Nikita Markaryan, Alexander Polishchuk, “Compatible Poisson Brackets Associated with Elliptic Curves in <nobr>$G(2,5)$</nobr>”, SIGMA, 2024, Volume 20,037, 19 pp.

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

Nikita Markaryan^a, Alexander Polishchuk^bc

^a Université de Strasbourg, France
^b National Research University Higher School of Economics, Moscow, Russia
^c Department of Mathematics, University of Oregon, Eugene, OR 97403, USA

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 $G(2,5)$.

Keywords: Poisson bracket, bi-Hamiltonian structure, elliptic curve, triple Massey products.

MSC: 14H52, 53D17

Received: December 5, 2023; in final form April 27, 2024; Published online May 7, 2024

Language: English

DOI: 10.3842/SIGMA.2024.037