Smooth rational curve
= Smooth rational curve
Over an algebraically closed field, a smooth projective rational curve is isomorphic to $\mathbb P^1$. Its <line bundles> are determined by degree, its degree-zero <line bundle> is trivial, and $H^1(\mathbb P^1,\mathcal O)=0$. An integral projective curve of <arithmetic genus> zero is automatically smooth and rational by the normalization genus formula.