{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,29]],"date-time":"2025-12-29T18:50:09Z","timestamp":1767034209461},"reference-count":27,"publisher":"American Mathematical Society (AMS)","issue":"341","license":[{"start":{"date-parts":[[2024,1,31]],"date-time":"2024-01-31T00:00:00Z","timestamp":1706659200000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>In this paper, we show how to compute, using Fitting ideals, the nonfree locus of the moduli space of arrangements of a rank <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"3\">\n  <mml:semantics>\n    <mml:mn>3<\/mml:mn>\n    <mml:annotation encoding=\"application\/x-tex\">3<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> 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\u2019s freeness criterion for multiarrangements. We use these computations to verify Terao\u2019s freeness conjecture for rank <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"3\">\n  <mml:semantics>\n    <mml:mn>3<\/mml:mn>\n    <mml:annotation encoding=\"application\/x-tex\">3<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> central arrangements with up to <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"14\">\n  <mml:semantics>\n    <mml:mn>14<\/mml:mn>\n    <mml:annotation encoding=\"application\/x-tex\">14<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> hyperplanes in any characteristic.<\/p>","DOI":"10.1090\/mcom\/3812","type":"journal-article","created":{"date-parts":[[2022,12,14]],"date-time":"2022-12-14T16:42:18Z","timestamp":1671036138000},"page":"1431-1452","source":"Crossref","is-referenced-by-count":7,"title":["Computing the nonfree locus of the moduli space of arrangements and Terao\u2019s freeness conjecture"],"prefix":"10.1090","volume":"92","author":[{"given":"Mohamed","family":"Barakat","sequence":"first","affiliation":[]},{"given":"Lukas","family":"K\u00fchne","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2023,1,31]]},"reference":[{"issue":"5","key":"1","doi-asserted-by":"publisher","first-page":"1430","DOI":"10.1016\/j.disc.2015.12.017","article-title":"Non-recursive freeness and non-rigidity","volume":"339","author":"Abe, T.","year":"2016","journal-title":"Discrete Math.","ISSN":"http:\/\/id.crossref.org\/issn\/0012-365X","issn-type":"print"},{"issue":"3-4","key":"2","doi-asserted-by":"publisher","first-page":"911","DOI":"10.1007\/s00209-013-1165-6","article-title":"Free arrangements and coefficients of characteristic polynomials","volume":"275","author":"Abe, Takuro","year":"2013","journal-title":"Math. Z.","ISSN":"http:\/\/id.crossref.org\/issn\/0025-5874","issn-type":"print"},{"issue":"2","key":"3","doi-asserted-by":"publisher","first-page":"1201","DOI":"10.1137\/19M1296744","article-title":"On the generation of rank 3 simple matroids with an application to Terao\u2019s freeness conjecture","volume":"35","author":"Barakat, Mohamed","year":"2021","journal-title":"SIAM J. Discrete Math.","ISSN":"http:\/\/id.crossref.org\/issn\/0895-4801","issn-type":"print"},{"issue":"1","key":"4","doi-asserted-by":"publisher","first-page":"691","DOI":"10.1016\/j.aim.2011.09.011","article-title":"Coxeter and crystallographic arrangements are inductively free","volume":"229","author":"Barakat, Mohamed","year":"2012","journal-title":"Adv. Math.","ISSN":"http:\/\/id.crossref.org\/issn\/0001-8708","issn-type":"print"},{"key":"5","unstructured":"[BK19] M. Barakat and L. K\u00fchne, \ud835\ude96\ud835\ude8a\ud835\ude9d\ud835\ude9b\ud835\ude98\ud835\ude92\ud835\ude8d\ud835\ude9c_\ud835\ude9c\ud835\ude99\ud835\ude95\ud835\ude92\ud835\ude9d_\ud835\ude99\ud835\ude9e\ud835\ude8b\ud835\ude95\ud835\ude92\ud835\ude8c \u2013 a database collection for rank 3 integrally split simple matroids, 2019, \\url{https:\/\/homalg-project.github.io\/pkg\/MatroidGeneration}."},{"key":"6","unstructured":"[BK21] M. Barakat and L. K\u00fchne, A mechanical proof of a statement about images of pullbacks in abelian categories, 2021, \\url{https:\/\/homalg-project.github.io\/nb\/ImageOfPullback\/}."},{"key":"7","unstructured":"[BKLH19] M. Barakat, T. Kuhmichel, and M. Lange-Hegermann, \ud835\ude89\ud835\ude8a\ud835\ude9b\ud835\ude92\ud835\ude9c\ud835\ude94\ud835\ude92\ud835\ude75\ud835\ude9b\ud835\ude8a\ud835\ude96\ud835\ude8e\ud835\ude9c \u2013 (co)frames\/locales of Zariski closed\/open subsets of affine, projective, or toric varieties, 2018\u20132019, \\url{https:\/\/homalg-project.github.io\/pkg\/ZariskiFrames}."},{"issue":"2","key":"8","doi-asserted-by":"publisher","first-page":"269","DOI":"10.1142\/S0219498811004562","article-title":"An axiomatic setup for algorithmic homological algebra and an alternative approach to localization","volume":"10","author":"Barakat, Mohamed","year":"2011","journal-title":"J. Algebra Appl.","ISSN":"http:\/\/id.crossref.org\/issn\/0219-4988","issn-type":"print"},{"issue":"333","key":"9","doi-asserted-by":"publisher","first-page":"451","DOI":"10.1090\/mcom\/3632","article-title":"An algorithmic approach to Chevalley\u2019s theorem on images of rational morphisms between affine varieties","volume":"91","author":"Barakat, Mohamed","year":"2021","journal-title":"Math. Comp.","ISSN":"http:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"10","unstructured":"[BLHLM20] M. Barakat, M. Lange-Hegermann, A. Lorenz, and O. Motsak, The \ud835\ude76\ud835\ude9b\ud835\ude8a\ud835\ude8d\ud835\ude8e\ud835\ude8d\ud835\ude7c\ud835\ude98\ud835\ude8d\ud835\ude9e\ud835\ude95\ud835\ude8e\ud835\ude9c package \u2013 a \ud835\ude91\ud835\ude98\ud835\ude96\ud835\ude8a\ud835\ude95\ud835\ude90 based package for the Abelian category of finitely presented graded modules over computable graded rings, 2008\u20132020, \\url{https:\/\/homalg-project.github.io\/pkg\/GradedModules}."},{"key":"11","unstructured":"[BLHP17] M. Barakat, M. Lange-Hegermann, and S. Posur, Elimination via saturation,  arXiv:1707.00925, 2017."},{"key":"12","unstructured":"[BS21] M. Barakat and K. Saleh, \ud835\ude78\ud835\ude97\ud835\ude9d\ud835\ude9b\ud835\ude92\ud835\ude97\ud835\ude9c\ud835\ude92\ud835\ude8c\ud835\ude76\ud835\ude9b\ud835\ude8a\ud835\ude8d\ud835\ude8e\ud835\ude8d\ud835\ude7c\ud835\ude98\ud835\ude8d\ud835\ude9e\ud835\ude95\ud835\ude8e\ud835\ude9c \u2013 finitely presented graded modules over computable graded rings allowing multiple presentations and the notion of elements, 2021, \\url{https:\/\/homalg-project.github.io\/pkg\/IntrinsicGradedModules}."},{"key":"13","unstructured":"[DGPS19] Wolfram Decker, Gert-Martin Greuel, Gerhard Pfister, and Hans Sch\u00f6nemann, Singular 4-1-2 \u2014 A computer algebra system for polynomial computations, 2019, \\url{http:\/\/www.singular.uni-kl.de}."},{"issue":"8","key":"14","doi-asserted-by":"publisher","first-page":"2445","DOI":"10.1016\/j.disc.2019.05.016","article-title":"Freeness for 13 lines arrangements is combinatorial","volume":"342","author":"Dimca, Alexandru","year":"2019","journal-title":"Discrete Math.","ISSN":"http:\/\/id.crossref.org\/issn\/0012-365X","issn-type":"print"},{"key":"15","series-title":"Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-5350-1","volume-title":"Commutative algebra","volume":"150","author":"Eisenbud, David","year":"1995","ISBN":"http:\/\/id.crossref.org\/isbn\/0387942688"},{"key":"16","unstructured":"[GPS18] S. Gutsche, S. Posur, and \u00d8. Skarts\u00e6terhagen, On the syntax and semantics of \ud835\ude72\ud835\ude70\ud835\ude7f, Proceedings of the Workshop Computer Algebra in the Age of Types (Hagenberg, Austria, 17 August 2018), 2018, \\url{http:\/\/ceur-ws.org\/Vol-2307\/}."},{"key":"17","unstructured":"[{hom}20] homalg project authors, The \ud835\ude91\ud835\ude98\ud835\ude96\ud835\ude8a\ud835\ude95\ud835\ude90 project \u2013 algorithmic homological algebra, 2003\u20132020, \\url{https:\/\/homalg-project.github.io\/prj\/homalg_{p}roject}."},{"key":"18","unstructured":"[Leu19] M. Leuner, \ud835\ude8a\ud835\ude95\ud835\ude8c\ud835\ude98\ud835\ude9f\ud835\ude8e \u2013 algebraic combinatorics package for \ud835\udda6\ud835\udda0\ud835\uddaf, 2013\u20132019, \\url{https:\/\/github.com\/martin-leuner\/alcove}."},{"key":"19","series-title":"Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-02772-1","volume-title":"Arrangements of hyperplanes","volume":"300","author":"Orlik, Peter","year":"1992","ISBN":"http:\/\/id.crossref.org\/isbn\/3540552596"},{"key":"20","series-title":"Oxford Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780198566946.001.0001","volume-title":"Matroid theory","volume":"21","author":"Oxley, James","year":"2011","ISBN":"http:\/\/id.crossref.org\/isbn\/9780199603398","edition":"2"},{"key":"21","doi-asserted-by":"publisher","first-page":"157","DOI":"10.1090\/conm\/769\/15417","article-title":"Methods of constructive category theory","author":"Posur, Sebastian","year":"[2021] \\copyright2021"},{"issue":"7","key":"22","doi-asserted-by":"publisher","first-page":"Paper No. 106994, 19","DOI":"10.1016\/j.jpaa.2021.106994","article-title":"On free abelian categories for theorem proving","volume":"226","author":"Posur, Sebastian","year":"2022","journal-title":"J. Pure Appl. Algebra","ISSN":"http:\/\/id.crossref.org\/issn\/0022-4049","issn-type":"print"},{"issue":"1","key":"23","doi-asserted-by":"publisher","first-page":"126","DOI":"10.1112\/S0024609304003704","article-title":"On the freeness of 3-arrangements","volume":"37","author":"Yoshinaga, Masahiko","year":"2005","journal-title":"Bull. London Math. Soc.","ISSN":"http:\/\/id.crossref.org\/issn\/0024-6093","issn-type":"print"},{"issue":"10","key":"24","first-page":"179","article-title":"Free arrangements over finite field","volume":"82","author":"Yoshinaga, Masahiko","year":"2006","journal-title":"Proc. Japan Acad. Ser. A Math. Sci.","ISSN":"http:\/\/id.crossref.org\/issn\/0386-2194","issn-type":"print"},{"issue":"3","key":"25","doi-asserted-by":"publisher","first-page":"745","DOI":"10.2307\/2160116","article-title":"Free and locally free arrangements with a given intersection lattice","volume":"118","author":"Yuzvinsky, Sergey","year":"1993","journal-title":"Proc. Amer. Math. Soc.","ISSN":"http:\/\/id.crossref.org\/issn\/0002-9939","issn-type":"print"},{"key":"26","doi-asserted-by":"publisher","first-page":"345","DOI":"10.1090\/conm\/090\/1000610","article-title":"Multiarrangements of hyperplanes and their freeness","author":"Ziegler, G\u00fcnter M.","year":"1989"},{"issue":"2","key":"27","doi-asserted-by":"publisher","first-page":"525","DOI":"10.2307\/2001687","article-title":"Matroid representations and free arrangements","volume":"320","author":"Ziegler, G\u00fcnter M.","year":"1990","journal-title":"Trans. Amer. Math. Soc.","ISSN":"http:\/\/id.crossref.org\/issn\/0002-9947","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2023-92-341\/S0025-5718-2023-03812-1\/S0025-5718-2023-03812-1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,2,17]],"date-time":"2023-02-17T18:28:58Z","timestamp":1676658538000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2023-92-341\/S0025-5718-2023-03812-1\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,1,31]]},"references-count":27,"journal-issue":{"issue":"341","published-print":{"date-parts":[[2023,5]]}},"alternative-id":["S0025-5718-2023-03812-1"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3812","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["0025-5718","1088-6842"],"issn-type":[{"value":"0025-5718","type":"print"},{"value":"1088-6842","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,1,31]]}}}