Containment problem and combinatorics

TL;DR

This paper investigates two line configurations with identical combinatorics, revealing that the containment relation between symbolic and ordinary powers of their ideals can differ, and constructs a rare rational example of non-containment.

Contribution

It demonstrates that combinatorial data alone does not determine the containment of symbolic powers in ordinary powers for point ideals from line arrangements.

Findings

One configuration satisfies $I^{(3)} subseteq I^2$, the other does not.

The non-containment configuration's parameter space is a rational curve.

Rational non-containment configurations are very rare.

Abstract

In this note we consider two configurations of twelve lines with nineteen triple points (i.e., points where three lines meet). Both of them have the same combinatorial features. In both configurations nine of twelve lines have five triple points and one double point, and the remaining three lines have four triple points and three double points. Taking the ideal of the triple points of these configurations we discover that, quite surprisingly, for one of the configurations the containment $I^{(3)} \subset I^2$ holds, while for the other it does not. Hence for ideals of points defined by configurations of lines the (non)containment of a symbolic power in an ordinary power is not determined alone by combinatorial features of the arrangement. Moreover, for the configuration with the non-containment $I^{(3)} \nsubseteq I^2$ we examine its parameter space, which turns out to be a rational curve, and thus establish the existence of a rational non-containment configuration of points. Such rational examples are very rare.