On the halphen transform of algebraic space curves

TL;DR

This paper extends the Halphen transform from plane curves to space curves, analyzing its properties such as birationality, degree, rank, class, and desingularization, thereby broadening its application in algebraic geometry.

Contribution

It introduces the concept of the Halphen transform for space curves and investigates its key geometric and algebraic properties, expanding the classical theory.

Findings

The Halphen transform of space curves is birational under certain conditions.

The degree, rank, and class of the transformed space curve are characterized.

The transform can be used for desingularization of space curves.

Abstract

The Halphen transform of a plane curve is the curve obtained by intersecting the tangent lines of the curve with the corresponding polar lines with respect to some conic. This transform has been introduced by Halphen as a branch desingularization method and has also been studied by Coolidge and by Josse. We extend this notion to Halphen transform of a space curve and study several of its properties (birationality, degree, rank, class, desingularization).