Let $C$ be an algebraic curve over an algebraically closed field $k$ of characteristic $0$, and let $\mathcal{L}$ be a base-point-free line bundle on $C$. Furthermore, let $p \in C$ be a smooth point, and fix an identification $\widehat{\mathcal{O}}_{C,p} \simeq k[[t]]$.
Let $\sigma_1, \dots, \sigma_m \in H^0(\mathcal{L})$ be global sections, and let $\sigma_1(t), \dots, \sigma_m(t) \in k[[t]]$ be the corresponding analytic-local germs. Write $\sigma_i(t) = \sum_{j \geq 0} b_{ij} t^j$ for each $i \in \{1, \dots, m\}$, let $I \subset k[\{x_{ij} : i \in \{1, \dots, m\}, j \geq 0\}]$ be an ideal in the polynomial ring over the symbols $x_{ij}$. Consider the following condition, denoted ($\star$): No nonzero element of $I$ vanishes when we take $x_{ij} = b_{ij}$ for all $i,j$.
Does condition ($\star$) hold for a "general" choice of the sections $\sigma_1, \dots, \sigma_m$? To make this question precise, fix a basis $e_1, \dots, e_n$ of $H^0(\mathcal{L})$. Does there exist an open subscheme $U$ of the affine space $\operatorname{Spec} k[\{y_{ij} : i \in \{1, \dots, m\}, j \in \{1, \dots, n\}\}]$ with the property that for every $k$-point $(\{a_{ij}\}) \in U$ we have that the condition ($\star$) holds for the sections $\sigma_i = \sum_{j = 1}^n a_{ij}e_j$?
(Note: in the above, I am assuming that $\mathcal{L}$ has enough global sections for this to be possible in the first place.)
