  Let $M(n,\xi)$ be the moduli space of stable vector bundles of rank $n\geq 3$
and fixed determinant $\xi$ over a smooth projective algebraic curve $X$ over
$\mathbb{C}$ of genus $g\geq 4.$ We use the gonality of the curve and $r$-Hecke
morphisms to describe a smooth open set and to compute the dimension of a
component of the Hilbert scheme $Hilb_{M(n,\xi)}$, of the scheme of morphisms
$Mor(\mathbb{G},M(n,\xi))$ and of the moduli space $ M_{X \times \mathbb{G}}$
of stable bundles over $X\times \mathbb{G},$ where $\mathbb{G}$ is the
Grassmannian $\mathbb{G}(n-r,\mathbb{C}^n)$. In particular, we prove that $\dim
Mor_P(\mathbb{P}^2,M(3,\xi))=8g-7$ and we give a sufficient condition for
$Mor_{2ns}(\mathbb{P}^1,M(n,\xi))$ to be non-empty with $s\geq 1.$
