Geometric point

P

1. a point ${\displaystyle P}$ of the underlying topological space;
2. a ${\displaystyle T}$-valued point of ${\displaystyle S}$ is a morphism from ${\displaystyle T}$ to ${\displaystyle S}$, for any scheme ${\displaystyle T}$;
3. a geometric point, where ${\displaystyle S}$ is defined over (is equipped with a morphism to) ${\displaystyle {\textrm {Spec}}(K)}$, where ${\displaystyle K}$ is a field, is a morphism from ${\displaystyle {\textrm {Spec}}({\overline {K}})}$ to ${\displaystyle S}$ where ${\displaystyle {\overline {K}}}$ is an algebraic closure of ${\displaystyle K}$.