The Stacks project

Definition 66.19.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$.

1. A geometric point of $X$ is a morphism $\overline{x} : \mathop{\mathrm{Spec}}(k) \to X$, where $k$ is an algebraically closed field. We often abuse notation and write $\overline{x} = \mathop{\mathrm{Spec}}(k)$.

2. For every geometric point $\overline{x}$ we have the corresponding “image” point $x \in |X|$. We say that $\overline{x}$ is a geometric point lying over $x$.