6533b860fe1ef96bd12c404e

RESEARCH PRODUCT

On a quadratic form associated with the nilpotent part of the monodromy of a curve

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Minor correction on the metadata of one of the authors. The rest is exactly the same; We study the nilpotent part of certain pseudoperiodic automorphisms of surfaces appearing in singularity theory. We associate a quadratic form $\tilde{Q}$ defined on the first (relative to the boundary) homology group of the Milnor fiber $F$ of any germ analytic curve on a normal surface. Using the twist formula and techniques from mapping class group theory, we prove that the form $\tilde{Q}$ obtained after killing ${\ker N}$ is definite positive, and that its restriction to the absolute homology group of $F$ is even whenever the Nielsen-Thurston graph of the monodromy automorphism is a tree. The form $\tilde{Q}$ is computable in terms of the Nielsen-Thurston or the dual graph of the semistable reduction, as illustrated with several examples. Numerical invariants associated to $\tilde{Q}$ are able to distinguish plane curve singularities with different topological types but same spectral pairs or Seifert form. Finally, we discuss a generic linear germ defined on a superisolated surface with not smooth ambient space.