Geometric Invariants of Plane and Space Curves

TL;DR

This paper generalizes classical geometric invariants like curvature and torsion to higher algebraic curvatures, demonstrating that these invariants uniquely determine each analytic branch of a complex algebraic curve.

Contribution

It introduces higher algebraic curvatures as a complete set of invariants for complex algebraic curve branches, extending classical differential geometry.

Findings

Higher algebraic curvatures uniquely identify analytic branches of complex curves

Classical invariants are insufficient for distinguishing curves, but the generalized invariants are complete

The approach enhances understanding of the geometric structure of algebraic curves

Abstract

The traditional study of plane and space algebraic curves by looking at their tangent vectors, curvatures and torsions provides geometric, but unfortunately not sufficient information about individual curves in order to be able to distinguish between two different ones. The aim of this note is to generalize the classical concept of curvature and torsion to so-called higher algebraic curvatures and to show that each analytic branch of a complex curve is already uniquely defined by them.