Symbolic powers of monomial ideals

Let K be a field and consider the standard grading on A = K[X₁, ... ,X_d]. Let I, J be monomial ideals in A. Let I_n(J) = (Iⁿ : J^∞) be the n^th symbolic power of I with respect to J. It is easy to see that the function f^I _J (n) = e₀(I_n(J)/Iⁿ) is of quasi-polynomial type, say of period g and degree c. For n ≫ 0 say

f^I_J (n) = a_c(n)n^c + a_c−1(n)n^c−1 + lower terms,

where for i = 0, ... , c, a_i : N → Q are periodic functions of period g and a_c ≠0. In [4, 2.4] we (together with Herzog and Verma) proved that dim I_n(J)/Iⁿ is constant for n ≫ 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 ≥ 2 then a_c−1(−) is also a constant.

Mathematics Subject Classification. Primary 13D40; Secondary 13H15.