PreprintCC BY-NC-ND 4.0arXiv
On the quadratic length of plane Cremona maps of degree 4
Abstract
Every non-linear plane Cremona map can be decomposed into quadratic maps, and the minimum number of quadratic maps required is called its quadratic length. It is known that plane Cremona maps of degree 3 have quadratic length either 2 or 3. In this paper, we study the quadratic length of plane Cremona maps φ of degree 4. Recall that either φ is de Jonquières, i.e. it has a base point of multiplicity 3, or it is not de Jonquières. In the latter case, φ has quadratic length 2, whereas in the former case we prove that φ has quadratic length 3, 4, or 5, and we classify those maps that reach the upper bound.
Related
- On plane Cremona maps of degree 4 and length 2
- 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{calabri2026quadratic,
title = {{On the quadratic length of plane Cremona maps of degree 4}},
author = {Alberto Calabri and Nguyen Thi Ngoc Giao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39943},
eprint = {2609.39943},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39943},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - On the quadratic length of plane Cremona maps of degree 4
AU - Alberto Calabri
AU - Nguyen Thi Ngoc Giao
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39943
UR - https://arxiv.org/abs/2609.39943
AB - Every non-linear plane Cremona map can be decomposed into quadratic maps, and the minimum number of quadratic maps required is called its quadratic length. It is known that plane Cremona maps of degree 3 have quadratic length either 2 or 3. In this paper, we study the quadratic length of plane Cremona maps φ of degree 4. Recall that either φ is de Jonquières, i.e. it has a base point of multiplicity 3, or it is not de Jonquières. In the latter case, φ has quadratic length 2, whereas in the former case we prove that φ has quadratic length 3, 4, or 5, and we classify those maps that reach the upper bound.
ER -