Compatible Feigin–Odesskii Poisson brackets

Abstract

We prove that several Feigin–Odesskii Poisson brackets associated with normal elliptic curves in $${{\mathbb {P}}}^n$$ are compatible if and only if they are contained in a scroll or in a Veronese surface in $${{\mathbb {P}}}^5$$ (with an exception of one case when $$n=3$$ ). In the case $$n=3$$ we determine the quartic corresponding to the Schouten bracket of two (non-compatible) Poisson brackets associated with normal elliptic curves $$E_1$$ and $$E_2$$ .