Compatible Feigin-Odesskii Poisson brackets

Alexander Polishchuk; Nikita Markarian

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Compatible Feigin-Odesskii Poisson brackets

Algebraic Geometry · Submitted 15 Jul 2022

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We prove that several Feigin-Odesskii Poisson brackets associated with normal elliptic curves in ${\mathbb P}^n$ are compatible if and only if they are contained in a scroll or in a Veronese surface in ${\mathbb P}^5$ (with an exception of one case when $n=3$).

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,443 input tokens.

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We prove that several Feigin-Odesskii Poisson brackets associated with normal elliptic curves in ${\mathbb P}^n$ are compatible if and only if they are contained in a scroll or in a Veronese surface in ${\mathbb P}^5$ (with an exception of one case when $n=3$). In the case $n=3$ we determine the quartic corresponding to the Schouten bracket of two (non-compatible) Poisson brackets associated with normal elliptic curves $E_1$ and $E_2$.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We prove that several Feigin-Odesskii Poisson brackets associated with normal elliptic curves in ${\mathbb P}^n$ are compatible if and only if they are contained in a scroll or in a Veronese surface in ${\mathbb P}^5$ (with an exception of one case when $n=3$).	result 0.99	0.94	0.92	asserted 1.00
s1	In the case $n=3$ we determine the quartic corresponding to the Schouten bracket of two (non-compatible) Poisson brackets associated with normal elliptic curves $E_1$ and $E_2$.	result 0.90	0.94	0.05	asserted 1.00

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Original abstract

We prove that several Feigin-Odesskii Poisson brackets associated with normal elliptic curves in ${\mathbb P}^n$ are compatible if and only if they are contained in a scroll or in a Veronese surface in ${\mathbb P}^5$ (with an exception of one case when $n=3$). In the case $n=3$ we determine the quartic corresponding to the Schouten bracket of two (non-compatible) Poisson brackets associated with normal elliptic curves $E_1$ and $E_2$.

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Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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Conversely, the fact that the FO brackets associated with anticanonical divisors on a scroll or on a Veronese surface in $^5$ are compatible follows from the results of [ref] (see [ref] ).

Elliptic bihamiltonian structures from relative shifted Poisson structures · Proof of Theorem B · 0.97

Cites as background 1 paper

Recall (see [ref] ) that for $x∈^1(L,)∖{0}$ and $s_1,s_2∈x^⊥H^0(L)$, the bivector $Π_x$ is given by [equation] where $MP(s_1,x,s_2)∈H^0(L)/s_1,s_2$ is the Massey product.

Elliptic bihamiltonian structures from relative shifted Poisson structures · Conormal Lie algebras · 0.88

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