Restrictions on Hilbert coefficients give depths of graded domains

Abstract

In this paper, we prove that if P is a homogeneous prime ideal inside a standard graded polynomial ring S with $\dim (S/P)=d$, and for $s\le d$, adjoining s general linear forms to the prime ideal changes the $(d-s)$-th Hilbert coefficient by 1, then $\text{depth}(S/P)=s-1$. This criterion also tells us about possible restrictions on the generic initial ideal of a prime ideal inside a polynomial ring.