The containment problem and a rational simplicial arrangement

Justyna Szpond; Grzegorz Malara

doi:10.3934/era.2017.24.013

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2017, Volume 24: 123-128. Doi: 10.3934/era.2017.24.013

This volume Previous Article Central limit theorems in the geometry of numbers Next Article

The containment problem and a rational simplicial arrangement

Justyna Szpond and
Grzegorz Malara

Department of Mathematics, Pedagogical University of Cracow, Podchorążych 2, PL-30-084 Kraków, Poland

Received: August 22, 2017

Revised: October 17, 2017

Published: August 2017

Research of Malara was partially supported by National Science Centre, Poland, grant 2016/21/N/ST1/01491. Research of Szpond was partially supported by National Science Centre, Poland, grant 2014/15/B/ST1/02197.

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Abstract

Since Dumnicki, Szemberg, and Tutaj-Gasińska gave in 2013 in [11] 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 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 [18, Theorem A, Problem 1]. In [14, 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 [22] will be published elsewhere [19].

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Mathematics Subject Classification: Primary: 13F20. Secondary: 14N20, 13A15, 52C35.

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Figure 1. Affine part of the simplicial arrangement $A(25, 2)$

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