1 Introduction

Let X be a smooth irreducible projective variety of dimension \(n \ge 1\), polarized by a very ample divisor H on X. The existence of vector bundles on X which are Ulrich with respect to has interested various authors.

For some specific classes of varieties such problem has being attacked, see for instance [1, 2, 9,10,11, 13]. Whenever such bundles do exist, since they are always semistable (in the sense of Gieseker-Maruyama, cf. also § 1 below) and also slope-semistable (cf. [6, Def. 2.7, Thm. 2.9-(a)]), one is interested in knowing if these bundles are also stable, equivalently slope-stable (cf. [6, Def. 2.7, Thm. 2.9-(c)]). Furthermore, from their semi-stability, such rank-r vector bundles give rise to points in a moduli space, say \(M:= M^{ss}(r; c_1, c_2, \ldots , c_k)\), where \(k:= \textrm{min} \{r,\,n\}\), parametrizing (S-equivalence classes of) semistable sheaves of given rank r and given Chern classes \(c_i\) on X, \(1 \le i \le k\) (cf. [6, p.  1250083-9]). Therefore, one is also interested e.g. in understanding: whether M contains at least an irreducible component, say , which is generically smooth, i.e. reduced, or even smooth; to which sheaf on X corresponds the general point of such a component ; what can be said about the birational geometry of , namely if it is perhaps rational/unirational; finally, if by chance M turns out to be also irreducible, that is, .

In this paper we are interested in some of the aforementioned properties for the moduli spaces of Ulrich vector bundles on a variety \(X_e\) which is a 3-fold scroll over a Hirzebruch surface \({\mathbb {F}}_e\), with \(e \ge 0\). More precisely on 3-fold scrolls \(X_e\) arising as embedding, via very-ample tautological line bundles , of projective bundles over \({\mathbb {F}}_e\), where are very-ample rank-2 vector bundles on \({\mathbb {F}}_e\) with Chern classes numerically equivalent to \(3 C_e + b_ef\) and , where \(C_e\) and f are, as customary, generators of \(\textrm{Num}({\mathbb {F}}_e)\) as in [14, V, Prop. 2.3] and where \(b_e\) and \(k_e\) are integers satisfying some natural numerical conditions. We will set the hyperplane line bundle of the embedded 3-fold scroll, which we will also call tautological polarization of \(X_e\), as .

The existence of Ulrich bundles on such threefolds \(X_e\) has been considered in [13], where it was proved that \(X_e\) does not support any Ulrich line bundle w.r.t. \(\xi \), unless \(e = 0\). As to Ulrich vector bundles of rank \(r \ge 2\), it was proved in [13] that the moduli space M, in the above sense, arising from rank-r vector bundles on \(X_{e\ge 0}\) which are Ulrich w.r.t. \(\xi \) and with first Chern class

is not empty and it contains a generically smooth component of dimension

The general point has been proved to correspond to a slope-stable vector bundle, of slope w.r.t. \(\xi \) given by (see Theorem 2.5 below, for more details).

As a consequence of such result and a natural one-to-one correspondence among rank-r vector bundles on \(X_e\), of the form , which are Ulrich w.r.t. \(\xi \) on \(X_e\), and rank-r vector bundles on \({\mathbb {F}}_e\), of the form , which are Ulrich w.r.t. , in [13] we have deduced Ulrichness results for vector bundles on the base surface \({\mathbb {F}}_e\) with respect to naturally associated very ample polarization , see Theorem 2.6 for more details.

By a result of Antonelli, [1, Theorem 1.2], if is a rank-r vector bundle on \({\mathbb {F}}_e\) which is Ulrich with respect to a very ample polarization of the form and with , then must fit into a short exact sequence of the form

where \(\gamma , \delta \) and \(\tau \) are suitably defined by \(r, \alpha , \beta , a, b, e\) (cfr. (3.1)). This fact will be useful in the present note to give further information about our modular components as in [13]. Our main results in this paper are the following

Theorem A

(cf. Theorem 3.2, below) For any integer \(e \ge 0\), let \({\mathbb {F}}_e\) be the Hirzebruch surface and let denote the line bundle \(\alpha C_e + \beta f\) on \({\mathbb {F}}_e\), where \(C_e\) and f are generators of \(\textrm{Num}({\mathbb {F}}_e)\) (cf. [14, V, Prop. 2.3]). Let \((X_e, \xi )\) be a 3-fold scroll over \({\mathbb {F}}_e\) as above, where \(\varphi : X_e \rightarrow {\mathbb {F}}_e\) denotes the scroll map. Then the moduli space of rank-\(r\ge 2\) vector bundles on \(X_e\) which are Ulrich w.r.t. \(\xi \) and with first Chern class

is not empty and it contains a generically smooth component , which is of dimension

(see Theorem 2.5) and which is moreover unirational.

For the moduli space of rank-\(r \ge 2\) bundles on \({\mathbb {F}}_e\), the base of the scroll \(X_e\), which are Ulrich w.r.t. the polarization , a stronger result holds; precisely

Theorem B

(cf. Theorem 3.1, below) Let be the moduli space of rank-r vector bundles on \({\mathbb {F}}_e\) which are Ulrich w.r.t. and with first Chern class

Then is generically smooth, of dimension

(see Theorem 2.6) and moreover it is irreducible and unirational.

The above theorems extend unirationality results in [1] and [9].

The paper is structured as follows. In Sect. 1 we fix notation and terminology. In Sect. 2 we recall some of the known results that we will use throughout the paper. In Sect. 3 we state and prove our new main results.

2 Notation and terminology

In this paper we work over \({\mathbb {C}}\). All schemes will be endowed with the Zariski topology. We will interchangeably use the terms rank-r vector bundle on a smooth, projective variety X and rank-r locally free sheaf. In particular, sometimes, to ease some formulas, with a small abuse of notation we identify divisor classes with the corresponding line bundles, interchangeably using additive and tensor-product notation. The dual bundle of a rank-r vector bundle on X will be denoted by ; thus, if L is of rank-1, i.e. it is a line bundle, we interchageably use \(L^{\vee }\) or \(-L\). If M is a moduli space, parametrizing objects modulo a given equivalence relation, and if Y is a representative of an equivalence class in M, we will denote by \([Y] \in M\) the point corresponding to Y. For non-reminded general terminology, we refer the reader to [14]).

Because our object will be Ulrich bundles, we recall their definition and basic properties.

Definition 1.1

Let \(X\subset {\mathbb {P}}^N\) be a smooth, irreducible, projective variety of dimension n and let H be a hyperplane section of X. A vector bundle on X is said to be Ulrich with respect to if

Definition 1.2

Let \(X\subset {\mathbb {P}}^N\) be a smooth, irreducible, projective variety of dimension n polarized by , where H is a hyperplane section of X, and let be a rank-2 vector bundle on X which is Ulrich with respect to . Then \({\mathcal {U}}\) is said to be special if .

For the reader’s convenience, we briefly remind facts concerning (semi)stability and slope-(semi)stability properties of Ulrich bundles as in [6, Def. 2.7]. Let X be a smooth, irreducible, projective variety and let be a vector bundle on X; recall that is said to be semistable (in the sense of Gieseker-Maruyama) if for every non-zero coherent subsheaf , with , the inequality holds true, where and are the Hilbert polynomials of the sheaves. Furthermore, is stable if strict inequality above holds. Similarly, recall that the slope of a vector bundle (w.r.t. a given polarization on X) is defined to be ; the bundle is said to be \(\mu \)-semistable, or even slope-semistable, if for every non-zero coherent subsheaf with , one has . The bundle is \(\mu \)-stable, or slope-stable, if strict inequality holds.

The two definitions of (semi)stability are in general related as follows (cf. e.g. [6, § 2]):

$$\begin{aligned} \text{ slope-stability } \Rightarrow \text{ stability } \Rightarrow \text{ semistability } \Rightarrow \text{ slope-semistability }. \end{aligned}$$

If is in particular a rank-r vector bundle which is Urlich w.r.t. , then is always semistable, so also slope-semistable (cf. [6, Thm. 2.9-(a)]); moreover, for the notions of stability and slope-stability coincide (cf. [6, Thm. 2.9-(c)]).

As for the projective variety which will be the support of Ulrich bundles we are interested in, throughout this work we will denote it by \(X_e\) and it will be a 3-dimensional scroll over the Hirzebruch surface , with \(e \ge 0\) an integer.

More precisely, let \(\pi _e: {\mathbb {F}}_e \rightarrow {\mathbb {P}}^1\) be the natural projection onto the base. Then, as in [14, V, Prop. 2.3], \(\textrm{Num}({\mathbb {F}}_e) = {\mathbb {Z}}[C_e] \oplus {\mathbb {Z}}[f],\) where:

  • \(f:= \pi _e^*(p)\), for any \(p \in {\mathbb {P}}^1\), whereas

  • \(C_e\) denotes either the unique section corresponding to the morphism of vector bundles on \({\mathbb {P}}^1\) , when \(e>0\), or the fiber of the other ruling different from that induced by f, when otherwise \(e=0\).

In particular

$$\begin{aligned} C_e^2 = - e, \; f^2 = 0, \; C_ef = 1. \end{aligned}$$

Let be a rank-2 vector bundle over \({\mathbb {F}}_e\) and let be its \(i^{th}\)-Chern class. Then , for some \( a, b \in {\mathbb {Z}}\), and . For the line bundle we will also use the notation .

From now on, we will consider the following:

Assumption 1.3

Let \(e \ge 0\), \(b_e\), \(k_e\) be integers such that

$$\begin{aligned} b_e-e< k_e< 2b_e-4e, \end{aligned}$$
(1.1)

and let be a rank-2 vector bundle over \({\mathbb {F}}_e\), with

which fits in the exact sequence

(1.2)

where \(A_e\) and \(B_e\) are line bundles on \({\mathbb {F}}_e\) such that

$$\begin{aligned} A_e \equiv 2 C_e + (2b_e-k_e-2e) f \;\; \textrm{and} \;\; B_e \equiv C_e + (k_e - b_e + 2e) f \end{aligned}$$
(1.3)

From (1.2), in particular, one has .

By results in [13], as above, turns out to be very ample on \({\mathbb {F}}_e\). Thus we take \(X_e\) to be the 3-fold scroll arising as embedding, via very-ample tautological line bundle , of the projective bundle .

3 Preliminaries

In this section, for the reader convenience, we state some of the known results that we will be using in the sequel.

The following Theorem 2.1, (cf. [12, Theorem 2.4]) states under which conditions an Ulrich bundle on the base of the scroll gives rise to a bundle on the scroll itself which is Ulrich w.r.t. the tautological polarization \(\xi \).

Theorem 2.1

([12, Theorem 2.4]) Let (S, H) be a polarized surface, with H a very ample line bundle, and let be a rank-2 vector bundle on S such that is (very) ample and spanned. Let be a rank-\(r \ge 1\) vector bundle on S. Let be a 3-fold scroll over S, where \(\xi \) is the tautological polarization, and let \(X \xrightarrow {\varphi } S\) denote the scroll map. Then the vector bundle is Ulrich with respect to \(\xi \) if and only if the bundle is such that

(2.1)

In particular, if is very ample on S, then the rank-r vector bundle on X, , is Ulrich with respect to \(\xi \) if and only if the rank-r vector bundle on S, , is Ulrich with respect to .

Viceversa, starting with a rank-r vector bundle on the 3-fold scroll \((X, \xi )\) which is Ulrich w.r.t. \(\xi \), satisfying suitable properties, we recall how to obtain an Ulrich vector bundle of the same rank on the base S of the scroll.

Let \(\varphi :X \rightarrow S\) be a 3-fold scroll over a surface S. Let us recall, see [5, Theorem 11.1.2.], that a general hyperplane section \({\widetilde{S}}\) of X has the structure of a blow-up of the base surface S at points and one can consider the following diagram:

(2.2)

where i is the inclusion and \(\varphi '\) is the blow-up map, where we denote by \(E_i\) the exceptional divisors of the latter map. More precisely, if \({\widetilde{S}} \in |\xi |\) is a general hyperplane section of X, then it corresponds to the vanishing locus of a general global section \({\widetilde{\sigma }} \in H^0(X, \xi )\); since one has , then \({\widetilde{\sigma }}\) bijectively corresponds to a global section \(\sigma \) of whose vanishing locus \(Z:= V (\sigma )\) is a zero-dimensional subscheme on S which is an element of . From [5, Theorem 11.1.2.], \({\widetilde{S}}\) turns out to be isomorphic to the blow-up of \(\varphi ': {\widetilde{S}} \rightarrow S\) at such points Z and, for any \(z \in Z\), the \(\varphi \)-fiber \(\varphi ^{-1}(z):= F_z\) of X is contained in \({\widetilde{S}}\) as the \(\varphi '\)-exceptional divisor \(E_z\) over the point z of such a blow-up \(\varphi '\).

With this set-up, in [12, Thm. 6.1, Prop. 6.2], the authors gave conditions to get bijective correspondences among rank-r bundles on X which are Ulrich w.r.t. the tautological polarization \(\xi \) and rank-r bundles on the base surface S which are Ulrich w.r.t. the naturally related polarization as in Theorem 2.1.

Theorem 2.2

( [12, Theorem 6.1]) Let \(\varphi : X \rightarrow S\) be a 3-fold scroll over a surface S and let be a rank-r vector bundle on X which is Ulrich with respect to the tautological polarization \(\xi \), i.e. . Let us suppose that is very ample on S. Assume that on the general fiber \(F=\varphi ^{-1}(s)\), \(s\in S\), the vector bundle splits as follows: . Then , with , is a rank-r vector bundle on S which is Ulrich w.r.t. .

In the following remark we comment on the hypotheses of Theorem 2.2, in order to better explain the aforementioned Ulrich-bundle bijective correspondence arising from Theorems 2.1 and 2.2 (cf. Proposition 2.4 below).

Remark 2.3

We like to point out that the assumption on the splitting-type of the vector bundle on the general fiber F of \(\varphi \) as as in Theorem 2.2 implies that such a splitting-type holds true for all \(\varphi \)-fibers \(\varphi ^{-1}(u):= F_u\), for u varying in a suitable open dense subset \(U \subseteq S\). Thus, from the previous description on the birational structure of a general hyperplane section \({\widetilde{S}} = V({\widetilde{\sigma }})\) of X as in (2.2), the main points to let the Ulrich-bundle bijective correspondence arise are first of all that the zero-dimensional scheme \(Z = V (\sigma )\), corresponding to \({\widetilde{S}} \in |\xi |\) general, is entirely contained in the open set \(U \subseteq S\) (so that, for any \(z \in Z\), the restriction of to \(F_z:= \varphi ^{-1}(z)\) is namely, from (2.2), , for any , where \(\sum _i\, E_i\) denotes the total exceptional divisor of the blow-up \(\varphi '\) of S along Z) and then the use of [8, Thm. 4.2].

Arguments described in Remark 2.3 are the principles used in [12] to get the following Proposition.

Proposition 2.4

([12, Prop. 6.2]) Let \(\varphi : X \rightarrow S\) be a 3-fold scroll over a surface S, where for some very ample rank-2 vector bundle on S. Assume that is very ample on S. Then there exists a bijection:

the bijection given by the maps

and

Because we are interested on moduli spaces of Ulrich bundles on threefolds scrolls \(X_e\) over \({\mathbb {F}}_e\), as well as on moduli spaces of Ulrich bundles on \({\mathbb {F}}_e\), we recall what was already proved in [13].

Theorem 2.5

([13, Main Theorem]) For any integer \(e \ge 0\), consider the Hirzebruch surface \({\mathbb {F}}_e\) and let denote the line bundle \(\alpha C_e + \beta f\) on \({\mathbb {F}}_e\), where \(C_e\) and f are generators of \(\textrm{Num}({\mathbb {F}}_e)\).

Let \((X_e, \xi )\) be a 3-fold scroll over \({\mathbb {F}}_e\) as in Assumption 1.3, where \(\varphi : X_e \rightarrow {\mathbb {F}}_e\) denotes the scroll map. Then:

(a) \(X_e\) does not support any Ulrich line bundle w.r.t. \(\xi \) unless \(e = 0\). In this latter case, the unique Ulrich line bundles on \(X_0\) are the following:

  1. (i)

    and ;

  2. (ii)

    for any integer \(t\ge 1\), and , which only occur for \(b_0=2t, k_0=3t\).

(b) Set \(e=0\) and let \(r \ge 2\) be any integer. Then the moduli space of rank-r vector bundles on \(X_0\) which are Ulrich w.r.t. \(\xi \) and with first Chern class

is not empty and it contains a generically smooth component of dimension

The general point corresponds to a slope-stable vector bundle, of slope w.r.t. \(\xi \) given by . If moreover \(r=2\), then is also special (cf. Def. 1.2 above).

(c) When \(e >0\), let \(r \ge 2\) be any integer. Then the moduli space of rank-r vector bundles on \(X_e\) which are Ulrich w.r.t. \(\xi \) and with first Chern class

is not empty and it contains a generically smooth component of dimension

The general point corresponds to a slope-stable vector bundle, of slope w.r.t. \(\xi \) given by . If moreover \(r=2\), then is also special.

We want to stress that in [13, Proof of Thm. 5.1] it has been proved that bundles \(L_1, L_2\) and , for any \(r \ge 2\), as in Theorem 2.5 split on any \(\varphi \)-fiber of \(X_e\) as requested in Theorem 2.2 and in Proposition 2.4, namely for any \(\varphi \)-fiber F, one has whereas (this is due to the iterative contructions in [13] of such bundles as deformations of iterative extensions). As a direct consequence of Theorem 2.5, Theorem 2.1 and the one–to–one correspondence in Proposition 2.4, in [13] we could prove the following result concerning moduli spaces of rank-r vector bundles on Hirzebruch surfaces \({\mathbb {F}}_e\), for any \(r \ge 1\) and any \(e \ge 0\), which are Ulrich w.r.t. the very ample line bundle , with \(b_e \ge 3e+2\) as it follows from Assumption 1.3 (the case \(r=1,2,3\) already known by [1, 2, 7]).

Theorem 2.6

([13, Theorem 5.1]) For any integer \(e \ge 0\), consider the Hirzebruch surface \({\mathbb {F}}_e\) and let denote the line bundle \(\alpha C_e + \beta f\) on \({\mathbb {F}}_e\), where \(C_e\) and f are generators of \(\textrm{Num}({\mathbb {F}}_e)\).

Consider the very ample polarization on \({\mathbb {F}}_e\), where \(b_e \ge 3e+2\). Then:

  1. (a)

    \({\mathbb {F}}_e\) does not support any Ulrich line bundle w.r.t. unless \(e = 0\). In this latter case, the unique line bundles on \({\mathbb {F}}_0\) which are Ulrich w.r.t. are

  2. (b)

    Set \(e=0\) and let \(r \ge 2\) be any integer. Then the moduli space of rank-r vector bundles on \({\mathbb {F}}_0\) which are Ulrich w.r.t. and with first Chern class

    is not empty and it contains a generically smooth component of dimension

    $$\begin{aligned} {\left\{ \begin{array}{ll} \frac{(r^2 -1)}{4}(6 b_0 -4), &{} \text{ if } \text{ r } \text{ is } \text{ odd }, \\ \frac{r^2}{4} (6b_0-4) +1, &{} \text{ if } \text{ r } \text{ is } \text{ even }. \end{array}\right. } \end{aligned}$$

    The general point of such a component corresponds to a slope-stable vector bundle.

  3. (c)

    When \(e >0\), let \(r \ge 2\) be any integer. Then the moduli space of rank-r vector bundles on \({\mathbb {F}}_e\) which are Ulrich w.r.t. and with first Chern class

    is not empty and it contains a generically smooth component of dimension

    $$\begin{aligned} {\left\{ \begin{array}{ll} \left( \frac{(r -3)^2}{4}+ 2 \right) (6 b_e - 9e -4) + \frac{9}{2}(r-3) (2b_e-3e), &{} \text{ if } r \text{ is } \text{ odd }, \\ \frac{r^2}{4} (6b_e- 9e-4) +1, &{} \text{ if } r \text{ is } \text{ even }. \end{array}\right. } \end{aligned}$$

    The general point of such a component corresponds to a slope-stable vector bundle.

4 Moduli spaces

Our aim in this section is to prove that the moduli space of Ulrich bundles on \({\mathbb {F}}_e\), \(e \ge 0\), as in Theorem 2.6 is irreducible, generically smooth and unirational, whereas that the generically smooth modular component of Ulrich bundles on \(X_e\), \(e \ge 0\), as in Theorem 2.5 is unirational.

Theorem 3.1

Let be the moduli space of rank-r vector bundles on \({\mathbb {F}}_e\) which are Ulrich w.r.t. and with first Chern class

(see Theorem 2.6). Then is generically smooth, irreducible, unirational and of dimension

Proof

From Theorem 2.6 we know that the moduli space is not empty.

Let be any irreducible component and let be its general point. So is of rank r and as in the statement of Theorem 3.1.

For simplicity let . By [1, Theorem 1.1] necessarily fits into the following short exact sequence

(3.1)

where \(\gamma =\alpha +\beta -r(2+b_e)-e(\alpha -3r)\),  \(\delta =\beta -r(b_e-1)-e(\alpha -3r)\), \(\tau =\alpha -2r\) which, after plugging in the value of \(\alpha \) and \(\beta \), become

\(\gamma =\frac{(b_e-2e+1)r-b_e+3}{2}\), \(\delta = \frac{(r-1)b_e}{2}-er\), \(\tau =\frac{3(r+1)}{2}\),   if r is odd,   and

\(\gamma = \frac{(b_e-2e+1)r}{2}\), \(\delta = \frac{(b_e-2e)r}{2}\), \(\tau =\frac{3r}{2}\), if r is even.

Thus is expressed as the cokernel of an injective map \(\phi \in \textrm{Hom}_{{\mathbb {F}}_e} ( {\mathscr {A}}, {\mathscr {B}})\), where and , with \(\gamma , \delta , \tau \) as above.

On the other hand, by [1, Theorem 1.3], if we take a general map \(\phi _{gen} \in \textrm{Hom}_{{\mathbb {F}}_e} ( {\mathscr {A}}, {\mathscr {B}})\) then \(\textrm{coker}(\phi _{gen})\) is a rank-r vector bundle on \({\mathbb {F}}_e\), in particular locally free, which is Ulrich w.r.t. , and with Chern classes \(c_1(\textrm{coker}(\phi _{gen}))\) and \(c_2(\textrm{coker}(\phi _{gen}))\) as those of . Since \( {\mathscr {A}}, {\mathscr {B}}\) are uniquely determined by r, e, \((3,b_e)\) and and since \(\textrm{Hom}_{{\mathbb {F}}_e} ( {\mathscr {A}}, {\mathscr {B}})\) is irreducible, it follows that , i.e. is therefore irreducible and moreover it is unirational, being dominated by \(\textrm{Hom}_{{\mathbb {F}}_e} ( {\mathscr {A}}, {\mathscr {B}})\).

The generic smoothness of and the formula for its dimension follow as they have already been proved in Theorem 2.6-(b), (c). \(\square \)

Theorem 3.2

For any integer \(e \ge 0\), let \({\mathbb {F}}_e\) be the Hirzebruch surface and let denote the line bundle \(\alpha C_e + \beta f\) on \({\mathbb {F}}_e\), where \(C_e\) and f are generators of \(\textrm{Num}({\mathbb {F}}_e)\).

Let \((X_e, \xi )\) be a 3-fold scroll over \({\mathbb {F}}_e\) as in Assumption 1.3, where \(\varphi : X_e \rightarrow {\mathbb {F}}_e\) denotes the scroll map. Then the moduli space of rank-\(r\ge 2\) vector bundles on \(X_e\) which are Ulrich w.r.t. \(\xi \) and with first Chern class as in Theorem 2.5 is not empty and it contains a generically smooth component which is unirational and of dimension

Proof

As we have seen in the proof of Theorem 3.1, a general , turns out to be , with \(\phi \) a general vector bundle morphism as in (3.1).

Now take , , \(\gamma , \delta \) and \(\tau \) as in the proof of Theorem 3.1; then for \(\phi \in \textrm{Hom}_{{\mathbb {F}}_e} ( {\mathscr {A}}, {\mathscr {B}})\) general, one has therefore

We first tensor this exact sequence by , then we pull it back via \(\varphi ^*\), where \(\varphi : X_e \rightarrow {\mathbb {F}}_e\) is the scroll map, and the sequence remains exact on the left since is locally free; subsequently we tensor the resulting short exact sequence with \(\xi \), the tautological polarization on \(X_e\), and thus we get the exact sequence

(3.2)

defining \({\overline{\phi }}\). Set and . Recall that the modular component \({\mathcal {M}}(r)\) as in Theorem 2.5 has an open dense subset parametrizing isomorphism classes of slope-stable, rank-r vector bundles , which are Ulrich w.r.t. the tautological polarization \(\xi \) of \(X_e\) and with Chern classes determined by the iterative constructions as in [13] (in particular, the first Chern class \(c_1\) is as reminded in Theorem 2.5); for general it has also been proved in [13, Proof of Thm. 5.1] that the bundle has in particular the splitting type requested by Proposition 2.4, namely , on any \(\varphi \)-fiber F. As a consequence of the bijective correspondence induced by Proposition 2.4, in [13] we deduced therefore that , with Ulrich w.r.t. on \({\mathbb {F}}_e\) as above.

Then the sequence (3.2) reads

(3.3)

In particular, for those morphisms \({\overline{\phi }} \in \textrm{Hom}_{X_e} ( {\overline{{\mathscr {A}}}}, {\overline{{\mathscr {B}}}})\) such that , one has that \(\textrm{coker}({\overline{\phi }}) \) is locally free, of rank r and it is moreover Ulrich on \(X_e\) w.r.t. the tautological polarization \(\xi \), with Chern classes , \(1 \le i \le 3\), computed by iterative constructions of the vector bundles as in [13] (e.g. \(c_1\) is reminded in Theorem 2.5 above).

Let \({{\overline{\phi }}_{gen}} \in \textrm{Hom}_{X_e} ({\overline{{\mathscr {A}}}},{\overline{{\mathscr {B}}}})\) be general; since

i.e. \({\mathscr {A}}^{\vee }\otimes {\mathscr {B}}\) is globally generated, so \({\overline{{\mathscr {A}}}}^{\vee } \otimes {\overline{{\mathscr {B}}}}\) is also globally generated. Therefore, by [3, Thm. 4.2], (cf. also [4, Thm. 2]) \({{\overline{\phi }}_{gen}}\) is injective and it gives rise to an exact sequence

Since \({\overline{\phi }} \in \textrm{Hom}_{X_e} ( {\overline{{\mathscr {A}}}}, {\overline{{\mathscr {B}}}})\) as in (3.3) is such that is locally free, then also \(\textrm{coker}({{\overline{\phi }}_{gen}})\) is locally free, as locally freeness is an open condition on the (irreducible) vector space \( \textrm{Hom}_{X_e} ( {\overline{{\mathscr {A}}}}, {\overline{{\mathscr {B}}}})\). Moreover, the rank of \(\textrm{coker}({{\overline{\phi }}_{gen}})\) is given by , with \(\gamma , \delta , \tau \) as in the proof of Theorem 3.1. Furthermore, once again from the irreducibility of \(\textrm{Hom}_{X_e} ({\overline{{\mathscr {A}}}},{\overline{{\mathscr {B}}}})\) and from the constancy of Chern classes in irreducible flat families of vector bundles of given rank (or even from the fact that and \(\textrm{coker}({{\overline{\phi }}_{gen}})\) are both locally free cokernels of injective vector bundle morphisms in \(\textrm{Hom}_{X_e} ( {\overline{{\mathscr {A}}}}, {\overline{{\mathscr {B}}}})\)) one has that

(3.4)

Finally since is Ulrich on \(X_e\) w.r.t. \(\xi \) we have

then by semicontinuity

$$\begin{aligned} h^i(\textrm{coker}({{\overline{\phi }}_{gen}})(-j\xi ))= 0\quad \text{ for } 0\le i \le 3 \text{ and } 1\le j \le 3; \end{aligned}$$

hence \(\textrm{coker}({{\overline{\phi }}}_{gen})\) is Ulrich w.r.t. \(\xi \).

The fact that \(\textrm{Hom}_{X_e} ({\overline{{\mathscr {A}}}},{\overline{{\mathscr {B}}}})\) is irreducible implies that it must dominate the modular component (as in Theorem 2.5) containing as its general point, which therefore implies that is unirational. The generic smoothness of as well as its dimension formula have already being proved in Theorem 2.5-(b), (c) (more precisely in [13, Main Theorem]). \(\square \)