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arXiv · 1901.05915

Jacobian syzygies and plane curves with maximal global Tjurina numbers

Alexandru Dimca·Gabriel Sticlaru
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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 $r\leq d-1$ 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. Finally we prove the existence of curves with maximal global Tjurina numbers for certain pairs $(d,r)$. Moreover, we conjecture that such curves exist for any pair $(d,r)$, and that, in addition, they may be chosen to be line arrangements when $r\leq d-2$. This conjecture is proved for degrees $d \leq 11$.

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Alexandru Dimca, Gabriel Sticlaru. 2019-01-17. Jacobian syzygies and plane curves with maximal global Tjurina numbers. https://arxiv.org/abs/1901.05915

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