Symplectic $T_7$, $T_8$ singularities and Lagrangian tangency orders

TL;DR

This paper classifies symplectic $T_7$ singularities using algebraic restrictions and introduces Lagrangian tangency orders as invariants to distinguish various classical singularities in symplectic geometry.

Contribution

It provides a new classification method for symplectic $T_7$ singularities and introduces Lagrangian tangency orders as discrete invariants for distinguishing symplectic singularities.

Findings

Classification of symplectic $T_7$ singularities.

Introduction of Lagrangian tangency orders as invariants.

Geometric descriptions of symplectic singularities.

Abstract

We study the local symplectic algebra of curves. We use the method of algebraic restrictions to classify symplectic $T_7$ singularities. We define discrete symplectic invariants - the Lagrangian tangency orders. We use these invariants to distinguish symplectic singularities of classical $A-D-E$ singularities of planar curves, $S_5$ singularity and $T_7$ singularity. We also give the geometric description of these symplectic singularities.