Comparing powers and symbolic powers of ideals

Comparing Powers and Symbolic Powers of Ideals

We develop tools to study the problem of containment of symbolic powers I(m) in powers I for a homogeneous ideal I in a polynomial ring k[P ] 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) ⊆ I. As corollaries, we show that I2 contains I(3) whenever S is a finite generic set of points in P2 (thereby ...

Generalized Test Ideals and Symbolic Powers

In [HH7], developing arguments in [HH5], Hochster and Huneke used classical tight closure techniques to prove a fine behavior of symbolic powers of ideals in regular rings. In this paper, we use generalized test ideals, which are a characteristic p analogue of multiplier ideals, to give a generalization of Hochster-Huneke's results.

Links of symbolic powers of prime ideals

In this paper, we prove the following. Let (R,m) be a Cohen-Macaulay local ring of dimension d ≥ 2. Suppose that either R is not regular or R is regular with d ≥ 3. Let t ≥ 2 be a positive integer. If {α1, . . . , αd} is a regular sequence contained in m, then (α1, . . . , αd) : m t ⊆ m. This result gives an affirmative answer to a conjecture raised by Polini and Ulrich.

Symbolic Powers of Ideals in k[PN]

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 i...