Syzygies and logarithmic vector fields along plane curves
Journal de l’École polytechnique — Mathématiques, Volume 1 (2014), pp. 247-267.

We investigate the relations between the syzygies of the Jacobian ideal of the defining equation for a plane curve C and the stability of the sheaf of logarithmic vector fields along C, the freeness of the divisor C and the Torelli properties of C (in the sense of Dolgachev-Kapranov). We show in particular that curves with a small number of nodes and cusps are Torelli in this sense.

Nous étudions les relations entre les syzygies de l’idéal jacobien associé à l’équation définissant une courbe plane C et la stabilité du faisceau des champs de vecteurs logarithmiques le long de C, la liberté du diviseur C et les propriétés de Torelli de C (au sens de Dolgachev-Kapranov). Nous montrons en particulier que les courbes ayant un petit nombre de points doubles et de cusps ont la propriété de Torelli.

Received:
Accepted:
Published online:
DOI: 10.5802/jep.10
Classification: 14C34, 14H50, 32S05
Keywords: Syzygy, plane curve, logarithmic vector fields, stable bundle, free divisor, Torelli property
Mot clés : Syzygie, courbe plane, champ de vecteurs logarithmique, fibré stable, diviseur libre, propriété de Torelli
Alexandru Dimca 1; Edoardo Sernesi 2

1 Université Nice Sophia Antipolis, CNRS, LJAD, UMR 7351 Parc Valrose, 06108 Nice, Cedex 02, France
2 Dipartimento di Matematica e Fisica, Università Roma Tre, Largo S. L. Murialdo 1, 00146 Roma, Italy
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{JEP_2014__1__247_0,
     author = {Alexandru Dimca and Edoardo Sernesi},
     title = {Syzygies and logarithmic vector fields along plane curves},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {247--267},
     publisher = {\'Ecole polytechnique},
     volume = {1},
     year = {2014},
     doi = {10.5802/jep.10},
     mrnumber = {3322789},
     zbl = {1327.14049},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.10/}
}
TY  - JOUR
AU  - Alexandru Dimca
AU  - Edoardo Sernesi
TI  - Syzygies and logarithmic vector fields along plane curves
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2014
SP  - 247
EP  - 267
VL  - 1
PB  - École polytechnique
UR  - https://jep.centre-mersenne.org/articles/10.5802/jep.10/
DO  - 10.5802/jep.10
LA  - en
ID  - JEP_2014__1__247_0
ER  - 
%0 Journal Article
%A Alexandru Dimca
%A Edoardo Sernesi
%T Syzygies and logarithmic vector fields along plane curves
%J Journal de l’École polytechnique — Mathématiques
%D 2014
%P 247-267
%V 1
%I École polytechnique
%U https://jep.centre-mersenne.org/articles/10.5802/jep.10/
%R 10.5802/jep.10
%G en
%F JEP_2014__1__247_0
Alexandru Dimca; Edoardo Sernesi. Syzygies and logarithmic vector fields along plane curves. Journal de l’École polytechnique — Mathématiques, Volume 1 (2014), pp. 247-267. doi : 10.5802/jep.10. https://jep.centre-mersenne.org/articles/10.5802/jep.10/

[1] E. Angelini - “Logarithmic bundles of hypersurface arrangements in n (2013), arXiv:1304.5709

[2] E. Arbarello, M. Cornalba, P. A. Griffiths & J. Harris - Geometry of algebraic curves. Vol. I, Grundlehren der Mathematischen Wissenschaften, vol. 267, Springer-Verlag, New York, 1985 | DOI | MR | Zbl

[3] V. I. Arnold, S. M. Guseĭn-Zade & A. N. Varchenko - Singularities of differentiable maps. Vol. II, Monographs in Mathematics, vol. 83, Birkhäuser Boston, Inc., Boston, MA, 1988 | DOI | MR

[4] R.-O. Buchweitz & A. Conca - “New free divisors from old”, J. Commut. Algebra 5 (2013) no. 1, p. 17-47, arXiv:1211.4327 | DOI | MR | Zbl

[5] A. Dimca - Topics on real and complex singularities, Advanced Lectures in Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 1987 | DOI | Zbl

[6] A. Dimca - “Syzygies of Jacobian ideals and defects of linear systems”, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 56(104) (2013) no. 2, p. 191-203 | MR | Zbl

[7] A. Dimca & M. Saito - “Graded Koszul cohomology and spectrum of certain homogeneous polynomials” (2012), arXiv:1212.1081

[8] A. Dimca & M. Saito - “Generalization of theorems of Griffiths and Steenbrink to hypersurfaces with ordinary double points” (2014), arXiv:1403.4563 | Zbl

[9] A. Dimca & M. Saito - “Some remarks on limit mixed Hodge structures and spectrum”, An. Ştiinţ. Univ. Ovidius Constanţa Ser. Mat. 22 (2014) no. 2, p. 69-78 | MR | Zbl

[10] A. Dimca & G. Sticlaru - “Koszul complexes and pole order filtrations”, Proc. Edinburgh Math. Soc. (2), to appear, arXiv:1108.3976 | MR | Zbl

[11] A. Dimca & G. Sticlaru - “Chebyshev curves, free resolutions and rational curve arrangements”, Math. Proc. Cambridge Philos. Soc. 153 (2012) no. 3, p. 385-397 | DOI | MR | Zbl

[12] A. Dimca & G. Sticlaru - “Syzygies of Jacobian ideals and weighted homogeneous singularities” (2014), arXiv:1407.0168 | Zbl

[13] I. Dolgachev & M. Kapranov - “Arrangements of hyperplanes and vector bundles on n , Duke Math. J. 71 (1993) no. 3, p. 633-664 | DOI | MR | Zbl

[14] M. Granger, D. Mond, A. Nieto-Reyes & M. Schulze - “Linear free divisors and the global logarithmic comparison theorem”, Ann. Inst. Fourier (Grenoble) 59 (2009) no. 2, p. 811-850 | DOI | Numdam | MR | Zbl

[15] R. Hartshorne - Algebraic Geometry, Graduate Texts in Math., vol. 52, Springer-Verlag, 1977 | Zbl

[16] K. Hulek - “Stable rank-2 vector bundles on 2 with c 1 odd”, Math. Ann. 242 (1979) no. 3, p. 241-266 | DOI | Zbl

[17] J. Kollár - “Singularities of pairs”, in Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, American Mathematical Society, Providence, RI, 1997, p. 221-287 | DOI | MR | Zbl

[18] L. Narváez Macarro - “Linearity conditions on the Jacobian ideal and logarithmic-meromorphic comparison for free divisors”, in Singularities I, Contemp. Math., vol. 474, Amer. Math. Soc., Providence, RI, 2008, p. 245-269 | DOI | MR | Zbl

[19] C. Okonek, M. Schneider & H. Spindler - Vector bundles on complex projective spaces, Progress in Math., vol. 3, Birkhäuser, Boston, Mass., 1980 | MR | Zbl

[20] P. Orlik & H. Terao - Arrangements of hyperplanes, Grundlehren der Mathematischen Wissenschaften, vol. 300, Springer-Verlag, Berlin, 1992 | DOI | MR | Zbl

[21] K. Saito - “Einfach-elliptische Singularitäten”, Invent. Math. 23 (1974), p. 289-325 | DOI | Zbl

[22] K. Saito - “Theory of logarithmic differential forms and logarithmic vector fields”, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 27 (1980) no. 2, p. 265-291 | MR | Zbl

[23] E. Sernesi - Deformations of algebraic schemes, Grundlehren der Mathematischen Wissenschaften, vol. 334, Springer-Verlag, Berlin, 2006 | MR | Zbl

[24] E. Sernesi - “The local cohomology of the Jacobian ring”, Doc. Math. 19 (2014), p. 541-565 | MR | Zbl

[25] A. Simis & Ş. O. Tohăneanu - “Homology of homogeneous divisors”, Israel J. Math. 200 (2014) no. 1, p. 449-487, arXiv:1207.5862 | DOI | MR | Zbl

[26] G. Sticlaru - “Free divisors versus stability and coincidence thresholds” (2014), arXiv:1401.1843 | DOI

[27] K. Ueda & M. Yoshinaga - “Logarithmic vector fields along smooth divisors in projective spaces”, Hokkaido Math. J. 38 (2009) no. 3, p. 409-415 | DOI | MR | Zbl

[28] J. Vallès - “Nombre maximal d’hyperplans instables pour un fibré de Steiner”, Math. Z. 233 (2000) no. 3, p. 507-514 | DOI | MR | Zbl

[29] J. M. Wahl - “Deformations of plane curves with nodes and cusps”, Amer. J. Math. 96 (1974), p. 529-577 | DOI | MR | Zbl

[30] M. Yoshinaga - “Freeness of hyperplane arrangements and related topics”, Ann. Fac. Sci. Toulouse Math. (6) 23 (2014) no. 2, p. 483-512, arXiv:1212.3523 | DOI | Numdam | MR | Zbl

Cited by Sources: