Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 037, 19 pages      arXiv:2310.18759      https://doi.org/10.3842/SIGMA.2024.037

Compatible Poisson Brackets Associated with Elliptic Curves in $G(2,5)$

Nikita Markarian ^a and Alexander Polishchuk ^bc
a) Université de Strasbourg, France
^b) Department of Mathematics, University of Oregon, Eugene, OR 97403, USA
^c) National Research University Higher School of Economics, Moscow, Russia

Received December 05, 2023, in final form April 27, 2024; Published online May 07, 2024

Abstract
We prove that a pair of Feigin-Odesskii Poisson brackets on ${\mathbb P}^4$ associated with elliptic curves given as linear sections of the Grassmannian $G(2,5)$ are compatible if and only if this pair of elliptic curves is contained in a del Pezzo surface obtained as a linear section of $G(2,5)$.

Key words: Poisson bracket; bi-Hamiltonian structure; elliptic curve; triple Massey products.

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References

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