Bounding Symbolic Powers via Asymptotic Multiplier Ideals

For a radical ideal I, the symbolic power I is the collection of elements that vanish to order at least p at each point of Zeros(I). If I is actually prime, then I is the I-associated primary component of I; if I is only radical, writing I = C1 ∩ · · · ∩Cs as an intersection of prime ideals, I = C (p) 1 ∩ · · · ∩ C (p) s . The inclusion I ⊆ I always holds, but the reverse inclusion holds only in some special cases, such as when I is a complete intersection. Swanson [Swa00] showed that for rings R satisfying a certain hypothesis, for each ideal I, there is an integer e = e(I) such that the symbolic power I ⊆ I for all r ≥ 0. Ein– Lazarsfeld–Smith [ELS01] showed that in a regular local ring R in equal characteristic 0 and for I a radical ideal, one can take e(I) = bight(I), the big height of I, which is the maximum of the codimensions of the irreducible components of the closed subset of zeros of I. In particular, bight(I) is at most the dimension of the ambient space, so e = dimR is a single value that works for all ideals. More generally, for any k ≥ 0, I ⊆ (I) for all r ≥ 1. Very shortly thereafter, Hochster–Huneke [HH02] generalized this result by characteristic p methods. It is natural to regard these results in the form I ⊆ I for m ≥ f(r) = er, e = bight(I). Replacing f(r) = er with a smaller function would give a stronger bound on symbolic powers (containment in I would begin sooner). So it is natural to ask, how far can one reduce the bounding function f(r) = er? Bocci–Harbourne [BH07] introduced the resurgence of I, ρ(I) = sup{m/r : I 6⊆ I}. Thus if m > ρ(I)r, I ⊆ I. The Ein–Lazarsfeld–Smith and Hochster–Huneke results show ρ(I) ≤ bight(I) ≤ dimR. It can be smaller. For example, if I is smooth or a reduced