Local symplectic algebra of quasi-homogeneous curves

Volume 204 / 2009

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

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 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 $(3,4,5)$, $(3,5,7)$, $(3,7,8)$.

Authors

  • Wojciech DomitrzFaculty of Mathematics and Information Science
    Warsaw University of Technology
    Plac Politechniki 1
    00-661 Warszawa, Poland
    and
    Institute of Mathematics
    Polish Academy of Sciences
    Śniadeckich 8
    P.O. Box 21
    00-956 Warszawa, Poland
    e-mail

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