  Fix a point $t_0$ in the circle $S^1$. The space $J^k(t_0, \mathbb{P}^1)$ of
$k$-jets at $t_0$ of $C^{\infty}$ maps from $S^1$ to the Riemann sphere
$\mathbb{P}^1$ is a $k+1$ dimensional complex algebraic manifold. We identify a
class of holomorphic sections of line bundles on $J^k(t_0, \mathbb{P}^1)$.
