Globally generated line bundle on reducible curve

Question

Let $X=Y+Z$ be a reducible complex projective curve with two smooth components $Y$ and $Z$ intersecting at a node $p$. Suppose $L$ is a line bundle on $X$ such that $L|_Y$ and $L|_Z$ are globally generated, then is $L$ itself globally generated? If not is there some additional condition we can impose to make $L$ globally generated?

Answer 1

Since the node is simple, the global sections of $L$ are just the codimension-1 subspace of $H^0(L|_Y) \oplus H^0(L|_Z)$ consisting of sections with the same value at $p$.

edit: To see this, note that because the node is simple, there is a short exact sequence $0 \to O_X \to O_Y \oplus O_Z \to k(p) \to 0$, where $k(p)$ is a skyscraper sheaf at $p$ (the residue field). Now tensor with $L$.

In particular, any global section $s_Y$ of $L|_Y$ extends to a global section $s$ of $L$: if $s_Y(p) = 0$, extend $s_Y$ by setting $s=0$ on $Z$. If $s_Y(p) \ne 0$, instead take any global section of $L|_Z$ not vanishing at $p$ (using the fact that $L|_Z$ is globally generated) and rescale it to make it match with $s_Y$.

Thus, $H^0(L)$ generates the fiber at each point $y \in Y$. Likewise, it generates the fiber at each point $z \in Z$.