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

Part of: (Co)homology theory Symplectic geometry, contact geometry Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  08 October 2025

Alexander Polishchuk Open the ORCID record for Alexander Polishchuk [Opens in a new window]  and
Eric Rains
Alexander Polishchuk*
Affiliation:
Department of Mathematics, University of Oregon , Eugene, OR 97403, USA and National Research University Higher School of Economics, Moscow, Russia
Eric Rains
Affiliation:
Department of Mathematics, California Institute of Technology , Pasadena, CA 91125 (rains@caltech.edu)
*
apolish@uoregon.edu
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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 $>4$.

MSC classification

Primary: 14J60: Vector bundles on surfaces and higher-dimensional varieties, and their moduli 14F05: Sheaves, derived categories of sheaves and related constructions 53D17: Poisson manifolds; Poisson groupoids and algebroids

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Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , First View , pp. 1 - 35
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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