On plane Cremona maps of degree 4 and length 2
Abstract
Plane Cremona maps of degree four are divided into two different classes according to their homaloidal type: either they are de Jonquières, that is, they have a base point of multiplicity 3, or they are not de Jonquières. The maps of the former class have length one, while the maps of the latter class have length two, according to the definition of length of a plane Cremona map given by Blanc and Furter. In this paper, we classify plane Cremona maps of degree four and length two, up to the action of automorphisms of the projective plane on the left-hand side and on the right-hand side of the map. This classification also allows us to compute the ordinary quadratic length of all such maps. In particular, we show that this ordinary quadratic length is at most 7 and we classify those maps which attain this upper bound.
Related
- On the quadratic length of plane Cremona maps of degree 4
- Plane Cremona maps preserving a Euclidean volume form
- A central limit theorem for the degree of a random product of Cremona transformations
- The derived length of subgroups of the Cremona group
- p-elementary non-cyclic subgroups of the Cremona group of the plane
BibTeX
@misc{calabri2026plane,
title = {{On plane Cremona maps of degree 4 and length 2}},
author = {Alberto Calabri and Nguyen Thi Ngoc Giao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08416},
eprint = {2610.08416},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08416},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - On plane Cremona maps of degree 4 and length 2
AU - Alberto Calabri
AU - Nguyen Thi Ngoc Giao
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08416
UR - https://arxiv.org/abs/2610.08416
AB - Plane Cremona maps of degree four are divided into two different classes according to their homaloidal type: either they are de Jonquières, that is, they have a base point of multiplicity 3, or they are not de Jonquières. The maps of the former class have length one, while the maps of the latter class have length two, according to the definition of length of a plane Cremona map given by Blanc and Furter. In this paper, we classify plane Cremona maps of degree four and length two, up to the action of automorphisms of the projective plane on the left-hand side and on the right-hand side of the map. This classification also allows us to compute the ordinary quadratic length of all such maps. In particular, we show that this ordinary quadratic length is at most 7 and we classify those maps which attain this upper bound.
ER -