Fix an algebraic curve $C$ of genus $g$, and positive integers $d, r$. The variety $W^r\_d$ parametrizes complete linear series of degree $d$ and dimension at least $r$ on $C$ and the variety $G^r\_d$ parametrizes linear series of degree $d$ and dimension $r$ on $C$. I am trying to understand, what are the scheme structures on $W^r\_d$ and $G^r\_d$ following the book [A,C,G,H] of Arbarello, Cornalba, Griffiths, Harris (see Chapter 4, $\S 3$).


Fix a Poincare line bundle $\mathcal{L}$ of degree $d$ for $C$ and an effective divisor $E$ on $C$ with $m:=\deg{E}\geq 2g-d-1$, and let $\Gamma=E\times \text{Pic}^d(C)$ the divisor on $C\times \text{Pic}^d(C)$. Denote by $\nu: C\times \text{Pic}^d(C) \to \text{Pic}^d(C)$ the projection. 


My first question is why $R^1\nu\_{\*}\mathcal{L}(\Gamma)=0$, $\nu\_{\*}\mathcal{L}(\Gamma)$ is locally free of rank $d+m-g+1$ and $\nu\_{\*}(\mathcal{L}(\Gamma)/\mathcal{L})$ is locally free of rank $m$? In [A,C,G,H] is written that it is a consequence of the base change in cohomology, but I can't figure it out.


Taking the direct image of the short exact sequence
$$0\to \mathcal{L}\to\mathcal{L}(\Gamma)\to \mathcal{L}(\Gamma)/\mathcal{L}\to 0$$
on $C\times \text{Pic}^d(C)$ we obtain the long exact sequence
$$0\to \nu\_{\*}\mathcal{L}\to \nu\_{\*}\mathcal{L}(\Gamma)\stackrel{\gamma}\to \nu\_{\*}(\mathcal{L}(\Gamma)/\mathcal{L})\to R^1\nu\_{\*}\mathcal{L}\to 0.$$
Then $W^r\_d$ is $(m+d-g-r)$th determinantal variety attached to $\gamma$ and $G^r\_d$ is the canonical blow-up of $W^r\_d$. I read the Chapter 2 of [A,C,G,H] and understood the constructions of $W^r\_d$ and $G^r\_d$, but I didn't find there, why $G^r\_d$ is a blow-up of $W^r\_d$ (with the centre $W^{r+1}\_d$ I guess?). So, it is exactly my second question.


Thanks.



On the construction of the varieties parametrizing special linear series on a curve
For your first question, look up Grauert's theorem in Hartshorne. The vanishing of $R^1 \nu\_\* \mathcal L (\gamma)$ holds because the line bundle on the fiber has degree $\geq 2g-1$. Similarly $R^1 \nu\_\ast (\mathcal L(\Gamma)/\mathcal L) = 0$ since when this sheaf is restricted to fibers it has zero dimensional support, so again Grauert's theorem gives local freeness. 


For your second question, the canonical blowup of a determinantal variety is defined in chapter 2.2 as an incidence correspondence. The description of $G\_d^r$ at the beginning of the section is just a definition; at the end of the section it is explained why this definition actually makes sense.


