Is the given set an open subset of the $\mathcal{G}^r_d(|L|)$ 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 $|L|$.
 A: 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 $\mathcal{G}^r_d(|L|_s)$.     
