EXCEPTIONAL PAIRS ON DEL PEZZO SURFACES AND SPACES OF COMPATIBLE FEIGIN-ODESSKII BRACKETS

Abstract We prove that for every relatively prime pair of integers $(d,r)$ with $r>0$ , there exists an exceptional pair $({\mathcal {O}},V)$ 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 $>4$ .