The following questions maybe elementary, but I can't find them in the literature.
Assume now everything I will write is defined over some algebraically closed field. Let $S$ be a (geometrically) ruled surface over a nonsingular projective curve $C$ with sujective projection $\pi:\, S \to C$. Denote by $\mathscr E$ a normalized locally free sheaf of rank $2$ on $C$ corresponding to $S$. Let $\mathfrak e = \bigwedge^2 \mathscr E$ with $deg(\mathfrak e) = -e$. Finally let $C_0$ a section of $S$ corresponding $\mathcal O_S(1)$ and denote by $f$ any fiber of $\pi$.
I have two questions to ask:
If $C = \mathbf P^1$ (so $e \ge 0$), then we know having a section $D \sim C_0 + nf$ with $n >0$ is equivalent to finding a surjective map $\mathcal O \oplus \mathcal O(-e) \to \mathcal O(n-e)$ which exists if and only if $n \ge e$ because we can find maps $\mathcal O \to \mathcal O(n-e)$ and $\mathcal O \to \mathcal O(n)$ corresponding to effective divisors of degree $n-e$ and $n$ that do not meet with each other.
My question is if this idea is valid for a general nonsingular projective curve $C$. I.e. suppose $\mathfrak d$ is a divisor on $C$ and linear systems $|\mathfrak d|$ and $|\mathfrak d + \mathfrak e|$ are base-point-free, can we say there are morphisms $\mathcal O_C \to \mathscr L(\mathfrak d)$ and $\mathcal O_C \to \mathscr L (\mathfrak d +\mathfrak e)$ such that $\mathscr E \to \mathscr L(\mathfrak d + \mathfrak e)$ is surjective.
Suppose the linear system $|C_0 + \mathfrak bf|$ on $S$ contains a curve $C'$. Is $C'$ a section? Here $\mathfrak bf$ means $\pi^*(\mathfrak b)$.
Thank you.
