Local symplectic algebra of quasi-homogeneous curves

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 sin- gularities of quasi-homogeneous curves. We prove that the space of algebraic restrictions of closed 2-forms to the germ of a K-analytic curve is a nite-dimensional vector space. We also show that the action of local dieomorphisms preserving the quasi-homogeneous curve on this vector space is determined by the innitesimal action of liftable vector elds. We apply these results to obtain a complete symplectic classication of curves with semigroups (3; 4; 5), (3; 5; 7), (3; 7; 8).