At Proposition 1.1.1. (1) of https://arxiv.org/pdf/0706.0494.pdf it is written that $H^0(X, L \otimes {\cal{J}}(X, ||L||)) = H^0(X, L)$ ensures that every global holomorphic section $s$ of $L$, i.e., $s \in H^0(X, L)$, vanishes at ${\cal{J}}(X, ||L||)$. I cannot make head or tail of what it means actually.
Suppose that $L = {\cal O}_X(D)$ and $M = {\cal O}_X(E)$ for chosen Cartier divisros $D, E$. The zero loci $D(s) \colon= \{x \in X \,|\, s(x) =0\}$ of the global holomorphic section $s \in H^0(X, L)$ is linearly equivalent to $D$. Then what follows from the condition that $H^0(X, {\cal O}_X(D)) = H^0(X, {\cal O}_X(D) \otimes_{\cal{O}_X} {\cal O}_X(E))$?