Projective normal flatness and Hilbert functions

Abstract

Projective normal flatness of a local ring R R along an ideal I I is defined to be the flatness of the morphism on the exceptional divisor induced by blowing up R R with center I I . It is shown that most of the important properties of normal flatness have an analogue for projective normal flatness. In particular, we study the local Hilbert function in connection with projective normal flatness. If R / I R/I is regular and R R projectively normally flat along I I , then we obtain the same inequality for the local Hilbert functions under blowing up as in the permissible case.