On Yoshinaga's arrangement of lines and the containment problem

TL;DR

This paper explores Yoshinaga's line arrangement to produce new examples demonstrating the non-containment of symbolic powers within ordinary powers of ideals, advancing understanding in algebraic geometry.

Contribution

It introduces a novel series of non-containment examples derived from Yoshinaga's specific line arrangement, expanding the known cases in algebraic geometry.

Findings

Yoshinaga's arrangement yields non-containment examples for I^{(3)} subseteq I^{2}

Provides a new, shorter series of such examples

Enhances understanding of symbolic and ordinary power relationships in ideals

Abstract

The main purpose of the note is to show that Yoshinaga's arrangement of $18$ lines having $48$ triple and $9$ double intersection points leads to a new (short) series of non-containment examples for $I^{(3)} \subset I^{2}$, the question studied by Harbourne and Huneke.