Is $O(1)$ on $\mathbb{R}P^1$ the mobius bundle?

Question

I saw an argument today that convinced me it was (one can more or less compute that parallel transporting around the circle one full time swaps a vector with its negative), but then someone pointed out that the Mobius bundle has no global sections, while $O(1)$ always does. Of course there are only two line bundles on the circle. So, what is going on?

Answer 1

Yes, $\mathcal O(1)$ is an algebraic line bundle on the algebraic variety $\mathbb P^1_\mathbb R$, whose algebraic global sections can be identified with the linear forms $ax+by$ on $\mathbb R^2$ .

The corresponding topological line bundle on $\mathbb P^1(\mathbb R)=S^1$ with its classical topology is indeed the Möbius bundle, the only non trivial topological line bundle on $S^1$.
That line bundle has of course many continuous non algebraic sections, some of them non-zero but with arbitrarily small support, say of arc length $0.01$ or even smaller .