Page "Projective variety" Paragraph 7

* Every irreducible closed subset of P < sup > n </ sup >( k ) of codimension one is a hypersurface ; i. e., the zero set of some homogeneous polynomial.

This is because the homogeneous coordinate ring of P < sup > n </ sup >( k ), namely, a polynomial ring is a unique factorization domain and in a UFD every minimal prime ideal is principal.