Exceptional pairs on del Pezzo surfaces and spaces of compatible Feigin-Odesskii brackets

We prove that for every relatively prime pair of integers $(d,r)$ with $r>0$, there exists an exceptional pair ( O,V) on any del Pezzo surface of degree 4, such that V is a bundle of rank r and degree d. As an application, we prove that every Feigin-Odesskii Poisson bracket on a projective space can be included into a 5-dimensional linear space of compatible Poisson brackets. We also construct new examples of linear spaces of compatible Feigin-Odesskii Poisson brackets of dimension $>5$, coming from del Pezzo surfaces of degree $>4$.