Syzygies and logarithmic vector fields along plane curves

Dimca, Alexandru; Sernesi, Edoardo

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

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.

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

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