Given an irreducible homogeneous polynomial $F \in \mathbb{Z}[x_1,\cdots,x_n]$ (which can be assumed to be irreducible over $\bar{\mathbb{Q}}$ if necessary),
Is it true that there exists arbitrarily large prime such that $F$ modulo $p$ is irreducible? What about geometrically irreducible?
Is there anything we can say about the smallest prime $p$ such that $F$ modulo $p$ is irreducible? Same question for geometric irreducibililty.
Motivation: I am trying to understand the Lang-Weil bound for varieties over finite field. It seems like one needs some kind of geometric irreducibility to apply the bound, so I would like to know the answer to especially question 2, as it is related to how I want to use the bound. Thanks!