Hilbertian Towers, Frattini Towers and Serre's O(pen) I(mage) T(heorem)

Consider any cover of normal varieties φ: WV, defined over a number field K. Suppose p∈W (an algebraic point) has full Galois closure over V: As in HITcovers.html, the degree of the Galois closure of K(p)/K(φ(p)) is the same as the degree of the Galois closure of φ.

The phrase "a dimension u tower (of varieties) over (a number field) K" (for simplicity take the varieties affine) we will mean a normal sequence
(*) …→ XkXk-1 →  … → X0  →  Au of finite maps of absolutely irreducible, dimension u, varieties with all varieties and maps over K.
Below we will refer to the maps from Xk to Au by ψk. Now consider these two properties of the tower (*).

It is an Hilbertian Tower – with respect to some fixed integer k0 – if the following holds:
(*2) If pk0 Xk0 has full Galois closure over Au, then so does any p' Xk for any kk0, and p' lying over pk0.
It is a (geometric) Frattini Tower – with respect to some fixed integer k0 – if the following holds:
(*3) For kk0, the geometric monodromy group G(ψk), of ψk, is a Frattini extension of G(ψk).

Thm. If (*) is an Hilbertian Tower then it is a (geometric) Frattini Tower (with respect to the same k0).

Proof: