# Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

## Local symplectic algebra of quasi-homogeneous curves

### Tom 204 / 2009

Fundamenta Mathematicae 204 (2009), 57-86 MSC: Primary 53D05; Secondary 14H20, 58K50, 58A10. DOI: 10.4064/fm204-1-4

#### Streszczenie

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 a $\mathbb K$-analytic curve is a finite-dimensional vector space. We also show that the action of local diffeomorphisms preserving the quasi-homogeneous curve on this vector space is determined by the infinitesimal action of liftable vector fields. We apply these results to obtain a complete symplectic classification of curves with semigroups $(3,4,5)$, $(3,5,7)$, $(3,7,8)$.

#### Autorzy

• Wojciech DomitrzFaculty of Mathematics and Information Science
Warsaw University of Technology
Plac Politechniki 1
00-661 Warszawa, Poland
and
Institute of Mathematics