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Portugaliae Mathematica


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Volume 76, Issue 3/4, 2019, pp. 311–325
DOI: 10.4171/PM/2038

Published online: 2020-06-30

On the Hilbert vector of the Jacobian module of a plane curve

Armando Cerminara[1], Alexandru Dimca[2] and Giovanna Ilardi[3]

(1) Università degli Studi di Napoli “Federico II”, Italy
(2) Université Côte d’Azur, Nice, France
(3) Università degli Studi di Napoli “Federico II”, Italy

We identify several classes of complex projective plane curves $C:f=0$, for which the Hilbert vector of the Jacobian module $N(f)$ can be completely determined, namely the 3-syzygy curves, the maximal Tjurina curves and the nodal curves, having only rational irreducible components. A result due to Hartshorne, on the cohomology of some rank 2 vector bundles on $\mathbb P^2$, is used to get a sharp lower bound for the initial degree of the Jacobian module $N(f)$, under a semistability condition.

Keywords: Jacobian syzygy, Milnor algebra, Jacobian module, global Tjurina number, nodal curves, rational curves

Cerminara Armando, Dimca Alexandru, Ilardi Giovanna: On the Hilbert vector of the Jacobian module of a plane curve. Port. Math. 76 (2019), 311-325. doi: 10.4171/PM/2038