## Tri-linear birational maps in dimension three

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Laurent Busé, Pablo González-Mazón and Josef Schicho
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## Abstract:

A tri-linear rational map in dimension three is a rational map $\phi : (\mathbb {P}_\mathbb {C}^1)^3 \dashrightarrow \mathbb {P}_\mathbb {C}^3$ defined by four tri-linear polynomials without a common factor. If $\phi$ admits an inverse rational map $\phi ^{-1}$, it is a tri-linear birational map. In this paper, we address computational and geometric aspects about these transformations. We give a characterization of birationality based on the first syzygies of the entries. More generally, we describe all the possible minimal graded free resolutions of the ideal generated by these entries. With respect to geometry, we show that the set $\mathfrak {Bir}_{(1,1,1)}$ of tri-linear birational maps, up to composition with an automorphism of $\mathbb {P}_\mathbb {C}^3$, is a locally closed algebraic subset of the Grassmannian of $4$-dimensional subspaces in the vector space of tri-linear polynomials, and has eight irreducible components. Additionally, the group action on $\mathfrak {Bir}_{(1,1,1)}$ given by composition with automorphisms of $(\mathbb {P}_\mathbb {C}^1)^3$ defines 19 orbits, and each of these orbits determines an isomorphism class of the base loci of these transformations.## References

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## Additional Information

**Laurent Busé**- Affiliation: Université Côte d’Azur, Inria, 2004 route des Lucioles, 06902 Sophia Antipolis,, France
- Email: laurent.buse@inria.fr
**Pablo González-Mazón**- Affiliation: Université Côte d’Azur, Inria, 2004 route des Lucioles, 06902 Sophia Antipolis,, France
- ORCID: 0000-0003-3403-3356
- Email: pablo.gonzalez-mazon@inria.fr
**Josef Schicho**- Affiliation: Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Altenbergerstraße 69, 4040 Linz, Austria
- MR Author ID: 332588
- Email: jschicho@risc.jku.at
- Received by editor(s): March 4, 2022
- Received by editor(s) in revised form: September 22, 2022
- Published electronically: February 28, 2023
- Additional Notes: The three authors were funded by the European Union’s Horizon 2020 Research and Innovation Programme, under the Marie Skłodowska-Curie grant agreement n$^\circ$ 860843.

All the examples, and multiple computations not explicitly included, were performed using the computer algebra software Macaulay2 \cite{M2}.

The second author is the corresponding author. - © Copyright 2023 American Mathematical Society
- Journal: Math. Comp.
**92**(2023), 1837-1866 - MSC (2020): Primary 14E05; Secondary 13D02, 13P99
- DOI: https://doi.org/10.1090/mcom/3804
- MathSciNet review: 4570344