On a quadratic form associated with a surface automorphism and its applications to Singularity Theory

We study the nilpotent part N′ of a pseudo-periodic automorphism h real oriented surface with boundary Σ. associate quadratic form Q defined on first homology group (relative to boundary) Using twist formula and techniques from mapping class theory, we prove that Q̃ obtained after killing kerN is positive definite if all screw numbers associated certain orbits annuli are positive. also restriction absolute Σ even whenever quotient Nielsen–Thurston graph under action tree. The case monodromy automorphisms Milnor fibers Σ=F germs curves normal singularities discussed in detail, aforementioned results specialized such situation. This determined by Seifert but can be much more easily computed. Moreover, computable terms dual resolution or semistable reduction graph, as illustrated several examples. Numerical invariants able distinguish plane curve different topological types same spectral pairs. Finally, discuss generic linear germ superisolated surface. In this plumbing not tree even.