bibkey: graffeogubbiotti2024cremona
authors: Michele Graffeo, Giorgio Gubbiotti
year: 2024
title: "Growth and Integrability of Some Birational Maps in Dimension Three"
doi: 10.1007/s00023-023-01339-5
url: https://arxiv.org/abs/2301.09129v2
claim: "The KHK discretisation of the Euler top decomposes as l2 o crem3 o l1 with two projectivities and the standard cubic Cremona map of P^3 (Lemma 2.6); Remark 7.7 conjectures that the KHK map of a quadratic three-dimensional Nambu system with positive definite Nambu-Hamiltonians is, up to conjugation by a projectivity, of the same type (A)-(i)."
strata_touched:
  - D5/S3/FluidDynamics/Solitons/NambuKahanCremonaDecomposition
license: citation-only
triage: anchor

Graffeo–Gubbiotti, growth and integrability of birational maps in dimension three

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DOI: 10.1007/s00023-023-01339-5

Source: https://arxiv.org/abs/2301.09129v2

Ann. Henri Poincaré 25 (2024), 1733–1793.

M. Graffeo and G. Gubbiotti, arXiv:2301.09129v2 [math.AG] (3 June 2023, the latest version); Ann. Henri Poincaré 25 (2024), 1733–1793. Quotations are from the v2 TeX source GG_cremona_v1_clean.tex.

The KHK discretisation (Definition def:khk):

$\frac{\overrightarrow{x^{\prime }}-\vec{x}}{h}=2\vec{f}\left(\frac{\overrightarrow{x^{\prime }}+\vec{x}}{2}\right)-\frac{\vec{f}\left(\overrightarrow{x^{\prime }}\right)+\vec{f}\left(\vec{x}\right)}{2}$

The Euler-top decomposition (Lemma lem:eulerdecomposition):

The KHK discretisation of the Euler top decomposes as ${\Phi }_h={\ell }_2\circ {\mathrm{crem}}_3\circ {\ell }_1$ where ${\ell }_i:{\mathbb{P}}^3\to {\mathbb{P}}^3$ are two projectivies

The Nambu systems and the conjecture (Remark ex:celledoni):

$\dot{\vec{x}}=\nabla H_1(\vec{x})\times \nabla H_2(\vec{x})$, where, for $i=1,2$, $H_i(\vec{x})={\vec{x}}^T{A}_i\vec{x}$ is a Nambu–Hamiltonian function. […] We conjecture that, up to conjugation by a projectivity, the map ${\Phi }_h^{(N)}$ is of type (A)-(i), i.e. it exists an analogue of Lemma 2.6 for the map ${\Phi }_h^{(N)}$.

The same remark records that the invariant net “is generic in the sense that its base locus consists of eight distinct points” and that “The matrices $A_1$ and $A_2$ are positive definite symmetric quadratic forms.”

Encoding and boundary

For a quadratic homogeneous vector field the KHK step is $x^{\prime }-x=h\bar{f}(x,x^{\prime })$ with the symmetric polarisation $\bar{f}$; for $f=4Ax\times Bx$ this is $x^{\prime }=(I-2h{L}_x{)}^{-1}x$ with $L_{x}v=Ax\times Bv+Av\times Bx$, and its homogeneous lift is $(\mathrm{adj}Mx,\det M)$ with $M=x_{4}I-2h{L}_x$. The repository proves the decomposition ${\Phi }_h^{(N)}={\ell }_2\circ {\mathrm{crem}}_3\circ {\ell }_1$ as an equality of homogeneous lifts for every positive definite pair whose generalised eigenvalues are distinct and every $h>0$; the source’s eight-point genericity implies that eigenvalue condition. Degenerate pencils, the converse of the genericity implication and the source’s growth classification are outside this conclusion.