The Tjurina number of a plane curve with two branches and high intersection multiplicity

TL;DR

The paper develops a framework to study the Tjurina number $\tau$ of plane curve singularities via the value sets of Kähler differentials. It generalizes Berger’s relation to $\tau=\mu-d(\overline{\Lambda},S)$ for reduced plane curves and uses this to derive a closed, topological formula for $\tau$. Focusing on curves with two equisingular branches, the authors show that, when the branches share a semigroup and their intersection multiplicity $I$ is large, $\tau$ is determined by topological data as $\tau=2I+c-1$ (or $\tau=2I+\mu(C_1)-1$), making it constant in the equisingularity class. They also demonstrate limitations for more than two branches and propose conjectures for minimal $\tau$ in broader settings, enriching the understanding of when $\tau$ remains constant and highlighting new topological invariants. Overall, the work bridges analytic invariants with combinatorial/topological descriptors and provides explicit formulas with potential applications in singularity theory and algebraic geometry.

Abstract

In this paper we provide a family of reduced plane curves with two branches that have a constant Tjurina number in their equisingularity class, along with a closed formula for it in terms of topological data.

Theorems & Definitions (38)

• Remark 2.1
• Definition 2.2
• Theorem 2.3
• Remark 2.4
• Remark 2.5
• Theorem 2.6
• Example 2.7
• Theorem 2.8
• Theorem 2.9
• proof