Compatible Feigin–Odesskii Poisson brackets

We prove that several Feigin–Odesskii Poisson brackets associated with normal elliptic curves in ℙ^n are compatible if and only if they are contained in a scroll or in a Veronese surface in 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 .