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

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.