AbstractLet X be any smooth prime Fano threefold of degree $$2g2$$
2
g

2
in $${\mathbb P}^{g+1}$$
P
g
+
1
, with $$g \in \{3,\ldots ,10,12\}$$
g
∈
{
3
,
…
,
10
,
12
}
. We prove that for any integer d satisfying $$\left\lfloor \frac{g+3}{2} \right\rfloor \leqslant d \leqslant g+3$$
g
+
3
2
⩽
d
⩽
g
+
3
the Hilbert scheme parametrizing smooth irreducible elliptic curves of degree d in X is nonempty and has a component of dimension d, which is furthermore reduced except for the case when $$(g,d)=(4,3)$$
(
g
,
d
)
=
(
4
,
3
)
and X is contained in a singular quadric. Consequently, we deduce that the moduli space of rank–two slope–stable ACM bundles $${\mathcal F}_d$$
F
d
on X such that $$\det ({\mathcal F}_d)={\mathcal O}_X(1)$$
det
(
F
d
)
=
O
X
(
1
)
, $$c_2({\mathcal F}_d)\cdot {\mathcal O}_X(1)=d$$
c
2
(
F
d
)
·
O
X
(
1
)
=
d
and $$h^0({\mathcal F}_d(1))=0$$
h
0
(
F
d
(

1
)
)
=
0
is nonempty and has a component of dimension $$2dg2$$
2
d

g

2
, which is furthermore reduced except for the case when $$(g,d)=(4,3)$$
(
g
,
d
)
=
(
4
,
3
)
and X is contained in a singular quadric. This completes the classification of rank–two ACM bundles on prime Fano threefolds. Secondly, we prove that for every $$h \in {\mathbb Z}^+$$
h
∈
Z
+
the moduli space of stable Ulrich bundles $${\mathcal E}$$
E
of rank 2h and determinant $${\mathcal O}_X(3h)$$
O
X
(
3
h
)
on X is nonempty and has a reduced component of dimension $$h^2(g+3)+1$$
h
2
(
g
+
3
)
+
1
; this result is optimal in the sense that there are no other Ulrich bundles occurring on X. This in particular shows that any prime Fano threefold is Ulrich wild.