Fat lines in P^3: powers versus symbolic powers

TL;DR

This paper investigates the relationship between symbolic and regular powers of ideals associated with special line configurations in projective 3-space, establishing conditions under which these powers coincide.

Contribution

It characterizes when symbolic and regular powers are equal for a family of line configurations, answering a question posed by Huneke.

Findings

I^(m) = I^m for all m if and only if I^(3) = I^3.

Provides a specific criterion for equality of powers in the studied configurations.

Answers a longstanding question about the equality of symbolic and regular powers.

Abstract

We study the symbolic and regular powers of ideals I for a family of special configurations of lines in P^3. For this family, we show that I^(m) = I^m for all integers m if and only if I^(3) = I^3. We use these configurations to answer a question of Huneke that asks whether I^(m) = I^m for all m if equality holds when m equals the big height of the ideal I.