# Nature of morphism between Brill Noether varieties

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.

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?

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.
• how do you say it has a unique component dominating $Pic^{g+2} C$. Is this uniqueness true for any $G^r_d$ with $d=g+r$? Oct 14, 2016 at 13:47
• Yes. There is a nonempty open subset $U\subset\mathrm{Pic}^{g+r}(C)$ such that $h^0(L)=r+1$ for $L\in U$; the map $p:G^r_{g+r}\rightarrow\mathrm{Pic}^{g+r}(C)$ induces an isomorphism of $p^{-1}(U)$ onto $U$. The closure of $U$ in $G^r_{g+r}$ is the only component dominating $\mathrm{Pic}^{g+r}$.
• I was thinking about your example. The fiber is not in the dominating component. The fiber doesn't contain any complete linear series. Is it possible that a component of $G^r_d$ does not contain any complete linear series? Nov 3, 2016 at 15:13