I could not decide if I should post this question in MO or Mathstackexchange, so feel free to downvote it if you think it does not belong here. I will delete my post and post it in MathSE in that case.


What I am wondering about is the following: Given a projective map $\pi:X\to B$, where $B$ is integral and $\mathscr{F}$ a coherent sheaf on $X$, which is flat over $Y$. Grauert's theorem tells us that there is an identification $$R^i\pi\_\*(\mathscr{F})\otimes k(b)\cong H^i(X\_b,\mathscr{F}\_b),$$ provided that the dimension of $H^i(X\_b,\mathscr{F}\_b)$ is the same for all $b\in B$. I want to know what happens if we relax the last condition. 


To this end, let us consider the following example: Let $C$ be a genus $g$ curve and $\mathscr{L}$ a Poincare line bundle on $C\times Pic^{2g-2}(C)$ and consider ${\pi\_2}\_\*\mathscr{L}$ on $Pic^{2g-2}(C)$. Now we know that over the open set $U$ of nonspecial bundles in $Pic^{2g-2}(C)$ ,we have (by Grauert's theorem) the identification that $${\pi\_2}\_\*\mathscr{L}\otimes k(L)\cong H^0(C,L)$$ for every $L\in U$. What happens in this case over special bundles? For instance, what is $${\pi\_2}\_\*\mathscr{L}\otimes k(K\_C)$$ concretely and how far is it from $H^0(C,K\_C)$? ($K\_C$ being of course the canonical bundle of $C$)



Fibers of pushforward of a bundle when the fiber dimension is not constant
This is an explanation and a correction of Mohan's comment. Let us use the base change isomorphism (of derived functors)
$$
Lj^\*R\pi\_{2\*}\mathcal{L} \cong R\Gamma(C,Li^\*\mathcal{L}),
$$
where $j$ is the embedding of the point $[K\_C]$ and $i$ is the embedding of the fiber $C$ over this point. Since $\mathcal{L}$ is locally free, the right hand side is just $H^\bullet(C,K\_C)$. It follows that 
$$
L\_0j^\*R^1\pi\_{2\*}\mathcal{L} \cong H^1(C,K\_C) \cong k
$$ 
(the base field). With a bit more work one can check that $R^1\pi\_{2\*}\mathcal{L} \cong \mathcal{O}\_{[K\_C]}$. Now note that 
$$
L\_ti^\*\mathcal{O}\_{[K\_C]} \cong k^{g \choose t}.
$$ 
Canceling out the contribution $k^g$ of $L\_1R^1$ from $H^0(C,K\_C)$, we see that the contribution of $L\_0R^0$ should be zero, hence $L\_0R^0$ should cancel out in the spectral sequence with $L\_2R^1$. This shows that its fiber is $k^{g \choose 2}$. But this also shows that the map from $L\_0R^0$ to $H^0(C,K\_C)$ is zero (this is the correction for the comment).


