Symmetric Galois groups under specialization

Given an irreducible bivariate polynomial f (t, x) ∈ ℚ[t, x], what groups H appear as the Galois group of (t0, for infinitely many t0 ℚ? How often does a above f(t0, x), We give answer large x-degree with alternating or symmetric over ℚ(t). This is done by determining low genus subcovers coverings $$\tilde X \to \mathbb{P}_\mathbb{C}^1$$ monodromy groups.