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

TL;DR

This paper constructs special vector bundles on degree 4 del Pezzo surfaces and explores the structure of compatible Feigin-Odesskii Poisson brackets, revealing new examples and inclusion properties.

Contribution

It proves the existence of exceptional pairs on del Pezzo surfaces and demonstrates how Feigin-Odesskii brackets can be embedded into higher-dimensional compatible spaces.

Findings

Existence of exceptional pairs on degree 4 del Pezzo surfaces.

Feigin-Odesskii brackets can be embedded into 5-dimensional compatible spaces.

New examples of higher-dimensional compatible brackets from del Pezzo surfaces.

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 $>4$.