Let $X$ be a smooth projective curve over $k=\bar{k}$, and $Pic^d$ the $d$-part of the Picard group of $X$ (isomorphism classes of line bundles of degree $d$ on $X$).

I'd like to define a scheme $G_d^r$ parametrizing $g_d^r$'s on our curve $X$, so with set theoretic support given by $$ \operatorname{Supp}(G_d^r) = \{\; (L,W) \in Pic^d\times G(r+1, H^0(L)) \;\}. $$

My question is: ###What is the most elegant way to define $G_d^r$ as a scheme?

My thoughts:

Let $\mathscr{L}$ denote the Poincaré line bundle on $X$ (assuming it does exist), i.e. a line bundle on $X\times Pic^d$ such that for every $L\in Pic^d$ we have $f^*\mathscr{L} = L$, where $f:*\to Pic^d$ is the inclusion of $L$ in $Pic^d$.

I think it's a good idea to consider the Grassmanian $ G(\mathscr{L}) := G(r+1, H^0(\mathscr{L})) $. Indeed $G_d^r$ can be seen as the subset of the product $$ Pic^d \times G(\mathscr{L}) $$ consisting of those couples $(L, \mathscr{W})$ such that $\;f^*\mathscr{W} \subset H^0(L)$.

I have the feeling that $G_d^r$ can be defined as a particular fiber product between $Pic^d$ and $G(\mathscr{L})$, but I don't understand how to build the right fiber diagram. Do you think this is indeed possible?

Remark: I believe we have $f^*H^0(\mathscr{L}) \cong H^0(L)$. Do you agree on this?