Uniqueness of Maximal Curve Systems on Punctured Projective Planes

TL;DR

This paper proves that the maximal 1-systems of loops on punctured projective planes are unique up to symmetry only when there are at most five punctures, establishing a precise classification.

Contribution

It establishes a complete characterization of the uniqueness of maximal loop systems on punctured projective planes based on the number of punctures.

Findings

Maximal 1-systems are unique up to the mapping class group for at most five punctures.

For more than five punctures, such systems are not unique.

The result provides a classification criterion based on puncture count.

Abstract

We show that the maximal $1$-system of loops in a punctured projective plane is unique up to the mapping class group action if and only if the number of punctures is at most five.