Lemma 10.47.6. Let $k$ be a field. Let $S$ be a $k$-algebra.
If $S$ is geometrically irreducible over $k$ so is every $k$-subalgebra.
If all finitely generated $k$-subalgebras of $S$ are geometrically irreducible, then $S$ is geometrically irreducible.
A directed colimit of geometrically irreducible $k$-algebras is geometrically irreducible.