# The decomposition groups of plane conics and plane rational cubics

## TL;DR

This paper studies the decomposition groups of irreducible plane curves, showing they are generated by low-degree elements for conics and rational cubics, thus revealing structural properties of these groups.

## Contribution

It proves that the decomposition groups of conics and rational cubics are generated by elements of degree at most 2, providing new insights into their algebraic structure.

## Key findings

- Decomposition groups of conics are generated by degree ≤ 2 elements.
- Decomposition groups of rational cubics are generated by degree ≤ 2 elements.
- Structural understanding of birational symmetries of plane curves.

## Abstract

The decomposition group of an irreducible plane curve $X\subset\mathbb P^2$ is the subgroup $\mathrm{Dec}(X)\subset\mathrm{Bir}(\mathbb P^2)$ of birational maps which restrict to a birational map of $X$. We show that $\mathrm{Dec}(X)$ is generated by its elements of degree $\leq2$ when $X$ is either a conic or rational cubic curve.