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Let $C$ be a smooth projective curve, $J$ its Jacobian (of degree $d$, parametrizing degree $d$ line bundles, $d \geq 0$). Let $W_d^r$ be the Brill--Noether variety parameterizing degree $d$ line bundles with at least $r+1$ linearly independent sections, i.e. set theoretically \begin{equation} W_d^r = [ L \in J \mid h^0(L) \geq r+1 ] \end{equation} Let $L$ be an element in $W_d^r$. Assume that $W_d^r$ is smooth at $L$. There is a cup product map $$ H^1(C, \mathcal{O}) \to \mathrm{Hom}(H^0(C, L), H^1(C, L)) $$ whose kernel is precisely the tangent space of $W_d^r$ at $L$. Identifying $H^1(C, \mathcal{O})$ with the tangent space of $J$ at $L$, we see that there is a homomorphism $$ N_{W^r_d/J, L} \to \mathrm{Hom}(H^0(C, L), H^1(C, L)) $$ where $N_{W^r_d/J, L}$ is the normal space of $W_d^r$ in $J$ at the point $L$.

The question is, is there any alternative way to get this homomorphism (without using the cohomological terms like cup product)? For instance, via the definition of $W^r_d$ using Fitting ideals, etc.

Moreover, when we have a family of smooth projective curves, and then there is a family of $W^r_d$'s, does this homomorphism fit in families?

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I gave the question an extra tag to make it more visible. –  Artie Prendergast-Smith Mar 19 '13 at 20:05

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