Let $C$ be a smooth curve over complex numbers. Consider the Brill Noether varieties. If $g$ is the genus of $C$. If $r,d$ are positive integers,

$$W^r_d=\{A\in Pic^d(C): h^0(A)\geq r+1\},$$ $$G^r_d=\{(A,V): A\in Pic^d(C), V\in G(r+1,H^0(C,A))\}.$$

We have the projection morphism $p:G^r_d\rightarrow W^r_d$, which is constructed by 'canonical blow up', which doesnt turn out to be blow up in usual sense.

Related question On the construction of the varieties parametrizing special linear series on a curve

Suppose that $W^r_d$ is irreducible, is it true that $G^r_d$ is irreducible as well? If $p$ is the blow up map in the usual sense, this is true. But in this case is this true?

For example if $d\geq r+ g$, then $W^r_d=Pic^d C$ by Riemann Roch. Then is $G^r_d$ irreducible?