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.
Given a scheme X that is of finite type over a field k, the following are equivalent:[1] The same statement also holds if "irreducible" is replaced with "reduced" and the separable closure is replaced by the perfect closure.
This algebraic geometry–related article is a stub.