The Max Noether Fundamental Theorem is Combinatorial

TL;DR

This paper reformulates the Max Noether Fundamental Theorem for plane curves of equal degree using Abstract Curve Combinatorics, weakening local conditions and drawing an analogy to matroids in hyperplane arrangements.

Contribution

It introduces Abstract Curve Combinatorics as a new combinatorial framework to generalize the Noether Fundamental Theorem for specific plane curves.

Findings

Reformulation of Noether's Theorem for equal degree curves

Introduction of Abstract Curve Combinatorics as a combinatorial tool

Weakened local conditions in the theorem's context

Abstract

In the present paper we give a reformulation of the Noether Fundamental Theorem for the special case where the three curves involved have the same degree. In this reformulation, the local Noether's Conditions are weakened. To do so we introduce the concept of Abstract Curve Combinatorics (ACC) which will be, in the context of plane curves, the analogue of matroids for hyperplane arrangements.