Reducibility mod p of hypersurfaces in projective spaces - An application of arithmetic Bezout

Given an absolutely irreducible horizontal hypersurface Z in a projective s pace over the ring of integers R of a number field, we give an explicit bou nd for the product of the norms of the prime ideals of R over which the fib re of Z becomes reducible. This bound is given as a function of a projectiv e height of Z and is obtained using arithmetic intersection theory, in part icular, an arithmetic Bezout theorem. (C) 2000 Academic Press.