A Theory of Duality for Algebraic Curves

TL;DR

This paper develops a refined duality theory for algebraic curves, providing a geometric interpretation of genus and generalizing Plucker's formulas across different characteristics.

Contribution

It introduces a new duality framework for algebraic curves and extends classical formulas to broader settings, including arbitrary characteristic fields.

Findings

Provides a geometric interpretation of the genus of algebraic curves

Generalizes Plucker's formulas using duality principles

Results applicable to algebraic curves over fields of any characteristic

Abstract

We formulate a refined theory of linear systems, using the methods of a previous paper, "A Theory of Branches for Algebraic Curves", and use it to give a geometric interpretation of the genus of an algebraic curve. Using principles of duality, we prove generalisations of Plucker's formulae for algebraic curves. The results hold for arbitrary characteristic of the base field L, with some occasional exceptions when characteristic(L)=2, which we observe in the course of the paper.