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Jacobian syzygies, Fitting ideals, and plane curves with maximal global Tjurina numbers

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Abstract

First we give a sharp upper bound for the cardinal m of a minimal set of generators for the module of Jacobian syzygies of a complex projective reduced plane curve C. Next we discuss the sharpness of an upper bound, given by A. du Plessis and C.T.C. Wall, for the global Tjurina number of such a curve C, in terms of its degree d and of the minimal degree \(r\le d-1\) of a Jacobian syzygy. We give a homological characterization of the curves whose global Tjurina number equals the du Plessis-Wall upper bound, which implies in particular that for such curves the upper bound for m is also attained. A second characterization of these curves in terms of the 0-th Fitting ideal of their Jacobian module is also given. Finally we prove the existence of curves with maximal global Tjurina numbers for certain pairs (d, r). We conjecture that such curves exist for any pair (d, r), and that, in addition, they may be chosen to be line arrangements when \(r\le d-2\). This conjecture is proved for degrees \(d \le 11\).

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Authors and Affiliations

  1. Université Côte d’Azur, CNRS, LJAD and INRIA, France and Simion Stoilow Institute of Mathematics, P.O. Box 1-764, 014700, Bucharest, Romania

    Alexandru Dimca

  2. Faculty of Mathematics and Informatics, Ovidius University, Bd. Mamaia 124, 900527, Constanta, Romania

    Gabriel Sticlaru

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  1. Alexandru Dimca
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  2. Gabriel Sticlaru
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Correspondence to Alexandru Dimca.

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A. Dimca: This work has been partially supported by the French government, through the \(\mathrm UCA^\mathrm{JEDI}\) Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-15-IDEX-01 and by the Romanian Ministry of Research and Innovation, CNCS—UEFISCDI, Grant PN-III-P4-ID-PCE-2020-0029, within PNCDI III.

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Dimca, A., Sticlaru, G. Jacobian syzygies, Fitting ideals, and plane curves with maximal global Tjurina numbers. Collect. Math. 73, 391–409 (2022). https://doi.org/10.1007/s13348-021-00325-6

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