Let $k$ be algebraically closed and $X$ a projective curve over $k$. Given a line bundle $\mathscr{L}$ on $X$ and sections $s_0, \dots, s_n \in H^0(X, \mathscr{L})$, I wish to define a closed subscheme $$ Z \subset \mathbf{P}^n \times X $$ whose closed points are of the form $([\lambda_0 : \dots : \lambda_n], x)$ where $\lambda_0 s_0 + \dots + \lambda_n s_n$ vanishes at $x$, i.e. is in the maximal ideal of $\mathscr{L}_x$.
I've worked this out for the case $n=0$. In this case $\mathbf{P}^0 \times X = \{\text{pt}\} \times X \cong X$, so we can take an affine open $U = \text{Spec} \, A$ where $\mathscr{L}|_U \cong \mathcal{O}_U$ and then $Z \cap U$ is just described by the vanishing locus of $s$ as a section of $\mathcal{O}_U$. This description is compatible on an affine open cover, so we can patch this together to get a closed subscheme $Z$.
In the general case, I suppose we want to do the same thing and take an affine open $D_+(T_i) \times U \subset \mathbf{P}^n_k \times X$. Then this open is the spectrum of $k[y_1, \dots, y_n] \otimes_{\mathbf{Z}} A$, and the ideal of functions vanishing on $Z$ is generated by the $s_i$ (the goal being to find an ideal sheaf corresponding to $Z$). But I'm getting a bit lost in the details and would appreciate some guidance. Thanks!
These are typically done as follows and works for more general situations than line bundles. The sections give you a map $O_X^{n+1}\to L$ or in other words, a map $L^{-1}\to O_X^{n+1}$ by dualizing. Then we can pull this back to $\pi:\mathbb{P}^n\times X\to X$ and compose with the natural map $O_{\mathbb{P}^n\times X}^{n+1}\to O_{\mathbb{P}^n\times X}(1)$, where this is the pull back of $O_{\mathbb{P}^n}(1)$ by the first projection. Thus, we get a map $\pi^*{L^{-1}}\to O_{\mathbb{P}^n\times X}(1)$. The image of this map after twisting by $O(-1)$ gives an ideal sheaf (of a Cartier divisor), which is precisely your $Z$.