The Stacks project

Remark 33.43.9. Let $k$ be a field. Let $X$ be a regular curve over $k$. By Lemmas 33.43.8 and 33.43.6 there exists a nonsingular projective curve $\overline{X}$ which is a compactification of $X$, i.e., there exists an open immersion $j : X \to \overline{X}$ such that the complement consists of a finite number of closed points. If $k$ is perfect, then $X$ and $\overline{X}$ are smooth over $k$ and $\overline{X}$ is a smooth projective compactification of $X$.