Fat lines in P3: Powers versus symbolic powers

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 ...

Symbolic Powers of Ideals in k[PN]

Lefschetz Principle Applied to Symbolic Powers

In this paper, an alternative proof is presented of the following result on symbolic powers due to Ein, Lazarsfeld and Smith (for the affine case over C) and to Hochster and Huneke (for the general case). Let A be a regular ring containing a field K. Let a be a radical ideal of A and let h be the maximum of the heights of its minimal primes. Then for all n, we have an inclusion a(hn) ⊂ an, wher...

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.