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

Alexander Polishchuk, Eric Rains

TL;DR

The paper establishes the existence of exceptional pairs $(\mathcal{O}_X,V)$ on degree-4 del Pezzo surfaces for every coprime $(d,r)$ with $r>0$, using a construction that passes through a Weyl-group action on the Picard lattice and embeddings of weighted-projective-line derived categories. It then links these exceptional pairs to Feigin–Odesskii Poisson brackets, showing that any FO bracket on projective space embeds into a 5-dimensional space of compatible FO brackets, with higher-dimensional families arising from higher-degree del Pezzo surfaces. The work develops a detailed correspondence between the derived categories of the weighted lines $\mathcal{C}_4$ and $\mathcal{C}_5$ and the perpendicular subcategories to exceptional collections on degree-4 X, enabling precise slope control and descent/translation of data. In higher-degree del Pezzo surfaces ($k\ge 5$), the authors provide necessary inequalities and realizability results for possible $(d,r)$, and extend the Poisson-bracket framework to broader settings, including nodal anticanonical divisors and bihamiltonian structures. Collectively, the results connect derived-category methods, lattice/combinatorial symmetries, and Poisson geometry to yield new linear spaces of compatible Feigin–Odesskii brackets and to illuminate the structure of exceptional bundles on 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 $>4$.

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

TL;DR

The paper establishes the existence of exceptional pairs on degree-4 del Pezzo surfaces for every coprime with , using a construction that passes through a Weyl-group action on the Picard lattice and embeddings of weighted-projective-line derived categories. It then links these exceptional pairs to Feigin–Odesskii Poisson brackets, showing that any FO bracket on projective space embeds into a 5-dimensional space of compatible FO brackets, with higher-dimensional families arising from higher-degree del Pezzo surfaces. The work develops a detailed correspondence between the derived categories of the weighted lines and and the perpendicular subcategories to exceptional collections on degree-4 X, enabling precise slope control and descent/translation of data. In higher-degree del Pezzo surfaces (), the authors provide necessary inequalities and realizability results for possible , and extend the Poisson-bracket framework to broader settings, including nodal anticanonical divisors and bihamiltonian structures. Collectively, the results connect derived-category methods, lattice/combinatorial symmetries, and Poisson geometry to yield new linear spaces of compatible Feigin–Odesskii brackets and to illuminate the structure of exceptional bundles on del Pezzo surfaces.

Abstract

We prove that for every relatively prime pair of integers with , there exists an exceptional pair on any del Pezzo surface of degree 4, such that is a bundle of rank and degree . 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 , coming from del Pezzo surfaces of degree .
Paper Structure (17 sections, 22 theorems, 103 equations)

This paper contains 17 sections, 22 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Exceptional objects on del Pezzo surfaces
  4. Action of the Weyl group on geometric markings
  5. Transformations of exceptional pairs $(V,{\mathcal{O}})$
  6. Exceptional bundles on the weighted projective lines
  7. Generalities on hyperelliptic weighted projective lines
  8. Omnipresent exceptional bundles on ${\mathcal{C}}_5$
  9. Exceptional pairs on del Pezzo surfaces of degree $4$
  10. Connection with the derived category of ${\mathcal{C}}_5$
  11. Connection with the derived category of ${\mathcal{C}}_4$
  12. Existence of exceptional pairs with a given slope
  13. Non-sporadic range
  14. Exceptional pairs on del Pezzo surfaces of degree $\ge 5$
  15. Application to bihamiltonian structures
  16. ...and 2 more sections

Key Result

Lemma 2.1

(i) Let $F$ be an exceptional object in $D^b(\operatorname{Coh} X)$. Then either $F\simeq V[n]$, where $V$ is an exceptional bundle, or $F\simeq {\mathcal{O}}_R(m)[n]$, where $R$ is a $(-1)$-curve. (ii) Let $\pi:X\to X'$ be the blow down of a collection of non-intersecting $(-1)$-curves, and let $V$

Theorems & Definitions (48)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Remark 2.4
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Lemma 3.3
  • ...and 38 more