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[PN] 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) is contained in 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.