The symplectic leaves for the elliptic Poisson bracket on projective space defined by Feigin-Odesskii and Polishchuk

Abstract

This paper determines the symplectic leaves for a remarkable Poisson structure on $\mathbb{C}{\mathbb{P}}^{n-1}$ discovered by Feigin and Odesskii, and, independently, by Polishchuk. The Poisson bracket is determined by a holomorphic line bundle of degree $n\ge 3$ on a compact Riemann surface of genus one or, equivalently, by an elliptic normal curve $E\subseteq \mathbb{C}{\mathbb{P}}^{n-1}$. The symplectic leaves are described in terms of higher secant varieties to $E$.

numerous changes throughout the paper, altering the structure of the main argument; 50 pages + references