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](https://mathoverflow.net/questions/121540/on-the-construction-of-the-varieties-parametrizing-special-linear-series-on-a-cu)


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?



Nature of morphism between Brill Noether varieties
No in general. For instance, take for $C$ a hyperelliptic cuve of even genus $g=2k$ with $k\geq 6$. Then $G^2\_{g+2}$ has a unique component (of dimension $g+2$) which dominates $\mathrm{Pic}^{g+2}(C)$, but $p^{-1}((k+1)g^1\_2)\cong \mathbb{G}(3,k+2)$ has dimension $3k-3>g+2$, so it cannot be contained in that component.


