Suppose $X$ a normal project variety over complex numbers, $D$ is an integral Cartier divisor on $X$, and $L$ is a line bundle on $X$. Suppose one know the natural map $H^0(X, L) \to H^0(D, L|\_D)$ is **surjective**, I want to know if the following statement holds:


For any effective divisor $G\_D \in |L|\_D|$, is there an effective divisor $G \in |L|$, such that $G|\_D = G\_D$? (Of course $G, D$ should not have common components in order to make sense of $G|\_D$.)


Actually, my feeling is that this holds when there exists a divisor $R \in |L|$ such that $D$ is not a component of $R$, but may not hold in general. Hence I try to find a counterexample by looking at the case when $|L|$ has fixed component, and $D$ is one such fixed component. However, I don't have too much such example on hand. 



Restriction of a linear system to a divisor
Any effective divisor in $|L\vert\_D|$ is the zero locus of a global section of $L\vert\_D$. If $s\_D$ is such a section, by the surjectivity assumption there is a global section $s$ of $L$ on $X$ that restricts to $s\_D$. If $G$ is its zero locus then $G\vert\_D = G\_D$. So, what you ask is a tautology.


