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

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 $\overline{x} = \mathop{\mathrm{Spec}}(k)$.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 $x$*.

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