# Dynamical Systems of Correspondences on the Projective Line I: Moduli Spaces
and Multiplier Maps

Rin Gotou Department of Mathematics, Graduate School of Osaka University
<EMAIL_ADDRESS>
###### Abstract

We consider moduli spaces of dynamical systems of correspondences over the
projective line as a generalization of moduli spaces of dynamical systems of
endomorphisms on the projective line. We obtain the rationality of the moduli
spaces. The rationality of the moduli space of degree $(d,e)$ correspondences
is obtained by constructing a projection to the one for the usual dynamical
systems of degree $d+e-1$ representation-theoretically. We also show that the
multiplier maps for the fixed points and the multiplier index theorem (Woods
Hole formula) are also reduced through the projection and obtain the reduced
form explicitly.

##  1 Introduction

Moduli spaces of dynamical systems over the projective line $\mathbb{P}^{1}$,
which parameterizes endomorphisms up to the conjugations by the automorphisms
on $\mathbb{P}^{1}$, are considered in [32] by using geometric invariant
theory (GIT, [25]).

Self-correspondence is a generalization of endomorphism. Some important
concepts on a dynamical system of endomorphism have natural generalization for
a dynamical system of (self-)correspondence. An example is Woods Hole formula,
which is originally stated for correspondence in [1] and [14] and have used
for dynamical system of self-maps from [36]. Other examples are the canonical
measure and the canonical height, they are originally stated for self-map and
generalized to correspondence in [5] and [15] respectively.

In this paper, we construct moduli spaces for dynamical systems of
correspondences on the projective line as an analogue of [32].

We firstly construct the moduli space $\operatorname{Corr}_{d,e}$ of
correspondences of degree $(d,e)$, which parameterizes the closed subschemes
$C\subset\mathbb{P}^{1}_{x}\times\mathbb{P}^{1}_{y}$ defined by an equation
$\sum_{i=0}^{d}\sum_{j=0}^{e}a_{ij}x^{i}y^{j}=0$. We consider the diagonal
action of $\operatorname{Aut}(\mathbb{P}^{1})\simeq\operatorname{PGL}_{2}$ on
$\mathbb{P}^{1}\times\mathbb{P}^{1}$, which is equivalent to the conjugation
action on the graph variety ([30]) and give characterization of stable points
and semistable points respectively. The loci were given in [32] for the case
$d=1$, that is, the case of moduli spaces of rational maps. We obtain a simple
generalization for our moduli spaces of correspondences.

######  Theorem 1.1.

The point of $\operatorname{Corr}_{d,e}$ which represents a correspondence $C$
is a stable point (resp. a semistable point) if and only if $C$ has no point
of multiplicity $\geq\frac{d+e}{2}$ (resp. of multiplicity $>\frac{d+e}{2}$)
on the diagonal of $\mathbb{P}^{1}\times\mathbb{P}^{1}$.

######  Corollary 1.2.

$\operatorname{Corr}_{d,e}^{ss}=\operatorname{Corr}_{d,e}^{s}$ if and only if
$d+e$ is odd.

######  Remark 1.3.

Geometric invariant theory (GIT, [25]) ensures the existence of a uniform
geometrical quotient of the stable locus and a compactification of the
quotient as a universal categorical quotient of the semistable locus.

The compactified moduli of dynamical systems
$\operatorname{Dyn}_{d,e}:=\operatorname{Corr}_{d,e}^{ss}\sslash\operatorname{PGL}_{2}$
is constructed as the projective spectrum of a graded invariant ring.
Computational invariant theory (CIT, [4]) gives precise information for an
invariant ring by treating degree-wise actions representation-theoretically
before taking the invariant subring. An application of CIT for the moduli
space of rational maps is done in [37].

In this paper, we construct (a $\mathbb{A}^{1}\setminus\\{0\\}$-family of)
rational maps

$\rho_{c}:\operatorname{Dyn}_{d,e}\dashrightarrow\operatorname{Dyn}_{1,d+e-1}$

from representation-theoretic (classically well-known, Clebsch-Gordan)
decomposition. The rationality of the moduli $\operatorname{Dyn}_{1,d+e-1}$ is
shown in [17] and argument in [17] also leads that these rational maps
$\rho_{c}$ are generically affine space bundle, therefore we can generalize
the rationality result for $\operatorname{Dyn}_{d,e}$.

######  Proposition 1.4.

(Proposition 5.12) $\operatorname{Dyn}_{d,e}$ is rational for $d,e\geq 1$ and
$(d,e)\neq(1,1)$.

An application of the moduli space is about an inverse problem, a problem
which asks the existence and the classification of dynamical systems with
proscribed invariants. An example of such parameter is the set of the
multipliers of periodic orbits.

For a dynamical system $f:\mathbb{P}^{1}\to\mathbb{P}^{1}$, we denote the
elementary symmetric polynomials of the fixed point multipliers by
$\sigma_{k}(f)$, that is,

$1+\sum_{i=1}^{d+1}\sigma_{k}(f)t^{k}=\prod_{x:f(x)=x}\left(1+f^{\prime}(x)t\right)$
(1)

for a formal variable $t$. The rational map

$\lambda_{1,(1,d)}:\operatorname{Dyn}_{1,d}\dashrightarrow\mathbb{P}^{d+1},\
\lambda_{1,(1,d)}([f]):=[1:\sigma_{1}(f):\cdots:\sigma_{d+1}(f)]$

is called the fixed point multiplier map, which is used to solve the
rationality of the moduli $\operatorname{Dyn}_{1,2}$, ([24], [32]), and also
to state and to consider inverse problems for multipliers
([7],[8],[13],[23],[35]).

A fundamental relation among multipliers was obtained as an application of the
Woods Hole Formula ([33],[36])

$\sum_{x:f(x)=x}\frac{1}{1-f^{\prime}(x)}=1,$ (2)

or equivalently,

$\sum_{i=0}^{d+1}(-1)^{i}(d-i)\sigma_{i}=0.$ (3)

In this paper, we construct multiplier maps for correspondence

$\lambda_{n,(d,e)}:\operatorname{Dyn}_{d,e}\dashrightarrow\mathbb{P}^{d+e},$

where we consider the fixed points as the points
$\\{z\in\mathbb{P}^{1}|f(z,z)=0\\}$ for a correspondence defined by
$f(x,y)=0$. The Woods Hole formula for correspondence is obtained in [1] and
[14], but the reduction using Clebsch-Gordan decomposition gives an
alternative proof. We can see that the fixed point multiplier map for
correspondences are also reduced through the rational maps
$\operatorname{Dyn}_{d,e}\dashrightarrow\operatorname{Dyn}_{1,d+e-1}$ used to
show the rationality of $\operatorname{Dyn}_{d,e}$, that is,

######  Proposition 1.5.

(Proposition 6.3) There exists a projective linear morphism
$A\in\operatorname{Aut}(\mathbb{P}^{d+e})=\operatorname{PGL}_{d+e}$ which
makes the following diagram commutative:

${\operatorname{Dyn}_{d,e}}$${\mathbb{P}^{d+e}}$${\operatorname{Dyn}_{1,d+e-1}}$${\mathbb{P}^{d+e}.}$$\scriptstyle{\lambda_{1,(d,e)}}$$\scriptstyle{A}$$\scriptstyle{\lambda_{1,(1,d+e-1)}}$

Representation theoretically, we can obtain the following formula about
resultant from the Woods Hole formula:

######  Theorem 1.6.

(Corollary 6.6) For an arbitrary field $k$ and any polynomials $f,g\in k[x]$
such that $\deg f\geq 3$ and $\deg f\geq\deg g+2$,

$\left.\frac{\partial}{\partial
t}\operatorname{res}_{x}(f(x),f^{\prime}(x)+tg(x))\right|_{t=0}=0.$

This paper is organized as follows: In Section 2, we set up notation and
terminology. In Section 3, we review representation theory of linear groups
including Clebsch-Gordan decomposition that we use later. In Section 4, we
construct moduli spaces of correspondences and rewrite the composition and
conjugation of correspondences as maps and actions on the moduli spaces. In
Section 5, we construct the moduli spaces of dynamical systems of
correspondences, characterize the stable and semistable loci of the
conjugation action and show the rationality of the moduli spaces. In Section
6, we construct multiplier maps and reformulate the multiplier index theorem
representation-theoretically.

### Acknowledgments

I would like to thank Seidai Yasuda for proofreading this paper, originally
the former half of my master thesis and for having supervised me in the master
course. I am grateful to Takehiko Yasuda for proofreading this paper, having
useful discussion and having accepted me as a master course and doctor course
student.

### Anacknowledgment

The author and this work are supported by JSPS KAKENHI Grant-in-Aid for
Research Fellow JP202122197.

##  2 Notation and Terminology

Throughout this paper, we follow [18] for the terminology of algebraic
geometry.

We fix a field $k$ of characteristic zero. Unless otherwise stated, we suppose
that every scheme is a scheme over $k$.

For a ring $R$ and a free $R$-module $M$ of finite rank, we denote by $R[M]$
the symmetric algebra
$\operatorname{Sym}^{\bullet}_{R}M:=\bigoplus_{n=0}^{\infty}M^{\otimes
n}/\langle v\otimes w-w\otimes v\rangle$. We note that if we take an $R$-basis
$\\{x_{1},\ldots,x_{r}\\}$ of $M$, $R[M]$ is canonically isomorphic to the
polynomial ring $R[x_{1},\ldots,x_{r}]$. If a group $G$ and a representation
$\rho:G\to\operatorname{Aut}_{R}(M)$ are also given, we write $R[M]^{G}$ for
the invariant ring.

We denote by $\operatorname{Sym}_{n}M$ the permutation-invariant part of the
$n$-th tensor power.

##  3 Representation Theory for $\operatorname{SL}_{2}$ and
$\operatorname{PGL}_{2}$

In this section, we provide representation-theoretic terminology. We only use
classically well-known operations, which are Cayley operators and Clebsch-
Gordan decomposition. Contents in this section are found in, for example, [3].

For the special linear group $\operatorname{SL}_{2}(k)$, we denote the trivial
representation on $k$ by $V_{0}$, the natural representation on $k^{2}$ by
$V_{1}$ and the symmetric tensor representation
$\operatorname{Sym}_{n}(V_{1})$ by $V_{n}$. From the construction, we have
$\dim V_{n}=n+1$.

######  Proposition 3.1.

If $n$ is an even number, then there exists an action of
$\operatorname{PGL}_{2}(k)$ on the vector space $V_{n}$ such that the pullback
action to $\operatorname{SL}_{2}(k)$ is $V_{n}$.

###### Proof.

We consider the representation $\Delta_{n}(-\frac{n}{2})$ of
$\operatorname{GL}_{2}(k)$ on $(k^{2})^{\otimes n}$ which is given by

$g\cdot(v_{1}\otimes v_{2}\otimes\cdots\otimes v_{n}):=(\det
g)^{-\frac{n}{2}}(gv_{1}\otimes gv_{2}\otimes\cdots\otimes gv_{n})\ (\text{for
}g\in\operatorname{GL}_{2}(k)).$

By $\Delta_{n}(-\frac{n}{2})$, any scalar matrix $\binom{c\ \ 0}{0\ \
c}\in\operatorname{GL}_{2}(k)$ acts trivially, therefore
$\Delta_{n}(-\frac{n}{2})$ is a representation of $\operatorname{PGL}_{2}(k)$.
Moreover, the action of $\operatorname{PGL}_{2}(k)$ by
$\Delta_{n}(-\frac{n}{2})$ commutes with the action of $\mathfrak{S}_{n}$ by
permuting components, therefore we can obtain a subrepresentation of
$\operatorname{PGL}_{2}(k)$ on $\operatorname{Sym}_{n}k^{2}$. The restriction
of the subrepresentation on $\operatorname{SL}_{2}(k)$ is the $n$-th symmetric
tensor of $V_{1}$, therefore this is a required representation. ∎

######  Proposition 3.2.

The dual representation of $V_{n}$ is isomorphic to $V_{n}$.

###### Proof.

Since $k$ is of characteristic $0$, it is enough to show the proposition for
$n=1$ (see Remark 3.3).

Let $[e_{1},e_{2}]$ be the basis of $V_{1}$ and $[f_{1},f_{2}]$ the dual basis
of $V_{1}^{*}$. For $A\in\operatorname{SL}_{2}(k)$, By the definition of dual
representation, $A\cdot f_{1}=f_{1}\circ A^{-1}$. Therefore the transpose
$(A^{-1})^{T}$ is the representation matrix of the action of $A$ by the dual
representation. We have $(A^{-1})^{T}=IAI^{-1}$ for $I=\binom{0\ -1}{1\ \ 0}$,
therefore the linear morphism such that $e_{1}\mapsto-f_{2},\ e_{2}\mapsto
f_{1}$ is an isomorphism. ∎

######  Remark 3.3.

The canonical morphism
$\operatorname{Sym}_{n}(V^{*})\to(\operatorname{Sym}_{n}V)^{*}$ induced from
the inclusion morphism

$\operatorname{Sym}_{n}V=(V^{\otimes n})^{\mathfrak{S}_{n}}\to V^{\otimes n}$

is an isomorphism because our field $k$ has characteristic 0 and the binomial
coefficients are invertible. In positive characteristic $p$, the canonical
morphism $\operatorname{Sym}_{n}(V^{*})\to(\operatorname{Sym}_{n}V)^{*}$ is
not isomorphism for $n\geq p$. In fact, the representations
$V_{n}=\operatorname{Sym}_{n}(V_{1})$ and
$V_{n}^{*}=\operatorname{Sym}_{n}(V_{1})^{*}$ are not isomorphic ([3]).

###  3.1 Weight

For a group $G$, we denote the abelian category of representations of $G$ over
$k$ by $\operatorname{Rep}_{k}(G)$. For a representation $V$ of a group $G$,
we denote by $\operatorname{Rep}_{k}(V)$ the minimal full subcategory of
$\operatorname{Rep}_{k}(G)$ includes $V$ as an object and whose objects are
closed by taking finite tensor products, finite direct sums, dual
representation and subrepresentations. The $K_{0}$-group of a small abelian
category $\mathcal{C}$ is defined by

$K_{0}(\mathcal{C}):=\bigoplus_{V\in\operatorname{objclass}(\mathcal{C})}\mathbb{Z}[V]{\bigg{/}}\binom{[V]-[V^{\prime}]-[V^{\prime\prime}]}{\text{for
exact }0\to V^{\prime}\to V\to V^{\prime\prime}\to 0}$

where $\operatorname{objclass}(\mathcal{C})$ is the set of isomorphism classes
of objects of $\mathcal{C}$. If the abelian category $\mathcal{C}$ is monoidal
and the tensor functor is exact in both arguments, $K_{0}(\mathcal{C})$
becomes a ring by $[V]\cdot[W]:=[V\otimes W]$. The induced product is
commutative if the monoidal product is symmetric, therefore
$K_{0}(\operatorname{Rep}_{k}(V))$ for any group $G$ and any representation
$V$ is a commutative ring.

######  Proposition 3.4.

Let $k$ be an infinite field. For the representation
$k(1):=(\operatorname{id}:k^{\times}\to\operatorname{GL}_{1}(k),k)$ of
$k^{\times}$, there exists an isomorphism
$w:K_{0}(\operatorname{Rep}_{k}(k(1)))\simeq\mathbb{Z}[q,q^{-1}]$ induced by
$[k(1)]\mapsto q$.

###### Proof.

We write $k(n)$ for $k(1)^{\otimes n}$ and $k(-n)$ for $k(n)^{*}$ for
nonnegative integer $n$. We define a ring morphism
$w^{-1}:\mathbb{Z}[q,q^{-1}]\to K_{0}(\operatorname{Rep}_{k}(k(1)))$ by
$q\mapsto[k(1)]$. The morphism $w^{-1}$ is well-defined by sending $q^{-1}$ to
$[k(-1)]$, because $k(0):=k(1)\otimes k(-1)$ is the trivial representation,
which is the multiplicative unit of the ring
$K_{0}(\operatorname{Rep}_{k}(k(1)))$.

We will show $w^{-1}$ is bijective.

(injectivity) For distinct integers $m\neq n$, there exists $a\in k^{\times}$
such that $a^{n}\neq a^{m}$ since $k$ is infinite. Therefore $k(n)$ and $k(m)$
are not isomorphic to each other. This leads to that $w^{-1}$ is injective.

(surjectivity) We show that every subrepresentation $W$ of a representation
$V:=\bigoplus_{i=1}^{m}k(n_{i})^{\oplus d_{i}}$ for distinct integers $n_{i}$
and positive integer $m$ is isomorphic to
$\bigoplus_{j=1}^{m^{\prime}}k(n^{\prime}_{j})^{\oplus d^{\prime}_{j}}$ for
some $n^{\prime}_{i}$s, $d^{\prime}_{i}$’s and $m^{\prime}$. Let $p_{i}:V\to
k(n_{i})^{\oplus d_{i}}$ be the projections. It is enough to show that for any
vector $v\in W$, $p_{i}(v)\in W$.

Let $v\in W$ be a vector. We denote the action of $a\in k^{\times}$ by $a\cdot
v$. Since $k$ is infinite, we can take $a\in k^{\times}$ as the map
$\\{n_{1},\ldots,n_{m}\\}\to k^{\times}:n_{i}\mapsto a^{n_{i}}$ is injective
(where $\\{n_{1},\ldots,n_{m}\\}$ is considered as a set). Then a linear map

$k^{\oplus
m}\to\sum_{i=1}^{m}kp_{i}(v):(b_{1},b_{2},\cdots,b_{n})\mapsto\sum_{i=1}^{m}b_{i}(a^{i-1}\cdot
v)$

is surjective because the representation matrix via the basis
$(b_{1},b_{2},\ldots,b_{n})$ and
$(p_{i_{1}}(v),p_{i_{2}}(v),\ldots,p_{i_{m^{\prime}}}(v))$ is full rank
because it is a submatrix of the Vandermonde matrix of
$\\{a^{n_{1}},\ldots,a^{n_{m}}\\}$, here
$(p_{i_{1}}(v),p_{i_{2}}(v),\ldots,p_{i_{m^{\prime}}}(v))$ is made by picking
nonzero components out of $(p_{1}(v),\ldots,p_{n}(v))$. Therefore we have
$p_{i}(v)=\sum_{i=1}^{m}b_{i}(a^{i-1}\cdot v)$ for some $b_{i}\in k$, that is
$p_{i}(v)\in W$ because $W$ is a subrepresentation. ∎

######  Definition 3.5.

For a group $G$, a group morphism $\lambda:k^{\times}\to G$ and a finite
dimensional $G$-representation $V$ such that the $k^{\times}$-representation
of $V$ induced from $V$ and $\lambda$ is in $\operatorname{Rep}_{k}(k(1))$, we
call the composition morphism

$K_{0}(\operatorname{Rep}_{k}(V))\to
K_{0}(\operatorname{Rep}_{k}(k(1)))\simeq\mathbb{Z}[q,q^{-1}]$

the weight morphism associated to $\lambda$ and denote by $w_{\lambda}$.

In the notation of the above definition, the Laurent polynomial
$w_{\lambda}([W])$ for any representation $W\in\operatorname{Rep}_{k}(V)$ has
positive coefficients.

We fix the group morphism

$c:k^{\times}\ni t\mapsto\binom{t\ \ \ 0\ }{0\ \
t^{-1}}\in\operatorname{SL}_{2}(k).$

It immediately follows that the $k^{\times}$-representation induced from
$V_{1}$ and $c$ is in $\operatorname{Rep}_{k}(k(1))$ and
$w_{c}(V_{1})=q+q^{-1}$.

###  3.2 Clebsch-Gordan Decomposition

For a finite dimensional vector space $V$, the space of $n$-ary forms, that
is, the vector space of the all degree $n$ homogeneous polynomials in $k[V]$
is naturally isomorphic to $(\operatorname{Sym}_{n}(V^{*}))^{*}$ in arbitrary
characteristic. Therefore the representation $V_{n}$ is identified with the
space of $n$-ary binomial forms in characteristic zero. The variables are the
standard basis of $V_{1}=k^{2}$ indeed.

######  Proposition 3.6.

Let $k$ be a field of characteristic zero. Then the representation $V_{n}$ is
irreducible.

###### Proof.

We note that $c(t)\cdot x_{0}^{n-i}x_{1}^{i}=t^{n-2i}x_{0}^{n-i}x_{1}^{i}$
under the identification between $V_{n}$ and the space of $n$-ary binomial
forms, where the pair $(x_{0},x_{1})$ is the standard basis of $k^{2}$.

Let $W$ be an arbitrary nonzero $\operatorname{SL}_{2}$-stable subspace of
$V_{n}$. By Proposition 3.4, we have

$W=\bigoplus_{i\in I_{W}}kx_{0}^{n-i}x_{1}^{i}$

for some nonempty $I_{W}\subset\\{0,1,\ldots,n\\}$. We take a monomial
$x_{0}^{n-i}x_{1}^{i}\in W$. Since $W$ is $\operatorname{SL}_{2}$-stable, we
have $\binom{1\ 1}{1\ 2}\cdot
x_{0}^{n-i}x_{1}^{i}=(x_{0}+x_{1})^{n-i}(x_{0}+2x_{1})^{i}\in W$. Therefore we
have $I_{W}\supset\\{0,\ldots,n\\}$, that is, $V=W$. ∎

Under the identification, if we have two binomial forms $f(x_{0},x_{1})$ and
$g(y_{0},y_{1})$ of degree $d$ and $e$ respectively, the Cayley operator
$\Omega_{xy}:=\partial_{x_{0}}\partial_{y_{1}}-\partial_{y_{0}}\partial_{x_{1}}$
gives a new binomial form

$\left.\left(\Omega_{xy}^{m}f(x_{0},x_{1})\cdot
g(y_{0},y_{1})\right)\right|_{(x_{0},x_{1})=(y_{0},y_{1})=(z_{0},z_{1})}$

of variables $(z_{0},z_{1})$ and degree $d+e-2m$, for $0\leq
m\leq\min(d,e)=\frac{1}{2}(d+e-|d-e|)$. This linear map is
$\operatorname{SL}_{2}$-equivariant, that is, we have a morphism of
representation

$\Omega^{m}:V_{d}\otimes V_{e}\to V_{d+e-2m}$

given by

$\Omega^{m}(f(x_{0},x_{1})\otimes
g(y_{0},y_{1})):=\left.\left(\Omega_{xy}^{m}f(x_{0},x_{1})\cdot
g(y_{0},y_{1})\right)\right|_{(x_{0},x_{1})=(y_{0},y_{1})=(z_{0},z_{1})}.$ (4)

######  Proposition 3.7.

(Clebsch-Gordan Decomposition, [3, Theorem 3.2.4]) If
$\operatorname{char}k=0$, the morphism

$\bigoplus_{i=0}^{|d-e|}\Omega^{i}:V_{d}\otimes V_{e}\to V_{d+e}\oplus
V_{d+e-2}\oplus\cdots\oplus V_{|d-e|+2}\oplus V_{|d-e|}$

is an isomorphism.

###### Proof.

It is enough to show that the morphism $\Omega^{m}:V_{d}\otimes V_{e}\to
V_{d+e-2m}$ is surjective for $0\leq m\leq\min(d,e)$. By Proposition 3.6, it
is sufficient to show that the morphism $\Omega^{m}:V_{d}\otimes V_{e}\to
V_{d+e-2m}$ is nonzero, which follows from the explicit calculation

$\Omega^{m}(x_{0}^{d}\otimes
y_{0}^{e-m}y_{1}^{m})=\frac{d!m!}{(d-m)!}z_{0}^{d+e-2m}.$

∎

##  4 Correspondence

Let $X$ and $Y$ be schemes. A closed subscheme of $X\times Y$ is called an
algebraic correspondence between $X$ and $Y$. We can regard a morphism $X\to
Y$ as an algebraic correspondence given by the graph of the morphism.
Therefore algebraic correspondence is a generalization of morphism.

###  4.1 The Moduli Space of Correspondences over $\mathbb{P}^{1}$

Over $\mathbb{P}^{1}\times\mathbb{P}^{1}$, we denote the line bundle
$p_{1}^{*}\mathcal{O}(d)\otimes p_{2}^{*}\mathcal{O}(e)$ by $\mathcal{O}(d,e)$
and the set of its global sections by $V_{d,e}$. We fix homogeneous
coordinates of each components, choosing bases $x_{0},x_{1}\in V_{1,0}$ and
$y_{0},y_{1}\in V_{0,1}$. Using these coordinates we have

$V_{d,e}=\left\\{\left.\sum_{0\leq i\leq d,\ 0\leq j\leq
e}a_{i,j}x_{0}^{d-i}x_{1}^{i}y_{0}^{e-j}y_{1}^{j}\right|a_{i,j}\in
k\right\\}.$

Later we will see that we can identify $V_{d,e}$ and $V_{d}\otimes V_{e}$
(Corollary 4.17).

######  Definition 4.1.

A nonzero element $f$ of $V_{d,e}$ is called a divisorial quasi-correspondence
of degree $(d,e)$. A $(d+1)\times(e+1)$-matrix $A=(a_{i,j})_{0\leq i\leq d,\\\
0\leq j\leq e}$ is called the coefficient matrix of $f$ if

$f=\sum_{0\leq i\leq d,\ 0\leq j\leq
e}a_{i,j}x_{0}^{d-i}x_{1}^{i}y_{0}^{e-j}y_{1}^{j}.$ (5)

We often abbreviate a divisorial quasi-correspondence (5) as
$f=\sum_{i,j}a_{ij}x^{i}y^{j}$ if the degree of correspondence $(d,e)$ is
apparent.

######  Definition 4.2.

A closed subscheme $C$ of $\mathbb{P}^{1}\times\mathbb{P}^{1}$ is said to be a
divisorial correspondence if
$\mathcal{O}_{C}=\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}/\mathcal{I}$
and $\mathcal{I}$ is a locally free sheaf of rank one. A divisorial
correspondence $C$ given by an ideal sheaf $\mathcal{I}$ is of degree $(d,e)$
if $\mathcal{I}$ is isomorphic to $\mathcal{O}(-d,-e)$ as an
$\mathcal{O}_{\mathbb{P}^{1}\times\mathbb{P}^{1}}$-module.

######  Remark 4.3.

We abbreviated the term “effective”, the condition what we required for
divisorial correspondence as a divisor on
$\mathbb{P}^{1}\times\mathbb{P}^{1}$.

######  Remark 4.4.

In a theory of quantum cohomology, quasimap is used to compactify moduli
spaces of maps ([2]). Divisorial quasi-correspondence is to divisorial
correspondence what quasimap is to map.

######  Proposition 4.5.

For the scheme $(X,\mathcal{O}_{X})$ and an invertible sheaf $\mathcal{L}$
over $X$, let $\operatorname{P}(\mathcal{L})$ be the set of the ideal sheaves
on $X$ isomorphic to $\mathcal{L}$ as an $\mathcal{O}_{X}$-module. Then we
have a bijection
$\phi_{X,\mathcal{L}}:\operatorname{P}(\mathcal{L})\simeq(\mathcal{L}^{\vee}(X)\setminus\\{0\\})/\mathcal{O}(X)^{\times}$
which is natural in the following sense:

For any morphism $f:Y\to X$ of schemes, the following diagram

${P(\mathcal{L})}$${(\mathcal{L}^{\vee}(X)\setminus\\{0\\})/\mathcal{O}(X)^{\times}}$${P(f^{*}\mathcal{L})}$${(f^{*}\mathcal{L}^{\vee}(Y)\setminus\\{0\\})/\mathcal{O}(Y)^{\times}}$$\scriptstyle{\phi_{X,\mathcal{L}}}$$\scriptstyle{\phi_{Y,f^{*}\mathcal{L}}}$

is commutative.

###### Proof.

We first construct a map
$\pi:\mathcal{L}^{\vee}(X)\setminus\\{0\\}\to\operatorname{P}(\mathcal{L})$.
For a global section $s\in\mathcal{L}^{\vee}(X)$, we denote by
$s_{f}:\mathcal{O}\to\mathcal{L}^{\vee}$ an injective sheaf morphism defined
by $1|_{U}\mapsto s|_{U}$ at every open $U\subset X$. By $\mathcal{I}_{f}$ we
denote the ideal sheaf defined by
$i_{s}\otimes_{\mathcal{O}}\mathcal{L}:\mathcal{L}\to\mathcal{O}$. This gives
a map
$\pi:\mathcal{L}^{\vee}(X)\setminus\\{0\\}\to\operatorname{P}(\mathcal{L})$.

We show $\pi$ is surjective. For a given ideal sheaf
$\mathcal{I}\to\mathcal{O}$, we fix an isomorphism
$\mathcal{L}\simeq\mathcal{I}$. By tensoring $\mathcal{L}^{\vee}$ to these
morphisms, we obtain an injective morphism
$\mathcal{O}\simeq\mathcal{L}^{\vee}\otimes\mathcal{I}\to\mathcal{L}^{\vee}$.
By taking the global sections, we obtain a $\mathcal{O}(X)$-linear map
$i:\mathcal{O}(X)\to\mathcal{L}^{\vee}(X).$ We set $s:=i(1)$ and then
$\mathcal{I}=\mathcal{I}_{s}$ follows immediately. Therefore $\pi$ is
surjective.

For
$c\in\mathcal{O}(X)^{\times}=\operatorname{Aut}_{\mathcal{O}_{X}}(\mathcal{O})$,
we have $\mathcal{I}_{s}=\mathcal{I}_{cs}$ as subsheaf of $\mathcal{O}$.
Therefore the map $\phi_{X,\mathcal{L}}$ is well-defined and injective. On the
other hand, if $\mathcal{I}_{s}=\mathcal{I}_{s}$ for
$s,s^{\prime}\in\mathcal{L}^{\vee}(X)$, then we have
$i_{s}(\mathcal{O})=i_{s}^{\prime}(\mathcal{O})$ as images on
$\mathcal{L}^{\vee}$. Therefore the images on global sections are equal, that
is, we have $\mathcal{O}(X)s=\mathcal{O}(X)s^{\prime}$ as a sub
$\mathcal{O}(X)$-module of $\mathcal{L}(X)$. Therefore, we have
$\pi(s)=\pi(s^{\prime})\Leftrightarrow\exists c\in\mathcal{O}(X)^{\times},\ \
s=cs^{\prime}$. Thus $\phi_{X,\mathcal{L}}$ is surjective.

To show the naturality, we have to show
$\phi_{X,\mathcal{L}}(s)\otimes_{\mathcal{O}_{X}}\mathcal{O}_{Y}=\phi_{Y,\mathcal{L}}(s\otimes
1)$, for any $s\in\mathcal{L}^{\vee}(X)\setminus\\{0\\}$. This is obvious from
the definition of the morphism of the tensor product functor. ∎

######  Corollary 4.6.

Let $\operatorname{Corr}_{d,e}$ be the fine moduli space of divisorial
correspondences of degree $(d,e)$. Then we have
$\operatorname{Corr}_{d,e}\simeq(\mathbb{A}^{(d+1)(e+1)}\setminus\\{0\\})/\mathbb{G}_{m}\simeq\mathbb{P}^{(d+1)(e+1)-1}$.

###  4.2 Resultant and Composition

For a Laurent polynomial $f$ in a variable $x$, we write $f[x^{i}]$ for the
coefficient of $x^{i}$.

######  Definition 4.7.

Let $R$ be a commutative ring $R$ and $x$ a variable. For $R[x]_{d}:=\\{f\in
R[x]\mid\deg_{x}f\leq d\\}$, the resultant
$\operatorname{res}_{x,(d,e)}:R[x]_{d}\times R[x]_{e}\to R$ is defined as the
determinant of Sylvester matrix

$\operatorname{res}_{x,(d,e)}(f,g):=\left|\begin{array}[]{ccccccccc}f[x^{0}]&f[x^{1}]&&\cdots&f[x^{d}]&&&\\\
&f[x^{0}]&f[x^{1}]&&\cdots&f[x^{d}]&&\\\ &&\ddots&\ddots&&\ddots&\ddots&\\\
&&&f[x^{0}]&f[x^{1}]&&\cdots&f[x^{d}]\\\
g[x^{0}]&g[x^{1}]&&\cdots&g[x^{e}]&&&\\\
&g[x^{0}]&g[x^{1}]&&\cdots&g[x^{e}]&&\\\ &&\ddots&\ddots&&\ddots&\ddots&\\\
&&&g[x^{0}]&g[x^{1}]&&\cdots&g[x^{e}]\end{array}\right|$

For homogeneous polynomials of two variables $x,y$,
$F(x,y)=x^{d}f(\frac{y}{x})$ and $G(x,y)=x^{e}f(\frac{y}{x})$ of degree $d$
and $e$ respectively, we define the homogeneous resultant

$\operatorname{res}_{[x,y]}(F(x,y),G(x,y)):=\operatorname{res}_{x,(d,e)}(f(x),g(x)).$

######  Remark 4.8.

([33, Chapter 2.4]) For an integral domain $R$, the homogeneous resultant of
two homogeneous polynomials on $R$ is $0$ if and only if the polynomials have
a common factor as homogeneous polynomials over $\operatorname{Frac}(R)$.

######  Proposition 4.9.

For any regular matrix $g=\binom{a\ b}{c\ d}\in GL_{2}(\mathbb{C})$ and two
homogeneous polynomials $F(x,y)$ and $G(x,y)$, we have

$\operatorname{res}_{[x,y]}(F(g\cdot(x,y)),G(g\cdot(x,y)))=(\det
g)^{\operatorname{deg}F\cdot\operatorname{deg}G}\operatorname{res}_{[x,y]}(F(x,y),G(x,y)),$

where $g\cdot(x,y):=(ax+by,cx+dy)$.

###### Proof.

See [9, Chapter 12.1]. ∎

In the theory of arithmetic dynamics, a homogeneous resultant is familiar as a
value which determines well-definedness of a rational self-map over
$\mathbb{P}^{1}$ and gives a Lipschitz constant with respect to the chordal
metric (see [33, Chapter 2.4]). In the rest of this section, we will see
composition of correspondences is also expressed by the resultant. This
expression roughly follows from the following observation: composition is
substitution, substitution is elimination and elimination is expressed by
(homogeneous) resultant.

######  Proposition 4.10.

Let $C=\operatorname{Spec}k[x,y]/(f(x,y))$ and
$D=\operatorname{Spec}k[x,y]/(g(x,y))$ be closed subschemes of
$\mathbb{A}^{2}$ and $i_{C}$ and $i_{D}$ their inclusion morphisms
respectively. Let $p:\widetilde{C\circ D}\to\mathbb{A}^{2}$ be the morphism
defined as the composition of the morphisms in upper row of the following
diagram

${\widetilde{C\circ D}}$${\mathbb{A}^{3}}$${\mathbb{A}^{2}}$${C\times
D}$${\mathbb{A}^{4}}$${\mathrm{p.b.}}$$\scriptstyle{\mathrm{id}\times\Delta\times\mathrm{id}}$$\scriptstyle{p_{13}}$$\scriptstyle{i_{C}\times
i_{D}}$

where $\Delta:\mathbb{A}^{1}\to\mathbb{A}^{2}$ is the diagonal morphism. Then
the scheme-theoretic image of $p$, denoted by $C\circ D\subset\mathbb{A}^{2}$,
is written as

$C\circ
D\simeq\operatorname{Spec}k[x,y]/(\operatorname{res}_{z,(d,e)}(f(x,z),g(z,y)))\to\mathbb{A}^{2}.$

###### Proof.

The following diagrams

${C}$${\mathbb{A}^{2}}$${D}$${\mathbb{A}^{2}}$${C\times\mathbb{A}^{1}}$${\mathbb{A}^{3}}$${\mathbb{A}^{1}\times
D}$${\mathbb{A}^{3}}$$\scriptstyle{i_{C}}$$\scriptstyle{\operatorname{id}\times(p_{2}\circ
i_{C})}$$\scriptstyle{\mathrm{id}\times\Delta}$$\scriptstyle{i_{D}}$$\scriptstyle{(p_{1}\circ
i_{D})\times\operatorname{id}}$$\scriptstyle{\Delta\times\operatorname{id}}$$\scriptstyle{i_{C}\times\operatorname{id}}$$\scriptstyle{\operatorname{id}\times
i_{D}}$

are pullback diagrams. On the other hand, we have a pullback diagram

${C\times D}$${C\times\mathbb{A}^{2}}$${\mathbb{A}^{2}\times
D}$${\mathbb{A}^{2}\times\mathbb{A}^{2}.}$$\scriptstyle{\operatorname{id}\times
i_{D}}$$\scriptstyle{i_{C}\times\operatorname{id}}$$\scriptstyle{\operatorname{id}\times
i_{C}}$$\scriptstyle{\operatorname{id}\times i_{D}}$ (6)

By base-changing (6) by
$\mathbb{A}^{3}\xrightarrow{\operatorname{id}\times\Delta\times\operatorname{id}}\mathbb{A}^{4}$,
we obtain the pullback diagram

${\widetilde{C\circ D}}$${C\times\mathbb{A}^{1}}$${\mathbb{A}^{1}\times
D}$${\mathbb{A}^{3}.}$${\mathrm{p.b.}}$$\scriptstyle{i_{C}\times\operatorname{id}}$$\scriptstyle{\operatorname{id}\times
i_{D}}$

This means $\widetilde{C\circ D}$ is equal to
$\operatorname{Spec}k[x,y,z]/(f(x,z),g(z,y))$. The assertion follows from the
property of resultant. ∎

######  Proposition 4.11.

Let $C,D\subset\mathbb{P}^{1}\times\mathbb{P}^{1}$ be divisorial
correspondences, $F=(f_{ij}),\ G=(g_{kl})$ be the coefficient matrices of $C$
and $D$ respectively. Let us consider the following diagram

${\widetilde{C\circ
D}}$${\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}}$${\mathbb{P}^{1}\times\mathbb{P}^{1}}$${C\times
D}$${\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}}$${\mathrm{p.b.}}$$\scriptstyle{\mathrm{id}\times\Delta\times\mathrm{id}}$$\scriptstyle{p_{13}}$$\scriptstyle{\subset\times\subset}$
(7)

Then the composition of the morphisms in upper row $\widetilde{C\circ
D}\to\mathbb{P}^{1}\times\mathbb{P}^{1}$ factors through the divisorial
correspondence given by the coefficient matrix $H=(h_{mn})$ such that

$\sum_{m,n}h_{mn}x_{0}^{dd^{\prime}-m}x_{1}^{m}y_{0}^{ee^{\prime}-n}y_{1}^{n}=\operatorname{res}_{[z_{0},z_{1}]}\left(\sum_{i,j}f_{ij}x_{0}^{d-i}x_{1}^{i}z_{0}^{e-j}z_{1}^{j},\sum_{k,l}g_{kl}z_{0}^{d^{\prime}-k}z_{1}^{k}y_{0}^{e^{\prime}-l}y_{1}^{l}\right)$

if $H\neq 0$.

###### Proof.

For a standard open covering of $\mathbb{P}^{1}$,
$\\{U_{0}=\mathbb{P}^{1}\setminus\\{0\\},U_{1}=\mathbb{P}^{1}\setminus\\{\infty\\}\\}$,
we denote $U_{\alpha_{1}\alpha_{2}\ldots\alpha_{n}}$ for the open subscheme
$U_{\alpha_{1}}\times\cdots\times U_{\alpha_{n}}$ of $(\mathbb{P}^{1})^{n}$.
The closed subscheme
$\operatorname{Im}(\operatorname{id}\times\Delta\times\operatorname{id})$ of
$(\mathbb{P}^{1})^{4}$ is covered by the open subschemes
$U_{\alpha\beta\beta\gamma}$ of $(\mathbb{P}^{1})^{4}$. Proposition 4.10 gives
the construction of the upper row of (7) over each
$U_{\alpha\beta\beta\gamma}\simeq\mathbb{A}^{4}$, where the codomain is
restricted to
$U_{\alpha\gamma}\simeq\mathbb{A}^{2}\subset\mathbb{P}^{1}\times\mathbb{P}^{1}$.
By combining these constructions, we obtain the assertion. ∎

######  Remark 4.12.

In the situation of Proposition 4.11, the properties of resultants gives that
we have $H=0$ if and only if $f(x,y)=\bar{f}(x,y)(ay_{0}+by_{1})$ and
$g(x,y)=(ax_{0}+bx_{1})\bar{g}(x,y)$ for some $a,b\in\bar{k}$ and some
divisorial correspondences $\bar{f}$ and $\bar{g}$.

######  Definition 4.13.

The composition map
$\circ:\operatorname{Corr}_{d,e}\times\operatorname{Corr}_{d^{\prime},e^{\prime}}\dashrightarrow\operatorname{Corr}_{dd^{\prime},ee^{\prime}}$
is the map that sends the pair of divisorial quasi-correspondence
$(f(x,y),g(x,y))$ to the quasi-correspondence $(f\circ
g)(x,y):=\operatorname{res}_{z}(f(x,z),g(z,y))$.

######  Proposition 4.14.

The composition map is associative, that is, the following diagram is
commutative:

${\operatorname{Corr}_{d,e}\times\operatorname{Corr}_{d^{\prime},e^{\prime}}\times\operatorname{Corr}_{d^{\prime\prime},e^{\prime\prime}}}$${\operatorname{Corr}_{dd^{\prime},ee^{\prime}}\times\operatorname{Corr}_{d^{\prime\prime},e^{\prime\prime}}}$${\operatorname{Corr}_{d,e}\times\operatorname{Corr}_{d^{\prime}d^{\prime\prime},e^{\prime}e^{\prime\prime}}}$${\operatorname{Corr}_{dd^{\prime}d^{\prime\prime},ee^{\prime}e^{\prime\prime}}}$$\scriptstyle{\circ\times\mathrm{id}}$$\scriptstyle{\mathrm{id}\times\circ}$$\scriptstyle{\circ}$$\scriptstyle{\circ}$
(8)

###### Proof.

By Remark 4.12, the compositions of the two diagonal paths in (8) are rational
map. By Proposition 4.11, the images through the two paths of a general point
of
$\operatorname{Corr}_{d,e}\times\operatorname{Corr}_{d^{\prime},e^{\prime}}\times\operatorname{Corr}_{d^{\prime\prime},e^{\prime\prime}}$,
which indicates the tuple of correspondences
$(C,C^{\prime},C^{\prime\prime})$, are both given by the upper row of the
following diagram:

${\widetilde{C\circ C^{\prime}\circ
C^{\prime\prime}}}$${(\mathbb{P}^{1})^{4}}$${(\mathbb{P}^{1})^{2}}$${C\times
C^{\prime}\times
C^{\prime\prime}}$${(\mathbb{P}^{1})^{6}}$${\mathrm{p.b.}}$$\scriptstyle{\mathrm{id}\times\Delta\times\Delta\times\mathrm{id}}$$\scriptstyle{p_{14}}$$\scriptstyle{\subset\times\subset\times\subset}$

Therefore (8) is commutative. ∎

######  Definition 4.15.

The iteration map
$\Psi_{n}:\operatorname{Corr}_{d,e}\dashrightarrow\operatorname{Corr}_{d^{n},e^{n}}$
is the map that sends the divisorial quasi-correspondence $f(x,y)$ to the
quasi-correspondence $(f\circ f\circ\cdots\circ f)(x,y)$.

######  Proposition 4.16.

Let $g\in\operatorname{PGL}_{2}\simeq\operatorname{Aut}(\mathbb{P}^{1})$ be a
morphism given by $g([x_{0}:x_{1}])=[dx_{0}+cx_{1}:bx_{0}+ax_{1}]$ and
$f(x,y)$ be a divisorial quasi-correspondence. Then we have

$(g\circ f\circ
g^{-1})(x,y)=f(dx_{0}+cx_{1},bx_{0}+ax_{1},dy_{0}+cy_{1},by_{0}+ay_{1}).$

###### Proof.

Let $g\in\operatorname{PGL}_{2}$ be the morphism given by
$g([x_{0}:x_{1}])=[dx_{0}+cx_{1}:bx_{0}+ax_{1}]$. The graph correspondence
$\Gamma_{g}$ is given by a divisorial quasi-correspondence

$G=G(x,y)=(bx_{0}+ax_{1})y_{0}-(dx_{0}+cx_{1})y_{1}.$

Let $f\in\operatorname{Corr}_{d,e}$ be a divisorial correspondence. We have

$\displaystyle\operatorname{res}_{[z_{0},z_{1}]}(f(x,z),G(z,y))$
$\displaystyle=\left|\begin{array}[]{cccc}f[y^{0}]&f[y^{1}]&\cdots&f[y^{d}]\\\
G[x^{0}]&G[x^{1}]&&\\\ &\ddots&\ddots&\\\
&&G[x^{0}]&G[x^{1}]\end{array}\right|$
$\displaystyle=\sum_{i=0}^{d}(-1)^{i-1}f[y^{i}]G[x^{1}]^{d-i}G[x^{0}]^{i}$
$\displaystyle=\sum_{i=0}^{d}f[y^{i}]G[x_{1}]^{d-i}(-G[x_{0}])^{i}$
$\displaystyle=\sum_{i=0}^{d}f[y^{i}](ay_{0}-cy_{1})^{d-i}(-by_{0}+dy_{1})^{i}$
$\displaystyle=f(x_{0},x_{1},ay_{0}-cy_{1},-by_{0}+dy_{1})$

and

$\displaystyle\operatorname{res}_{[z_{0},z_{1}]}(G(x,z),f(z,y))$
$\displaystyle=\left|\begin{array}[]{cccc}G[y^{0}]&G[y^{1}]&&\\\
&\ddots&\ddots&\\\ &&G[y^{0}]&G[y^{1}]\\\ f[x^{0}]&f[x^{1}]&\cdots&f[x^{d}]\\\
\end{array}\right|$
$\displaystyle=\sum_{i=0}^{d}(-1)^{d+i-1}f[x^{i}]G[y^{1}]^{d-i}G[y^{0}]^{i}$
$\displaystyle=\sum_{i=0}^{d}f[x^{i}](-G[y_{1}])^{d-i}G[y_{0}]^{i}$
$\displaystyle=\sum_{i=0}^{d}f[x^{i}](dx_{0}+cx_{1})^{d-i}(bx_{0}+ax_{1})^{i}$
$\displaystyle=f(dx_{0}+cx_{1},bx_{0}+ax_{1},y_{0},y_{1}).$

For the inverse element $g^{-1}\in\operatorname{PGL}_{2}$, the graph variety
$\Gamma_{g^{-1}}$ is given by a divisorial quasi-correspondence

$G^{-1}(x,y):=(-bx_{0}+dx_{1})y_{0}-(ax_{0}-cx_{1})y_{1}.$

Therefore we have

$\displaystyle(g\circ f\circ g^{-1})(x,y)$ $\displaystyle=(g\circ
f)(x_{0},x_{1},dy_{0}+cy_{1},by_{0}+ay_{1})$
$\displaystyle=f(dx_{0}+cx_{1},bx_{0}+ax_{1},dy_{0}+cy_{1},by_{0}+ay_{1}).$

∎

######  Corollary 4.17.

For the conjugation action

$\operatorname{Corr}_{d,e}\times\operatorname{PGL}_{2}\to\operatorname{Corr}_{d,e}:(f,g)\mapsto
g\circ f\circ g^{-1},$ (9)

the induced action of $\operatorname{SL}_{2}$ on
$\operatorname{Corr}_{d,e}\simeq\mathbb{P}(V_{d,e})$ is given by a
representation on $V_{d,e}$ and the representation is isomorphic to
$V_{d}\otimes V_{e}$.

###### Proof.

By Proposition 4.16 the action of $\operatorname{SL}_{2}$ on $V_{d,e}$ is
isomorphic to the one on the tensor space of the space of $d$-ary forms and
the space of $e$-ary forms,
$\operatorname{Sym}_{d}(V^{*}_{1})^{*}\otimes\operatorname{Sym}_{e}(V^{*}_{1})^{*}$.
By Proposition 3.2, it is isomorphic to $V_{d}\otimes V_{e}$. ∎

##  5 Fundamental Properties of The Moduli of Correspondence

In this section, we give simple generalizations of the results in [32] and
[17], a characterization of the stable/semistable locus of the group action
and the rationality of the moduli spaces.

###  5.1 Stability of Group Action

From now on until Theorem 5.9, we briefly review the geometric invariant
theory [25].

######  Definition 5.1.

([25, Definition 1.6]) Let $G$ be a reductive group scheme and $X$ a scheme
with $G$-action $\sigma:G\times X\to X$. For an invertible sheaf $\mathcal{L}$
over $X$, an isomorphism $\phi:\sigma^{*}\mathcal{L}\simeq
p_{2}^{*}\mathcal{L}$ is said to be $G$-linearization if $\phi$ satisfies the
cocycle condition

$p_{23}^{*}\phi\circ(\operatorname{id}_{G}\times\sigma)^{*}\phi=(\mu\times\operatorname{id}_{X})^{*}\phi\text{
(on }G\times G\times X).$

######  Remark 5.2.

If $\mathcal{L}$ is very ample and $G$ is affine, then $G$-linearization is
described as the $G(\mathcal{O}(X))$-action on $\mathcal{L}(X)$ compatible
with $\sigma$.

######  Remark 5.3.

For a $G$-linearization $\phi$ of an invertible sheaf $\mathcal{L}$ over a
normal scheme $X$, $\phi^{n}:\sigma^{*}\mathcal{L}^{n}\simeq
p_{2}^{*}\mathcal{L}^{n}$ is a $G$-linearization of $\mathcal{L}^{n}$.

######  Remark 5.4.

([25, Proposition 1.4]) If there exists no surjective homomorphism
$G\to\mathbb{G}_{m}$ of group schemes and $X\times_{k}\bar{k}$ is normal,
$G$-linearization $\phi$ of an invertible sheaf $\mathcal{L}$ is unique if
exists.

For a given action and a given invertible sheaf, $G$-linearization may not be
unique, for instance, if the action is trivial, any regular homomorphism
$G\to\operatorname{Aut}(\mathcal{L})$ gives a $G$-linearization.

######  Definition 5.5.

([25, Definition 1.7]) Let $G$ be a reductive group, $X$ an algebraic variety
with $G$-action and $P$ a geometric point of $X$.

  1. (i)

$P$ is said to be pre-stable if the stabilizer group of $P$ is finite and
there exists a $G$-stable affine open neighborhood of $P$.

Moreover, we suppose that $\mathcal{L}$ be an ample invertible sheaf over $X$
with $G$-linearization.

  1. (i)

$P$ is said to be $\mathcal{L}$-semistable if there exists $f\in
H^{0}(X,\mathcal{L}^{n})^{G}$ such that $f(P)\neq 0$ and $X_{f}$ is affine.

  2. (ii)

$P$ is said to be (proper) $\mathcal{L}$-stable if $P$ is
$\mathcal{L}$-semistable and pre-stable.

The set of pre-stable (resp. $\mathcal{L}$-semistable, $\mathcal{L}$-stable)
geometric points is the set of geometric points of an open subscheme of $X$
called pre-stable (resp. $\mathcal{L}$-semistable, $\mathcal{L}$-stable)
locus. We denote the loci by $X^{s}(\operatorname{Pre})$ (resp.
$X^{ss}(\mathcal{L})$, $X^{s}(\mathcal{L})$).

######  Remark 5.6.

For a $G$-variety $X$ which is isomorphic to a projective space
$\mathbb{P}(V)$, we sometimes write $X^{s}$ and $X^{ss}$ for the stable locus
and semistable locus of any $\mathcal{O}(n)$ with $G$-linearization.

######  Remark 5.7.

([25, Converse 1.12]) If the categorical (resp. the geometric) quotient of $X$
by $G$ exists, then $X=X^{ss}(\mathcal{L})$ (resp. $X=X^{s}(\mathcal{L})$) for
some ample invertible sheaf $\mathcal{L}$ over $X$ with $G$-linearization.

######  Theorem 5.8.

([25, Theorem 1.1]) Let $G$ be a reductive group, $X=\operatorname{Spec}R$ an
affine algebraic variety with $G$-action. Then the categorical quotient
$X\sslash G$ is constructed as $\operatorname{Spec}R^{G}$.

######  Theorem 5.9.

([25, p.40]) Let $G$ be a reductive group, $X$ a proper algebraic variety with
$G$-action, $\mathcal{L}$ a very ample invertible sheaf with
$G$-linearization. Then the categorical quotient $X^{ss}(\mathcal{L})\sslash
G$ is constructed as
$\operatorname{Proj}\bigoplus_{i=0}^{\infty}H^{0}(\mathcal{L}^{i},X)^{G}.$

By Proposition 4.6 and Corollary 4.17, the moduli space of correspondences
$\operatorname{Corr}_{d,e}$ is isomorphic to $\mathbb{P}(V_{d}\otimes V_{e})$
equivariantly with respect to $\operatorname{SL}_{2}$-actions. By Remark 5.2,
an $\operatorname{SL}_{2}$-linearization is given by the natural
representation over $H^{0}(\mathbb{P}(V_{d}\otimes
V_{e}),\mathcal{O}(1))\simeq(V_{d}\otimes V_{e})^{*}$, the dual representation
of $V_{d}\otimes V_{e}$. The last representation is isomorphic to
$V_{d}\otimes V_{e}$ by Proposition 3.2. By Theorem 5.9, the uniform
categorical quotient
$\operatorname{Corr}_{d,e}^{ss}\sslash\operatorname{SL}_{2}$ is constructed as
$\operatorname{Proj}k[V_{d}\otimes V_{e}]^{\operatorname{SL}_{2}}$.

The reductive group $\operatorname{PGL}_{2}$ also has a $G$-linearization on
$\mathcal{O}(\frac{2}{\gcd(2,d+e)})$ by Proposition 3.7 and Corollary 4.17. In
fact, we have $k[V]^{\operatorname{SL}_{2}}=k[V]^{\operatorname{PGL}_{2}}$ for
any finite dimensional representation $V$ of $\operatorname{PGL}_{2}$,
therefore
$\operatorname{Corr}_{d,e}\sslash\operatorname{PGL}_{2}\simeq\operatorname{Proj}\bigoplus_{i=0}^{\infty}k[V_{d}\otimes
V_{e}]_{2i}^{\operatorname{PGL}_{2}}\simeq\operatorname{Proj}\bigoplus_{i=0}^{\infty}k[V_{d}\otimes
V_{e}]^{\operatorname{SL}_{2}}$.

######  Theorem 5.10.

A divisorial correspondence $C$ given by a quasi-divisorial correspondence
$\sum_{i=0}^{d}\sum_{j=0}^{e}a_{ij}x^{i}y^{j}$ is _not_ a stable point (resp.
_not_ a semistable point) of $\operatorname{Corr}_{d,e}$ if and only if there
exists an $\operatorname{SL}_{2}$-conjugate of the coefficient matrix
$(b_{ij})$ such that $b_{ij}=0$ for all $i+j<\frac{d+e}{2}$ (resp.
$i+j\leq\frac{d+e}{2}$).

###### Proof.

By [32], any maximal subtorus of $\operatorname{SL}_{2}(k)$ is conjugate to
$c:\mathbb{G}_{m}\to\operatorname{SL}_{2}(k)$ such that $c(t):=\binom{t\ \ 0\
}{0\ t^{-1}}$. For a divisorial quasi-correspondence $f$, we have

$c(t)\cdot
f=f(tx_{0},t^{-1}x_{1},ty_{0},t^{-1}y_{1})=\sum_{i=0}^{d}\sum_{j=0}^{e}t^{(d+e)-2(i+j)}a_{ij}x^{i}y^{j}.$

Therefore by the numerical criterion ([25, Theorem 2.1]), we obtain the claim.
∎

###  5.2 Rationality

Let $V=V^{\prime}\oplus V^{\prime\prime}$ be a representation of a reductive
group $G$. Then we have the inclusion morphism $k[V^{\prime}]\to k[V]$. This
morphism is $G$-equivariant by definition, therefore leads to the morphism
$k[V^{\prime}]^{G}\to k[V]^{G}$ and the rational map
$\mathbb{P}(V^{*})^{ss}\sslash
G\dashrightarrow\mathbb{P}((V^{\prime})^{*})^{ss}\sslash G$. If the action of
the group $G$ on $(V^{\prime})^{*}$ is free for general point, then the fiber
of general point of $\mathbb{P}((V^{\prime})^{*})^{ss}\sslash G$ via
$\mathbb{P}(V^{*})^{ss}\sslash
G\dashrightarrow\mathbb{P}((V^{\prime})^{*})\sslash G$ is naturally isomorphic
to $(V/V^{\prime})^{*}$. Therefore we have the following proposition.

######  Proposition 5.11.

For a representation of a reductive group $G$ and a representation $V$,
$\mathbb{P}(V^{*})\sslash G$ is rational if there exists a subrepresentation
$V^{\prime}\subset V$ such that the action of $G$ on $V^{\prime}$ is
generically free and $\mathbb{P}((V^{\prime})^{*})\sslash G$ is rational.

We fix a infinity field $k$. Let $1\leq d\leq e$ be positive integers. Then
the Clebsch-Gordan decomposition (Proposition 3.7) gives

$\dim_{k}\operatorname{Hom}_{\operatorname{Rep}_{k}(\operatorname{SL}_{2})}(V_{d+e-1}\otimes
V_{1},V_{d}\otimes V_{e})=2.$

We fix an injective morphism

$\rho_{1}:V_{d+e-1}\otimes V_{1}\to V_{d}\otimes V_{e}.$

The morphism $\rho_{1}$ gives rational maps

$\rho_{1}:\operatorname{Corr}_{d,e}\dashrightarrow\operatorname{Corr}_{1,d+e-1}$

and

$\bar{\rho_{1}}:\operatorname{Dyn}_{d,e}\dashrightarrow\operatorname{Dyn}_{1,d+e-1}.$

######  Proposition 5.12.

$\operatorname{Dyn}_{d,e}$ is rational for $d,e\geq 1$ and $(d,e)\neq(1,1)$.

###### Proof.

For the case $d=1$, this is shown in [17]. In the same paper, it is also shown
$\operatorname{SL}_{2}(k)$ acts generically free on the representation
$V_{D}\otimes V_{1}$ for $D\geq 3$. Therefore the general case follows from
Proposition 5.11. ∎

######  Remark 5.13.

In [19], [20], [21], [22] and [31], the rationality of
$\operatorname{Corr}_{d,e}\sslash\operatorname{SL}_{2}\times\operatorname{SL}_{2}$
is shown for some $(d,e)$’s. The rationality of $\operatorname{SL}_{2}$ leads
to the rationality of $\operatorname{Corr}_{d,e}$ for these cases.

##  6 Multiplier Map

###  6.1 Construction

Let $f$ be a divisorial quasi-correspondence

$f(x_{0},x_{1},y_{0},y_{1})=\sum_{0\leq i\leq d,\ 0\leq j\leq
e}a_{ij}x_{0}^{d-i}x_{1}^{i}y_{0}^{e-j}y_{1}^{j}.$

We fix affine coordinates $x=\frac{x_{1}}{x_{0}}$ and $y=\frac{y_{1}}{y_{0}}$.
Then the restriction of the divisorial quasi-correspondence given by $f$ on
the affine chart
$\mathbb{A}_{x}\times\mathbb{A}_{y}=U_{\mathbb{P}^{1}_{x}}^{+}(x_{0})\times
U_{\mathbb{P}^{1}_{y}}^{+}(y_{0})$ of
$\mathbb{P}^{1}_{x}\times\mathbb{P}^{1}_{y}$ is given by $V(\bar{f}(x,y))$,
where

$\bar{f}(x,y):=\sum_{0\leq i\leq d,\ 0\leq j\leq e}a_{ij}x^{i}y^{j}.$

Intuitively, from implicit function theorem, the derivative (we denote by)
$\frac{dy}{dx}$ on $V(\bar{f}(x,y))$ around a point $(\bar{x},\bar{y})\in
V(\bar{f}(x,y))$ is given by

$\frac{dy}{dx}=-\left.\frac{\partial_{x}\bar{f}(x,y)}{\partial_{y}\bar{f}(x,y)}\right|_{(x,y)=(\bar{x},\bar{y})}.$
(10)

if $\left.\partial_{y}\bar{f}(x,y)\right|_{(x,y)=(\bar{x},\bar{y})}\neq 0$. In
fact, from the universality of differential sheaf, if we change affine
coordinates of each components equivariantly, the value $\frac{dy}{dx}$ does
not changes. We use (10) as the definition of the derivative $\frac{dy}{dx}$
of $f$ at $(\bar{x},\bar{y})\in V(f)\setminus V(\partial_{y}f)$. We note that
here

$\displaystyle x_{0}^{d}y_{0}^{e}\partial_{x}\bar{f}(x,y)$
$\displaystyle=\partial_{x_{1}}f(x,y)=d\cdot f(x,y)-\partial_{x_{0}}f(x,y)\
\text{and}$ $\displaystyle x_{0}^{d}y_{0}^{e}\partial_{y}\bar{f}(x,y)$
$\displaystyle=\partial_{y_{1}}f(x,y)=d\cdot f(x,y)-\partial_{y_{0}}f(x,y)$

as rational functions of $x_{0},x_{1},y_{0}$ and $y_{1}$. Therefore, at a
point $(\bar{x},\bar{y})\in V(f)\setminus V(\partial_{y}f)$, we have

$\displaystyle\frac{dy}{dx}=-\frac{\partial_{x_{1}}f(\bar{x},\bar{y})}{\partial_{y_{1}}f(\bar{x},\bar{y})}=-\frac{-\partial_{x_{0}}f(\bar{x},\bar{y})}{-\partial_{y_{0}}f(\bar{x},\bar{y})}.$

For convenience, we define rational functions $\partial_{x}f(x,y)$ and
$\partial_{y}f(x,y)$ as

$\displaystyle x\partial_{x}f(x,y)$
$\displaystyle=x_{1}\partial_{x_{1}}f(x,y)-x_{0}\partial_{x_{0}}f(x,y)\text{
and }$ $\displaystyle y\partial_{y}f(x,y)$
$\displaystyle=y_{1}\partial_{y_{1}}f(x,y)-y_{0}\partial_{y_{0}}f(x,y).$ (11)

We construct the symmetric form of the fixed point multipliers $\sigma_{i}(f)$
by

$1+\sum_{i=1}^{d+e}(-1)^{i}\sigma_{i}(f)t^{i}=\prod_{z:f(z,z)=0}\left(1+\frac{\partial_{x}f(z,z)}{\partial_{y}f(z,z)}t\right).$
(12)

The map $\operatorname{Corr}_{d,e}\dashrightarrow\mathbb{A}^{d+e}$ given by
$f\mapsto(\sigma_{i}(f))_{i=1,\ldots d+e}$ is a morphism if $d=1$ and $f$ is a
morphism, but is just a rational map for $d\geq 2$ because there are points
such that $\partial_{y}f(z)=0$. To incorporate $y$-critical points, then we
prefer to consider the following homogenized form of the multiplier map:

######  Definition 6.1.

The fixed point multiplier map is the rational map

$\lambda_{1,(d,e)}:\operatorname{Corr}_{d,e}\ni
f\mapsto[\operatorname{res}_{z}(f(z,z),\partial_{x}f(z,z)dx+\partial_{y}f(z,z)dy)]\in\mathbb{P}(D_{d+e}),$

where we regard $dx$ and $dy$ as just a variables and $D_{n}$ be the space of
$n$-ary forms in them.

The $n$-th multiplier map is the rational map
$\lambda_{n,(d,e)}:=\lambda_{1,(d^{n},e^{n})}\circ\Psi_{n}$.

######  Remark 6.2.

By some elementary transformation of Sylvester matrix, we have a fundamental
property of resultant

$\operatorname{res}_{x}(f(x),g(x)+af(x))=\operatorname{res}_{x}(f(x),g(x))$

for any $f,g\in k[x]$ and $a\in k$. For any divisorial quasi-correspondence
$f(x,y)$, this property of resultant and
$x_{1}\partial_{x_{1}}f(x,y)=df(x,y)-x_{0}\partial_{x_{0}}f(x,y)$ shows the
equality

$[\operatorname{res}_{z}(f(z,z),\partial_{x}f(z,z)dx+\partial_{y}f(z,z)dy)]=[\operatorname{res}_{z}(f(z,z),z_{0}\partial_{x_{0}}f(z,z)dx+z_{0}\partial_{y_{0}}f(z,z)dy)]$

of points of $\mathbb{P}(D_{d+e})$.

###  6.2 Resultant Form of Woods Hole Formula

Let $f$ be a divisorial quasi-correspondence of degree $(d,e)$. Then we have

$\displaystyle z(\partial_{x}f)(z,z)$
$\displaystyle=\sum_{i,j}(d-2i)a_{ij}z^{i+j}\text{ and }$ $\displaystyle
z(\partial_{y}f)(z,z)$ $\displaystyle=\sum_{i,j}(e-2j)a_{ij}z^{i+j}$

from (11) and

$\displaystyle z\partial_{z}(\Omega^{0}f)(z)$
$\displaystyle=\sum_{i,j}(d+e-2i-2j)a_{ij}z^{i+j}\text{ and }$ $\displaystyle
z_{0}z_{1}(\Omega^{1}f)(z)$ $\displaystyle=\sum_{i,j}(-ei+dj)a_{ij}z^{i+j}$

from the definition of Cayley operator (4). This leads to

$\left(\begin{matrix}z\partial_{z}(\Omega^{0}f)(z)\\\
z_{0}z_{1}(\Omega^{1}f)(z)\end{matrix}\right)=\left(\begin{matrix}1&1\\\
\frac{e}{2}&\frac{-d}{2}\end{matrix}\right)\left(\begin{matrix}z(\partial_{x}f)(z,z)\\\
z(\partial_{y}f)(z,z)\end{matrix}\right)$

and

$\displaystyle\left(\begin{matrix}z(\partial_{x}f)(z,z)\\\
z(\partial_{y}f)(z,z)\end{matrix}\right)$
$\displaystyle=\frac{-2}{d+e}\left(\begin{matrix}\frac{-d}{2}&-1\\\
\frac{-e}{2}&1\end{matrix}\right)\left(\begin{matrix}z\partial_{z}(\Omega^{0}f)(z)\\\
z_{0}z_{1}(\Omega^{1}f)(z)\end{matrix}\right)$
$\displaystyle=\left(\begin{matrix}\frac{d}{d+e}&\frac{2}{d+e}\\\
\frac{e}{d+e}&-\frac{2}{d+e}\end{matrix}\right)\left(\begin{matrix}z\partial_{z}(\Omega^{0}f)(z)\\\
z_{0}z_{1}(\Omega^{1}f)(z)\end{matrix}\right).$

This coordinate change gives the following proposition.

######  Proposition 6.3.

There exists a projective linear morphism
$A\in\operatorname{Aut}(\mathbb{P}^{d+e})=\operatorname{PGL}_{d+e}(k)$ which
makes the following diagram commutative:

${\operatorname{Dyn}_{d,e}}$${\mathbb{P}(D_{d+e})}$${\operatorname{Dyn}_{1,d+e-1}}$${\mathbb{P}(D_{d+e}).}$$\scriptstyle{\lambda_{1,(d,e)}}$$\scriptstyle{\bar{\rho}_{1}}$$\scriptstyle{A}$$\scriptstyle{\lambda_{1,(1,d+e-1)}}$

###### Proof.

By the definition of $\rho_{1}$, we have
$\Omega^{i}\rho_{1}(f)=c_{i}\cdot\Omega^{i}f$ for any
$f\in\operatorname{Corr}_{d,e}$ and each $i=0,1$, where $c_{i}\in k^{\times}$
are given. Therefore for a correspondence $f\in\operatorname{Corr}_{d,e}$ we
have

$\displaystyle\lambda_{1,(1,d+e-1)}\circ\rho_{1}(f)=\operatorname{res}_{z}(\Omega^{0}f(z),$
$\displaystyle
c_{0}z\partial_{z}(\Omega^{0}f)(z)\left(\frac{1}{d+e}dx+\frac{d+e-1}{d+e}dy\right)$
$\displaystyle+c_{1}z_{0}z_{1}\Omega^{1}f(z)\left(\frac{2}{d+e}dx-\frac{2}{d+e}dy\right))$

and

$\displaystyle\lambda_{1,(d,e)}(f)=\operatorname{res}_{z}(\Omega^{0}f(z),$
$\displaystyle
z\partial_{z}(\Omega^{0}f)(z)\left(\frac{d}{d+e}dx+\frac{e}{d+e}dy\right)$
$\displaystyle+z_{0}z_{1}\Omega^{1}f(z)\left(\frac{2}{d+e}dx-\frac{2}{d+e}dy\right))$

for any correspondence $f$. We take the basis
$(dz^{0}_{(d^{\prime},e^{\prime})},dz_{d^{\prime}+e^{\prime}}^{1})$ of
$D_{1}=k\cdot dx\oplus k\cdot dy$ for each pair of positive integers
$d^{\prime},e^{\prime}$ by

$dz^{0}_{(d^{\prime},e^{\prime})}:=\frac{d^{\prime}}{d^{\prime}+e^{\prime}}dx+\frac{e^{\prime}}{d^{\prime}+e^{\prime}}dy\text{
and
}dz_{d^{\prime}+e^{\prime}}^{1}:=\frac{2}{d^{\prime}+e^{\prime}}dx-\frac{2}{d^{\prime}+e^{\prime}}dy.$
(13)

The automorphism $A$ on $\mathbb{P}(D_{d+e})$ is induced by the linear map

$\left(\begin{matrix}dz^{0}_{(d,e)}\\\
dz_{d+e}^{1}\end{matrix}\right)\mapsto\left(\begin{matrix}c_{0}dz^{0}_{(1,d+e-1)}\\\
c_{1}dz_{d+e}^{1}\end{matrix}\right).$

∎

######  Remark 6.4.

For a correspondence $f=\sum_{i,j}a_{ij}x^{i}y^{j}$, we have

$\lambda_{1,(d,e)}(f)=[\operatorname{res}_{z}(\Omega^{0}f(z),z\partial_{z}(\Omega^{0}f)(z)dz^{0}_{(d,e)}+z_{0}z_{1}\Omega^{1}f(z)dz^{1}_{d+e})]\in\mathbb{P}(D_{d+e}).$

From the definition of resultant, we have each coefficient is divisible by
$a_{00}a_{de}$. As a point of $\mathbb{P}(D_{d+e})$, we have

$\lambda_{1,(d,e)}(f)=[\operatorname{res}_{z}(\Omega^{0}f(z),z\partial_{z}(\Omega^{0}f)(z)dz^{0}_{(d,e)}+z_{0}z_{1}\Omega^{1}f(z)dz^{1}_{d+e})/a_{00}a_{de}].$
(14)

We write $[(dz^{0})^{d+e-i}(dz^{1})^{i}]$ for the homogeneous coordinate of
$\mathbb{P}(D_{d+e})$ given by taking the coefficient of
$(dz^{0}_{d,e})^{d+e-i}(dz^{1}_{d+e})^{i}$ for any nonzero element of
$D_{d+e}$ as a polynomial of variables $dz^{0}_{d,e}$ and $dz^{1}_{d+e}$
defined in (13). In this form, we have the coordinate $[(dz^{1}_{d+e})^{d+e}]$
of (14) is

$\operatorname{res}_{z}(\Omega^{0}f(z),z_{0}z_{1}\Omega^{1}f(z))/a_{00}a_{de}=\operatorname{res}_{z}(\Omega^{0}f(z),\Omega^{1}f(z)),$

therefore $\operatorname{SL}_{2}$-invariant by Proposition 4.9. From the
invariance of the multiplier map, the other coordinate functions of (14) are
$\operatorname{SL}_{2}$-invariant on $\operatorname{Corr}_{d,e}$ of degree
$2(d+e-1)$.

######  Theorem 6.5.

The image of
$\lambda_{1,(d,e)}:\operatorname{Dyn}_{d,e}\to\mathbb{P}(D_{d+e})$ is the
hyperplane defined by

$[(dz^{0})^{d+e-1}(dz^{1})^{1}]=0,$

where $[(dz^{0})^{d+e-1}dz^{1}]$ is the coordinate function of
$(dz_{(d,e)}^{0})^{d+e-1}dz_{d+e}^{1}$.

###### Proof.

By Proposition 6.3, it is enough to show the assertion for $e=1$, the case of
the moduli of maps. We show the index theorem (3) is equivalent to the
hyperplane $[(dz^{0})^{d}(dz^{1})^{1}]=0$ by the coordinate change. The
coordinate change is induced by the coordinate change

$\left(\begin{matrix}dz^{0}\\\
dz^{1}\end{matrix}\right)=\left(\begin{matrix}\frac{d}{d+1}&\frac{2}{d+1}\\\
\frac{1}{d+1}&-\frac{2}{d+1}\end{matrix}\right)\left(\begin{matrix}dx\\\
dy\end{matrix}\right)$

on $D_{1}$, which induces a coordinate change

$\left(\begin{matrix}[dz^{0}]\\\
[dz^{1}]\end{matrix}\right)=\left(\begin{matrix}\frac{d}{d+1}&\frac{2}{d+1}\\\
\frac{1}{d+1}&-\frac{2}{d+1}\end{matrix}\right)^{-1}\left(\begin{matrix}[dx]\\\
[dy]\end{matrix}\right)=\left(\begin{matrix}1&1\\\
\frac{1}{2}&\frac{-d}{2}\end{matrix}\right)\left(\begin{matrix}[dx]\\\
[dy]\end{matrix}\right)$

on $D_{1}^{*}$.

We note that $D_{n}$ is naturally isomorphic to
$(\operatorname{Sym}_{n}(D_{1}^{*}))^{*}$ and therefore $D_{n}^{*}$ is
naturally isomorphic to $\operatorname{Sym}_{n}(D_{1}^{*})$. The vector space
of $n$-ary form of $D_{1}^{*}$ is isomorphic to the vector space
$(\operatorname{Sym}_{n}(D_{1}^{*})^{*})^{*}\simeq\operatorname{Sym}_{n}(D_{1})^{*}$
and the canonical morphism
$\beta_{n}:\operatorname{Sym}_{n}(D_{1}^{*})\to(\operatorname{Sym}_{n}D_{1})^{*}$
is given by $\beta_{n}([(ds)^{i}(dt)^{n-i}])=\binom{n}{i}[ds]^{i}[dt]^{n-i}$
for any basis $\\{ds,dt\\}$ of $D_{1}$ (Remark 3.3). Therefore we have

$\displaystyle\beta_{d+1}([(dz^{0})^{d}(dz^{1})])$
$\displaystyle=\binom{d+1}{1}[dz^{0}]^{d}[dz^{1}]$
$\displaystyle=(d+1)([dx]+[dy])^{d}(\frac{1}{2}[dx]-\frac{d}{2}[dy])$
$\displaystyle=\frac{d+1}{2}\left(\sum_{i=0}^{d+1}\left(d\cdot\binom{d}{i-1}-\binom{d}{i}\right)[dx]^{i}[dy]^{d+1-i}\right)$
$\displaystyle=\frac{d+1}{2}\left(\sum_{i=0}^{d+1}(i-1)\binom{d+1}{i}[dx]^{i}[dy]^{d+1-i}\right)$
$\displaystyle=\beta_{d+1}\left(\frac{d+1}{2}\sum_{i=0}^{d+1}(i-1)[(dx)^{i}(dy)^{d+1-i}]\right),$

where we used

$\displaystyle d\cdot\binom{d}{i-1}-\binom{d}{i}$
$\displaystyle=\binom{d+1}{i}\left(d\frac{i}{d+1}-\frac{d+1-i}{d+1}\right)=(i-1)\binom{d+1}{i}$

and $\binom{d}{i}=0$ for $i<0,d<i$.

Therefore the hyperplane defined by $[(dz^{0})^{d}(dz^{1})]=0$ is the one
defined by $\sum_{i=0}^{d+1}(i-1)[(dx)^{i}(dy)^{d+1-i}]=0$. By (12) and
Definition 6.1, this is which the index theorem (3) defines.

The surjectivity of the multiplier map to the hyperplane is shown in [11]. ∎

This reduction is a representationally simplified form of Woods Hole Formula.

######  Corollary 6.6.

For an arbitrary field $K$ and any polynomials $f,g\in K[x]$ of degree $\deg
f\geq 3$ and $\deg f\geq\deg g+2$,

$\left.\frac{\partial}{\partial
t}\operatorname{res}_{x}(f(x),f^{\prime}(x)+tg(x))\right|_{t=0}=0.$

###### Proof.

We put $d:=\deg f$. By Proposition 3.7, there exists a divisorial quasi-
correspondence $\widetilde{f}(x,y)$ of degree $(d-1,1)$ such that
$\Omega^{0}\widetilde{f}(z)=f(z)$ and $\Omega^{1}\widetilde{f}(z)=g(z)$. By
Remark 6.4 and Theorem 6.5, we have the $k$-coefficient of $t$ of the
polynomial $\operatorname{res}_{z}(f(z),zf^{\prime}(z)+tzg(z))\in k[t]$ is
$0$. Since resultant is $\mathbb{Z}$-polynomial of the coefficients of $f$ and
$g$, the $K[z]$-coefficient of $t$ is $0$ in arbitrary characteristics. This
leads the assertion. ∎

## References

  * [1] M. F. Atiyah, R. Bott, A Lefschetz fixed point formula for elliptic differential operators. Bull. Amer. Math. Soc. 72 (1966), 245-250.
  * [2] A. Braverman, Spaces of quasi-maps into the flag varieties and their applications. International Congress of Mathematicians. Vol. II, 1145-1170, Eur. Math. Soc., Zürich, 2006.
  * [3] M. Cavallin, Representations of SL2(K), Master Project, Ecole Polytechnique Fédérale de Lausanne, 2012.
  * [4] Harm Derksen, Gregor Kemper, Computational invariant theory. Second enlarged edition. With two appendices by Vladimir L. Popov, and an addendum by Norbert A’Campo and Popov. Encyclopaedia of Mathematical Sciences, 130. Invariant Theory and Algebraic Transformation Groups, VIII. Springer, Heidelberg, 2015. xxii+366 pp. ISBN: 978-3-662-48420-3; 978-3-662-48422-7
  * [5] Tien-Cuong Dinh, Lucas Kaufmann, Hao Wu, Dynamics of holomorphic correspondences on Riemann surfaces. (English summary) Internat. J. Math. 31 (2020), no. 5, 2050036, 21 pp.
  * [6] Laura DeMarco, The moduli space of quadratic rational maps. J. Amer. Math. Soc. 20 (2007), no. 2, 321-355.
  * [7] Masayo Fujimura, Projective moduli space for the polynomials. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 13 (2006), no. 6, 787-801.
  * [8] Masayo Fujimura, The moduli space of rational maps and surjectivity of multiplier representation (English summary) Comput. Methods Funct. Theory 7 (2007), no. 2, 345-360.
  * [9] I. M. Gel’fand, M. M. Kapranov, A. V. Zelevinsky Discriminants, resultants, and multidimensional determinants. Mathematics: Theory & Applications. Birkhäuser Boston, Inc., Boston, MA, 1994. x+523 pp. ISBN: 0-8176-3660-9
  * [10] I. Gorbovickis, Algebraic Independence of Multipliers of Periodic Orbits in the Space of Rational Maps of the Riemann Sphere. Mosc. Math. J. 15 (2015), no. 1, 73-87.
  * [11] I. Gorbovickis, Algebraic independence of multipliers of periodic orbits in the space of polynomial maps of one variable. Ergodic Theory Dynam. Systems 36 (2016), no. 4, 1156-1166.
  * [12] Phillip Griffiths, Joseph Harris, Principles of algebraic geometry. Reprint of the 1978 original. Wiley Classics Library. John Wiley & Sons, Inc., New York, 1994. xiv+813 pp. ISBN: 0-471-05059-8 14-01
  * [13] B. Hutz and M. Tepper, Multiplier Spectra and the Moduli Space of Degree 3 Morphisms on $\mathbb{P}^{1}$. JP Journal of Algebra, Number Theory and Applications 29 (2013), no. 2, 189-206.
  * [14] Luc Illusie, ed. (1977). Séminaire de Géométrie Algébrique du Bois Marie - 1965-66 - Cohomologie l-adique et Fonctions L - (SGA 5). Lecture Notes in Mathematics (in French). 589. Berlin; New York: Springer-Verlag.
  * [15] Patrick Ingram, Canonical heights for correspondences. (English summary) Trans. Amer. Math. Soc. 371 (2019), no. 2, 1003-1027.
  * [16] P. I. Katsylo, Rationality of the moduli spaces of hyperelliptic curves, Izv. Akad. Nauk SSSR. 48 (1984), 705-710.
  * [17] Alon Levy, The space of morphisms on projective space. Acta Arith. 146 (2011), no. 1, 13-31.
  * [18] Qing Liu, Algebraic geometry and arithmetic curves. Translated from the French by Reinie Erné. Oxford Graduate Texts in Mathematics, 6. Oxford Science Publications. Oxford University Press, Oxford, 2002. xvi+576 pp. ISBN: 0-19-850284-2
  * [19] Shouhei Ma, The rationality of the moduli spaces of trigonal curves of odd genus. J. Reine. Angew. Math. 683 (2013), 181-187.
  * [20] Shouhei Ma, Rationality of fields of invariants for some representations of $\operatorname{SL}_{2}\times\operatorname{SL}_{2}$. Compositio Math. 149 (2013) no.7, 1225-1234.
  * [21] Shouhei Ma, The rationality of the moduli spaces of trigonal curves. Int. Math. Res. Not. IMRN 2015, no. 14, 5456-5472.
  * [22] Shouhei Ma, Rationality of some tetragonal loci. Algebr. Geom. 1 (2014), no. 3, 271-289.
  * [23] C. McMullen, Families of rational maps and iterative root-finding algorithms, Ann. of Math. 125 (1987), 467-493.
  * [24] J. Milnor, Geometry and dynamics of quadratic rational maps. Experiment. Math. 2 (1993), 37-83. With an appendix by the author and Lei Tan
  * [25] D. Mumford, J. Fogarty, F. Kirwan, Geometric invariant theory. Third edition. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], 34. Springer-Verlag, Berlin, 1994. xiv+292 pp. ISBN: 3-540-56963-4
  * [26] Marc Olive, About Gordan’s algorithm for binary forms. (English summary) Found. Comput. Math. 17 (2017), no. 6, 1407-1466.
  * [27] Peter J. Olver, Classical invariant theory, volume 44 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1999.
  * [28] Fedor Pakovich, Recomposing rational functions. (English summary) Int. Math. Res. Not. IMRN 2019, no. 7, 1921-1935.
  * [29] V. Ramírez, The Woods Hole trace formula and indices for vector fields and foliations on $\mathbb{C}^{2}$, arXiv preprint, arXiv:1608.05321, 2016
  * [30] Johannes Schmitt, A compactification of the moduli space of self-maps of CP1 via stable maps. Conform. Geom. Dyn. 21 (2017), 273-318.
  * [31] N. I. Shepherd-Barron, The rationality of certain spaces associated to trigonal curves, in: Algebraic geometry (Brunswick 1985), Proc. Sympos. Pure Math. 46, Part 1, American Mathematical Society, Providence (1987), 165-171.
  * [32] Joseph H. Silverman, The space of rational maps on $\mathbb{P}^{1}$, Duke Math. J. Volume 94, Number 1 (1998), 41-77.
  * [33] Joseph H. Silverman, The arithmetic of dynamical systems. Graduate Texts in Mathematics, 241. Springer, New York, 2007. x+511 pp. ISBN: 978-0-387-69903-5
  * [34] Joseph H. Silverman, Moduli spaces and arithmetic dynamics. CRM Monograph Series, 30. American Mathematical Society, Providence, RI, 2012. viii+140 pp. ISBN: 978-0-8218-7582-7
  * [35] Toshi Sugiyama, The moduli space of polynomial maps and their fixed-point multipliers. Adv. Math. 322 (2017), 132-185.
  * [36] Tetsuo Ueda, Complex Dynamics on Projective Spaces – Index Formula for Fixed Poinds, Proceedings of the International Conference on Dynamical Systems and Chaos, Tokyo, Japan, 23-27 May 1994 ; vol. 1, 252-259.
  * [37] Lloyd William West, The Moduli Space of Rational Maps. Thesis (Ph.D.)-City University of New York. 2015. 88 pp. ISBN: 978-1339-02142-3