The containment problem and a rational simplicial arrangement

TL;DR

This paper reports the discovery of a new set of 49 rational points in the complex projective plane that exhibits a specific algebraic non-containment property, expanding known examples and addressing an open question in algebraic geometry.

Contribution

The authors present a new example of rational points with a non-containment property, previously known only for a single case, thus advancing understanding in the containment problem.

Findings

Found a set of 49 rational points with the non-containment property

Established the existence computationally

A conceptual proof will be published separately

Abstract

Since Dumnicki, Szemberg and Tutaj-Gasi\'nska gave in 2013 in [9] the first example of a set of points in the complex projective plane such that for its homogeneous ideal I the containment of the third symbolic power in the second ordinary power fails, there has been considerable interest in searching for further examples with this property and investigating into the nature of such examples. Many examples, defined over various fields, have been found but so far there has been essentially just one example found of 19 points defined over the rationals, see [16, Theorem A, Problem 1]. In [12, Problem 5.1] the authors asked if there are other rational examples. This has motivated our research. The purpose of this note is to flag the existence of a new example of a set of 49 rational points with the same non-containment property for powers of its homogeneous ideal. Here we establish the existence and justify it computationally. A more conceptual proof, based on Seceleanu's criterion [20] will be published elsewhere [17].