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 $\{(C,(A,V))|C\in |L|_s,(A,V)$ 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?
Notation: By $C\in |L|_s$, we mean $C$ is a smooth curve in $|L|$.