Comparing powers and symbolic powers of ideals

Abstract

We develop tools to study the problem of containment of symbolic powers I ( m ) I^{(m)} in powers I r I^r for a homogeneous ideal I I in a polynomial ring k [ P N ] k[\textbf {P}^N] in N + 1 N+1 variables over an arbitrary algebraically closed field k k . We obtain results on the structure of the set of pairs ( r , m ) (r,m) such that I ( m ) ⊆ I r I^{(m)}\subseteq I^r . As corollaries, we show that I 2 I^2 contains I ( 3 ) I^{(3)} whenever S S is a finite generic set of points in P 2 \textbf {P}^2 (thereby giving a partial answer to a question of Huneke), and we show that the containment theorems of Ein–Lazarsfeld–Smith [Invent. Math. 144 (2001), pp. 241–252] and Hochster–Huneke [Invent. Math. 147 (2002), pp. 349–369] are optimal for every fixed dimension and codimension.