Smooth projective curve

Definition

A smooth projective curve over a field $k$ is an algebraic curve $X$ whose structure morphism $X\to\operatorname{Spec}k$ is smooth and which admits a closed immersion into some projective space over $k$. Equivalently, its structure morphism is both smooth of relative dimension $1$ and projective.

In the geometric Langlands setting one commonly assumes that $k$ is algebraically closed of characteristic $0$ and that $X$ is connected. The automorphic stack $\operatorname{Bun}_G(X)$ and the spectral stack of $\widehat G$-local systems both depend on this curve.