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Freeness versus maximal global Tjurina number for plane curves

Published online by Cambridge University Press:  21 September 2016

ALEXANDRU DIMCA*
Affiliation:
Université Côte d'Azur, CNRS, LJAD, France. e-mail: [email protected]

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
Copyright
Copyright © Cambridge Philosophical Society 2016 

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References

REFERENCES

[1] Choudary, A. D. R. and Dimca, A.. Koszul complexes and hypersurface singularities. Proc. Amer. Math. Soc. 121 (1994), 10091016.CrossRefGoogle Scholar
[2] Dimca, A.. Syzygies of Jacobian ideals and defects of linear systems. Bull. Math. Soc. Sci. Math. Roumanie Tome 56 (104) (2013), 191203.Google Scholar
[3] Dimca, A.. Freeness versus maximal degree of the singular subscheme for surfaces in P 3 . Geom. Dedicata 183 (2016), 101112.CrossRefGoogle Scholar
[4] Dimca, A. and Popescu, D.. Hilbert series and Lefschetz properties of dimension one almost complete intersections. Comm. Algebra 44 (2016), 44674482.CrossRefGoogle Scholar
[5] Dimca, A. and Sernesi, E.. Syzygies and logarithmic vector fields along plane curves. J. de l'École polytechnique-Mathématiques 1 (2014), 247267.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), 259266.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), 117132.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), 265291.Google Scholar
[15] Schenck, H.. Elementary modifications and line configurations in P 2 . Comm. Math. Helv. 78 (2003), 447462.CrossRefGoogle Scholar
[16] Sernesi, E.. The local cohomology of the jacobian ring. Documenta Mathematica, 19 (2014), 541565.CrossRefGoogle Scholar
[17] Simis, A. and Tohăneanu, S.O.. Homology of homogeneous divisors. Israel J. Math. 200 (2014), 449487.CrossRefGoogle Scholar
[18] Yoshinaga, M.. Freeness of hyperplane arrangements and related topics. Annales de la Faculté des Sciences de Toulouse 23 (2014), 483512.Google Scholar