles
which are Ulrich w.r.t. $c_{1}(\mathcal{E}_{e})=3C_{e}+b_{e}f$ is proved in
[4, Thm. 2.1]; however, this fact is also a direct consequence of Main
Theorem–(a) and Theorem 1.7.

For $e=0$ observe that the line bundles $L_{1}$ and $L_{2}$ as in Main
Theorem–(a), which are Ulrich w.r.t. $\xi$ on $X_{0}$, give rise, by push
forward and by Theorem 1.7, to Ulrich line bundles w.r.t
$c_{1}(\mathcal{E}_{0})=3C_{0}+b_{0}f$ on $\mathbb{F}_{0}$. Indeed

$\displaystyle
L_{1}\otimes\xi^{\vee}=\varphi^{*}\mathcal{O}_{\mathbb{F}_{0}}(2,-1)\,\,\mbox{and}\,\,L_{2}\otimes\xi^{\vee}=\varphi^{*}\mathcal{O}_{\mathbb{F}_{0}}(-1,b_{0}-1),$

thus

$\displaystyle\varphi_{*}(L_{1}\otimes\xi^{\vee})=\mathcal{O}_{\mathbb{F}_{0}}(2,-1)\,\,\mbox{and}\,\,\varphi_{*}(L_{2}\otimes\xi^{\vee})=\mathcal{O}_{\mathbb{F}_{0}}(-1,b_{0}-1),$

which give

$\displaystyle\mathcal{L}_{1}:=\varphi_{*}(L_{1}\otimes\xi^{\vee})\otimes\mathcal{O}_{\mathbb{F}_{0}}(c_{1}(\mathcal{E}_{0}))=\mathcal{O}_{\mathbb{F}_{0}}(5,b_{0}-1)$
$\displaystyle\mathcal{L}_{2}:=\varphi_{*}(L_{2}\otimes\xi^{\vee})\otimes\mathcal{O}_{\mathbb{F}_{0}}(c_{1}(\mathcal{E}_{0}))=\mathcal{O}_{\mathbb{F}_{0}}(2,2b_{0}-1)$

that are the only Ulrich line bundles on $\mathbb{F}_{0}$ w.r.t.
$c_{1}(\mathcal{E}_{e})=3C_{0}+b_{0}f$, according to [12, Example 2.3] and [2,
Proposition 4.4].

(b) As for any rank $r\geqslant 2$ in the case $e=0$, observe that vector
bundles $\mathcal{U}_{r}$ on $X_{0}$ as in Main Theorem–(b) when restricted to
a general fibre $F$ of $\varphi:X_{0}\to\mathbb{F}_{0}$ are such that
$\mathcal{U}_{r}|_{F}\cong\mathcal{O}_{\mathbb{P}^{1}}(1)^{\oplus r}$. To see
this, one proceeds by induction. By (3.1),
$L_{1}|_{F}\cong\mathcal{O}_{\mathbb{P}^{1}}(1)\cong L_{2}|_{F}$, thus
$\mathcal{F}_{1}|_{F}\in{\rm
Ext}^{1}(\mathcal{O}_{\mathbb{P}^{1}}(1),\mathcal{O}_{\mathbb{P}^{1}}(1))=H^{1}(\mathcal{O}_{\mathbb{P}^{1}})=(0)$,
hence $\mathcal{F}_{1}|_{F}\cong\mathcal{O}_{\mathbb{P}^{1}}(1)^{\oplus 2}$,
which is the most balanced splitting. Since $\mathcal{U}_{2}=\mathcal{U}$ as
in Theorem 3.1 is a deformation of $\mathcal{F}_{1}$ then
$\mathcal{U}_{2}|_{F}\cong\mathcal{O}_{\mathbb{P}^{1}}(1)^{\oplus 2}$. Assume
by induction, that for some $r\geqslant 3$ one has
$\mathcal{U}_{r-1}|_{F}\cong\mathcal{O}_{\mathbb{P}^{1}}(1)^{\oplus r-1}$ for
$[\mathcal{U}_{r-1}]\in\mathcal{M}(r-1)$ general. Then by (4.18), one has

$\displaystyle
0\to\mathcal{U}_{r-1}|_{F}=\mathcal{O}_{\mathbb{P}^{1}}(1)^{\oplus
r-1}\to\mathcal{F}_{r}|_{F}\to
L_{\epsilon_{r}}|_{F}=\mathcal{O}_{\mathbb{P}^{1}}(1)\to 0$

and, once again, since ${\rm
Ext}^{1}(\mathcal{O}_{\mathbb{P}^{1}}(1),\mathcal{O}_{\mathbb{P}^{1}}(1)^{\oplus
r-1})=0$ then
$\mathcal{F}_{r}|_{F}\cong\mathcal{O}_{\mathbb{P}^{1}}(1)^{\oplus r}$.

Thus by [23, Theorem 6.1], using the diagram

$\textstyle{\tilde{S}\,\,\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\scriptstyle{\varphi^{\prime}}$$\textstyle{X_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{{\mathbb{F}_{0}}}$

therein, where $\varphi^{\prime}:\tilde{S}\to\mathbb{F}_{0}$ is the blow-up at
$c_{2}(\mathcal{E}_{0})$ points on $\mathbb{F}_{0}$ and where $\sum E_{i}$ is
the $\varphi^{\prime}$-exceptional divisor, one has that
$\varphi_{*}(\mathcal{U}_{r}\otimes i_{*}(\mathcal{O}_{\tilde{S}}(\sum
E_{i})))$ is a rank $r$ vector bundle on $\mathbb{F}_{0}$ which is Ulrich
w.r.t. $c_{1}(\mathcal{E}_{0})=3C_{0}+b_{0}f$. More precisely [23, Proposition
6.2] gives rise to a component of the moduli space of Ulrich bundles of rank
$r$ on $\mathbb{F}_{0}$, which are Ulrich w.r.t.
$c_{1}(\mathcal{E}_{0})=3C_{0}+b_{0}f$, of first Chern class

$\displaystyle c_{1}(\varphi_{*}(\mathcal{U}_{r}\otimes
i_{*}(\mathcal{O}_{\tilde{S}}(\sum E_{i})))).$

On the other hand, if we set
$\mathcal{H}_{r}:=\varphi_{*}(\mathcal{U}_{r}\otimes
i_{*}(\mathcal{O}_{\tilde{S}}(\sum E_{i})))$, the one-to-one correspondence in
[23, Proposition 6.2] asserts that
$\mathcal{U}_{r}\cong\xi\otimes\varphi^{*}(\mathcal{H}_{r}(-c_{1}(\mathcal{E}_{0})))$.
From Main Theorem-(b) one knows $c_{1}(\mathcal{U}_{r}\otimes\xi^{\vee})$ and,
since
$c_{1}(\mathcal{U}_{r}\otimes\xi^{\vee})=c_{1}(\mathcal{H}_{r}(-c_{1}(\mathcal{E}_{0})))=c_{1}(\mathcal{H}_{r})-rc_{1}(\mathcal{E}_{0})$,
it follows that $c_{1}(\mathcal{H}_{r})$ is as stated; pairs
$(r,c_{1}(\mathcal{H}_{r}))$ are Ulrich admissible pairs w.r.t.
$3C_{0}+b_{0}f$ in the sense of [2, Def. 5.1].

By the one-to-one correspondence in [23, Proposition 6.2] and Main
Theorem-(b), we have therefore the existence of moduli spaces
$\mathcal{M}_{\mathbb{F}_{0}}(r)$ of Ulrich bundles on $\mathbb{F}_{0}$ w.r.t.
$3C_{0}+b_{0}f$, of any rank $r\geqslant 2$, of first Chern class and
dimension as stated. Moreover since
$\mathcal{U}_{r}\cong\xi\otimes\varphi^{*}(\mathcal{H}_{r}(-c_{1}(\mathcal{E}_{0})))$
then

$\displaystyle h^{j}(X_{0},\mathcal{U}_{r}\otimes\mathcal{U}_{r}^{\vee})$
$\displaystyle=$ $\displaystyle
h^{j}(X_{0},\varphi^{*}(\mathcal{H}_{r}(-c_{1}(\mathcal{E}_{0})))\otimes\varphi^{*}(\mathcal{H}_{r}(-c_{1}(\mathcal{E}_{0})))^{\vee})$
$\displaystyle=$ $\displaystyle
h^{j}(X_{0},\varphi^{*}(\mathcal{H}_{r}\otimes\mathcal{H}_{r}^{\vee}))=h^{j}(\mathbb{F}_{0},\mathcal{H}_{r}\otimes\mathcal{H}_{r}^{\vee}),$

i.e. $h^{2}(\mathbb{F}_{0},\mathcal{H}_{r}\otimes\mathcal{H}_{r}^{\vee})=0$
and $h^{0}(\mathbb{F}_{0},\mathcal{H}_{r}\otimes\mathcal{H}_{r}^{\vee})=1$,
namely the component $\mathcal{M}_{\mathbb{F}_{0}}(r)$ is also generically
smooth and its general point
$[\mathcal{H}_{r}]\in\mathcal{M}_{\mathbb{F}_{0}}(r)$ corresponds to a simple
bundle.

It is also clear that $\mathcal{H}_{r}$ is slope-stable w.r.t.
$3C_{0}+b_{0}f$: if not, by the one-to-one correspondence given by [23,
Proposition 6.2], any destabilizing rank-$k$ Ulrich sub-bundle
$\mathcal{D}_{k}$ of $\mathcal{\mathcal{H}}_{r}$ general, for some $1\leqslant
k\leqslant r-1$, would give rise to a rank-$k$ vector bundle
$\xi\otimes\varphi^{*}(\mathcal{D}_{k}(-c_{1}(\mathcal{E}_{0}))$ which is
Ulrich on $X_{0}$ w.r.t. $\xi$, by Theorem 1.7, and which would be a
destabilizing sub-bundle on $\mathcal{U}_{r}$ general in $\mathcal{M}(r)$ on
$X_{0}$, contradicting Main Theorem–(b).

When $r=2,3$, the previous arguments are in accordance with [2, Prop. 6.4],
for the polarization $3C_{0}+b_{0}f$, and $c_{1}$ as stated.

(c) Similar arguments as in (b), but for the case for $e>0$, are obtained by
using Main Theorem–(c), Theorem 1.7 and the one-to-one correspondence in [23,
Proposition 6.2]. ∎

From Theorem 5.1 it follows that the pairs $(\mathbb{F}_{e},3C_{e}+b_{e}f)$
are Ulrich wild and this is in accordance with [12, Lemma 5.2].

We want to stress that, when $r=2,\;3$, we cannot deduce the irreducibility of
the moduli spaces of bundles $\mathcal{U}_{r}$ as in Main Theorem–(b) or (c)
from the correspondence in [23, Proposition 6.2] and irreducibility results on
$\mathbb{F}_{e}$, $e\geqslant 0$, given by [17, Theorem 4.7] (cf. also [2,
Propositions 6.1, 6.4]). Indeed, in principle, there could exist other
components, different from ours $\mathcal{M}(r)$, $r=2,3$, where the general
Ulrich bundle therein does not split as
$\mathcal{O}_{\mathbb{P}^{1}}(1)^{\oplus r}$ on the general $\varphi$–fiber.

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# BadODD: Bangladeshi Autonomous Driving Object Detection Dataset

Mirza Nihal Baig∗ , Rony Hajong∗ , Mahdi Murshed Patwary∗ , Mohammad Shahidur
Rahman∗,
Husne Ara Chowdhury∗

∗ CSE, Shahjalal University of Science and Technology, Sylhet, Bangladesh

###### Abstract

We propose a comprehensive dataset for object detection in diverse driving
environments across 9 districts in Bangladesh. The dataset, collected
exclusively from smartphone cameras, provided a realistic representation of
real-world scenarios, including day and night conditions. Most existing
datasets lack suitable classes for autonomous navigation on Bangladeshi roads,
making it challenging for researchers to develop models that can handle the
intricacies of road scenarios. To address this issue, the authors proposed a
new set of classes based on characteristics rather than local vehicle names.
The dataset aims to encourage the development of models that can handle the
unique challenges of Bangladeshi road scenarios for the effective deployment
of autonomous vehicles. The dataset did not consist of any online images to
simulate real-world conditions faced by autonomous vehicles. The
classification of vehicles is challenging because of the diverse range of
vehicles on Bangladeshi roads, including those not found elsewhere in the
world. The proposed classification system is scalable and can accommodate
future vehicles, making it a valuable resource for researchers in the
autonomous vehicle sector.

Key Words: Computer Vision, Object Detection, Autonomous Vehicle, Dataset,
Vehicle Classification.

##  1 Introduction

Autonomous navigation is advancing towards incredible technology in our daily
lives. Many companies integrate autonomous technologies into their products in
various regions and try to achieve level 5 autonomy. Large-scale datasets have
contributed significantly to the progress of autonomous navigation. However,
in many parts of the world, this technology still has a long way to go. A key
challenge is to obtain a diverse set of data to account for extreme corner
cases. Most of the algorithms developed for autonomous vehicles are not
benchmarked on unstructured and congested data from different parts of the
world. In addition, existing datasets do not account for classes that are
available on the Indian subcontinent, which is a major disadvantage in
advancing autonomous technology.

In this paper, we propose BDOR, a novel dataset for autonomous navigation of
Bangladeshi roads that addresses the problems of existing autonomous vehicle
datasets. Our dataset consists of 9825 images with 78,943 objects of a
bangladesh road under various lightning conditions with 13 classes. As the
number of objects per image is higher in Bangladesh roads, this dataset
consists of the most common objects that are seen on BD roads.

Furthermore, we propose a new set of classes for classifying vehicles in
Bangladesh that are scalable and can be used as a heuristic for autonomous
cars. There are a diverse set of vehicles, many of which are manufactured
locally in unique shapes and customized to the needs of people. These vehicles
are not globally acknowledged by any universal name; rather, local names are
provided. This creates a problem for setting class names that are aligned with
the existing datasets and account for all of the vehicles. Our proposed
classes solved this problem by classifying vehicles based on their
characteristics rather than their local names. This set of classes can be
scalable to new types of vehicles that may be manufactured locally or imported
from other countries.

Another potential use case for this new set of classes is that the
characteristics of the vehicle can be used for faster decision-making for
autonomous cars. For the existing dataset, each class has the same weight as
autonomous vehicles. Our proposed classes have different characteristics;
therefore, autonomous vehicles can precompute decisions that can help make
accurate decisions in real time. Each characteristic will cater to a different
decision-making process for navigation. This will be helpful for creating more
scalable dataset in the future.

We provide a detailed analysis of the class distribution in BDOR dataset.
While some classes are seen heavily on road such as Person, Autorickshaw,
Three Wheeler other classes are rarely seen such as wheel chair, train,
construction vehicle. For this reason, the dataset is imbalance for the
disproponate ratio of the vehicles seen on road. Furthermore, the unstructure
nature of the road scenario and congested objects poses certain problems like
occlusion in detecting the object.

Our main contributions are the following:

  * •

We released a 2D Object Detection autonomous for autonomous driving on
Bangladesh Roads

  * •

A new set of scalable classes suitable for classifying bangladeshi vehicles

  * •

We trained and benchmarked existing baseline models and made a comparative
analysis of them.

Feature SODA10M BDD100K Waymo Open Dataset Size 10M (unlabeled) + 20K
(labeled) images 100K video frames 1.28 TB Types of Data Images (front and
rear cameras) Images (various cameras) LiDAR, Camera, Radar Object Categories
6 10 19 Scenarios Diverse weather, time, location Urban, rural, various
weather, day/night Highway, urban Annotations Bounding boxes Bounding boxes
Bounding boxes, 3D object points Challenges Occlusions, scale variations, low
resolution Occlusions, night-time, diverse weather Proprietary format, limited
access Focus Self/semi-supervised learning General 2D object detection
Autonomous driving research Publicly Available? Yes Yes Partially

Table 1: Comparison of 2D Object Detection Datasets for Autonomous Vehicles

##  2 Dataset

###  2.1 Data Collection

The data were collected from 9 districts in Bangladesh: Sylhet, Dhaka,
Rajshahi, Mymensingh, Maowa, Chittagong, Sirajganj, Sherpur, and Khulna. These
cities have different road scenarios and different types of vehicles in
various proportions. There are urban and rural areas, highways, and expressway
roads with different types of traffic. We collected a dataset from the front
of the car. Sometimes, there was car glass in front of the camera, and others
time we collected without glass in front. Day and nighttime datasets were
collected for different types of roads. To ensure a real-world road scenario,
videos were collected while the driver was driving. No online images were
collected to ensure the driving road scenery and quality of the dataset.

###  2.2 Frame Selection

The method by which we sampled images from video sources was carefully planned
to capture the dynamic quality of various urban environments. Because traffic
densities vary in different settings, we used a flexible frame-rate sampling
strategy to provide a representative and varied dataset.

The frame rate on highways and expressways, which is frequently associated
with reduced traffic volumes, was set at one frame per second. This decision
was made after a thorough manual review of the film with the goal of
eliminating repetition in less dynamic situations while retaining crucial
details.

However, we chose a lower frame rate of one frame every two seconds in highly
crowded urban regions with increased traffic and pedestrian activity. The
necessity of guaranteeing dataset diversity by recording a wider range of
circumstances in high-density traffic zones motivated this decision.

The changing frame rates are largely determined by observing the video
content. This made it possible for us to customize the sample plan to the
unique features of every site, producing a dataset that faithfully captured
the dynamic character of both densely populated areas and those with little
traffic.

In addition to improving the representativeness of the dataset, this adaptive
sampling technique advances our understanding of the traffic patterns in
various metropolitan environments. Frame rates were carefully considered in
accordance with our dedication to capturing the subtleties of real-world
surroundings, providing a strong basis for reliable analysis and model
training.

###  2.3 Redefining Classes & Annotation

Most of the existing object detection datasets have similar classes that do
not represent the vehicles on Bangladesh roads. As there are diverse types of
vehicles, labelling the vehicles by their local or globally acknowledged names
will increase the number of classes, and if new vehicles are manufactured, new
classes must be added. Our approach to redefining labels based on the
characteristics of vehicles solves this problem.

Class Wheel Number Driving Force Size Car 4 wheeler Diesel, Gas, Fossil Fuel
Medium Three Wheeler 3 Wheeler Paddle Small Autorickshaw 3 Wheeler Gas,
Electric Medium Priority Vehicle Any vehicle with Siren on Top Fossil fuel or
paddle Any size Bus 4 Wheeler Diesel Big Truck 4 or 8 wheeler fossil fuel or
electric Medium and Small Cart Car 3 or 2 Wheeler Human or Animal Supported
(No paddle) Small Construction Vehicle Construction Related Vehicle Diesel,
Fossil Fuel Medium, Big Train Runs on Railway Track Diesel Big Wheelchair 2
Wheeler with Priority Human Support Small Motorbike 2 Wheeler Oil, Electric
Medium Bicycle 2 Wheeler Paddle Small

Table 2: Comparison with existing Datasets

###  2.4 Statistical Analysis of Dataset

In this section, we present a comprehensive statistical analysis of the
dataset used for training, validation, and testing our object detection model.
The dataset comprised 9,825 images distributed among 13 distinct classes. The
dataset was split into training, validation, and testing sets in a 60:20:20
ratio, ensuring that each split included night images.

Dataset Overview

Dataset | Number of Images
---|---
Training Set | 5,896
Validation Set | 1,964
Testing Set | 1,965
Table 3: Dataset Overview

Class Distribution

Class | Training Set | Validation Set | Testing Set
---|---|---|---
Person | 18,010 (38.22%) | 6,202 (38.69%) | 6,104 (38.64%)
Three-Wheeler | 5,710 (12.12%) | 1,929 (12.04%) | 1,961 (12.41%)
Motorbike | 3,749 (7.96%) | 1,218 (7.60%) | 1,254 (7.94%)
Auto Rickshaw | 10,614 (22.53%) | 3,638 (22.70%) | 3,489 (22.09%)
Car | 3,785 (8.03%) | 1,305 (8.14%) | 1,233 (7.81%)
Truck | 2,296 (4.87%) | 777 (4.85%) | 782 (4.95%)
Bus | 1,885 (4.00%) | 589 (3.67%) | 613 (3.88%)
Bicycle | 673 (1.43%) | 256 (1.60%) | 241 (1.53%)
Priority Vehicle | 229 (0.49%) | 70 (0.44%) | 64 (0.41%)
Cart Vehicle | 141 (0.30%) | 38 (0.24%) | 44 (0.28%)
Construction Vehicle | 23 (0.05%) | 6 (0.04%) | 11 (0.07%)
Wheelchair | 2 (0.00%) | 0 (0.00%) | 1 (0.01%)
Train | 1 (0.00%) | 0 (0.00%) | 0 (0.00%)
Table 4: Total Class Distribution in Each Split Figure 1: Sample Dataset

##  3 Model

In this section, we elucidate the intricacies of the training regimen and
architectural configurations for YOLOv5 and YOLOv8. Our primary objective is
to draw a comparative analysis of their hyperparameters and performance, as
gauged by the mean Average Precision (mAP) metric on a designated object
detection dataset.

###  3.1 YOLOv5

The YOLOv5 model stands as a vanguard in the realm of object detection,
esteemed for its swift processing capabilities and commendable accuracy. Its
architecture is founded upon a deep neural network, striking a judicious
equilibrium between speed and efficacy.

####  3.1.1 Hyperparameters

Our training process for the YOLOv5 model adhered to a set of meticulously
chosen hyperparameters, as delineated in Table 6.

Hyperparameter | Value
---|---
Batch Size | 64
Learning Rate | 0.001
Number of Epochs | 10
Optimizer | Adam
Weight Decay | $1\times 10^{-4}$
Input Image Size | 416x416
Table 5: Hyperparameters employed for training YOLOv5.

####  3.1.2 Results

Upon completion of our training regimen, the YOLOv5 model manifested a
commendable mAP score of 0.6 when evaluated on our designated object detection
dataset.

###  3.2 YOLOv8

####  3.2.1 Hyperparameters

Hyperparameter | Value
---|---
Batch Size | 64
Learning Rate | 0.001
Number of Epochs | 10
Optimizer | Adam
Weight Decay | $1\times 10^{-4}$
Input Image Size | 416x416
Table 6: Hyperparameters employed for training YOLOv8.

####  3.2.2 Results

Upon completion of our training regimen, the YOLOv8 model manifested a
commendable mAP score of 0.7 when evaluated on our designated object detection
dataset.

###  3.3 Comparison

We present a juxtaposition of the hyperparameters and mAP results for YOLOv5
and YOLOv8 in Table 7.

Model | mAP | Training Time
---|---|---
YOLOv5 | 0.6 | 1 hour
YOLOv8 | 0.7 | 1 hour
Table 7: Comparative analysis of YOLOv5 and YOLOv8.

##  4 Conclusion

The research discusses a unique and complex task of object detection in
diverse driving environments across 9 districts in Bangladesh. The dataset,
collected exclusively from smartphone cameras, provides a comprehensive
representation of real-world scenarios, encompassing both day and night
conditions. The limitations of existing datasets available online don’t have
vehicles that are available on Bangladesh roads or suitable for autonomous
navigation in Bangladesh. The challenges in object detection require robust
algorithms capable of identifying and localizing objects in dynamic and
challenging environments, such as varying lighting conditions, different road
types, and the unique characteristics of each district. The dataset was
collected using smartphone cameras to ensure authenticity and to simulate
real-world conditions faced by autonomous vehicles. The classification of
vehicles was challenging due to the many types of vehicles that cannot be seen
elsewhere in the world and have local vehicle names. A new set of classes
based on characteristics rather than local vehicle names was proposed to
overcome this problem. The dataset will set a new benchmark for developing
better models on Bangladesh roads and will open up many research opportunities
for the autonomous vehicle sector.

Key highlights of our work:

  * •

Object detection in diverse Bangladesh driving environments

  * •

Exclusive dataset from smartphone cameras for real-world scenarios

  * •

9 districts covered, including day and night conditions

  * •

Limitations of existing datasets addressed

  * •

Challenges: robust algorithms for dynamic environments

  * •

Data collection: smartphone cameras for authenticity

  * •

New class system for scalable vehicle classification

  * •

Benchmark for better models on Bangladesh roads

  * •

Research opportunities for the autonomous vehicle sector.

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