Unirationality of the universal moduli space of semistable bundles over smooth curves

We construct explicit dominant, rational morphisms from projective bundles over rational varieties to relevant moduli spaces, showing their unirationality. These constructions work for $U_{r,d,g}$; for all ranks, degrees and genus $2\le g\le 9$. Furthermore, the arguments presented also show that a similar conclusion can be made for $U_{r,\mathcal{L},g}$ for all $r,d$ and unirational $M_g$