Local symplectic algebra of quasi-homogeneous curves

TL;DR

This paper classifies symplectic singularities of quasi-homogeneous curves using algebraic restrictions, providing a complete symplectic classification for specific curve semigroups.

Contribution

It introduces a method to classify symplectic singularities of curves via algebraic restrictions and completes the classification for certain semigroups.

Findings

Finite dimensional vector space of algebraic restrictions.

Action of diffeomorphisms determined by liftable vector fields.

Complete classification for curves with semigroups (3,4,5), (3,5,7), (3,7,8).

Abstract

We study the local symplectic algebra of parameterized curves introduced by V. I. Arnold. We use the method of algebraic restrictions to classify symplectic singularities of quasi-homogeneous curves. We prove that the space of algebraic restrictions of closed 2-forms to the germ of an analytic curve is a finite dimensional vector space. We also show that the action of local diffeomorphisms preserving the curve on this vector space is determined by the infinitesimal action of liftable vector fields. We apply these results to obtain the complete symplectic classification of curves with the semigroups (3,4,5), (3,5,7), (3,7,8).