# Compatible Feigin-Odesskii Poisson brackets

Nikita Markarian  and  Alexander Polishchuk

###### 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 $E_{2}$.

A.P. is supported in part by the NSF grant DMS-2001224, and within the
framework of the HSE University Basic Research Program and by the Russian
Academic Excellence Project ‘5-100’.

##  1\. Introduction

We work over an algebraically closed field $\mathbf{k}$ of characteristic $0$.

This work is a contribution to the study of a remarkable class of Poisson
brackets on projective spaces associated with elliptic curves, that were
introduced by Feigin and Odesskii (see [3]). Namely, for every normal elliptic
curve $E\subset{\mathbb{P}}^{n}$, there is an associated Poisson bracket
$\Pi_{E}$ defined up to rescaling (we refer to these as FO brackets; sometimes
they are denoted as $q_{n+1,1}(E)$). Setting $L={\mathcal{O}}(1)|_{E}$ one can
identify ${\mathbb{P}}^{n}$ with the projective space of extensions of $L$ by
${\mathcal{O}}$. Then the bracket $\Pi_{E}$ can be defined and studied in
terms of geometry of such extensions on $E$.

Recall that two Poisson structures $\Pi_{1},\Pi_{2}$ on the same space $X$ are
called compatible if every linear combination
$\lambda_{1}\Pi_{1}+\lambda_{2}\Pi_{2}$ is still a Poisson structure. This is
equivalent to the identity $[\Pi_{1},\Pi_{2}]=0$, where we use the Schouten
bracket of bivectors. More generally, one can consider larger linear subspaces
of Poisson bivectors. In the work [7] Odesskii and Wolf gave a construction of
$9$-dimensional subspaces of Poisson structures on ${\mathbb{P}}^{n}$ whose
general member is a bracket $\Pi_{E}$ for some $E\subset{\mathbb{P}}^{n}$. In
[5] this construction was interpreted geometrically in terms of families of
anticanonical divisors on Hirzebruch surfaces, and extended to give new
examples of compatible Poisson structures. In the present work we show that
these constructions are the only way to produce compatible FO brackets of type
$q_{n+1,1}(E)$, with one exception occurring for $n=3$.

First, let us consider the case of FO brackets associated with normal elliptic
curves in ${\mathbb{P}}^{3}$.

Theorem A. For a collection of normal elliptic curves $(E_{i})_{i\in I}$ in
${\mathbb{P}}^{3}={\mathbb{P}}V$, the brackets $(\Pi_{E_{i}})$ are compatible
if and only if either all $E_{i}$ lie on one quadric surface (possibly
singular), or there exists a $3$-dimensional subspace $W\subset S^{2}V^{*}$
such that every $E_{i}$ is the intersection of two quadrics in $W$.

Theorem A has a natural generalization to FO brackets in higher-dimensional
projective spaces. Let us fix $n\geq 4$. With every normal elliptic curve $E$
in ${\mathbb{P}}^{n}$ we can associate a $1$-dimensional family $\SS_{E}$ of
rational normal scrolls $S(r,r)\subset{\mathbb{P}}^{n}$ if $n=2r+1$, with
several exceptional members of type $S(r-1,r+1)$ (resp., $S(r-1,r)$ if
$n=2r$), which are parametrized by points of $\operatorname{Pic}_{2}(E)$, see
Sec. 2.2 for details. In the case of $n=5$, each normal elliptic curve $E$ in
${\mathbb{P}}^{5}$ is contained in four Veronese surfaces, corresponding to
choices of a square root of the line bundle ${\mathcal{O}}(1)|_{E}$ of degree
$6$.

Theorem B. For a collection of normal elliptic curves $(E_{i})_{i\in I}$ in
${\mathbb{P}}^{n}$, where $n\geq 4$, the brackets $(\Pi_{E_{i}})$ are
compatible if and only if

  * •

either the corresponding families $(\SS_{E_{i}})$ have an element in common,

  * •

or $n=5$ and all $E_{i}$ are contained in a Veronese surface
${\mathbb{P}}^{2}\subset{\mathbb{P}}^{5}$.

The family of elliptic curves contained in a scroll is precisely the family of
anticanonical divisors producing compatible Poisson brackets of Odesskii-Wolf
(see [5, Sec. 5.3]). The fact that elliptic curves contained in a Veronese
surface in ${\mathbb{P}}^{5}$ give rise to compatible Poisson brackets was
observed in [5, Ex. 4.6].

Let us call an FO-subspace any linear subspace of compatible Poisson brackets
on ${\mathbb{P}}^{n}$, whose generic point is of the form $\Pi_{E}$ for some
normal elliptic curve $E$. Odesskii-Wolf conjectured that their examples of
$9$ compatible Poisson brackets on ${\mathbb{P}}^{n}$ give maximal linear
subspaces of Poisson brackets. We prove that this is true if $n$ is even,
while for odd $n$ we prove maximality of these subspaces among FO-subspaces.

Corollary C. (a) Given a rational normal scroll $S=S(s,r)$ in
${\mathbb{P}}^{n}$, with $0<s\leq r\leq s+2$, consider the $9$-dimensional FO-
subspace $V_{S}$ of Poisson brackets on ${\mathbb{P}}^{n}$ coming from
anticanonical divisors on $S$. Then $V_{S}$ is a maximal subspace of Poisson
brackets provided $n$ is even (resp., a maximal FO-subspace if $n$ is odd).

(b) The maximal dimension of a linear subspace of Poisson brackets on
${\mathbb{P}}^{n}$, containing some FO bracket $\Pi_{E}$, is $9$ provided $n$
is even and $n\geq 4$. For odd $n$, the maximal dimension of an FO-subspace of
Poisson brackets on ${\mathbb{P}}^{n}$ is $9$, for $n\geq 3$, $n\neq 5$, and
is $10$ for $n=5$.

The construction of compatible Poisson brackets in [5] also produces some
families of compatible FO brackets of type $q_{n,k}(E)$ (these are associated
with higher rank stable bundles on elliptic curves). It would be interesting
to find criteria for compatibility of brackets of type $q_{n,k}(E)$ similar to
our Theorems A and B.

Let us return to FO brackets on ${\mathbb{P}}^{3}$. In the case when
$\Pi_{E_{1}}$ and $\Pi_{E_{2}}$ are not compatible, the Schouten bracket
$[\Pi_{E_{1}},\Pi_{E_{2}}]$ is a section of
${\bigwedge}^{3}T_{{\mathbb{P}}^{3}}\simeq{\mathcal{O}}_{{\mathbb{P}}^{3}}(4)$,
so it gives a quartic hypersurface in ${\mathbb{P}}^{3}$. We will compute this
hypersurface in terms of defining pairs of quadrics for $E_{1}$ and $E_{2}$.
Let ${\mathbb{P}}^{3}={\mathbb{P}}V$, where $V$ is a $4$-dimensional vector
space. Consider a natural linear map

$\Phi:{\bigwedge}^{4}(S^{2}V^{*})\to\det(V^{*})\otimes S^{4}(V^{*})$

that associates with $Q_{1}\wedge Q_{2}\wedge Q_{3}\wedge Q_{4}$ the
polynomial map of degree $4$ with values in $\det(V^{*})$,

$v\mapsto(\partial_{v}Q_{1})\wedge(\partial_{v}Q_{2})\wedge(\partial_{v}Q_{3})\wedge(\partial_{v}Q_{4}).$

On the other hand, the Poisson bivector associated with $Q_{1}=Q_{2}=0$
canonically is an element of $\det(V^{*})\otimes
H^{0}({\mathbb{P}}V,{\bigwedge}^{2}T_{{\mathbb{P}}V})$, hence the Schouten
bracket of two such bivectors is a section of

$(\det V^{*})^{\otimes 2}\otimes{\bigwedge}^{3}T_{{\mathbb{P}}^{3}}\simeq(\det
V^{*})^{\otimes
2}\otimes\det(V)\otimes{\mathcal{O}}_{{\mathbb{P}}V}(4)\simeq\det(V^{*})\otimes{\mathcal{O}}_{{\mathbb{P}}V}(4).$

Theorem D. One has an equality in $\det(V^{*})\otimes S^{4}(V^{*})$,

$[\Pi_{Q_{1}=Q_{2}=0},\Pi_{Q_{3}=Q_{4}=0}]=4\cdot\Phi(Q_{1}\wedge Q_{2}\wedge
Q_{3}\wedge Q_{4}).$

The quartic surface $X_{E_{1},E_{2}}\subset{\mathbb{P}}^{3}$ given by
$[\Pi_{E_{1}},\Pi_{E_{2}}]=0$ can be characterized geometrically as follows.
First, with a normal elliptic curve $E\subset{\mathbb{P}}^{3}$ and a point
$p\in{\mathbb{P}}^{3}$, which is not the vertex of one of the four singular
quadrics through $E$, we can associate a line
$L_{E}(p)\subset{\mathbb{P}}^{3}$ called the polar line of $p$ with respect to
$E$: if $E$ is given by $Q_{1}=Q_{2}=0$, then $L_{E}(p)$ is the intersection
of polar planes to $p$ with respect to $Q_{1}=0$ and $Q_{2}=0$ (it does not
depend on a choice of $Q_{1},Q_{2}$). Now $X_{E_{1},E_{2}}$ is the closure of
the set of points $p$ such that the polars $L_{E_{1}}(p)$ and $L_{E_{2}}(p)$
have a nontrivial intersection (this quartic also contains vertices of
singular quadrics through $E_{1}$ and $E_{2}$). Another definition of the
polar line to a point $p$ with respect to $E$ (for a generic $p$) is the
following. Take two secant lines to $E$, $L_{1}=\overline{p_{1}q_{1}}$ and
$L_{2}=\overline{p_{2}q_{2}}$, passing through $p$ (they coincide with two
lines through $p$ on the unique quadric $Q$ passing through $E$ and $p$). On
each of these two lines $L_{i}$, choose a point $r_{i}$, such that
$(p,p_{i},q_{i},r_{i})$ is a harmonic quadruple (have double ratio $-1$). Then
the polar line $L_{E}(p)$ is the line passing through $r_{1}$ and $r_{2}$.

Acknowledment. This work was done while both authors were staying at the IHES.
We thank this institution for hospitality and excellent working environment.

##  2\. Some facts about FO brackets

###  2.1. Conormal Lie algebras

Recall that for any point $x$ of a smooth Poisson variety $(X,\Pi)$, there is
a natural Lie algebra structure on the space

${\mathfrak{g}}_{\Pi,x}:=\operatorname{ker}(\Pi_{x}:T_{x}^{*}X\to
T_{x}X)\subset T_{x}^{*}X.$

Namely, if we lift
$a,b\in{\mathfrak{g}}_{x}\subset{\mathfrak{m}}_{x}/{\mathfrak{m}}_{x}^{2}$ to
some local functions $\widetilde{a},\widetilde{b}\in{\mathfrak{m}}_{x}$, then
$\\{\widetilde{a},\widetilde{b}\\}\in{\mathfrak{m}}_{x}$ and

$[a,b]:=\\{\widetilde{a},\widetilde{b}\\}\operatorname{mod}{\mathfrak{m}}_{x}^{2}.$

Now let $\Pi=\Pi_{E}$ denote the FO bracket on
${\mathbb{P}}^{n}={\mathbb{P}}\operatorname{Ext}^{1}(L,{\mathcal{O}})$,
associated with a normal elliptic curve $E\subset{\mathbb{P}}^{n}$ (so
$L={\mathcal{O}}(1)|_{E}$). It is known (see [5, Prop. 2.3]) that the rank of
$\Pi_{x}$ is equal to $n+1-\dim\operatorname{End}(V)$, where $V$ is the
extension of $L$ by ${\mathcal{O}}$ corresponding to $x$. The next result
realizes this equality by an isomorphism of natural Lie algebras.

######  Proposition 2.1.1.

For $x\in\operatorname{Ext}^{1}(L,{\mathcal{O}})\setminus\\{0\\}$, which is
the class of an extension

$0\to{\mathcal{O}}\rTo{i}V\rTo{p}L\to 0,$ (2.1)

the conormal Lie algebra ${\mathfrak{g}}_{\Pi,x}$ for the FO-bracket
$\Pi=\Pi_{E}$ is naturally isomorphic to the Lie algebra
$\operatorname{End}(V)/\langle\operatorname{id}\rangle$.

###### Proof.

Recall (see [5, Lem. 2.1]) that for
$x\in\operatorname{Ext}^{1}(L,{\mathcal{O}})\setminus\\{0\\}$ and
$s_{1},s_{2}\in\langle x\rangle^{\perp}\subset H^{0}(L)$, the bivector
$\Pi_{x}$ is given by

$s_{1}\wedge s_{2}\mapsto\langle MP(s_{1},x,s_{2}),x\rangle,$

where $MP(s_{1},x,s_{2})\in H^{0}(L)/\langle s_{1},s_{2}\rangle$ is the Massey
product. Note that we can identify $\langle x\rangle^{\perp}\subset H^{0}(L)$
with the kernel of the connecting homomorphism $H^{0}(L)\to
H^{1}({\mathcal{O}})$ associated with (2.1), hence, $\langle x\rangle^{\perp}$
consists of sections liftable to $H^{0}(V)$. By definition,
$MP(s_{1},x,s_{2})$ is the $\langle s_{1},s_{2}\rangle$-coset represented by
$\widetilde{s}_{1}(\widetilde{s}_{2})$, where $\widetilde{s}_{1}:V\to L$ is a
map extending $s_{1}:{\mathcal{O}}\to L$ and $\widetilde{s}_{2}\in H^{0}(V)$
is a lifting of $s_{2}$.

Let us consider the map

$\operatorname{End}(V)/\langle\operatorname{id}\rangle\to H^{0}(L):A\mapsto
p\circ A\circ i.$

We claim that this map is injective. Indeed, if $p\circ A\circ i=0$ for
$A\in\operatorname{End}(V)$, then $A\circ i=c\cdot i$ for some constant $c$.
Hence, $(A-c\operatorname{id})\circ i=0$, so $A-c\operatorname{id}$ factors
through a map $L\to V$. But any such map is zero since the extension (2.1)
does not split, so $A\in\langle\operatorname{id}\rangle$, which proves our
claim.

Now suppose $s_{1}=p\circ A\circ i$, for some $A\in\operatorname{End}(V)$.
Then we can take $\widetilde{s}_{1}=p\circ A$. Hence,
$\widetilde{s}_{1}(\widetilde{s}_{2})=p(A(\widetilde{s}_{2}))$ which lies in
$\langle x\rangle^{\perp}$ since it is liftable to $H^{0}(V)$. Thus, we have
an inclusion

$\operatorname{End}(V)/\langle\operatorname{id}\rangle\hookrightarrow\operatorname{ker}(\Pi_{x})\subset\langle
x\rangle^{\perp}.$

Since the dimensions are equal, in fact, we get an identification of vector
spaces

$\operatorname{End}(V)/\langle\operatorname{id}\rangle\rTo{\sim}{\mathfrak{g}}_{\Pi,x}=\operatorname{ker}(\Pi_{x}).$

It remains to check compatibility of the brackets. Let us fix a pair of
endomorphisms $A,A^{\prime}\in\operatorname{End}(V)$, and let $s=p\circ A\circ
i$ and $s^{\prime}=p\circ A^{\prime}\circ i$ be the corresponding elements of
$\operatorname{ker}(\Pi_{x})$. Let us also fix a vector
$v\in\operatorname{Ext}^{1}(L,{\mathcal{O}})\setminus\langle x\rangle$,
representing a nontrivial tangent direction to $\langle x\rangle$ in the
projective space. Let us consider the corresponding line $x(t)=x+tv$ in
$\operatorname{Ext}^{1}(L,{\mathcal{O}})$, and the induced line $\langle
x(t)\rangle$ in the projective space. In order to compute
$\langle[s,s^{\prime}],v\rangle$ we have to extend $s$ and $s^{\prime}$ to
sections $s(t)$, $s^{\prime}(t)$ of the cotangent bundle along the line
$\langle x(t)\rangle$. Then by definition, we have

$\langle[s,s^{\prime}],v\rangle=\frac{d}{dt}\langle\Pi,s(t)\wedge
s^{\prime}(t)\rangle|_{t=0}.$

Furthermore, it is enough to work over dual numbers, where $t^{2}=0$. Thus, we
can pick any elements $u,u^{\prime}\in H^{0}(L)$, such that

$\langle x,u\rangle+\langle v,s\rangle=0,\ \langle x,u^{\prime}\rangle+\langle
v,s^{\prime}\rangle=0,$

and set $s(t)=s+tu$, $s^{\prime}(t)=s^{\prime}+tu^{\prime}$. Then

$\langle x(t),s(t)\rangle=\langle x+tv,s+tu\rangle=0,\ \ \langle
x(t),s^{\prime}(t)\rangle=0,$

so $s(t)$ and $s^{\prime}(t)$ are in sections of the cotangent bundle. Now we
have to calculate the derivative at $0$ (i.e., the coefficient of $t$)of

$\langle\Pi,s(t)\wedge s^{\prime}(t)\rangle=\langle
MP(s+tu,x+tv,s^{\prime}+tu^{\prime}),x+tv\rangle,$

To calculate this we use Cech representatives. We cover $E$ with two affine
opens $U_{1}$ and $U_{2}$, set $U_{12}=U_{1}\cap U_{2}$, and realize $x$ and
$v$ by Cech $1$-cocycles $x\in L^{-1}(U_{12})$, $v\in L^{-1}(U_{12})$. The
extension $V=V_{x}$ of $L$ by ${\mathcal{O}}$ associated with $x$ is equipped
with splitting $\sigma_{a}:L\to V$ over each $U_{a}$, $a=1,2$, such that over
$U_{12}$ we have

$(\sigma_{2}-\sigma_{1})(\phi)=i(x\cdot\phi),$

where $\phi\in L(U_{12})$.

If we have a section $s\in\langle x\rangle^{\perp}\subset H^{0}(L)$, then
there exist $f_{a}\in{\mathcal{O}}(U_{a})$, $a=1,2$, such that

$x\cdot s=f_{1}-f_{2}.$ (2.2)

Then we can extend $s$ to a section
$\alpha=\alpha(s,f_{\bullet}):{\mathcal{O}}\to V$ and to a morphism
$\beta=\beta(s,f_{\bullet}):V\to L$ such that $p(\alpha)=s=\beta\circ i$, and

$\alpha|_{U_{a}}=i(f_{a})+\sigma_{a}(s),\ \
\beta|_{U_{a}}(\sigma_{a}(\phi))=-f_{a}\cdot\phi.$

It is easy to see that an operator $A\in\operatorname{End}(V)$ with
$\operatorname{tr}(A)=0$, is determined by the section $s=p\circ A\circ i$,
together with functions $f_{a}\in{\mathcal{O}}(U_{a})$ and sections $A_{a}\in
L^{-1}(U_{a})$, $a=1,2$, satisfying (2.2) and

$A_{2}-A_{1}=(f_{1}+f_{2})\cdot x.$ (2.3)

Namely, the operator $A:V\to V$ is given over $U_{i}$ by

$Ai(1)=f_{a}+\sigma_{a}(s),\ \
A\sigma_{a}(\phi)=i(A_{a}\cdot\phi)-f_{a}\cdot\sigma_{a}(\phi),$

for $\phi\in L(U_{a})$.

Next, we need to consider the extension $V_{x+tv}$ of $L$ by ${\mathcal{O}}$
corresponding to $x+tv$. We denote by $\sigma_{a}(t):L\to V_{x+tv}$ the
splittings over $U_{a}$, $a=1,2$, such that

$(\sigma_{2}(t)-\sigma_{1}(t))(\phi)=i((x+tv)\cdot\phi).$

We start with two traceless operators
$A,A^{\prime}\in\operatorname{End}_{0}(V)$ and consider the corresponding
sections $s,s^{\prime}\in\langle x\rangle^{\perp}\subset H^{0}(L)$. Thus, we
can assume that we have the corresponding functions
$f_{a},f^{\prime}_{a}\in{\mathcal{O}}(U_{a})$ and sections
$A_{a},A^{\prime}_{a}\in L^{-1}(U_{a})$ satisfying (2.2), (2.3) and the
similar relations involving $s^{\prime}$, $f^{\prime}_{a}$ and
$A^{\prime}_{a}$. We also choose functions
$g_{a},g^{\prime}_{a}\in{\mathcal{O}}(U_{a})$ such that

$v\cdot s+x\cdot u=g_{1}-g_{2},\ \ v\cdot s^{\prime}+x\cdot
u^{\prime}=g^{\prime}_{1}-g^{\prime}_{2}.$ (2.4)

Then we have the following deformed version of (2.2) over $U_{12}$:

$(x+tv)\cdot(s+tu)=(f_{1}+g_{1}t)-(f_{2}+g_{2}t).$

By definition, the Massey product $MP(s+tu,x+tv,s^{\prime}+tu^{\prime})$ is
the class of the composition $\beta\circ\alpha^{\prime}$, where
$\alpha^{\prime}:{\mathcal{O}}\to V_{x+tv}$ is the global section projecting
to $s^{\prime}+tu^{\prime}$, while $b:V_{x+tv}\to L$ is the morphism extending
$s+tu$. As we have seen before, we can define $\alpha^{\prime}$ and $\beta$ so
that over $U_{a}$ we have

$a^{\prime}=i(f^{\prime}_{a}+tg^{\prime}_{a})+\sigma_{a}(s^{\prime}+tu^{\prime}),$
$\beta(i(f)+\sigma_{a}(\phi))=f(s+tu)-(f_{a}+tg_{a})\phi.$

Hence,

$\displaystyle
MP=MP(s+tu,x+tv,s^{\prime}+tu^{\prime})=(s+tu)(f^{\prime}_{a}+tg^{\prime}_{a})-(s^{\prime}+tu^{\prime})(f_{a}+tg_{a})=$
$\displaystyle
sf^{\prime}_{a}-s^{\prime}f_{a}+t(sg^{\prime}_{a}+uf^{\prime}_{a}-s^{\prime}g_{a}-u^{\prime}f_{a}),$

and therefore, considering the coefficient of $t$ in $\langle MP,x+tv\rangle$
we get

$[s,s^{\prime}]=v\cdot(sf^{\prime}_{a}-s^{\prime}f_{a})+x\cdot(sg^{\prime}_{a}+uf^{\prime}_{a}-s^{\prime}g_{a}-u^{\prime}f_{a})\operatorname{mod}\operatorname{im}(\delta),$

where $a=1$ or $2$, and we view the right-hand side as a Cech cohomology class
$H^{1}({\mathcal{O}})\simeq\mathbf{k}$ (the answers for $a=1$ and $a=2$ differ
by a cocycle). Therefore, we have

$2[s,s^{\prime}]=v\cdot[s(f^{\prime}_{1}+f^{\prime}_{2})-s^{\prime}(f_{1}+f_{2})]+C\operatorname{mod}\operatorname{im}(\delta),$

where

$C=x\cdot[s(g^{\prime}_{1}+g^{\prime}_{2})+u(f^{\prime}_{1}+f^{\prime}_{2})-s^{\prime}(g_{1}+g_{2})-u^{\prime}(f_{1}+f_{2})]\operatorname{mod}\operatorname{im}(\delta).$

Note that $u$ and $u^{\prime}$ are global sections, while $x(f_{1}+f_{2})$ and
$x(f^{\prime}_{1}+f^{\prime}_{2})$ are coboundaries due to the relation (2.3).
Hence,

$C=xs(g^{\prime}_{1}+g^{\prime}_{2})-xs^{\prime}(g_{1}+g_{2})=(f_{1}-f_{2})(g^{\prime}_{1}+g^{\prime}_{2})-(f^{\prime}_{1}-f^{\prime}_{2})(g_{1}+g_{2})\equiv
f_{1}g^{\prime}_{2}-f_{2}g^{\prime}_{1}-f^{\prime}_{1}g_{2}+f^{\prime}_{2}g_{1}\operatorname{mod}\operatorname{im}(\delta),$

where we removed the coboundary terms $f_{1}g^{\prime}_{1}$, etc.

On the other hand, multiplying the first of the relations (2.4) with
$f^{\prime}_{1}+f^{\prime}_{2}$, we get

$f^{\prime}_{2}g_{1}-f^{\prime}_{1}g_{2}\equiv(f^{\prime}_{1}+f^{\prime}_{2})(g_{1}-g_{2})=(f^{\prime}_{1}+f^{\prime}_{2})vs+(f^{\prime}_{1}+f^{\prime}_{2})xu\equiv(f^{\prime}_{1}+f^{\prime}_{2})vs\operatorname{mod}\operatorname{im}(\delta).$

Similarly,

$f_{2}g^{\prime}_{1}-f_{1}g^{\prime}_{2}\equiv(f_{1}+f_{2})vs^{\prime}\operatorname{mod}\operatorname{im}(\delta).$

Hence, we obtain

$C\equiv v\cdot[s(f^{\prime}_{1}+f^{\prime}_{2})-s^{\prime}(f_{1}+f_{2})],$

which gives

$\langle[s,s^{\prime}],v\rangle=v\cdot[s(f^{\prime}_{1}+f^{\prime}_{2})-s^{\prime}(f_{1}+f_{2})]\operatorname{mod}\operatorname{im}(\delta).$

Finally, using (2.2), we get

$(f_{1}-f_{2})\cdot s^{\prime}=x\cdot s\cdot
s^{\prime}=(f^{\prime}_{1}-f^{\prime}_{2})\cdot s,$

or equivalently,

$sf^{\prime}_{1}-s^{\prime}f_{1}=sf^{\prime}_{2}-s^{\prime}f_{2}.$

Hence, we can rewrite the above formula as

$\langle[s,s^{\prime}],v\rangle=2v\cdot(sf^{\prime}_{1}-s^{\prime}f_{1}).$

It remains to compare this with $p(AA^{\prime}-A^{\prime}A)i(1)$. We can
compute this over $U_{1}$:

$p(AA^{\prime}-A^{\prime}A)i(1)=pA[i(f^{\prime}_{1})+\sigma_{1}(s^{\prime})]-pA^{\prime}[i(f_{1})+\sigma_{1}(s)]=2(f^{\prime}_{1}s-f_{1}s^{\prime}).$

Comparing this with the formula above, we get

$\langle
p(AA^{\prime}-A^{\prime}A)i(1),v\rangle=\langle[s,s^{\prime}],v\rangle$

which proves our assertion. ∎

######  Remark 2.1.2.

Proposition 2.1.1 is a part of a broader picture, involving the notion of a
symplectic groupoid (see [13]). Given a symplectic groupoid $M$ with the space
of objects $X$ (in the category of smooth schemes), then $X$ has a natural
Poisson structure. In this situation one can also consider the algebraic stack
${\mathcal{X}}$ associated with the groupoid $M$. Then the Lie algebra of
automorphisms in $M$ of an object $x\in X$ (or equivalently, of the
corresponding point of ${\mathcal{X}}$) is naturally identified with the
coisotropic Lie algebra at $x\in X$ of the Poisson structure on $X$ (see [6,
ch. 2, Lem. 1.2]).

In the case of the Feigin–Odesskii Poisson structure on
$X={\mathbb{P}}\operatorname{Ext}^{1}(L,{\mathcal{O}})$, the symplectic
groupoid $M$ can be constructed as follows. We have a natural map
$f:X\to\operatorname{Bun}_{L}$, where $\operatorname{Bun}_{L}$ is the stack of
rank $2$ bundles on $E$ with the determinant $L$, and $f$ associates the
bundle $V$ with an extension (2.1). Then we define $M$ to be the $2$-fibered
product

$M:=X\times_{\operatorname{Bun}_{L}}X.$

The first part of the proof of Prop. 2.1.1 shows that the map $f$ induces a
surjection on tangent spaces, hence, $M$ is a smooth algebraic space. From the
fibered product structure we get a groupoid structure on $M$. The
corresponding stack ${\mathcal{X}}$ is equipped with a fully faithful map
${\mathcal{X}}\to\operatorname{Bun}_{L}$ (see [12, Lem. 93.16.1], so
automorphisms of points are the same. One can show that in fact $M$ has a
structure of a symplectic groupoid that induces the Feigin–Odesskii Poisson
structure on $X$ (the details will appear elsewhere). This gives a more
conceptual proof of the identification of Lie algebras in Prop. 2.1.1.

It is well known that the FO-bracket $\Pi_{E}$ vanishes at any $p\in E$.
Hence, in this case we get a Lie algebra structure on
$T_{p}^{*}{\mathbb{P}}^{n}$.

######  Corollary 2.1.3.

For $p\in E\subset{\mathbb{P}}^{n}$, the conormal Lie algebra
${\mathfrak{g}}=T_{p}^{*}{\mathbb{P}}^{n}$ admits a basis
$x_{1},\ldots,x_{n-1},y$, such that $\langle
x_{1},\ldots,x_{n-1}\rangle=(T_{x}E)^{\perp}$ is an abelian subalgebra, and
$[y,x_{i}]=x_{i}$.

###### Proof.

In this case $V={\mathcal{O}}(p)\oplus L(-p)$ and the algebra
$\operatorname{End}(V)/\langle\operatorname{id}\rangle$ has an abelian
subalgebra $A$ corresponding to endomorphisms of the form
$V\to{\mathcal{O}}(p)\to L(-p)\to V$. The $1$-dimensional complement to $A$ is
spanned by the idempotent $e$ corresponding to the summand $L(-p)$, and for
any $a\in A$ we have $ea=a$, $ae=0$, so $[e,a]=a$. ∎

Using this description of the conormal Lie algebras, we get the following
constraints on Poisson brackets compatible with $\Pi_{E}$.

######  Proposition 2.1.4.

Let $E\subset{\mathbb{P}}^{n}$ be a normal elliptic curve, and let $\Pi$ be a
Poisson bracket on ${\mathbb{P}}^{n}$. Assume that $[\Pi_{E},\Pi]=0$. Then for
every $p\in E$, one has $\Pi|_{p}=v_{1}\wedge v_{2}$, where $v_{1}\in T_{p}E$.

###### Proof.

We will calculate $[\Pi_{E},\Pi]|_{p}$ for $p\in E$, and then equate it to
zero. We have $\Pi_{E}|_{p}=0$, and the linear part of $\Pi_{E}$ is

$\Pi_{E}^{lin}:=\sum_{i}x_{i}\partial_{x_{i}}\wedge\partial_{y},$

where $x_{1},\ldots,x_{n-1},y$ is a basis of $T_{p}{\mathbb{P}}^{n}$ chosen as
in Corollary 2.1.3. The bracket $[\Pi_{E},\Pi]|_{p}=[\Pi_{E}^{lin},\Pi]|_{p}$
depends only on $\Pi|_{p}$. We have

$[\Pi_{E}^{lin},\partial_{x_{i}}\wedge\partial_{y}]=0,$
$[\Pi_{E}^{lin},\partial_{x_{i}}\wedge\partial_{x_{j}}]=\pm
2\partial_{x_{i}}\wedge\partial_{x_{j}}\wedge\partial_{y}.$

Hence, $[\Pi_{E},\Pi]|_{p}=0$ if and only if $\Pi|_{p}=v\wedge\partial_{y}$,
for some $v\in T_{p}{\mathbb{P}}^{n}$. Since
$\langle\partial_{y}\rangle=T_{p}E$, our assertion follows. ∎

###  2.2. Secant variety of $E$ and its fibration in scrolls

Assume $n\geq 4$, and let $E\subset{\mathbb{P}}^{n}$ be a normal elliptic
curve, $L={\mathcal{O}}(1)|_{E}$. Consider the ${\mathbb{P}}^{1}$-bundle
$p:X\to E^{[2]}$ over the symmetric square of $E$, with the fiber
${\mathbb{P}}H^{0}(L|_{D})^{*}\subset{\mathbb{P}}H^{0}(E,L)^{*}={\mathbb{P}}^{n}$
over a point $D\in E^{[2]}$. We have $X={\mathbb{P}}({\mathcal{V}})$, for the
following vector bundle ${\mathcal{V}}$ on $E^{[2]}$. Consider the natural map
$\pi:E\times E\to E^{[2]}$, then ${\mathcal{V}}=(\pi_{*}p_{1}^{*}L)^{\vee}$,
where $p_{1}:E^{2}\to E$ is the projection to the first factor.

We have a well defined morphism $\phi:X\to{\mathbb{P}}^{n}$, embedding
linearly each line $p^{-1}(D)\subset X$ into ${\mathbb{P}}^{n}$, so that its
image is precisely the corresponding secant line corresponding to $D$. More
precisely, the natural morphism $H^{0}(L)\otimes{\mathcal{O}}_{E^{2}}\to
p_{1}^{*}L$ induces a surjection
$H^{0}(L)\to\pi_{*}p_{1}^{*}L={\mathcal{V}}^{\vee}$. Hence, we get a closed
embedding $X={\mathbb{P}}({\mathcal{V}})\hookrightarrow
E^{[2]}\times{\mathbb{P}}H^{0}(L)^{*}$, and $\phi$ is obtained by the second
projection.

######  Lemma 2.2.1.

Assume that $n\geq 4$. The map $\phi:X\to{\mathbb{P}}^{n}$ induces a
surjective birational morphism $X\to\operatorname{Sec}^{2}(E)$ which is an
isomorphism over $\operatorname{Sec}^{2}(E)\setminus E$.

###### Proof.

This is a well known consequence of the fact that $L$ “separates $4$ points on
$E$” since $\deg(L)\geq 5$ (see Terracini’s Lemma as stated in [1, Lem. 1.2,
Lem. 1.4]). ∎

Next, we consider the Abel-Jacobi map
$a:E^{[2]}\to\operatorname{Pic}_{2}(E):D\mapsto{\mathcal{O}}(D)$. For each
$M\in\operatorname{Pic}_{2}(E)$, we set $S_{M}:=\pi^{-1}(a^{-1}(M))\subset X$.
Note that $\phi(S_{M})\subset{\mathbb{P}}^{n}$ is the union of secant lines
$\overline{pp^{\prime}}$ over all pairs $p,p^{\prime}\in E$ such that
${\mathcal{O}}_{E}(p+p^{\prime})\simeq M$.

We refer to [2] or [10] for background on rational normal scrolls.

######  Lemma 2.2.2.

Assume that $n\geq 4$. Then the restriction
$\phi|_{S_{M}}:S_{M}\to{\mathbb{P}}^{n}$ is an embedding. If $n=2r$, then
$\phi(S_{M})$ is the scroll of type $S(r-1,r)$. If $n=2r+1$, then
$\phi(S_{M})$ is the scroll of type $S(r,r)$ for $M$ such that
$M^{r+1}\not\simeq L$, and it is of type $S(r-1,r+1)$ for $M$ such that
$M^{r+1}\simeq L$.

###### Proof.

Let us compute the restriction of the bundle ${\mathcal{V}}$ to the projective
line $a^{-1}(M)\subset E^{[2]}$. By the base change formula,

${\mathcal{V}}|_{a^{-1}(M)}\simeq\pi_{M*}(p_{1}^{*}L|_{E_{M}}),$

where $E_{M}\subset E^{2}$ is the preimage of $a^{-1}(M)$ under $\pi:E^{2}\to
E^{[2]}$, $\pi_{M}:E_{M}\to a^{-1}(M)$ is the projection. Thus,
$E_{M}=\\{(x,y)\in E^{2}\ |\ {\mathcal{O}}(x+y)\simeq M\\}$. Note that the
projection $p_{1}:E_{M}\to E$ is an isomorphism, and the composition
$\pi_{M}(p_{1}|_{E_{M}})^{-1}:E\to a^{-1}(M)$ is the double cover that can be
identified with the map $f:E\to{\mathbb{P}}^{1}$ given by the linear system
$|M|$. We want to compute the splitting type of $f_{*}L$.

Assume first that $n=2r$, so $\deg(L)=2r+1$. We claim that in this case
$f_{*}L\simeq{\mathcal{O}}(r-1)\oplus{\mathcal{O}}(r)$. Indeed, this follows
from the fact that $\deg(f_{*}L)=\deg(L)-2=2r-1$ and from the vanishing
$H^{0}(f_{*}L(-r-1))=H^{0}(L\otimes M^{-r-1})=0$ (since $\deg(L\otimes
M^{-r-1})=-1$).

Now assume that $n=2r+1$, so $\deg(L)=2r+2$. If $L\not\simeq M^{r+1}$ then
$H^{0}(f_{*}L(-r-1))=H^{0}(L\otimes M^{-r-1})=0$, so
$f_{*}L\simeq{\mathcal{O}}(r)\oplus{\mathcal{O}}(r)$. On the other hand, if
$L\simeq M^{r+1}$, then $H^{0}(f_{*}(-r-1))$ is $1$-dimensional, so
$f_{*}L\simeq{\mathcal{O}}(r-1)\oplus{\mathcal{O}}(r+1)$.

This proves that $S_{M}$ is isomorphic to the projectivization of the rank 2
vector bundle over $a^{-1}(M)\simeq{\mathbb{P}}^{1}$ of one of the types
${\mathcal{O}}(n_{1})\oplus{\mathcal{O}}(n_{2})$ described above. The fact
that the corresponding morphism $S_{M}\to{\mathbb{P}}^{n}$ is an embedding is
the standard fact (using that $n_{1}>0$ and $n_{2}>0$, which is the case since
$n\geq 4$). ∎

######  Corollary 2.2.3.

Assume that $n\geq 4$. There is a well defined morphism
$\operatorname{Sec}^{2}(E)\setminus E\to\operatorname{Pic}_{2}(E)\simeq E$,
whose fibers are of the form $\phi(S_{M})\setminus E$, where $\phi(S_{M})$ is
the scroll of type $S(r-1,r)$ for $n=2r$, or of type $S(r,r)$ or $S(r-1,r+1)$
for $n=2r+1$.

###  2.3. Rank $2$ locus

Fix a normal elliptic curve $E\subset{\mathbb{P}}^{n}$, where $n\geq 4$, and
let $\Pi_{E}$ be the corresponding FO-bracket on ${\mathbb{P}}^{n}$. We want
to describe the locus of points in ${\mathbb{P}}^{n}$ where the rank of
$\Pi_{E}$ is $\leq 2$.

Let $L:={\mathcal{O}}(1)|_{E}$, so we can identify the embedding with
$E\to{\mathbb{P}}H^{0}(L)^{*}\simeq{\mathbb{P}}\operatorname{Ext}^{1}(L,{\mathcal{O}})$.
We will use the following well known fact (see [5, Prop. 2.3]): let
$x\in\operatorname{Ext}^{1}(L,{\mathcal{O}})\setminus\\{0\\}$ be the class of
an extension (2.1). Then the rank of $(\Pi_{E})_{x}$ is equal to
$\deg(L)-\dim\operatorname{End}(V)$.

In the case $n=5$, i.e., $L:={\mathcal{O}}(1)|_{E}$ has degree $6$, for each
$M\in\operatorname{Pic}_{3}(E)$ such that $M^{2}\simeq L$ (there are four such
$M$), we define a Veronese surface $P_{M}\subset{\mathbb{P}}^{n}$ as follows.
We observe that the natural map

$S^{2}H^{0}(E,M)\to H^{0}(E,M^{2})\simeq H^{0}(E,L)$

is an isomorphism, and define $P_{M}$ as the image of the Veronese embedding

${\mathbb{P}}H^{0}(M)^{*}\to{\mathbb{P}}H^{0}(S^{2}H^{0}(M)^{*})\simeq{\mathbb{P}}H^{0}(L)^{*}={\mathbb{P}}^{n}.$

######  Proposition 2.3.1.

(i) Assume $n=4$ or $n\geq 6$. Then for $x\in{\mathbb{P}}^{n}$, one has
$\operatorname{rk}(\Pi_{E})_{x}\leq 2$ if and only if
$x\in\operatorname{Sec}^{2}(E)$.

(ii) Assume $n=5$. Then for for $x\in{\mathbb{P}}^{n}$, one has
$\operatorname{rk}(\Pi_{E})_{x}\leq 2$ if and only if
$x\in\operatorname{Sec}^{2}(E)$ or $x\in P_{M}\subset{\mathbb{P}}^{5}$, for
some $M\in\operatorname{Pic}_{3}(E)$ such that $M^{2}\simeq L$.

###### Proof.

(i) Let $\Pi=\Pi_{E}$. By [5, Prop. 2.3], we have
$\operatorname{rk}\Pi_{x}\leq 2$ if and only if
$\dim\operatorname{End}(V)\geq\deg(L)-2$, where $\deg(L)=n+1$. Assume first
that $n\geq 6$. Then we get $\dim\operatorname{End}(V)\geq 5$, so $V$ cannot
be semistable. Hence, $V$ is isomorphic to a direct sum of line bundles:
$V=L_{1}\oplus L_{2}$, where $\deg(L_{1})<\deg(L_{2})$. Furthermore, we have

$\dim\operatorname{End}(V)=2+\deg(L_{2})-\deg(L_{1})\geq\deg(L)-2=\deg(L_{1})+\deg(L_{2})-2,$

so $\deg(L_{1})\leq 2$. The composed map ${\mathcal{O}}\to V\to L_{1}$ cannot
be zero, so it vanishes on an effective divisor $D$ of degree $\leq 2$. Since
$V$ is a nontrivial extension, we have $\deg(D)\geq 1$. If $\deg(D)=1$ then
our extension splits under ${\mathcal{O}}\to{\mathcal{O}}(p)$ for some point,
so $x$ lies on the elliptic curve $E\subset{\mathbb{P}}^{n}$. If $\deg(D)=2$
then our extension splits under ${\mathcal{O}}\to{\mathcal{O}}(D)$, so it lies
on the chord of $E$ in ${\mathbb{P}}^{n}$ corresponding to $D$.

Conversely, if $x\in\operatorname{Sec}^{2}(E)$, then there exists an effective
degree $2$ divisor $D$ on $E$ such that the extension associated with $x$
splits under ${\mathcal{O}}\to{\mathcal{O}}(D)$. If $x\in E$ then the
extension splits already under ${\mathcal{O}}\to{\mathcal{O}}(q)$ for some
$q\in E$. Then $V\simeq{\mathcal{O}}(q)\oplus L(-q)$, so
$\dim\operatorname{End}(V)=\deg(L)$, which means that $\Pi_{x}=0$. Otherwise,
if $x\in\operatorname{Sec}^{2}(E)\setminus E$, then we get an exact sequence

$0\to L(-D)\to V\to{\mathcal{O}}(D)\to 0.$

Since $\deg(L)\geq 5$, we have $\deg(L(-D))=3>2=\deg({\mathcal{O}}(D))$, so
$V\simeq{\mathcal{O}}(D)\oplus L(D)$. Hence,
$\deg\operatorname{End}(V)=\deg(L)-2$ and $\operatorname{rk}\Pi_{x}=2$.

Assume now that $n=4$. Then the extensions $V$ corresponding points where
$\operatorname{rk}\Pi_{x}\leq 2$ satisfy $\dim\operatorname{End}(V)\geq 3$.
Since $\deg(V)=5$, this means that $V$ is unstable, so $V=L_{1}\oplus L_{2}$
where $(\deg(L_{1}),\deg(L_{2}))$ is either $(1,4)$ or $(2,3)$. We finish the
proof in the same way as before.

(ii) Now the condition on $V$ (which has degree $6$) is that
$\dim\operatorname{End}(V)\geq 4$. This means that either $V$ is unstable, or
$V=M\oplus M$, for some line bundle $M$ of degree $3$, such that $M^{2}\simeq
L$. In the former case, we use the same argument as in part (i) to show that
the correspond point lies on $\operatorname{Sec}^{2}(C)$. We claim that
extensions with $V\simeq M\oplus M$ correspond to points of $P_{M}\setminus
E$, for the Veronese surface $P_{M}\subset{\mathbb{P}}H^{0}(L)^{*}$. Indeed,
an extension of the form

$0\to{\mathcal{O}}\to M\oplus M\rTo{p}L\to 0$

corresponds to a base point free pencil $\langle s_{1},s_{2}\rangle\subset
H^{0}(M)$, which corresponds to a point $\ell:=\langle
s_{1},s_{2}\rangle^{\perp}\subset H^{0}(M)^{*}$ of the projective plane
${\mathbb{P}}H^{0}(M)^{*}$. The line in ${\mathbb{P}}H^{0}(L)^{*}$
corresponding to our extension is given by the hyperplane $H\subset H^{0}(L)$,
obtained as the image of the map $H^{0}(M)^{2}\to H^{0}(L)$ induced by
$p:M\oplus M\to L$. In other words,

$H=s_{1}\cdot H^{0}(M)+s_{2}\cdot H^{0}(M)\subset H^{0}(L).$

It remains to observe that $H$ is exactly the space of quadrics on
${\mathbb{P}}H^{0}(M)^{*}$ vanishing at the point $\ell$. Hence, it is exactly
the image of $\ell$ under the Veronese embedding
${\mathbb{P}}H^{0}(M)^{*}\to{\mathbb{P}}S^{2}H^{0}(M)^{*}\simeq{\mathbb{P}}H^{0}(L)^{*}$.
∎

######  Lemma 2.3.2.

Assume $n\geq 4$, and let $x\in\operatorname{Sec}^{2}(E)\setminus E$. Let
$D\subset E$ be the corresponding effective divisor of degree $2$. Then we can
view $x$ as an element of the projective line
${\mathbb{P}}H^{0}(L^{-1}(D)|_{D})\subset{\mathbb{P}}H^{1}(L^{-1})$, and its
lifting $\widetilde{x}\in H^{0}(L^{-1}(D)|_{D})$ gives a trivialization of
$L^{-1}(D)|_{D}$. The tangent space to $\operatorname{Sec}^{2}(E)$ at $x$ can
be identified with $H^{0}(L^{-1}(2D)|_{2D})/\langle\widetilde{x}\rangle$, so
that the tangent map to the morphism $\operatorname{Sec}^{2}(E)\setminus
E\to\operatorname{Pic}_{2}(E)$ is given by the composition

$H^{0}(L^{-1}(2D)|_{2D})/\langle\widetilde{x}\rangle\to
H^{0}(L^{-1}(2D)|_{D})\rTo{\widetilde{x}^{-1}}H^{0}({\mathcal{O}}(D)|_{D})\to
H^{1}({\mathcal{O}}).$

###### Proof.

This is a standard calculation related to Terracini Lemma (see [1, Sec. 1]). ∎

Recall that for $n\geq 4$, the secant variety $\operatorname{Sec}^{2}(E)$ has
a fibration in scrolls $S_{M}$ over $E$, described in Sec. 2.2.

######  Lemma 2.3.3.

(i) Assume $n\geq 4$. For $x\in\operatorname{Sec}^{2}(E)\setminus E$, one has

$\operatorname{im}((\Pi_{E})_{x}:T^{*}_{x}{\mathbb{P}}^{n}\to
T_{x}{\mathbb{P}}^{n})=T_{x}S_{M},$

where $S_{M}$ is the scroll associated with $M\in\operatorname{Pic}_{2}(E)$,
passing through $x$.

(ii) Assume $n=3$. Let $\Sigma_{E}\subset{\mathbb{P}}^{3}$ denote the set of
four vertices of singular quadrics passing through $E$. For
$x\in{\mathbb{P}}^{3}\setminus(E\cup\Sigma_{E})$, one has

$\operatorname{im}((\Pi_{E})_{x}:T^{*}_{x}{\mathbb{P}}^{n}\to
T_{x}{\mathbb{P}}^{n})=T_{x}Q,$

where $Q$ is the unique smooth quadric passing through $E$ and $x$.

###### Proof.

(i) Set $\Pi=\Pi_{E}$. We have
$\operatorname{im}(\Pi_{x})=\operatorname{ker}(\Pi_{x})^{\perp}$, and as we
have seen in the proof of Proposition 2.1.1, $\operatorname{ker}(\Pi_{x})$ is
precisely the image of the map

$\operatorname{End}(V)/\langle\operatorname{id}\rangle\to\langle
x\rangle^{\perp}\subset H^{0}(L):A\mapsto p\circ A\circ i,$ (2.5)

where $x$ corresponds to an extension (2.1).

As we have seen in the proof of Proposition 2.3.1,
$x\in\operatorname{Sec}^{2}(E)\setminus E$ if and only if
$V\simeq{\mathcal{O}}(D)\oplus L(-D)$, where $D$ is an effective divisor of
degree $2$ and the embedding $i:{\mathcal{O}}\to{\mathcal{O}}(D)\oplus L(-D)$
is given by $(1,s)$, for some section $s\in H^{0}(L(-D))$ not vanishing on
$D$. The projection $p:{\mathcal{O}}(D)\oplus L(-D)\to L$ is given by
$(s,-1)$. Thus, $\operatorname{End}(V)/\langle\operatorname{id}\rangle$ is
generated by the idempotent $e$ corresponding to the factor
${\mathcal{O}}(D)$, and by nilpotent endomorphisms induced by maps
$t:{\mathcal{O}}(D)\to L(-D)$. The image of $e$ under the embedding (2.5) is
$s\in H^{0}(L(-D))\subset H^{0}(L)$, while the image of $t\in H^{0}(L(-2D))$
is $-t\in H^{0}(L(-2D))\subset H^{0}(L)$. In other words,
$\operatorname{ker}(\Pi_{x})$ is identified with the preimage of the line $L$
spanned by $s|_{D}\in H^{0}(L(-D)|_{D})\subset H^{0}(L|_{2D})$ under the
projection $H^{0}(L)\to H^{0}(L|_{2D})$. It follows that
$\operatorname{im}(\Pi_{x})=\operatorname{ker}(\Pi_{x})^{\perp}$ is the image
of $L^{\perp}\in H^{0}(L^{-1}(2D)|_{2D})$ under the natural homomorphism
$H^{0}(L^{-1}(2D)|_{2D})\to H^{1}(L^{-1})/\langle x\rangle$.

Note that the restriction $s|_{D}$ gives a trivialization of $L(-D)|_{D}$. We
claim that its inverse $(s|_{D})^{-1}\in H^{0}(L^{-1}(D)|_{D})$ induces the
class of our extension under the connecting homomorphism
$H^{0}(L^{-1}(D)|_{D})\to H^{1}(L^{-1})$. Indeed, this is equivalent to
checking that our extension (2.1) is obtained as the pullback of the standard
extension

$0\to{\mathcal{O}}\to{\mathcal{O}}(D)\to{\mathcal{O}}_{D}(D)\to 0$

under the map $(s|_{D})^{-1}:L\to{\mathcal{O}}_{D}(D)$. But this immediately
follows from the existence of the following morphism of exact sequences where
the middle arrow is the projection onto the summand ${\mathcal{O}}(D)\subset
V$.

Hence, by Lemma 2.3.2, the tangent map to $\operatorname{Sec}^{2}(E)\setminus
E\to\operatorname{Pic}_{2}(E)$ at $x$ can be identified with the composition

$H^{0}(L^{-1}(2D)|_{2D})/\langle\widetilde{x}\rangle\to
H^{0}(L^{-1}(2D)|_{D})\rTo{s|_{D}}H^{0}({\mathcal{O}}(D)|_{D})\to
H^{1}({\mathcal{O}}).$

But this composition is precisely the pairing with $s|_{D}\in H^{0}(L|_{2D})$,
so $L^{\perp}$ goes to zero.

(ii) This immediately follows from the well known fact that the quadrics
passing through $E$ are Poisson subvarieties of ${\mathbb{P}}^{3}$. ∎

######  Remark 2.3.4.

Lemma 2.3.3 also follows easily from the fact that the symplectic leaves of
the FO brackets correspond to fixing the isomorphism class of an extension, as
mentioned in Remark 2.1.2.

##  3\. Proofs of the main results

###  3.1. Proof of Theorem A

Consider first the case of two brackets. Assume $[\Pi_{E_{1}},\Pi_{E_{2}}]=0$.
By Proposition 2.1.4, at every point $p\in
E_{1}\setminus(E_{2}\cup\Sigma_{E_{2}})$ (where $\Sigma_{E_{2}}$ is the set of
four vertices of singular quadrics through $E_{2}$), one has
$(\Pi_{E_{2}})_{p}=v_{1}\wedge v_{2}$, where $v_{1}\in T_{p}E_{1}$. Note also
that the subspace $\langle v_{1},v_{2}\rangle\subset T_{p}{\mathbb{P}}^{3}$ is
exactly the image of $(\Pi_{E_{2}})_{p}:T_{p}^{*}{\mathbb{P}}^{3}\to
T_{p}{\mathbb{P}}^{3}$, which is contained in $T_{p}Q$ where $Q$ is the unique
quadric passing through $p$ and $E_{2}$ (see Lemma 2.3.3(ii)). Hence, we get
an inclusion $T_{p}E_{1}\subset T_{p}Q$. It follows that the restriction to
$E_{1}\setminus E_{2}$ of the map ${\mathbb{P}}^{3}\setminus
E_{2}\to{\mathbb{P}}^{1}$ given by quadrics through $E_{2}$, has zero tangent
map at every point. Hence, this map is constant, which means that $E_{1}$ is
contained in a single quadric through $E_{2}$.

For the case of a general collection $(E_{i})$, let $(L_{i}=L(E_{i}))$ denote
the corresponding collection of projective lines in the projective space of
all quadrics (where $L(E_{i})$ is the pencil of quadrics passing through
$E_{i}$). Then $(\Pi_{E_{i}})$ are compatible if and only if every pair of
lines $L_{i}$ and $L_{j}$ has nontrivial intersection. It is well known that
this happens if and only if either all lines pass through one point, or all
are contained in a projective plane.

Conversely, the fact that the FO brackets associated with anticanonical
divisors on a quadric in ${\mathbb{P}}^{3}$ are compatible follows from the
results of [5] (see [5, Ex. 4.8]). ∎

###  3.2. Proof of Theorem B

Consider first the case when we have two normal elliptic curves
$E_{1},E_{2}\subset{\mathbb{P}}^{n}$ such that $[\Pi_{E_{1}},\Pi_{E_{2}}]=0$.
By Proposition 2.1.4, at every point $p\in E_{1}$, one has
$(\Pi_{E_{2}})_{p}=v_{1}\wedge v_{2}$, where $v_{1}\in T_{p}E_{1}$. In
particular, $\operatorname{rk}(\Pi_{E_{2}})_{x}\leq 2$, so by Proposition
2.3.1, we have an embedding $E_{1}\subset\operatorname{Sec}^{2}(E_{2})$, or
(in the case $n=5$), $E_{1}\subset P_{M}$, where
$P_{M}\subset{\mathbb{P}}^{5}$ is the Veronese surface associated with some
square root $M$ of $L$. In the latter case we are done, so we can assume that
$E_{1}\subset\operatorname{Sec}^{2}(E_{2})$. Using Lemma 2.3.3(i), for $p\in
E_{1}\setminus E_{2}$, we obtain the inclusion $T_{p}E_{1}\subset T_{p}S_{M}$,
where $S_{M}\subset\operatorname{Sec}^{2}(E_{2})$ is some scroll, which is the
(closure of the) fiber of the map $\operatorname{Sec}^{2}(E_{2})\setminus
E_{2}\to E_{2}$. Thus, the restriction of the latter map to $E_{1}$ is
constant, so $E_{1}$ is contained in some $S_{M}$, as claimed.

Next, consider any family $(E_{i})$ such that $(\Pi_{E_{i}})$ are compatible.
Consider a pair $E_{1}\neq E_{2}$ in this family, and let $E_{i}$ be any other
elliptic curve in the family. By the first part of the proof, there exists a
rational surface $S\subset{\mathbb{P}}^{n}$ (either a scroll or a Veronese
surface), such that $E_{1}$ and $E_{2}$ are both anticanonical divisors on
$S$. Note that the linear combinations of $\Pi_{E_{1}}+\lambda\Pi_{E_{2}}$ are
the FO brackets associated with elliptic curves $E_{\lambda}$ in the pencil of
anticanonical divisors on $S$, generated by $E_{1}$ and $E_{2}$. Hence, for
each $E_{\lambda}$ the brackets $\Pi_{E_{\lambda}}$ and $\Pi_{E_{i}}$ are
compatible. By the first part of the proof, this implies that each
$E_{\lambda}$ is contained in $\operatorname{Sec}^{2}(E_{i})$. But the surface
$S$ is the closure of the union $\cup_{\lambda}E_{\lambda}$, hence we get the
inclusion $S\subset\operatorname{Sec}^{2}(E_{i})$. Since $S$ is rational, the
composed map $S\setminus E_{i}\to\operatorname{Sec}^{2}(E_{i})\to E_{i}$ is
constant. Therefore, $S$ is contained in the closure $S^{\prime}$ of the fiber
of the map $\operatorname{Sec}^{2}(E_{i})-E_{i}\to E_{i}$, which is one of the
scrolls in $\SS_{E_{i}}$. Since $S^{\prime}$ is an irreducible surface, we get
$S=S^{\prime}$, as claimed.

Conversely, the fact that the FO brackets associated with anticanonical
divisors on a scroll or on a Veronese surface in ${\mathbb{P}}^{5}$ are
compatible follows from the results of [5] (see [5, Ex. 4.6, 4.8]). ∎

###### Proof of Corollary C.

First, we recall that by [9, Cor. 1.2], if $n$ is even then for any FO-bracket
$\Pi_{E}$ on ${\mathbb{P}}^{n}$, there exists a Zariski open neighborhood of
$\Pi_{E}$ in the variety of Poisson brackets consisting of FO-brackets. Hence,
for even $n$, any linear subspace of Poisson brackets containing $\Pi_{E}$ is
an FO-subspace. Thus, we only need to prove our assertions for FO-subspaces.
Now part (b) is immediate from Theorem B. Part (a) also follows Theorem B,
since if all elliptic curves lying on $S$, which is either or scroll or a
Veronese surface, are contained in another irreducible surface $S^{\prime}$,
then $S=S^{\prime}$. ∎

###  3.3. Proof of Theorem D

Let ${\mathbb{P}}^{3}={\mathbb{P}}V$, where $V$ is a $4$-dimensional vector
space. Since normal elliptic curves in ${\mathbb{P}}^{3}$ are intersections of
pairs of quadrics, we can view the construction of the FO Poisson bracket as
associating with a pair of generic quadratic forms $Q_{1},Q_{2}\in
S^{2}V^{*}$, a Poisson bivector $\Pi_{Q_{1}=Q_{2}=0}\in
H^{0}({\mathbb{P}}V,{\bigwedge}^{2}T)$ defined up to rescaling. More
precisely, the choice of quadrics $Q_{1},Q_{2}$ and of a volume form on $V$
gives a nonvanishing differential on the elliptic curve
$E=(Q_{1}=0)\cap(Q_{2}=0)$, which is used in the construction of
$\Pi(Q_{1},Q_{2})$. We claim that in fact there is a linear
$\operatorname{GL}(V)$-equivariant map

$\Psi:{\bigwedge}^{2}(S^{2}V^{*})\to\det(V^{*})\otimes
H^{0}({\mathbb{P}}V,{\bigwedge}^{2}T)$

such that for $Q_{1}$ and $Q_{2}$ defining an elliptic curve,
$\Pi(Q_{1},Q_{2})$ coincides with $\Psi(Q_{1}\wedge Q_{2})$ up to a factor.
Indeed, it is well known that the FO brackets associated with the elliptic
curve $Q_{1}=Q_{2}=0$ are induced by the quadratic Poisson bracket on
$S(V^{*})$ given by

$\\{\ell_{1},\ell_{2}\\}=d\ell_{1}\wedge d\ell_{2}\wedge dQ_{1}\wedge
dQ_{2}/\operatorname{vol}_{V}^{*},$

where $\operatorname{vol}_{V}^{*}$ is a generator of $\det(V^{*})$ (see e.g.,
[11, Sec. 2.1]). This gives the required linear map $\Psi$ which should be
viewed as a normalized version of the bivectors $\Pi_{Q_{1}=Q_{2}=0}$.

Hence, taking the Schouten bracket of two FO brackets corresponds to a linear
$\operatorname{GL}(V)$-invariant map

$\Psi^{(2)}:({\bigwedge}^{2}(S^{2}V^{*}))^{\otimes 2}\to\det(V^{*})^{2}\otimes
H^{0}({\mathbb{P}}V,{\bigwedge}^{3}T)\simeq\det(V^{*})\otimes S^{4}(V^{*}).$

We claim that this map factors through the multiplication map in the exterior
algebra of $S^{2}V^{*}$,

$\mu:({\bigwedge}^{2}(S^{2}V^{*}))^{\otimes 2}\to{\bigwedge}^{4}(S^{2}V^{*}).$

The proof will be based on the fact that by Theorem A,
$\Psi^{(2)}((Q_{1}\wedge Q_{2})\otimes(Q_{3}\wedge Q_{4}))=0$ whenever the
planes $\langle Q_{1},Q_{2}\rangle$ and $\langle Q_{3},Q_{4}\rangle$ in
$S^{2}(V^{*})$ have nonzero intersection (since the corresponding FO brackets
are compatible). Set $W:=S^{2}(V^{*})$. Namely, we can interpret the dual map
$(\Psi^{(2)})^{\vee}$ as a section of the vector bundle $\det(V^{*})\otimes
S^{4}(V^{*})\otimes{\mathcal{O}}(1,1)$ on the square of the Grassmannian
variety $G(2,W)^{2}$, which vanishes on the subvariety $Z\subset G(2,W)^{2}$
consisting of pairs of planes $P_{1},P_{2}\subset W$ with $P_{1}\cap P_{2}\neq
0$. It is well known that the subspace

$H^{0}(G(2,W)^{2},{\mathcal{I}}_{Z}(1,1))\subset
H^{0}(G(2,W)^{2},{\mathcal{O}}(1,1))$

is precisely the image of the map

$\mu^{\vee}:{\bigwedge}^{4}(W)^{*}\to({\bigwedge}^{2}(W)^{*})^{\otimes 2}$

dual to the multiplication. Hence, $(\Psi^{(2)})^{\vee}$ belongs to the
subspace $\det(V^{*})\otimes
S^{4}(V^{*})\otimes\operatorname{im}(\mu^{\vee})$, or equivalently,
$\Psi^{(2)}$ factors through $\mu$.

Thus, we get $\Psi^{(2)}=\Phi^{\prime}\circ\mu$ for some nonzero
$\operatorname{GL}(V)$-invariant map

$\Phi^{\prime}:{\bigwedge}^{4}(S^{2}V^{*})\to\det(V^{*})\otimes S^{4}(V^{*}).$

But $\det(V^{*})\otimes S^{4}(V^{*})$ is an irreducible
$\operatorname{GL}(V)$-representation occurring with multiplicity one in
${\bigwedge}^{4}(S^{2}V^{*})$, Hence, $\Phi^{\prime}=c\Phi$ for some constant
$c$.

Finally, to find the constant $c$, we calculate both sides for

$Q_{1}=x_{1}^{2},\ Q_{2}=x_{1}x_{2},\ Q_{3}=x_{1}x_{3},\ Q_{4}=x_{1}x_{4}.$

We use coordinates $y_{i}=x_{i}/x_{1}$, $i=2,3,4$ on the affine part of
${\mathbb{P}}^{3}$. The Poisson structure associated with $(Q_{1},Q_{2})$ has
the only nonzero bracket between the coordinates

$\\{y_{3},y_{4}\\}_{1}=2$

(where we trivialize $\det(V^{*})$ using $\operatorname{vol}=x_{1}\wedge
x_{2}\wedge x_{3}\wedge x_{4}$), so

$\Pi_{1}=2\partial_{y_{3}}\wedge\partial_{y_{4}}.$

The Poisson structure associated with $(Q_{3},Q_{4})$ has

$\\{y_{2},y_{3}\\}=2y_{3},\ \\{y_{2},y_{4}\\}=2y_{4},\ \\{y_{3},y_{4}\\}=0,$

so

$\Pi_{2}=2y_{3}\cdot\partial_{y_{2}}\wedge\partial_{y_{3}}+2y_{4}\cdot\partial_{y_{2}}\wedge\partial_{y_{4}}.$

Therefore,

$[\Pi_{1},\Pi_{2}]=8\partial_{y_{2}}\wedge\partial_{y_{3}}\wedge\partial_{y_{4}},$

which corresponds to the quartic $8x_{1}^{4}$. On the other hand the quartic
polynomial associated with $Q_{1}\wedge Q_{2}\wedge Q_{3}\wedge Q_{4}$ sends
$e_{1}$ to $2\cdot\operatorname{vol}$. Hence, $c=4$. ∎

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