Group Actions, Divisors, and Plane Curves

TL;DR

This paper introduces elementary moduli spaces of divisors and 1-cycles on projective spaces, analyzing their structure under group actions, with implications for understanding geometric objects and symmetries.

Contribution

It provides foundational descriptions of two key moduli spaces involving divisors and 1-cycles under projective group actions.

Findings

Description of the moduli space of divisors on P^1 with finite stabilizer

Description of the moduli space of 1-cycles on P^2 with finite stabilizer

Elementary introduction to orbifolds and group actions in algebraic geometry

Abstract

After a general discussion of group actions, orbifolds, and "weak orbifolds" this note will provide elementary introductions to two basic moduli spaces over the real or complex numbers: First the moduli space of effective divisors with finite stabilizer on the projective space ${\mathbb P}^1$ modulo the group ${\rm PGL}_2$ of projective transformations of ${\mathbb P}^1$; and then the moduli space of effective 1-cycles with finite stabilizer on ${\mathbb P}^2$ modulo the group ${\rm PGL}_3$ of projective transformations of ${\mathbb P}^2$.