The decomposition groups of plane conics and plane rational cubics

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 $\le 2$ when $X$ is either a conic or rational cubic curve.

12 pages, 6 figures. Comments are welcome