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arXiv · 2601.08583

Graded Betti numbers of the Jacobian algebra and total Tjurina numbers of plane curves

Alexandru Dimca·Gabriel Sticlaru
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Abstract

In this paper we compute an explicit closed formula for the total Tjurina number $\tau(C)$ of a reduced projective plane curve $C$ in terms of the graded Betti numbers of the corresponding Jacobian algebra. This formula allows a completely new view point on the classical upper bounds for the total Tjurina number $\tau(C)$ of a plane curve $C$ given by A. du Plessis and C. T. C. Wall. This approach yields in particular a new necessary condition for a set of positive integers to be the graded Betti numbers of the Jacobian algebra of a reduced plane curve.

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Alexandru Dimca, Gabriel Sticlaru. 2026-01-13. Graded Betti numbers of the Jacobian algebra and total Tjurina numbers of plane curves. https://arxiv.org/abs/2601.08583

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