Geometrically irreducible variety Let $X\subseteq \mathbb{C}^n$ be an irreducible variety defined over $\mathbb{Q}$. I would like to show that for all but finitely many prime $p$ the variety $X(\mathbb{F}_p)$, defined over $\mathbb{F}_p$, is geometrically irreducible, i.e., $X(\overline{\mathbb{F}}_p)$ is irreducible. Can someone give me a hint or a reference?
Thanks
 A: I agree with the OP that there is no need to use fancy and sophisticated language and tools. (Maybe what he calls variety is nowadays better called algebraic set.)
(a) The case that $X$ is a hypersurface is well known, some people call it the Bertini-Noether Theorem. Here one has to show that an absolutely irreducible polynomial $f(\bf x)\in{\mathbb Q}[\bf x]$ remains absolutely irreducible modulo all but finitely many primes. That follows from a straightforward application of Hilbert's Nullstellensatz, see e.g. Chapter VIII, $\S$5, Prop. 7 (page 157) of Lang's Diophantine Geometry or
Chapter IX, $\S$5, Prop. 5.3 (page 241) of Lang's Fundamentals of Diophantine Geometry.
(b) The general case can be reduced to the case of a hypersurface because $X$ is birationally equivalent to a hypersurface (this uses the primitive element theorem). Indeed, a generalization of the OP's question is Prop. 10.4.2 in the second and third edition of Field Arithmetic by Fried and Jarden.
A: Note that there is some small ambiguity here, as to talk about the reduction of $X/\mathbb{Q}$ modulo a prime $p$, one needs to choose a model $X$. i.e., a scheme $\mathcal{X} \to \mathbb{Z}$ whose generic fibre is isomorphic to $X$.
Anyway, if $X/\mathbb{Q}$ is geometrically integral, then the same holds over $\mathbb{F}_p$ for all but finitely many primes $p$. This follows from fact that being geometrically integral is a constructible property (see Section 9 of EGAIV). One applies this to the morphism $\mathcal{X} \to \mathbb{Z}$.
Precise references:
Definition of a constructible property: EGAIV Définition (9.2.1)
Proof that being geometrically integral is a constructible property: EGAIV Théorème (9.7.7).
You can also find a nice treatment of this in Chapter 10 of the book
Ulrich Görtz, Torsten Wedhorn - Algebraic Geometry: Part I: Schemes. With Examples and Exercises
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