What line bundle pulls back to the trivial line bundle

Let $X$ be an abelian surface. $C$ be a curve in $X$. Consider the projective bundle $\pi:\mathbb{P}^1_C\longrightarrow C$. This is a projective morphism. I have two questions :

1) Can we find an effective divisor $D\neq 0$ on $C$ such that the line bundle corresponding to it pulls back to the trivial line bundle, that is $\pi^*\mathcal{O}_C(D)=\mathcal{O}_{\mathbb{P}^1_C}$? I suppose that if $D$ is a divisor linearly equivalent to the zero divisor, then this happens. But is it true that every degree 0 line bundle, that is an element of $Pic^O(C)$ pulls back to the trivial line bundle?

2) While I was trying this I was wondering if the degree of the line bundle is preserved by pullback. And what the degree of the morphism $\pi$ is.

There are several things I don't understand here. What role does the abelian surface play? (None that I can see.) What is the projective bundle? Is it just $\mathbf P^1 \times C$? (You didn't specify any rank 2 vector bundle on $C$ to be projectivised).
• Thank you! By $\mathbb{P}^1_C$, I mean $\mathbb{P}^1\times C$ only, is that not $\mathbb{P}(\mathcal{O}^{\oplus 2})$? – gradstudent Aug 18 '15 at 16:08
• @poorna: yes, that's right.It was just confusing to me that you denoted it $\mathbf P^1_C$ and used the term "projective bundle" rather than just calling it the product. – Schemer Aug 19 '15 at 8:12