Reducibility mod p of Hypersurfaces in Projective Spaces—An Application of Arithmetic Bézout Original Research Article

Given an absolutely irreducible horizontal hypersurface Z in a projective space over the ring of integers R of a number field, we give an explicit bound for the product of the norms of the prime ideals of R over which the fibre of Z becomes reducible. This bound is given as a function of a projective height of Z and is obtained using arithmetic intersection theory, in particular, an arithmetic Bézout theorem.