Let $X$ be a smooth projective surface over $\mathbb{C}$. Let $L$ be a very ample line bundle on $X$. We have a variety $\mathcal{G}^r\_d(|L|\_s)$ associated to the linear system of curves $|L|$. The support of this variety is given by 


Supp$\ \mathcal{G}^r\_d(|L|\_s)=\{(C,(A,V))\ | C\in |L|\_s, (A,V)$ is a $g^r\_d$ on $C\}.$


By definition $(A,V)$ is a $g^r\_d$ on $C$ means that $A\in Pic^d(C)$ and $V$ is a linear subspace of $H^0(C,A)$ of dimension $r+1$.


My question is as follows.


Consider $B= \{(C,(A,V))|C\in |L|\_s,(A,V)$ is a basepoint free $g^r\_d$ on $C\}\subset\mathcal{G}^r\_d(|L|\_s)$. Is this an open subset of $\mathcal{G}^r\_d(|L|\_s)$. How can we see this?


Can we say something at least for $\mathcal{G}^1\_d(|L|\_s)$ and $\mathcal{G}^2\_d(|L|\_s)$?


If $\mathcal{G}^1\_d(|L|\_s)$ is nonempty then the dimension of $B$ will be strictly lesser? 


Notation: By $C\in |L|\_s$, we mean $C$ is a smooth curve in $|L|$.



Is the given set an open subset of the $\mathcal{G}^r_d(|L|)$
Denote by $\mathcal{C} \to |L|\_s$ the universal curve. This is faithfully flat, finitely presented, and even smooth. Denote by $\mathcal{G}$ the fiber product $\mathcal{C}\times\_{|L|\_s} \mathcal{G}^r\_d(|L|\_s)$, so that also $\mathcal{G}\to \mathcal{G}^r\_d(|L|\_s)$ is faithfully flat and finitely presented. For the fiber product $\mathcal{C}\_{\mathcal{G}} = \mathcal{C} \times\_{|L|\_s} \mathcal{G}$ with its natural projection, $\pi:\mathcal{C}\_{\mathcal{G}} \to \mathcal{G}$, the diagonal $\Delta\_{\mathcal{C}/|L|\_s}$ defines a section, $s:\mathcal{G} \to \mathcal{C}\_{\mathcal{G}}$. In particular, this implies that there is a Poincare invertible sheaf $\mathcal{P}$ on $\mathcal{C}\_{\mathcal{G}}$ such that $s^\*\mathcal{P}$ is $\mathcal{O}\_{\mathcal{G}}$ and such that for every geometric point of $\mathcal{G}$ mapping to a point $(C,A,V)$ of $\mathcal{G}^r\_d(|L|\_s)$, the restriction of $\mathcal{P}$ to the fiber $C$ of $\pi$ over this point equals $A$. Moreover, the linear systems $V$ define a locally free sheaf $\mathcal{V}$ on $\mathcal{G}$ of rank $r+1$ and a universal surjection $$\phi':R^1\pi\_\*(\omega\_\pi\otimes\_{\mathcal{O}} \mathcal{P}^\vee) \to \mathcal{V}^\vee.$$ Of course I am writing things in terms of Serre duality to get a formulation that is compatible with arbitrary base change. By adjunction, $\phi'$ is equivalent to a morphism $$\phi:\pi^\* \mathcal{V} \to \mathcal{P}.$$ For your subset $B$ of $\mathcal{G}^r\_d(|L|\_s)$, the inverse image of $B$ in $\mathcal{G}$ equals the complement in $\mathcal{G}$ of the image under $\pi$ of the support of the cokernel of $\phi$. Since $\phi$ is a homomorphism of coherent sheaves on a Noetherian scheme, this support is a closed subset. Since $\pi$ is proper, the image of this support in $\mathcal{G}$ is a closed subset. Thus the inverse image of $B$ in $\mathcal{G}$ is an open subset. Since $\mathcal{G}$ is faithfully flat and finitely presented over $\mathcal{G}^r\_d(|L|\_s)$, the image of this open subset of $\mathcal{G}$ is an open subset of $\mathcal{G}^r\_d(|L|\_s)$, i.e., $B$ is an open subset of $\mathcal{G}^r\_d(|L|\_s)$. 


