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 × k k ¯ = X × Spec k Spec k ¯ {\displaystyle X\times _{k}{\overline {k}}=X\times _{\operatorname {Spec} k}{\operatorname {Spec} {\overline {k}}}} is irreducible, where k ¯ {\displaystyle {\overline {k}}} denotes an algebraic closure of k.
X × k k s {\displaystyle X\times _{k}k_{s}} is irreducible for a separable closure k s {\displaystyle k_{s}} of k.
X × k F {\displaystyle 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.
References
Sources Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
