PreprintCC BY 4.0arXiv
Birational cubic fourfolds via an Enriques Cremona transformation
Abstract
We describe a Cremona transformation on ℙ5 given by the linear system of quintics that are double along the Fano model of an Enriques surface S. Using this Cremona transform, we construct a birational equivalence between the very general cubic fourfold X of discriminant 44 and its unique Fourier--Mukai partner. We also classify birational maps between smooth, non-isomorphic cubic fourfolds resolved by blowing up a smooth surface on each side: the centres must be either Fano models of Enriques surfaces, as in our construction, or Veronese surfaces, as studied by Fan--Lai.
Related
- Classification of Automorphism Groups of Smooth Cubic Threefolds and Fourfolds
- On the Supersingular reduction of Cubic fourfolds with Complex Multiplication
- Derived Torelli for Kuznetsov components of cubic fourfolds
- Remarks on Lattice Actions Corresponding to Automorphism Groups of Smooth Cubic Fourfolds
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BibTeX
@misc{brooke2026birational,
title = {{Birational cubic fourfolds via an Enriques Cremona transformation}},
author = {Corey Brooke and Lisa Marquand},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.10353},
eprint = {2609.10353},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10353},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Birational cubic fourfolds via an Enriques Cremona transformation
AU - Corey Brooke
AU - Lisa Marquand
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2609.10353
UR - https://arxiv.org/abs/2609.10353
AB - We describe a Cremona transformation on ℙ^5 given by the linear system of quintics that are double along the Fano model of an Enriques surface S. Using this Cremona transform, we construct a birational equivalence between the very general cubic fourfold X of discriminant 44 and its unique Fourier--Mukai partner. We also classify birational maps between smooth, non-isomorphic cubic fourfolds resolved by blowing up a smooth surface on each side: the centres must be either Fano models of Enriques surfaces, as in our construction, or Veronese surfaces, as studied by Fan--Lai.
ER -