A Theory of Harmonic Variations

TL;DR

This paper introduces a geometric theory of harmonic variations for nonsingular curves, revealing a relationship between class and genus, and explores potential applications in understanding covalent bonding through shared electrons.

Contribution

It develops a new geometric framework for harmonic variations on nonsingular curves and links class and genus, with implications for chemical bonding models.

Findings

Established a relationship between class and genus of curves.

Analyzed the distribution of class points in harmonic variations.

Suggested applications to covalent bonding and shared electrons.

Abstract

We consider a class of "harmonic variations" for nonsingular curves, obtained as asymptotic degenerations along bitangents. On a geometric level, we obtain an attractive relationship between the class and the genus of $C$. The distribution of class points in pairs across nonsingular curves with such variations, further suggests applications to understanding covalent bonding in terms of shared electrons.