# Feigin-Odesskii brackets, syzygies, and Cremona transformations

Alexander Polishchuk  University of Oregon, National Research University
Higher School of Economics, and Korea Institute for Advanced Study

###### Abstract.

We identify Feigin-Odesskii brackets $q_{n,1}(C)$, associated with a normal
elliptic curve of degree $n$, $C\subset{\mathbb{P}}^{n-1}$, with the skew-
symmetric $n\times n$ matrix of quadratic forms introduced by Fisher in [7] in
connection with some minimal free resolutions related to the secant varieties
of $C$. On the other hand, we show that for odd $n$, the generators of the
ideal of the secant variety of $C$ of codimension $3$ give a Cremona
transformation of ${\mathbb{P}}^{n-1}$, generalizing the quadro-cubic Cremona
transformation of ${\mathbb{P}}^{4}$. We identify this transformation with the
one considered in [14] and find explict formulas for the inverse
transformation. We also find polynomial formulas for Cremona transformations
from [14] associated with higher rank bundles on $C$.

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

Let $C\subset{\mathbb{P}}^{n-1}={\mathbb{P}}(V)$ be an elliptic normal curve
of degree $n\geq 3$ over a field $k$ of characteristic $0$. With $C$ one can
associate canonically (up to rescaling) a Poisson bracket $q_{n,1}(C)$ on the
ambient projective space ${\mathbb{P}}(V)$. Namely, if
$L={\mathcal{O}}(1)|_{C}$, a line bundle of degree $n$ on $C$, then we can
identify ${\mathbb{P}}(V)$ with

${\mathbb{P}}H^{0}(C,L)^{*}\simeq{\mathbb{P}}\operatorname{Ext}^{1}(L,{\mathcal{O}})$

(where we used Serre duality on $C$). Now there is a natural Poisson bracket
on the projective space ${\mathbb{P}}\operatorname{Ext}^{1}(L,{\mathcal{O}})$
described in [5] and [14].

The Poisson brackets $q_{n,1}(C)$ correspond to quadratic brackets on $V$
arising as semi-classical limit of Feigin-Odesskii elliptic algebras (see
[4]). The Poisson geometry of $q_{n,1}(C)$ has been well studied (see e.g.,
[5], [13], [14], [10], [12], [15], [8], [9]). In particular, it is known that
for odd $n$ the stratification of ${\mathbb{P}}^{n-1}$ by the rank of this
Poisson bracket is related to the secant varieties of $C$.

The first goal of the present work is to establish a direct relation of
$q_{n,1}(C)$ with a certain skew-symmetric matrix of quadratic forms $\Omega$
related to the secant varieties of $C$, introduced and studied by Fisher in
[7].

Assume first that $n$ is odd and let $r=(n-1)/2$. Then as is well known the
secant variety $\operatorname{Sec}^{r}C$ is a hypersurface of degree $n$ in
${\mathbb{P}}^{n-1}$. Let $F(x_{1},\ldots,x_{n})=0$ be its equation (of degree
$n$). Then Fisher showed in [7] that the module of syzygies between the
partial derivatives $(\frac{\partial F}{\partial x_{1}},\ldots,\frac{\partial
F}{\partial x_{n}})$ is generated by the relations

$\sum_{i}\frac{\partial F}{\partial x_{i}}\Omega_{ij}=0,$ (1.1)

for a unique (up to rescaling) skew-symmetric $n\times n$ matrix of quadratic
forms $\Omega$ (see Sec. 2.3 for details).

In the case when $n$ is even, we set $r=(n-2)/2$. Then the secant variety
$\operatorname{Sec}^{r}C$ is a complete intersection of codimension $2$ given
by $F_{1}=F_{2}=0$, where $\deg(F_{1})=\deg(F_{2})=r+1$. In this case the
$n\times n$ skew-symmetric matrix of quadratic forms $\Omega$ is characterized
by the condition that the relations

$\sum_{i}\frac{\partial F_{a}}{\partial x_{i}}\Omega_{ij}=0,\text{ where
}a=1,2,$ (1.2)

generate the module of syzygies between the columns of the $2\times n$ matrix
$(\frac{\partial F_{a}}{\partial x_{j}})$.

Theorem A. (i) The formula $\\{x_{i},x_{j}\\}=\Omega_{ij}$ defines a Poisson
bracket on $V$ such that the induced Poisson bracket on ${\mathbb{P}}(V)$ is
$q_{n,1}(C)$ (up to rescaling).

(ii) If $n$ is odd then $\\{x_{i},F\\}=0$ for every $i$. If $n$ is even then
$\\{x_{i},F_{1}\\}=\\{x_{i},F_{2}\\}=0$ for every $i$.

Note that part (ii) is stated in [11, Sec. 5] (see also [10]).

Assume now that $n$ is odd and $n\geq 5$. Our second result gives an explicit
formula for the Poisson birational transformation from
$({\mathbb{P}}^{n-1},q_{n,1}(C))$ to $({\mathbb{P}}^{n-1},q_{n,(n-1)/2}(C))$
constructed in [14]. Recall that for any relatively prime $(n,k)$, the Feigin-
Odesskii bracket $q_{n,k}(C)$ is a natural Poisson bracket on
${\mathbb{P}}^{n-1}={\mathbb{P}}\operatorname{Ext}^{1}(V,{\mathcal{O}})$,
where $V$ is a stable bundle of rank $k$ and degree $n$ on $C$ (see [5],
[14]).

Let us start with a line bundle $L={\mathcal{O}}(1)|_{C}$ of degree $n$, and
let $V_{0}$ be the unique stable bundle of rank $2$ with $\det(V_{0})=L$ (it
exists since $n$ is odd). In [14] we constructed a natural birational map (a
Cremona transformation)

$\phi:{\mathbb{P}}^{n-1}={\mathbb{P}}\operatorname{Ext}^{1}({\mathcal{O}},L)\dashrightarrow{\mathbb{P}}H^{0}(C,V_{0})={\mathbb{P}}^{n-1}$

by observing that for a generic extension

$0\to{\mathcal{O}}\to E\to L\to 0$

the bundle $E$ is stable, so there is an isomorphism $E\simeq V_{0}$, unique
up to rescaling. Hence, from the above extension we get a nonzero section of
$V_{0}$, well defined up to rescaling. Furthermore, we showed in [14] that
$\phi$ is compatible with Poisson structures, where
${\mathbb{P}}H^{0}(C,V_{0})$ is equipped with a natural Poisson bracket $q$,
such that there exists a Poisson isomorphism

$({\mathbb{P}}H^{0}(C,V_{0}),q)\simeq({\mathbb{P}}\operatorname{Ext}^{1}(V,{\mathcal{O}}),q_{n,r}(C)),$

where $V$ is a stable bundle of degree $n$ and rank $r=(n-1)/2$.

Now let us look at the variety $\operatorname{Sec}^{r-1}C$ of codimension $3$
in ${\mathbb{P}}^{n-1}$. It is known that its homogeneous ideal is generated
by $n$ forms $(p_{1},\ldots,p_{n})$ of degree $r$.

Theorem B. The above birational map
$\phi:{\mathbb{P}}^{n-1}\dashrightarrow{\mathbb{P}}^{n-1}$ is given by

$(x_{1}:\ldots:x_{n})\mapsto(p_{1}(x):\ldots:p_{n}(x)).$

The inverse is given by
$(y_{1}:\ldots:y_{n})\mapsto(f_{1}(y):\ldots:f_{n}(y))$, where $f_{i}$ are
certain homogeneous polynomials of degree $n-2$.

For example, when $n=5$, we have $r=2$ and $(p_{1},\ldots,p_{5})$ are the
quadrics generating the ideal of $C$ in ${\mathbb{P}}^{4}$. The corresponding
Cremona transformation of ${\mathbb{P}}^{4}$ is well known classically as
quadro-cubic Cremona transformation (see [17]). The polynomials
$(p_{1},\ldots,p_{n})$ can be computed as (signed) submaximal pfaffians of a
skew-symmetric $n\times n$ matrix of linear forms $\Phi$, called the Klein
matrix in [6]. For a Heisenberg invariant curve $C$, there is a simple formula
for $\Phi$ (see [6, Prop. 3.7]). In the course of the proof of Theorem B we
also give a recipe for computing the polynomials $f_{i}$ in terms of $\Phi$.

Corollary C. For $r\geq 2$, any hypersurface of degree $r$ in
${\mathbb{P}}^{2r}$ containing $\operatorname{Sec}^{r-1}C$, where
$C\subset{\mathbb{P}}^{2r}$ is an elliptic normal curve, is rational.

Indeed, the corresponding birational map $\phi$ induces a birational map from
such a hypersurface to a hyperplane in ${\mathbb{P}}^{2r}$.

Finally, we find polynomial formulas for similar Cremona transformations
associated with stable bundles of higher rank on $C$ (these transformations
were defined in [14]). Namely, for $k>0$ relatively prime to $n$, let $V_{k}$
be the unique stable vector bundle of rank $k$ on $C$ with $\det V_{k}=L$
(where $\deg L=n$). Assume that $k+1$ is still relatively prime to $n$. Then
we can also consider the bundle $V_{k+1}$ of rank $k+1$. Now all three spaces
$H^{0}(C,V_{k+1})=\operatorname{Hom}({\mathcal{O}},V_{k+1})$,
$\operatorname{Hom}(V_{k+1},V_{k})$ and
$\operatorname{Ext}^{1}(V_{k},{\mathcal{O}})$ have dimension $n$. As in [14],
let us consider the closed subvariety

$Z\subset{\mathbb{P}}\operatorname{Hom}({\mathcal{O}},V_{k+1})\times{\mathbb{P}}\operatorname{Hom}(V_{k+1},V_{k})\times{\mathbb{P}}\operatorname{Ext}^{1}(V_{k},{\mathcal{O}})$

which is the closure of the locus of $(\alpha,\beta,\gamma)$ such that

${\mathcal{O}}\rTo{\alpha}V_{k+1}\rTo{\beta}V_{k}\rTo{\gamma}{\mathcal{O}}[1]$

is an exact triangle. Then the projection from $Z$ to any of the factors is
birational, so we get three Cremona transormations

$\displaystyle\phi:{\mathbb{P}}\operatorname{Ext}^{1}(V_{k},{\mathcal{O}})\dashrightarrow{\mathbb{P}}\operatorname{Hom}(V_{k+1},V_{k}),\
\
\psi:{\mathbb{P}}\operatorname{Hom}(V_{k+1},V_{k})\dashrightarrow{\mathbb{P}}H^{0}(C,V_{k+1}),$
$\displaystyle\rho:{\mathbb{P}}H^{0}(C,V_{k+1})\dashrightarrow{\mathbb{P}}\operatorname{Ext}^{1}(V_{k},{\mathcal{O}}),$

such that $\rho\psi\phi=\operatorname{id}$.

Theorem D. Let us define numbers $l$ and $m$, where $0<l<n$, $0<m<n$, by

$l\equiv-(k+1)^{-1}\operatorname{mod}(n),\ \
m\equiv-1-k^{-1}\operatorname{mod}(n).$

Then there exist linear maps

$\displaystyle\widetilde{\phi}:S^{l}(\operatorname{Ext}^{1}(V_{k},{\mathcal{O}}))\to\operatorname{Hom}(V_{k+1},V_{k}),\
\ \widetilde{\psi}:S^{k}(\operatorname{Hom}(V_{k+1},V_{k}))\to
H^{0}(C,V_{k+1}),$
$\displaystyle\widetilde{\rho}:S^{m}(H^{0}(C,V_{k+1}))\to\operatorname{Ext}^{1}(V_{k},{\mathcal{O}}),$

inducing $\phi$, $\psi$ and $\rho$.

We give an explicit construction for the polynomial map $\widetilde{\psi}$,
and then we construct $\widetilde{\phi}$ and $\widetilde{\rho}$ using
autoequivalences of the derived category of $C$. Note that in the case $k=1$,
the map $\widetilde{\psi}$ is a linear isomorphism, while the polynomials
giving $\widetilde{\phi}$ and $\widetilde{\rho}$ have the same degrees as the
polynomial maps of Theorem B. In fact, it is easy to see that in this case
$\widetilde{\phi}$ coincides with $(p_{1},\ldots,p_{n})$, and we expect
$\widetilde{\rho}$ to coincide with $(f_{1},\ldots,f_{n})$ constructed in the
proof of Theorem B (see Remark 3.3.1).

Acknowledgment. This paper was inspired by a discussion with Volodya Rubtsov
in 2017 and by his paper [16], where the Feigin-Odesskii brackets $q_{5,1}(C)$
and the corresponding quadro-cubic Cremona transformations of
${\mathbb{P}}^{4}$ are studied. I am very grateful to Volodya for sharing his
insights on Feigin-Odesskii brackets and for drawing my attention to the works
[6], [7] on secant varieties of normal elliptic curves.

##  2\. Preliminaries

###  2.1. Quadratic Poisson brackets

Let $V$ be a vector space of dimension $n$. By a quadratic Poisson bracket on
$V$ we mean a Poisson structure on the algebra $S(V^{*})$ compatible with the
grading, i.e., such that the Poisson brackets of linear forms are given by
homogeneous quadratic forms. Equivalently, the corresponding bivector on $V$
has to be homogeneous (i.e., preserved by the natural
${\mathbb{G}}_{m}$-action). It is well known that every such bracket induces a
Poisson bracket on the projective space ${\mathbb{P}}(V)$ and that every
Poisson bracket on ${\mathbb{P}}(V)$ comes from a quadratic Poisson bracket on
$V$, not necessarily unique (see [1], [13]).

Different liftings of a bivector on ${\mathbb{P}}(V)$ to a homogeneous
bivector on $V$ differ by bivectors on $V$ of the form $\langle
x,y\rangle=A(x)y-A(y)x$, where $x,y\in V^{*}$, for some linear operator
$A:V^{*}\to V^{*}$. This easily implies that different liftings of a Poisson
bracket $\Pi$ on ${\mathbb{P}}(V)$ to quadratic Poisson brackets on $V$ are
numbered by Poisson vector fields for $\Pi$ on ${\mathbb{P}}(V)$, i.e., global
vector fields on ${\mathbb{P}}(V)$ preserving $\Pi$.

The following simple criterion is helpful in identifying a lifting of a
Poisson bracket on ${\mathbb{P}}(V)$ to a quadratic Poisson bracket on $V$.

######  Lemma 2.1.1.

Let $\\{\cdot,\cdot\\}$ be a homogeneous bivector on $V$, inducing a Poisson
bracket on ${\mathbb{P}}(V)$. Assume that there exists a nonzero homogeneous
polynomial $F\in S^{d}V^{*}$ of positive degree $d$ such that $\\{x,F\\}=0$
for every $x\in V^{*}$. Then $\\{\cdot,\cdot\\}$ is a Poisson bracket, i.e.,
it satisfies the Jacobi identity.

###### Proof.

Set $J(x,y,z):=\\{\\{x,y\\},z\\}+\\{\\{y,z\\},x\\}+\\{\\{z,x\\},y\\}$. First,
let us check that $J(x,y,z)=0$ for all $x,y,z\in S^{d}V^{*}$. Indeed, $x/F$,
$y/F$ and $z/F$ are local functions on ${\mathbb{P}}(V)$, so we know that

$J(\frac{x}{F},\frac{y}{F},\frac{z}{F})=0.$

Furthermore, since $\\{x,F\\}=\\{y,F\\}=\\{z,F\\}=0$, we get
$\\{\frac{x}{F},\frac{y}{F}\\}=\frac{\\{x,y\\}}{F^{2}}$, etc., hence,

$J(\frac{x}{F},\frac{y}{F},\frac{z}{F})=\frac{J(x,y,z)}{F^{3}}.$

Thus, we get that $J(x,y,z)=0$ for all $x,y,z\in S^{d}V^{*}$.

Hence, if $x,y,z$ are in $V^{*}$ then we have $J(x^{d},x^{d-1}y,x^{d-1}z)=0$.
Now the vanishing of $J(x,y,z)$ follows from the formal identity

$J(x^{d},x^{d-1}y,x^{d-1}z)=dx^{3d-3}J(x,y,z).$

∎

###  2.2. Formula for Feigin-Odesskii bracket

Let us recall the formula for the Poisson bivector $\Pi=q_{n,1}(C)$ on
${\mathbb{P}}\operatorname{Ext}^{1}(L,{\mathcal{O}})\simeq{\mathbb{P}}H^{0}(C,L)^{*}$
in terms of Szegö kernel, established in [9] (equivalent formulas are also
stated in [10] and in [12]).

Let $p\in C$ be a point, and let us fix a trivialization of $\omega_{C}$. The
Szegö kernel is the unique section $S\in H^{0}(C\times
C,{\mathcal{O}}(p)\boxtimes{\mathcal{O}}(p)(\Delta))$ such that
$\operatorname{Res}_{\Delta}(S)=1$ and $S(y,x)=-S(x,y)$. If $C$ is identified
with a plane cubic $y^{2}=P(x)$ and the trivializing global differential is
$dx/2y$, then one has

$S=\frac{y_{1}+y_{2}}{x_{2}-x_{1}}$

(see [9, Sec. 5.1.2]).

To give a Poisson bivector $\Pi$ we need to specify for each $\phi\in
H^{0}(L)^{*}$ a skew-symmetric form $\Pi_{\phi}$ on the cotangent space

$T^{*}_{\phi}{\mathbb{P}}H^{0}(L)^{*}\simeq\operatorname{ker}(\phi)\subset
H^{0}(L).$

This form is given by the formula (see [9, Lem. 2.1, Prop. 5.8])

$\Pi_{\phi}(s_{1}\wedge
s_{2})=\pm\langle\widetilde{\phi}\otimes\widetilde{\phi},S\cdot(s_{1}\boxtimes
s_{2}-s_{2}\boxtimes s_{1})\rangle,$ (2.1)

for $s_{1},s_{2}\in\operatorname{ker}(\phi)$. Here we consider $H^{0}(L)$ as a
subspace in $H^{0}(L(p))$ and denote by $\widetilde{\phi}$ any extension of
$\phi$ to a functional on $H^{0}(L(p))$. Further, we view $s_{1}\boxtimes
s_{2}-s_{2}\boxtimes s_{1}$ as a section of $L\boxtimes L$ on $C\times C$,
vanishing on the diagonal. Hence, the product $S\cdot(s_{1}\boxtimes
s_{2}-s_{2}\boxtimes s_{1})$ is a global section of $L(p)\boxtimes L(p)$.

The fact that the right-hand side of (2.1) does not depend on a choice of
$\widetilde{\phi}$ follows from the existence of a linear operator
$D:H^{0}(L)\to H^{0}(L(p))$ such that for any $s_{1},s_{2}\in H^{0}(L)$ one
has

$S\cdot(s_{1}\boxtimes s_{2}-s_{2}\boxtimes s_{1})+D\\{s_{1},s_{2}\\}\in
H^{0}(L)\otimes H^{0}(L),\text{ where}$ $D\\{s_{1},s_{2}\\}:=s_{1}\boxtimes
D(s_{2})+D(s_{2})\boxtimes s_{1}-s_{2}\boxtimes D(s_{1})-D(s_{1})\boxtimes
s_{2}$

(so $D$ cancels the poles at $p$). The existence of such $D$ was first
observed in [10] (see also [12] and [9, Sec. 5.3]). For
$s_{1},s_{2}\in\operatorname{ker}(\phi)$ this allows to rewrite the formula
(2.1) as

$\Pi_{\phi}(s_{1}\wedge s_{2})=\pm\langle\phi\otimes\phi,S\cdot(s_{1}\boxtimes
s_{2}-s_{2}\boxtimes s_{1}+D\\{s_{1},s_{2}\\})\rangle.$ (2.2)

###  2.3. Secant varieties and syzygies

Let us recall some results of Fisher in [6] and [7] on secant varieties of a
normal elliptic curve $C\subset{\mathbb{P}}^{n-1}$.

First, assume that $n$ is odd, and set $r=(n-1)/2$. The variety
$\operatorname{Sec}^{r}C$ is a hypersurface in ${\mathbb{P}}^{n-1}$ given by
$F=0$, where $\deg(F)=n$. The first result, [7, Thm. 1.1(i)] gives a form of
the minimal free resolution of the ideal $(\frac{\partial F}{\partial
x_{1}},\ldots,\frac{\partial F}{\partial x_{n}})$. Namely, there is a unique
skew-symmetric $n\times n$ matrix $\Omega$ of quadratic forms, such that the
following complex is exact, where $R=k[x_{1},\ldots,x_{n}]$:

$0\to R(-2n)\rTo{\nabla^{T}}R(-n-1)^{n}\rTo{\Omega}R(-n+1)^{n}\rTo{\nabla}R,$
(2.3)

where $\nabla=(\frac{\partial F}{\partial x_{1}},\ldots,\frac{\partial
F}{\partial x_{n}})$. Note that the fact that this is a complex corresponds to
the identity (1.1).

In the case when $n$ is even, let us set $r=(n-2)/2$. Then the variety
$\operatorname{Sec}^{r}C$ is a complete intersection of codimension $2$ given
by $F_{1}=F_{2}=0$, where $\deg(F_{i})=r+1$ (see [2, Sec. 8], [7, Thm. 9.1]).
Now setting

$\nabla=\left(\begin{matrix}\frac{\partial F_{1}}{\partial
x_{1}}&\ldots&\frac{\partial F_{1}}{\partial x_{n}}\\\ \frac{\partial
F_{2}}{\partial x_{1}}&\ldots&\frac{\partial F_{2}}{\partial
x_{n}}\end{matrix}\right),$

the statement of [7, Thm. 1.1(ii)] is that there is a unique skew-symmetric
$n\times n$ matrix $\Omega$ of quadratic forms, such that the following
complex is exact:

$0\to
R(-n)^{2}\rTo{\nabla^{T}}R(-r-1)^{n}\rTo{\Omega}R(-r)^{n}\rTo{\nabla}R^{2}.$
(2.4)

Again, the fact that this is a complex corresponds to the identities (1.2).

Next, in the case when $n$ is odd, we will use the following description of
the minimal free resolution of the ideal of $\operatorname{Sec}^{r-1}C$ in
$R[x_{1},\ldots,x_{n}]$, where $r=(n-1)/2$ (see [6, Sec. 4]). This ideal is
generated by $n$ linearly independent forms of degree $r$,
$(p_{1},\ldots,p_{n})$, and there exists a unique skew-symmetric $n\times n$
matrix of linear forms $\Phi$ such that the following complex is exact:

$0\to R(-n)\rTo{p^{T}}R(-r-1)^{n}\rTo{\Phi}R(-r)^{n}\rTo{p}R,$ (2.5)

where $p=(p_{1},\ldots,p_{n})$.

Furthermore, we will use the following description of $\Phi$, which is called
Klein matrix in [6]. Let $V_{0}$ be the unique stable bundle on $C$ with
$\det(V_{0})\simeq L={\mathcal{O}}(1)|_{C}$. Then identifying the space of
linear forms on ${\mathbb{P}}^{n-1}$ with $H^{0}(C,L)$, the skew-symmetric
matrix $\Phi$ corresponds to the natural $H^{0}(C,L)$-valued skew-symmetric
pairing on $H^{0}(C,V_{0})$,

$\Phi:{\bigwedge}^{2}H^{0}(C,V_{0})\to H^{0}(C,\det(V_{0}))\simeq H^{0}(C,L).$
(2.6)

######  Remark 2.3.1.

Both complexes (2.3) and (2.5) for odd $n$ are examples of Buchsbaum-Eisenbud
pfaffian presentations of the corresponding Gorenstein ideals of height $3$
(see [3]). In particular, the generators of the ideal ($\frac{\partial
F}{\partial x_{i}}$ in the former case and $p_{i}$ in the latter case), are
equal to the (signed) submaximal pfaffians of the corresponding skew-symmetric
matrix ($\Omega$ in the former case and $\Phi$ in the latter case).

######  Remark 2.3.2.

The fact that the submaximal pfaffians $(p_{i})$ vanish on
$\operatorname{Sec}^{r-1}C\subset{\mathbb{P}}H^{0}(C,L)^{*}$ means that for
$\xi\in\operatorname{Sec}^{r-1}C$ the skew-symmetric form $\Phi_{\xi}$ on
$H^{0}(C,V_{0})$ has rank $\leq n-3$. This can be explained geometrically as
follows. A point $\xi\in\operatorname{Sec}^{r-1}C$ corresponds to a functional
that factors as $H^{0}(C,L)\to H^{0}(C,L|_{D})\to k$, where $D$ is an
effective divisor of degree $r-1$ on $C$. But then the $3$-dimensional
subspace $H^{0}(C,V_{0}(-D))\subset H^{0}(C,V_{0})$ is in the kernel of
$\Phi_{\xi}$.

###  2.4. Skew-symmetric matrices of linear forms

Let $A$ and $B$ be a pair of $n$-dimensional vector spaces, and let
$\Phi\in{\bigwedge}^{2}A^{*}\otimes B$. Typically one views $\Phi$ as a skew-
symmetric matrix of linear forms on $B^{*}$. However, $\Phi$ also gives a
linear map $\nu_{\Phi}:A\to B\otimes A^{*}$, so we can view it as an $n\times
n$ matrix of linear forms on $A$.

More precisely, consider the polynomial algebra $R=S^{*}(A^{*})$. Then we can
view $\nu_{\Phi}$ as an $R$-valued linear map $A\to B\otimes R$, and consider
the corresponding matrix of minors,

${\bigwedge}^{n-1}(\nu_{\Phi}):{\bigwedge}^{n-1}A\to{\bigwedge}^{n-1}B\otimes
R$

with entries in $S^{n-1}(A^{*})\subset R$.

######  Lemma 2.4.1.

(i) Consider the composed map

$\displaystyle\sigma(\Phi):A^{*}\rTo{\sim}\det(A)^{*}\otimes{\bigwedge}^{n-1}(A)\rTo{\operatorname{id}\otimes{\bigwedge}^{n-1}(\nu_{\Phi})}\det(A)^{*}\otimes{\bigwedge}^{n-1}(B)\otimes
R$ $\displaystyle\simeq\det(A)^{*}\otimes\det(B)\otimes B^{*}\otimes R.$

Then for every $\xi\in A^{*}$, the element $\sigma(\Phi)(\xi)$ (of the target
$R$-module) is divisible by $\xi$.

(ii) There exists a unique element

$\sigma_{n-2}(\Phi)\in\det(A)^{*}\otimes\det(B)\otimes B^{*}\otimes
S^{n-2}(A^{*}),$

such that

$\sigma(\Phi)(\xi)=\sigma_{n-2}(\Phi)\cdot\xi$

(where the product is in $R$).

(iii) Viewing $\sigma_{n-2}(\Phi)$ as a $\det(A)^{*}\otimes\det(B)\otimes
R$-valued linear form on $B$, we have

$\sigma_{n-2}(\Phi)\circ\nu_{\Phi}=0$

as a $\det(A)^{*}\otimes\det(B)\otimes R$-valued linear form on $A$.

(iv) Assume that $\Phi(a,?):A\to B$ has rank $\geq n-1$ for some $a\in A$.
Then $\sigma_{n-2}(\Phi)(a)\neq 0$.

###### Proof.

(i) Let us think of $\nu_{\Phi}$ as a $\operatorname{Hom}(A,B)$-valued
function on $A$. Then $\sigma(\Phi)(\xi)$ is given by the
$(n-1)\times(n-1)$-minors of the corresponding
$\operatorname{Hom}(\langle\xi\rangle^{\perp},B)$-valued function on $A$,

$\nu_{\Phi,\xi}:\langle\xi\rangle^{\perp}\to B\otimes A^{*},$

where we use the restriction from $A$ to the hyperplane
$\langle\xi\rangle^{\perp}\subset A$. The condition that all $n\times
n$-minors of $\nu_{\Phi,\xi}$ are divisible by $\xi$ is equivalent to the
condition that for every $a\in\langle\xi\rangle^{\perp}$, the induced linear
map

$\nu_{\Phi,\xi,a}:\langle\xi\rangle^{\perp}\to B$

has rank $<n-1$. It is enough to check that for $a\neq 0$, the map
$\nu_{\Phi,\xi,a}$ has nonzero kernel. But this follows from the identity

$\nu_{\Phi,\xi,a}(a)=0$

which holds by the skew-symmetry of $\Phi$.

(ii) Note that the target of $\sigma(\Phi)$ is a free $R$-module. Thus, it is
enough to prove that if a linear map $\sigma:A^{*}\to R=S(A^{*})$ has the
property that $\sigma(\xi)$ is divisible by $\xi$ for every $\xi\in A^{*}$,
then $\sigma(\xi)=f\cdot\xi$ for a fixed polynomial $f\in R$.

Indeed, assume that for every $\xi\neq 0$, we have

$\sigma(\xi)=f_{\xi}\cdot\xi$

for some $f_{\xi}\in R$. We need to check that $f_{\xi_{1}}=f_{\xi_{2}}$ for a
linearly independent pair $\xi_{1},\xi_{2}\in A^{*}$. For any $c\in k^{*}$ we
have

$\sigma(\xi_{1}+c\xi_{2})=f_{\xi_{1}}\xi_{1}+f_{\xi_{2}}\xi_{2}=f_{c}(\xi_{1}+c\xi_{2}),$

where $f_{c}:=f_{\xi_{1}+c\xi_{2}}$. This implies that
$(f_{c}-f_{\xi_{1}})\xi_{1}$ is divisible by $\xi_{2}$, so we can write

$f_{c}=f_{\xi_{1}}+f\xi_{2}.$

From this we get

$f_{\xi_{2}}-f_{\xi_{1}}=(c^{-1}x_{1}+x_{2})\cdot f,$

so $f_{\xi_{2}}-f_{\xi_{1}}$ is divisible by $c^{-1}x_{1}+x_{2}$. Since $k$ is
infinite, this implies that $f_{\xi_{2}}-f_{\xi_{1}}=0$.

(iii) We use again the fact that for a nonzero $\xi\in A^{*}$,
$\sigma(\Phi)(\xi)$, which is an $R$-valued functional on $B$, is given by
${\bigwedge}^{n-1}(\Phi|_{\langle\xi\rangle^{\perp}})$, where we consider the
restriction

$\Phi|_{\langle\xi\rangle^{\perp}}:\langle\xi\rangle^{\perp}\to B\otimes R,$

where $\dim\langle\xi\rangle^{\perp}=n-1$. It follows that the composition

$\langle\xi\rangle^{\perp}\rTo{\Phi}B\otimes
R\rTo{\sigma(\Phi)(\xi)}\det(A)^{*}\otimes\det(B)\otimes R$

is zero. By part (ii) this implies that the composition

$\sigma_{n-2}(\Phi)\circ\Phi:A\to\det(A)^{*}\otimes\det(B)\otimes R$

has zero restriction to $\langle\xi\rangle^{\perp}$ for every $\xi\neq 0$.
Hence, this composition is zero.

(iv) The assumption implies that $\sigma(\Phi)(a)\neq 0$. Now the assertion
follows from (ii). ∎

##  3\. Proofs

###  3.1. Proof of Theorem A

Working over an algebraic closure of $k$, we can assume that
$L={\mathcal{O}}(np)$ for a point $p\in C$. In this case Fisher gives the
following explicit description of the matrix

$\Omega:{\bigwedge}^{2}H^{0}(C,L)\to S^{2}H^{0}(C,L)$

in [7, Sec. 5]. For $s_{1},s_{2}\in H^{0}(C,L)$ and $\phi\in H^{0}(C,L)^{*}$,
one has

$\frac{1}{n}\Omega(s_{1},s_{2})(\phi)=[S\cdot(s_{1}\boxtimes
s_{2}-s_{2}\boxtimes
s_{1})](\phi,\phi)+{\mathcal{D}}\\{s_{1},s_{2}\\}(\phi,\phi),$

for some operator ${\mathcal{D}}:H^{0}(C,{\mathcal{O}}(np))\to
H^{0}(C,{\mathcal{O}}((n+1)p))$ (which is given up to rescaling by $f\mapsto
df/\omega$, where $\omega$ is a global differential on $C$).

Comparing the above formula for $\Omega$ with (2.2) we immediately deduce that
the bivector on ${\mathbb{P}}^{n-1}$ induced by $\Omega$ is $q_{n,1}(C)$, up
to rescaling.

In the case $n$ is odd we have

$\\{F,x_{j}\\}=\sum_{i}\frac{\partial F}{\partial
x_{i}}\\{x_{i},x_{j}\\}=\sum_{i}\frac{\partial F}{\partial
x_{i}}\Omega_{ij}=0$

by (1.1). Hence, the assertion of the theorem follows in this case from Lemma
2.1.1.

Similarly, in the case $n$ is even, we deduce from (1.2) that
$\\{F_{1},x_{j}\\}=\\{F_{2},x_{j}\\}=0$.

###  3.2. Proof of Theorem B

We start with the following description of

$\phi^{-1}:{\mathbb{P}}H^{0}(C,V_{0})\dashrightarrow{\mathbb{P}}\operatorname{Ext}^{1}({\mathcal{O}},L)\simeq{\mathbb{P}}H^{0}(C,L)^{*}.$

Given a generic global section $s\in H^{0}(C,V_{0})$, the corresponding map
$s:{\mathcal{O}}\to V_{0}$ is an embedding of a subbundle, and we have a
canonical identification of $V_{0}/s({\mathcal{O}})$ with $\det(V_{0})=L$.
Hence, we get an extension of $L$ by ${\mathcal{O}}$. Note that the Serre
duality isomorphism $\operatorname{Ext}^{1}({\mathcal{O}},L)\simeq
H^{0}(C,L)^{*}$ associates with an extension

$0\to{\mathcal{O}}\to E\to L\to 0$

the corresponding coboundary homomorphism $H^{0}(C,L)\to
H^{1}(C,{\mathcal{O}})\simeq k$. The exact sequence of cohomology shows that
the kernel of this homomorphism coincides with the image of the map
$H^{0}(C,E)\to H^{0}(C,L)$.

Thus, $\phi^{-1}$ associates with a generic $s\in H^{0}(C,V_{0})$ the unique
functional $\xi\in H^{0}(C,L)^{*}$ (up to rescaling), such that
$\operatorname{ker}(\xi)$ is equal to the image of the map

$d_{s}:H^{0}(C,V_{0})\to H^{0}(C,V_{0}/s({\mathcal{O}}))\simeq H^{0}(C,L).$

Recall that an isomorphism $\alpha_{s}:V_{0}/s({\mathcal{O}})\rTo{\sim}L$
induces an isomorphism

$\beta_{s}:\det(V_{0})\rTo{\sim}L:s\wedge x\mapsto\alpha(x).$

Note that $\beta_{s}$ does not depend on $s$ up to rescaling.

Thus, up to rescaling, the map $d_{s}$ can be identified with the map

$\Phi(s,?):H^{0}(C,V_{0})\to H^{0}(C,L),$

where $\Phi$ is the Klein matrix (2.6). In other words, $\xi=\phi^{-1}(s)$ is
characterized by the condition

$\operatorname{ker}(\xi)=\operatorname{im}\Phi(s,?).$

Let us set $A:=H^{0}(C,V_{0})$ and $B:=H^{0}(C,L)$ for brevity (note that $B$
is the space of linear forms on our projective space
${\mathbb{P}}^{n-1}={\mathbb{P}}H^{0}(C,L)^{*}$), so that $\Phi$ can be viewed
as a skew-symmetric linear map

$\Phi:A\to A^{*}\otimes B.$

We showed above that the birational map

$\phi^{-1}:{\mathbb{P}}A\dashrightarrow{\mathbb{P}}B^{*}$

sends a generic $a$ to the unique functional $b^{*}$ (up to rescaling) such
that $b^{*}$ vanishes on the image of $\Phi(a,?):A\to B$. Now we recall that
with $\Phi$ we can associate canonically (up to rescaling) an element
$\sigma_{n-2}(\Phi)\in B^{*}\otimes S^{n-2}(A^{*})$, such that the composition

$A\rTo{\Phi(a,?)}B\rTo{\sigma_{n-2}(\Phi)(a)}k$

is zero (see Lemma 2.4.1). Furthermore, since $\Phi(a,?)$ has rank $n-1$ for
generic $a$, we have $\sigma_{n-2}(\Phi)(a)\neq 0$ for such $a$ (see Lemma
2.4.1(iv)). This implies that

$\phi^{-1}(a)=\sigma_{n-2}(\Phi)(a)$

in ${\mathbb{P}}B^{*}$. If we choose a basis in $B$, then $\sigma_{n-2}(\Phi)$
is given by $n$ polynomials $(f_{1},\ldots,f_{n})$, where $f_{i}\in
S^{n-2}(A^{*})$, and we have

$\phi^{-1}(a)=(f_{1}(a):\ldots:f_{n}(a)).$

It remains to prove the formula for $\phi:{\mathbb{P}}B^{*}\to{\mathbb{P}}A$.
We can rewrite the complex (2.5) as

$0\to R(-n)\rTo{p^{T}}A\otimes R(-r-1)\rTo{\Phi}A^{*}\otimes R(-r)\rTo{p}R$

where $R=S^{*}(B)$. This complex shows that for each $b^{*}\in B^{*}$ the
specialization $\Phi_{b^{*}}:A\to A^{*}$ satisfies

$\Phi_{b^{*}}(p(b^{*})^{T})=0,$ (3.1)

where $p(b^{*})^{T}\in A$.

We need to check that $\phi(b^{*})=p(b^{*})^{T}$. Equivalently, setting

$a:=p(b^{*})^{T},$

we need to prove that

$\phi^{-1}(a)=b^{*}.$

By the above description of $\phi^{-1}$, it is enough to check that $b^{*}$
annihilates the image of $\Phi(a,?):A\to B$ (since the latter image is a
hyperplane in $B$). In other words, we need to check that
$b^{*}\circ\Phi(a,?)=0$. But we have

$b^{*}\circ\Phi(a,?)=\Phi_{b^{*}}(a)=0$

by (3.1). This ends the proof.

###  3.3. Proof of Theorem D

First, let us construct
$\widetilde{\psi}:S^{k}(\operatorname{Hom}(V_{k+1},V_{k}))\to
H^{0}(C,V_{k+1})$. A generic morphism $f:V_{k+1}\to V_{k}$ fits into an exact
sequence

$0\to{\mathcal{O}}\rTo{s}V_{k+1}\rTo{f}V_{k}\to 0$

and our goal is to find a degree $k$ polynomial formula for $s\in
H^{0}(C,V_{k+1})$ in terms of $f$. Let us consider the induced map

$\alpha_{f}:\det(V_{k+1})\otimes
V_{k+1}^{\vee}\simeq{\bigwedge}^{k}V_{k+1}\stackrel{{\scriptstyle{\bigwedge}^{k}f}}{{\longrightarrow}}{\bigwedge}^{k}V_{k}\simeq\det(V_{k}).$

Since $\det(V_{k})=\det(V_{k+1})$, we can view the dual map
$\alpha_{f}^{\vee}$ as a map ${\mathcal{O}}\to V_{k+1}$ and we have a well
known identity $f\circ\alpha_{f}^{\vee}=0$ (which follows from the vanishing
of ${\bigwedge}^{k+1}(f)$). Thus,

$\widetilde{\psi}(f)=\alpha_{f}^{\vee}\in H^{0}(C,V_{k+1})$

is the required degree $k$ polynomial map.

To find $\widetilde{\phi}$, we note that it should associate to a generic
$e\in\operatorname{Ext}^{1}(V_{k},{\mathcal{O}})$ a unique (up to rescaling)
$f\in\operatorname{Hom}(V_{k+1},V_{k})$ such that $e\circ f=0$. Now we can
find an autoequivalence $\Phi$ of $D^{b}(C)$ such that

$\Phi(V_{k+1})={\mathcal{O}},\ \ \Phi(V_{k})=V_{l+1},\ \
\Phi({\mathcal{O}}[1])=V_{l},$

where $l\equiv-(k+1)^{-1}\operatorname{mod}(n)$, $0<l<n$. Indeed, this is
essentially a statement about the action of
$\operatorname{SL}_{2}(\mathbb{Z})$ on the pairs of primitive vectors in
$\mathbb{Z}^{2}$ (by looking at the action of autoequivalences on pairs
$(\deg,\operatorname{rk})$). To find $l\operatorname{mod}(n)$ one uses the
invariant $\alpha(v_{1},v_{2})\in(\mathbb{Z}/\det(v_{1},v_{2}))^{*}$ of such a
pair defined by $v_{1}\equiv\alpha(v_{1},v_{2})\cdot
v_{2}\operatorname{mod}\det(v_{1},v_{2})\mathbb{Z}^{2}$ (see [14, Sec. 4]).

The above autoequivalence $\Phi$ identifies the composition map

$\operatorname{Hom}(V_{k+1},V_{k})\otimes\operatorname{Ext}^{1}(V_{k},{\mathcal{O}})\to\operatorname{Ext}^{1}(V_{k+1},{\mathcal{O}})$

we are interested in with the composition map

$\operatorname{Hom}({\mathcal{O}},V_{l+1})\otimes\operatorname{Hom}(V_{l+1},V_{l})\to\operatorname{Hom}({\mathcal{O}},V_{l}).$

Now we can apply our construction of $\widetilde{\psi}$ to the latter
composition map, and view it via the above isomorphisms as the needed degree
$l$ map $\widetilde{\phi}$ from $\operatorname{Ext}^{1}(V_{k},{\mathcal{O}})$
to $\operatorname{Hom}(V_{k+1},V_{k})$.

Similarly, to find $\widetilde{\rho}$, we find an autoequivalence $\Psi$ of
$D^{b}(C)$ such that

$\Psi(V_{k}[-1])={\mathcal{O}},\ \ \Psi({\mathcal{O}})=V_{m+1},\ \
\Psi(V_{k+1})=V_{m},$

where $m\equiv-1-k^{-1}\operatorname{mod}(n)$, $0<m<n$. As above, this gives
the required degree $m$ map $\widetilde{\psi}$ from $H^{0}(C,V_{k+1})$ to
$\operatorname{Ext}^{1}(V_{k},{\mathcal{O}})$.

######  Remark 3.3.1.

It is easy to see that if two sets of homogeneous polynomials of degree $d$,
$(f_{1},\ldots,f_{n})$ and $(\widetilde{f}_{1},\ldots,\widetilde{f}_{n})$
induce the same birational maps
${\mathbb{P}}^{n-1}\dashrightarrow{\mathbb{P}}^{n-1}$ and if
$(f_{1},\ldots,f_{n})$ have no common factor, then there exists a constant $c$
such that $\widetilde{f}_{i}=cf_{i}$. Indeed, the corresponding rational
function $\widetilde{f}_{1}/f_{1}=\ldots=\widetilde{f}_{n}/f_{n}$ on
${\mathbb{P}}^{n-1}$ has to be regular, hence, constant. Since the polynomials
$(p_{1},\ldots,p_{n})$ from Theorem B clearly have no common factor (as they
define a codimension $3$ variety), it follows that in the case $k=1$, our map
$\widetilde{\phi}$ agrees with $(p_{1},\ldots,p_{n})$.

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