Mathematical Research Letters

Volume 26 (2019)

Number 1

The decomposition groups of plane conics and plane rational cubics

Pages: 35 – 52



Tom Ducat (School of Mathematics, University of Bristol, United Kingdom; and the Heilbronn Institute for Mathematical Research, Bristol, United Kingdom)

Isac Hedén (Mathematics Institute, University of Warwick, Coventry, United Kingdom)

Susanna Zimmermann (Département de mathématiques, Université d’Angers, France)


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

This work was done whilst the first and second named authors were JSPS fellows supported by Grant-in-Aid for JSPS Fellows Number 15F15771 and 15F15751 respectively, and last named author was supported by the Swiss National Science Foundation grant P2BSP2 168743 at the University of Toulouse.

Received 23 August 2017

Accepted 15 January 2018

Published 7 June 2019