Let $X$ smooth projective curve over $\mathbb{C}$, fix a line bundle $L$ of degree $d$, and let $\mathcal{M}(r,d)$ denote the moduli space of semistable vector bundles of rank $r$ and degree $d$. It is known to be a normal projective variety.

There is a determinant map $det: \mathcal{M}(r,d)\rightarrow Pic^d(X)$, where $Pic^d(X)$ denotes the isomorphism class of line bundles of degree $d$.

Is it true that the $det$ map is a fibre bundle? I recently heard this statement, but I can't seem to prove it. Can someone give some idea, or give some reference if possible? Thanks in advance!