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

TL;DR

This paper derives a closed-form formula linking the total Tjurina number of a plane curve to the graded Betti numbers of its Jacobian algebra, offering new insights into classical bounds and algebraic conditions.

Contribution

It provides a novel explicit formula connecting Tjurina numbers and Betti numbers, and introduces a new necessary condition for Betti number sequences of plane curves.

Findings

Explicit formula for total Tjurina number in terms of Betti numbers

New perspective on classical upper bounds for Tjurina numbers

A necessary condition for Betti number sequences of Jacobian algebras

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.