Persistence of Galois property of hypersurfaces over algebraic integers across other characteristics

In this paper, we investigate hypersurfaces defined over a ring of algebraic integers, and show that if the projection from a point induces a Galois extension over either a number field or the residue field associated with a prime ideal satisfying certain conditions, then the Galois property persists under reduction modulo the residue field associated with all but finitely many such prime ideals. Furthermore, for quartic hypersurfaces, we provide necessary and sufficient conditions for the Galois group to be given by a projective linear group, depending on the characteristic of the base field.