On the minimal value of global Tjurina numbers for line arrangements
We show that a general lower bound for the global Tjurina number of a reduced complex projective plane curve, given by Andrew A. du Plessis and Charles T. C. Wall, can be improved when the curve is a line arrangement. This fact is in sharp contrast to a conjecture saying that the general upper bound...
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| Published in | European journal of mathematics Vol. 6; no. 3; pp. 817 - 828 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Cham
Springer International Publishing
01.09.2020
Springer Nature B.V Springer |
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| Abstract | We show that a general lower bound for the global Tjurina number of a reduced complex projective plane curve, given by Andrew A. du Plessis and Charles T. C. Wall, can be improved when the curve is a line arrangement. This fact is in sharp contrast to a conjecture saying that the general upper bound for the global Tjurina number of a reduced complex projective plane curve, also given by du Plessis and Wall, is realized by line arrangements in practically all cases. |
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| AbstractList | We show that a general lower bound for the global Tjurina number of a reduced complex projective plane curve, given by Andrew A. du Plessis and Charles T. C. Wall, can be improved when the curve is a line arrangement. This fact is in sharp contrast to a conjecture saying that the general upper bound for the global Tjurina number of a reduced complex projective plane curve, also given by du Plessis and Wall, is realized by line arrangements in practically all cases. We show that a general lower bound for the global Tjurina number of a reduced complex projective plane curve, given by A. A. du Plessis and C.T.C. Wall, can be improved when the curve is a line arrangement. |
| Author | Dimca, Alexandru |
| Author_xml | – sequence: 1 givenname: Alexandru orcidid: 0000-0001-9679-9870 surname: Dimca fullname: Dimca, Alexandru email: dimca@unice.fr organization: Université Côte d’Azur, CNRS, LJAD, Simion Stoilow Institute of Mathematics |
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| Cites_doi | 10.1017/S0305004116000803 10.4310/MRL.2017.v24.n4.a5 10.1017/S0305004198003302 10.1007/s00014-003-0762-0 10.1307/mmj/1490639821 10.4171/PRIMS/54-1-6 10.1080/00927872.2015.1087535 10.1080/00927872.2012.667466 10.5802/afst.1413 10.1142/S0129167X18500556 10.5802/jep.10 10.4007/annals.2003.158.473 10.1007/s10801-018-0859-6 10.1007/978-3-319-56221-6 10.4171/RLM/863 10.1016/j.disc.2015.07.004 10.1007/s10711-019-00485-7 |
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| Keywords | Tjurina number 32S22 Milnor number 14H50 Line arrangement 14B05 |
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| References | Dimca (CR6) 2017; 163 CR19 Dimca, Sticlaru (CR15) 2019 CR17 Artal Bartolo, Gorrochategui, Luengo, Melle-Hernández, Decker, Pfister, Schulze (CR3) 2017 CR16 Tohăneanu (CR24) 2016; 339 Dimca, Sticlaru (CR14) 2019; 30 Dimca, Popescu (CR10) 2016; 44 Orlik, Terao (CR21) 1992 Dimca, Sticlaru (CR12) 2017; 24 Abe, Dimca (CR1) 2018; 29 Yoshinaga (CR25) 2014; 23 Dimca, Papadima (CR9) 2003; 158 CR2 Dimca, Sticlaru (CR13) 2018; 54 Schenck (CR22) 2003; 78 CR8 CR7 Dimca (CR5) 2017 CR20 du Plessis, Wall (CR18) 1999; 126 Dimca (CR4) 2017; 66 Tohăneanu (CR23) 2013; 41 Dimca, Sernesi (CR11) 2014; 1 373_CR2 A Dimca (373_CR6) 2017; 163 AA du Plessis (373_CR18) 1999; 126 P Orlik (373_CR21) 1992 373_CR8 373_CR7 A Dimca (373_CR9) 2003; 158 E Artal Bartolo (373_CR3) 2017 A Dimca (373_CR5) 2017 A Dimca (373_CR15) 2019 A Dimca (373_CR10) 2016; 44 ŞO Tohăneanu (373_CR24) 2016; 339 A Dimca (373_CR11) 2014; 1 M Yoshinaga (373_CR25) 2014; 23 T Abe (373_CR1) 2018; 29 A Dimca (373_CR14) 2019; 30 373_CR20 A Dimca (373_CR4) 2017; 66 A Dimca (373_CR13) 2018; 54 373_CR16 373_CR17 373_CR19 ŞO Tohăneanu (373_CR23) 2013; 41 HK Schenck (373_CR22) 2003; 78 A Dimca (373_CR12) 2017; 24 |
| References_xml | – volume: 163 start-page: 161 issue: 1 year: 2017 end-page: 172 ident: CR6 article-title: Freeness versus maximal global Tjurina number for plane curves publication-title: Math. Proc. Cambridge Philos. Soc. doi: 10.1017/S0305004116000803 contributor: fullname: Dimca – ident: CR2 – ident: CR16 – volume: 24 start-page: 1023 issue: 4 year: 2017 end-page: 1042 ident: CR12 article-title: Free divisors and rational cuspidal plane curves publication-title: Math. Res. Lett. doi: 10.4310/MRL.2017.v24.n4.a5 contributor: fullname: Sticlaru – volume: 126 start-page: 259 year: 1999 end-page: 266 ident: CR18 article-title: Application of the theory of the discriminant to highly singular plane curves publication-title: Math. Proc. Cambridge Philos. 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Math. doi: 10.4007/annals.2003.158.473 contributor: fullname: A Dimca – volume-title: Hyperplane Arrangements year: 2017 ident: 373_CR5 doi: 10.1007/978-3-319-56221-6 contributor: fullname: A Dimca – ident: 373_CR20 – volume: 41 start-page: 2916 issue: 8 year: 2013 ident: 373_CR23 publication-title: Comm. Algebra doi: 10.1080/00927872.2012.667466 contributor: fullname: ŞO Tohăneanu – volume: 54 start-page: 163 issue: 1 year: 2018 ident: 373_CR13 publication-title: Publ. Res. Inst. Math. Sci. doi: 10.4171/PRIMS/54-1-6 contributor: fullname: A Dimca – volume: 66 start-page: 347 issue: 2 year: 2017 ident: 373_CR4 publication-title: Michigan Math. J. doi: 10.1307/mmj/1490639821 contributor: fullname: A Dimca – volume: 1 start-page: 247 year: 2014 ident: 373_CR11 publication-title: J. É. Polytech. Math. doi: 10.5802/jep.10 contributor: fullname: A Dimca – start-page: 1 volume-title: Singularities and Computer Algebra year: 2017 ident: 373_CR3 contributor: fullname: E Artal Bartolo – volume: 78 start-page: 447 issue: 3 year: 2003 ident: 373_CR22 publication-title: Comment. Math. Helv. doi: 10.1007/s00014-003-0762-0 contributor: fullname: HK Schenck |
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| Snippet | We show that a general lower bound for the global Tjurina number of a reduced complex projective plane curve, given by Andrew A. du Plessis and Charles T. C.... We show that a general lower bound for the global Tjurina number of a reduced complex projective plane curve, given by A. A. du Plessis and C.T.C. Wall, can be... |
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| SubjectTerms | Algebraic Geometry Curves Lower bounds Mathematics Mathematics and Statistics Research Article Upper bounds |
| Title | On the minimal value of global Tjurina numbers for line arrangements |
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