I am reading a paper in which it is claimed that $H^1(\mathcal O(-k),\mathcal O)=0$, where $k\geqslant 1$. Moreover, the argument also requires that $H^2(\mathcal O(-k),\mathcal O)=0$.
Here $\mathcal O(-k)$ refers to the line bundle over $\mathbb CP^1$ with first Chern class $-k$ and I am assuming that $\mathcal O$ refers to the structure sheaf on the total space of $\mathcal O(-k)$, $\operatorname{Tot}(\mathcal O(-k))$.
Is there an easy way to see this claim?
Edit. I think that $\mathcal O(-k)$ can be covered by two open sets with acyclic intersection, so that $H^2(\mathcal O(-k),\mathcal F)=0$ for any coordinates. My question about $H^1$ remains.