Symbolic powers of monomial ideals

Abstract

Let $A=K[X_1,\dots ,X_d]$ and let $I$, $J$ be monomial ideals in $A$. Let $I_n(J)=(I^n:J^{\infty })$ be the $n^{th}$ symbolic power of $I$ \wrt \ $J$. It is easy to see that the function $f_J^I(n)=e_0(I_n(J)/I^n)$ is of quasi-polynomial type, say of period $g$ and degree $c$. For $n\gg 0$ say \[ f^I_J(n) = a_c(n)n^c + a_{c-1}(n)n^{c-1} + \text{lower terms}, \] where for $i=0,\dots ,c$, $a_i \colon \mathbb{N} \rt \mathbb{Z}$ are periodic functions of period $g$ and $a_c\ne 0$. In an earlier paper we (together with Herzog and Verma) proved that $\dim I_n(J)/I^n$ is constant for $n\gg 0$ and $a_c(-)$ is a constant. In this paper we prove that if $I$ is generated by some elements of the same degree and height $I\ge 2$ then $a_{c-1}(-)$ is also a constant.