Symplectic singularity of curves with semigroups (4,5,6,7), (4,5,6) and (4,5,7)

TL;DR

This paper investigates the local symplectic algebra of specific curve semigroups using algebraic restrictions, introducing a new invariant to distinguish symplectic orbits of quasi-homogeneous curves.

Contribution

It introduces a new discrete invariant for algebraic restrictions and applies it to classify symplectic orbits of curves with given semigroups.

Findings

New invariant effectively distinguishes symplectic orbits.

Method applied successfully to curves with semigroups (4,5,6,7), (4,5,6), and (4,5,7).

Enhanced understanding of symplectic singularities of these curves.

Abstract

We study the local symplectic algebra of curves with semigroups $(4,5,6,7)$, $(4,5,6)$ and $(4,5,7)$. We use the method of algebraic restrictions to parameterized curves as in \cite{D1}. A new discrete invariant for algebraic restrictions to parameterized quasi-homogeneous curves is introduced. This invariant together with the method of algebraic restriction can distinguish different symplectic orbits of quasi-homogeneous curves.