Geometrically (algebraic geometry)

In algebraic geometry, especially in scheme theory, a property is said to hold geometrically over a field if it also holds over the algebraic closure of the field. In other words, a property holds geometrically if it holds after a base change to a geometric point. For example, a smooth variety is a variety that is geometrically regular.

Geometrically irreducible and geometrically reduced
Given a scheme X that is of finite type over a field k, the following are equivalent:
 * X is geometrically irreducible; i.e., $$X \times_k \overline{k} := X \times_{\operatorname{Spec} k} {\operatorname{Spec} \overline{k}}$$ is irreducible, where $$\overline{k}$$ denotes an algebraic closure of k.
 * $$X \times_k k_s$$ is irreducible for a separable closure $$k_s$$ of k.
 * $$X \times_k F$$ is irreducible for each field extension F of k.

The same statement also holds if "irreducible" is replaced with "reduced" and the separable closure is replaced by the perfect closure.