## Computing the nonfree locus of the moduli space of arrangements and Terao’s freeness conjecture

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Mohamed Barakat and Lukas Kühne
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**92**(2023), 1431-1452

## Abstract:

In this paper, we show how to compute, using Fitting ideals, the nonfree locus of the moduli space of arrangements of a rank $3$ simple matroid, i.e., the subset of all points of the moduli space which parametrize nonfree arrangements. Our approach relies on the so-called Ziegler restriction and Yoshinaga’s freeness criterion for multiarrangements. We use these computations to verify Terao’s freeness conjecture for rank $3$ central arrangements with up to $14$ hyperplanes in any characteristic.## References

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## Additional Information

**Mohamed Barakat**- Affiliation: Department of mathematics, University of Siegen, 57068 Siegen, Germany
- MR Author ID: 706483
- ORCID: 0000-0003-3642-4190
- Email: mohamed.barakat@uni-siegen.de
**Lukas Kühne**- Affiliation: Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, 04103 Leipzig, Germany; and Fakultät für Mathematik, Universität Bielefeld, Bielefeld, Germany
- Email: lukas.kuehne@math.uni-bielefeld.de
- Received by editor(s): January 24, 2022
- Received by editor(s) in revised form: September 30, 2022
- Published electronically: January 31, 2023
- Additional Notes: An extended abstract of this paper appeared in the Oberwolfach workshop report 5/2021 and in the Computeralgebra Rundbrief Ausgabe 68.
- © Copyright 2023 by the authors.
- Journal: Math. Comp.
**92**(2023), 1431-1452 - MSC (2020): Primary 05B35, 52C35, 32S22, 14Q20
- DOI: https://doi.org/10.1090/mcom/3812
- MathSciNet review: 4550333