Jacobian syzygies and plane curves with maximal global Tjurina numbers

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. 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 $r\le d-2$. This conjecture is proved for degrees $d\le 11$.

v5: Corollary 1.3 and subsection 4.13 are new

References in corpus (2)

• Quasi-complete intersections in P2 and syzygies
• The 0-th Fitting ideal of the Jacobian module of a plane curve

Cited by in corpus (1)

• Quasi-complete intersections in P2 and syzygies