Generic symmetry defect set of an algebraic curve

TL;DR

This paper introduces an algebraic framework for the generic symmetry defect set of algebraic varieties and analyzes its singularities specifically for generic curves of degree d in complex two-dimensional space.

Contribution

It defines the algebraic version of the symmetry defect set and computes its singularities for generic algebraic curves in the complex plane.

Findings

The algebraic symmetry defect set generalizes classical notions.

Singularities of the defect set are characterized for generic curves.

Results apply to complex algebraic geometry and symmetry analysis.

Abstract

Let $X \subset \mathbb{C}^{2n}$ be an $n$-dimensional algebraic variety. We define the algebraic version of the generic symmetry defect set (Wigner caustic) of $X$. Moreover, we compute its singularities for $X_d$ being a generic curve of degree $d$ in $\mathbb{C}^2$.