COMPARING POWERS AND SYMBOLIC POWERS OF IDEALS

We develop tools to study the problem of containment of symbolic powers I-(m) in powers I-r for a homogeneous ideal I in a polynomial ring k[P-N] in N + 1 variables over an arbitrary algebraically closed field k. We obtain results on the structure of the set of pairs (r, m) such that I-(m) subset of I-r. As corollaries, we show that I-2 contains I-(3) whenever S is a finite generic set of points in 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.