Winter 2022 New phenomena in the containment problem for simplicial arrangements
Marek Janasz, Magdalena Lampa-Baczyńska, Grzegorz Malara
J. Commut. Algebra 14(4): 571-581 (Winter 2022). DOI: 10.1216/jca.2022.14.571


We consider two simplicial arrangements of lines and ideals I of intersection points of these lines. There are 127 intersection points in both cases and the numbers ti of points lying on exactly i configuration lines (points of multiplicity i) coincide. We show that in one of these examples the containment I(3)I2 holds, whereas it fails in the other. We also show that the containment fails for a subarrangement of 21 lines. The interest in the containment relation between I(3) and I2 for ideals of points in 2 is motivated by a question posted by Hochster and Huneke in 2002. Configurations of points with I(3)I2 are quite rare. Our example reveals two particular features: All points are defined over and all intersection points of lines are involved. In examples studied by now only points with multiplicity i3 were considered. The novelty of our arrangements lies in the geometry of the element in I(3) which witness the noncontainment in I2. In all previous examples such an element was a product of linear forms. Now, in both cases there is an irreducible curve of higher degree involved.


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Marek Janasz. Magdalena Lampa-Baczyńska. Grzegorz Malara. "New phenomena in the containment problem for simplicial arrangements." J. Commut. Algebra 14 (4) 571 - 581, Winter 2022.


Received: 28 February 2019; Revised: 28 November 2020; Accepted: 28 November 2020; Published: Winter 2022
First available in Project Euclid: 15 November 2022

MathSciNet: MR4509408
zbMATH: 1502.13013
Digital Object Identifier: 10.1216/jca.2022.14.571

Primary: 13A15 , 13F20 , 14N20 , 52C35

Keywords: Arrangements of lines , containment problem , simplicial arrangements , symbolic powers

Rights: Copyright © 2022 Rocky Mountain Mathematics Consortium


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Vol.14 • No. 4 • Winter 2022
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