Let $p:C\rightarrow S$ be a family of curves with $S=\text{Spec}\, R$ where $R$ is a DVR. Suppose that $C$ is smooth and the generic fiber $X_n$ of $p$ is smooth and the special fibre $X_0$ is a reducible nodal curve. More specifically $X_0=Y+Z$ where $Y$ and $Z$ are smooth curves intersecting transversally at a point, i.e. $Y\cdot Z=1$. Suppose there is a family $L$ of $g^r_d$-s on $C\setminus X_0$, then there is a concept of limit linear series which will enable us extend this to $X_0$ by means of a pair $(L_Y,V_Y)$ and $(L_Z,V_Z)$ on $X_0$.
1) Suppose the family $L$ of $g^r_d$-s is base point free, is there any notion of having a base point free limit linear series on $X_0$. That is I would like $L_Y$ and $L_Z$ to be globally generated, and so their restrictions to $Y$ and $Z$ will be globally generated as well. 2) I know that $L_Y$restricted to $Z$ and $L_Z$ restricted to $Y$ have degree zero. Can these be trivial line bundles?
