Syzygies and logarithmic vector fields along plane curves

Alexandru Dimca; Edoardo Sernesi

doi:10.5802/jep.10

Syzygies and logarithmic vector fields along plane curves
[Syzygies et champs de vecteurs logarithmiques le long de courbes planes]

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

Journal de l’École polytechnique — Mathématiques, Tome 1 (2014), pp. 247-267.

Résumé
Abstract

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.

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.

Reçu le : 2014-02-16
Accepté le : 2014-08-05
Publié le : 2014-08-26

MR Zbl

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

Affiliations des auteurs :

Alexandru Dimca ¹ ; Edoardo Sernesi ²

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     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/}
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Alexandru Dimca; Edoardo Sernesi. Syzygies and logarithmic vector fields along plane curves. Journal de l’École polytechnique — Mathématiques, Tome 1 (2014), pp. 247-267. doi : 10.5802/jep.10. https://jep.centre-mersenne.org/articles/10.5802/jep.10/

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