Duality of planar and spacial curves: new insight

TL;DR

This paper explores the duality between planar and spatial algebraic curves, revealing new conditions for the regularity of equiclassical families and extending these ideas to higher-dimensional spaces.

Contribution

It introduces novel sufficient conditions for the regularity of equiclassical stratifications using duality, and extends the concept to higher-dimensional projective spaces and Grassmannians.

Findings

Duality transformation yields new regularity conditions.

Conditions apply to both planar and spatial curves.

Extension of duality concepts to higher-dimensional spaces.

Abstract

We study the geometry of equiclassical strata of the discriminant in the space of plane curves of a given degree, which are families of curves of given degree, genus and class (degree of the dual curve). Our main observation is that the use of duality transformation leads to a series of new sufficient conditions for a regular behavior of the equiclassical stratification. We also discuss duality of curves in higher-dimensional projective spaces and in Grassmannians with focus on similar questions of the regularity of equiclassical families of spacial curves.