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Freeness versus maximal global Tjurina number for plane curves
Published online by Cambridge University Press: 21 September 2016
Abstract
We give a characterisation of nearly free plane curves in terms of their global Tjurina numbers, similar to the characterisation of free curves as curves with a maximal Tjurina number, given by A. A. du Plessis and C.T.C. Wall. It is also shown that an irreducible plane curve having a 1-dimensional symmetry is nearly free. A new numerical characterisation of free curves and a simple characterisation of nearly free curves in terms of their syzygies conclude this paper.
- Type
- Research Article
- Information
- Mathematical Proceedings of the Cambridge Philosophical Society , Volume 163 , Issue 1 , July 2017 , pp. 161 - 172
- Copyright
- Copyright © Cambridge Philosophical Society 2016
References
REFERENCES
[1]
Choudary, A. D. R. and Dimca, A.. Koszul complexes and hypersurface singularities. Proc. Amer. Math. Soc.
121 (1994), 1009–1016.CrossRefGoogle Scholar
[2]
Dimca, A.. Syzygies of Jacobian ideals and defects of linear systems. Bull. Math. Soc. Sci. Math. Roumanie Tome
56
(104) (2013), 191–203.Google Scholar
[3]
Dimca, A.. Freeness versus maximal degree of the singular subscheme for surfaces in P
3
. Geom. Dedicata
183 (2016), 101–112.CrossRefGoogle Scholar
[4]
Dimca, A. and Popescu, D.. Hilbert series and Lefschetz properties of dimension one almost complete intersections. Comm. Algebra
44 (2016), 4467–4482.CrossRefGoogle Scholar
[5]
Dimca, A. and Sernesi, E.. Syzygies and logarithmic vector fields along plane curves. J. de l'École polytechnique-Mathématiques
1 (2014), 247–267.CrossRefGoogle Scholar
[6]
Dimca, A. and Sticlaru, G.. Free divisors and rational cuspidal plane curves. arXiv:1504.01242.Google Scholar
[7]
Dimca, A. and Sticlaru, G.. Nearly free divisors and rational cuspidal curves. arXiv:1505.00666.Google Scholar
[8]
Dimca, A. and Sticlaru, G.. Free and nearly free surfaces in ℙ3. arXiv:1507.03450.Google Scholar
[9]
Dimca, A. and Sticlaru, G.. On the exponents of free and nearly free projective plane curves. arXiv:1511.08938.Google Scholar
[10]
du Pleseis, A.A. and Wall, C.T.C.. Application of the theory of the discriminant to highly singular plane curves. Math. Proc. Camb. Phil. Soc.
126 (1999), 259–266.CrossRefGoogle Scholar
[11]
du Pleseis, A.A. and Wall, C.T.C.. Curves in P
2(ℂ) with 1-dimensional symmetry. Revista Mat Complutense
12 (1999), 117–132.Google Scholar
[12]
Eisenbud, D.. The Geometry of Syzygies: A Second Course in Algebraic Geometry and Commutative Algebra. Graduate Texts in Math. vol. 229, (Springer, Berlin, Heidelberg, New York, 2005).CrossRefGoogle Scholar
[13]
Orlik, P. and Terao, H.. Arrangements of Hyperplanes. (Springer-Verlag, Berlin, Heidelberg, New York, 1992).CrossRefGoogle Scholar
[14]
Saito, K.. Theory of logarithmic differential forms and logarithmic vector fields. J. Fac. Sci. Univ. Tokyo Sect. IA Math.
27 (1980), 265–291.Google Scholar
[15]
Schenck, H.. Elementary modifications and line configurations in P
2
. Comm. Math. Helv.
78 (2003), 447–462.CrossRefGoogle Scholar
[16]
Sernesi, E.. The local cohomology of the jacobian ring. Documenta Mathematica, 19 (2014), 541–565.CrossRefGoogle Scholar
[17]
Simis, A. and Tohăneanu, S.O.. Homology of homogeneous divisors. Israel J. Math.
200 (2014), 449–487.CrossRefGoogle Scholar
[18]
Yoshinaga, M.. Freeness of hyperplane arrangements and related topics. Annales de la Faculté des Sciences de Toulouse
23 (2014), 483–512.Google Scholar
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