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arXiv · 2301.09129

Growth and integrability of some birational maps in dimension three

Michele Graffeo·Giorgio Gubbiotti
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Abstract

Motivated by the study of the Kahan--Hirota--Kimura discretisation of the Euler top, we characterise the growth and integrability properties of a collection of elements in the Cremona group of a complex projective 3-space using techniques from algebraic geometry. This collection consists of maps obtained by composing the standard Cremona transformation $\mathrm{c}_3\in\mathrm{Bir}(\mathbb{P}^3)$ with projectivities that permute the fixed points of $\mathrm{c}_3$ and the points over which $\mathrm{c}_3$ performs a divisorial contraction. More specifically, we show that three behaviour are possible: (A) integrable with quadratic degree growth and two invariants, (B) periodic with two-periodic degree sequences and more than two invariants, and (C) non-integrable with submaximal degree growth and one invariant.

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Michele Graffeo, Giorgio Gubbiotti. 2023-01-22. Growth and integrability of some birational maps in dimension three. https://arxiv.org/abs/2301.09129

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