Let
S
U
C
(
2
)
\mathcal {SU}_C(2)
be the moduli space of rank 2 semistable vector bundles with trivial determinant on a smooth complex algebraic curve
C
C
of genus
g
>
1
g>1
. We assume
C
C
nonhyperellptic if
g
>
2
g>2
. In this paper we construct large families of pointed rational normal curves over certain linear sections of
S
U
C
(
2
)
\mathcal {SU}_C(2)
. This allows us to give an interpretation of these subvarieties of
S
U
C
(
2
)
\mathcal {SU}_C(2)
in terms of the moduli space of curves
M
0
,
2
g
\mathcal {M}_{0,2g}
. In fact, there exists a natural linear map
S
U
C
(
2
)
→
P
g
\mathcal {SU}_C(2) \to \mathbb {P}^g
with modular meaning, whose fibers are birational to
M
0
,
2
g
\mathcal {M}_{0,2g}
, the moduli space of
2
g
2g
-pointed genus zero curves. If
g
>
4
g>4
, these modular fibers are even isomorphic to the GIT compactification
M
0
,
2
g
G
I
T
\mathcal {M}_{0,2g}^{GIT}
. The families of pointed rational normal curves are recovered as the fibers of the maps that classify extensions of line bundles associated to some effective divisors.