Suppose $C$ is a smooth conic in $\mathbb{P^2}$, that is, $C$ is a hypersurface of degree two in the projective plane.
Assume that the base field is complex numbers. By the genus degree formula, we can check that the genus of $C$ is zero, so $C$ is isomorphic to a $\mathbb{P}^1$. So the Picard group of $C$ is isomorphic to $\mathbb{Z}$.
My question:
Is this generated by $O_C(1)$?
This $O_C(1)$ is degree 2 line bundle on $C$ I suppose. If this is the generator, then it seems to mean there are no degree one divisors on $C$. This seems very strange. I think I am going wrong somewhere. Can anyone point this out? Thanks.