Is there a name for the following important group of line bundles with rational sections, given by Vakil in FOAG 15.4.3, page 436?
15.4.3. The important group of "line bundles with rational sections". Now consider the set $\{(\mathscr{L}, \mathrm{s})\}$ of pairs of line bundles $\mathscr{L}$ with rational sections $s$ of $\mathscr{L}$, not the zero section on any irreducible component of $X$, up to isomorphism. (An isomorphism $(\mathscr{L}, s) \cong\left(\mathscr{L}^{\prime}, s^{\prime}\right)$ means an isomorphism of line bundles under which $s$ is sent to $s^{\prime}$.) This set (after taking quotient by isomorphism) forms an abelian group under tensor product $\otimes$, with identity $\left(\mathscr{O}_X, 1\right)$. (Tricky question: what is the inverse of $(\mathscr{L}, s)$ in this group?)
It is important to notice that if $t$ is an invertible function on $X$, then multiplication by $t$ gives an isomorphism $$ (\mathscr{L}, s) \stackrel{\sim}{\longleftrightarrow}(\mathscr{L}, st) . $$ Similarly, $(\mathscr{L}, s) /(\mathscr{L}, u)=(\mathscr{O}, s / u)$. Here $s / u$ is a rational function. The map $\operatorname{div}$ yields a group homomorphism $$ \operatorname{div}:\{(\mathscr{L}, s)\} / \text { isomorphism } \longrightarrow \operatorname{Weil}X$$
Thank you in advance.