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:[1]

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.[2]

References

Hartshorne, Ch II, Exercise 3.15. (a)
Hartshorne, Ch II, Exercise 3.15. (b)

Sources

Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157

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[1] Hartshorne, Ch II, Exercise 3.15. (a)

[2] Hartshorne, Ch II, Exercise 3.15. (b)