On the Hilbert scheme of the moduli space of vector bundles over an algebraic curve

Let $M(n,\xi )$ be the moduli space of stable vector bundles of rank $n\ge 3$ and fixed determinant $\xi$ over a smooth projective algebraic curve $X$ over $\mathbb{C}$ of genus $g\ge 4.$ 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 $Hil{b}_{M(n,\xi )}$, 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}$^2,M(3,\xi))=8g-7$andwegiveasufficientconditionfor$Mor_{2ns}($\mathbb{P}$^1,M(n,\xi))$tobenon-emptywith$s\geq 1.$