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

Alexandru Dimca, Gabriel Sticlaru

TL;DR

This work connects the total Tjurina number $\tau(C)$ of a reduced plane curve $C$ to the graded Betti numbers of its Jacobian (Milnor) algebra $M(f)$. By introducing the curve type $t(C)$ and an associated ordered partition $\pi_C$, the authors derive a closed formula for $\tau(C)$ in terms of the exponents $d_i$ and the partition data $\epsilon_j$, encapsulated in a single expression with a nonnegative remainder $R$. This yields a natural decomposition of curves into $2^{t-1}$ classes $tT_{\pi}$ and clarifies how classical bounds (du Plessis–Wall) arise from the Betti-structure; the approach is illustrated with $t(C)=4$, including eight explicit classes and concrete Milnor algebra resolutions. The results provide a unified framework to study maximal, nearly maximal, and free/near-free curves, offering sharper bounds and a constructive path to new examples and proofs of known characterizations.

Abstract

In this paper we compute explicit closed formulas for the total Tjurina number $τ(C)$ of a reduced projective plane curve $C$ in terms of the graded Betti numbers of the corresponding Jacobian algebra. This yields in particular a completely new view point on the classical upper bounds for the total Tjurina number $τ(C)$ of a plane curve $C$ given by A. du Plessis and C. T. C. Wall.

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

TL;DR

This work connects the total Tjurina number of a reduced plane curve to the graded Betti numbers of its Jacobian (Milnor) algebra . By introducing the curve type and an associated ordered partition , the authors derive a closed formula for in terms of the exponents and the partition data , encapsulated in a single expression with a nonnegative remainder . This yields a natural decomposition of curves into classes and clarifies how classical bounds (du Plessis–Wall) arise from the Betti-structure; the approach is illustrated with , including eight explicit classes and concrete Milnor algebra resolutions. The results provide a unified framework to study maximal, nearly maximal, and free/near-free curves, offering sharper bounds and a constructive path to new examples and proofs of known characterizations.

Abstract

In this paper we compute explicit closed formulas for the total Tjurina number of a reduced projective plane curve in terms of the graded Betti numbers of the corresponding Jacobian algebra. This yields in particular a completely new view point on the classical upper bounds for the total Tjurina number of a plane curve given by A. du Plessis and C. T. C. Wall.
Paper Structure (5 sections, 7 theorems, 65 equations)

This paper contains 5 sections, 7 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Graded Betti numbers and partitions
  3. Tjurina numbers for curves via graded Betti numbers of their Jacobian algebra
  4. Examples for curves of type $t(C)=4$
  5. On a result by A. du Plessis and C.T.C. Wall

Key Result

Theorem 3.1

Let $C:f=0$ be a reduced plane curve of degree $d$ and type $t(C)=t\geq 1$ with exponents $(d_{1},d_{2},\ldots,d_{m})$. Let $\pi_C=(\epsilon_1,\dots,\epsilon_n)$ be the corresponding ordered partition of $t$, with $n=m-2$. Then one has where In particular, $R=0$ if and only if $m=3$.

Theorems & Definitions (14)

  • Conjecture 2.1
  • Theorem 3.1
  • Remark 3.2
  • Example 3.3
  • Corollary 4.1
  • Example 4.2
  • Theorem 5.1
  • Corollary 5.2
  • Remark 5.3
  • Corollary 5.4
  • ...and 4 more