# The integral Chow ring of $\mathcal{M}_{0}(\mathbb{P}^{r},d)$, for $d$ odd

Renzo Cavalieri  and  Damiano Fulghesu  Renzo Cavalieri, Department of
Mathematics, Colorado State University, 1874 Campus Delivery, Fort Collins,
CO, 80523-1874, U.S.A<EMAIL_ADDRESS>Damiano Fulghesu, Department
of Mathematics, Minnesota State University, 1104 7th Ave South, Moorhead, MN
56563, U.S.A<EMAIL_ADDRESS>
###### Abstract.

For any odd integer $d$, we give a presentation for the integral Chow ring of
the stack $\mathcal{M}_{0}(\mathbb{P}^{r},d)$, as a quotient of the polynomial
ring $\mathbb{Z}[c_{1},c_{2}]$. We describe an efficient set of generators for
the ideal of relations, and compute them in generating series form. The paper
concludes with explicit computations of some examples for low values of $d$
and $r$, and a conjecture for a minimal set of generators.

##  1\. Introduction

###  1.1. Results and strategy of proof

The main goal of this work is to compute the integral Chow ring of the open
set of the space of rational stable maps of odd degree to projective space,
consisting of maps from irreducible source curves. We exhibit such ring as the
quotient of a polynomial ring in two variables, and provide an efficient set
of generators for the ideal of relations. We state here the complete result,
which is proven in several steps throughout the paper.

######  Main Theorem (Thm.3.8, Thm. 4.3, Thm. 5.1, Thm. 6.1).

For $r$ a positive integer, and $d$ an odd positive integer, a presentation
for the integral Chow ring of the stack $\mathcal{M}_{0}(\mathbb{P}^{r},d)$ is
given by:

$A^{\ast}(\mathcal{M}_{0}(\mathbb{P}^{r},d))=\frac{\mathbb{Z}[c_{1},c_{2}]}{(\alpha^{r,d}_{i,k})_{i,k}},$

where $c_{i}$ is a graded variable of degree $i$ and $\alpha_{i,k}^{r,d}$ is
homogeneous of degree $ir+k$ (Thm. 3.8); in order to generate the ideal of
relations it is sufficient to consider the following values for $i,k$ (Thm.
4.3):

  * •

$i=1$ and $k=0,1$.

  * •

$i$ is a prime power and $k=0$.

The generating relations are exhibited in generating function form, where:

$\mathcal{A}_{i,k}(d):=\sum_{r=0}^{\infty}\alpha_{i,k}^{r,d}.$

Relations of the form $\alpha^{r,d}_{1,k}$ are explicitly computed in Theorem
5.1:

$\mathcal{A}_{1,0}(d)=\frac{d}{(1+\frac{d-1}{2}c_{1})(1-\frac{d+1}{2}c_{1})+d^{2}c_{2}}$
$\mathcal{A}_{1,1}(d)=\frac{1+\frac{d-1}{2}c_{1}}{(1+\frac{d-1}{2}c_{1})(1-\frac{d+1}{2}c_{1})+d^{2}c_{2}}-1.$

Relations of the form $\alpha^{r,d}_{i,0}$ are computed in Theorem 6.1 as:

$\mathcal{A}_{i,0}(d)=\displaystyle\sum_{j=0}^{i}\frac{(-1)^{j}}{i!(l_{1}-l_{2})^{i}}\binom{i}{j}\left[\left(1-\frac{\prod_{k=0}^{d}\left(\frac{c_{1}}{2}+\left(k-\frac{d}{2}\right)(l_{1}-l_{2})\right)}{\prod_{k=j}^{d-i+j}\left(\frac{c_{1}}{2}+\left(k-\frac{d}{2}\right)(l_{1}-l_{2})\right)}\right)^{-1}-1\right],$

where $-l_{1},-l_{2}$ are the Chern roots of the variables $c_{1},c_{2}$, i.e.
$c_{1}=-l_{1}-l_{2}$ and $c_{2}=l_{1}l_{2}$.

The integral Chow rings of the spaces $\mathcal{M}_{0}(\mathbb{P}^{r},d)$
exhibit a remarkable combinatorial structure. In Corollary 6.5 it is shown
that for fixed $i,r$ the relations of the form $\alpha_{i,0}$ are polynomial
in $d$, while in Corollary 6.4 one sees that, for fixed $d,r$, all relations
of the form $\alpha_{i,0}$ may be extracted from a single two-variable
monomial via the action of a differential operator, and an appropriate
Hadamard product of multi-variate power series.

The proof of the Main Theorem is organized into three main stages: the first
one consists of producing a presentation of
$A^{\ast}(\mathcal{M}_{0}(\mathbb{P}^{r},d))$, where the ideal of relations
has a very redundant generating set. The second consists in eliminating a
large number of redundant generators, and in the third we compute the
remaining relations.

To obtain a presentation of $A^{\ast}(\mathcal{M}_{0}(\mathbb{P}^{r},d))$, we
exhibit $\mathcal{M}_{0}(\mathbb{P}^{r},d)$ as a global quotient stack
$[\widehat{U}_{r,d}/\text{GL}_{2}]$ of an open set in affine space by the
action of the group $\text{GL}_{2}$, thus reducing the task to computing the
$\text{GL}_{2}$ equivariant Chow ring of $\widehat{U}_{r,d}$. It is at this
moment that the hypothesis of $d$ being odd becomes necessary. By considering
$\widehat{U}_{r,d}$ as a $\mathbb{G}_{m}$-bundle over its image in projective
space, we reduce the computation of the equivariant Chow ring of
$\widehat{U}_{r,d}$ to that of its image $U_{r,d}$: this is an open set in
projective space, and its complement is covered by a
$\text{GL}_{2}$-equivariant envelope, whose connected components
$Z_{i}\cong\mathbb{P}^{i}\times\mathbb{P}^{N}$ are isomorphic to products of
projective spaces. Using the standard excision sequence, we exhibit the Chow
ring of $\mathcal{M}_{0}(\mathbb{P}^{r},d)$ as the quotient of the Chow ring
of the ambient projective space by the ideal generated by the images of the
pushfoward maps from the components of the envelope.

Elementary arguments suffice to show that the pushforward
${\pi_{i}}_{\ast}(h_{i}^{k})$ of powers of the hyperplane classes from the
left factor of the $Z_{i}$’s, with degree bounded by $i$, generate the ideal
of relations. In order to further reduce the set of necessary generators, a
more subtle analysis is needed. One may replace the powers $h_{i}^{k}$ in the
$i$-th component of the envelope with suitably chosen monic polynomials in
$h_{i}$. In many cases, such polynomials may be seen as arising from classes
from lower envelopes, leading to an inductive argument that allows to narrow
down the indispensable generators to those mentioned in the Main Theorem. The
generating set we obtain is still not minimal, as can be seen from the
computations in Section 7.2. Experimental computations led to the following
conjecture.

######  Conjecture 7.2.

Consider the presentation of $A^{\ast}(\mathcal{M}_{0}(\mathbb{P}^{r},d))$
from Theorem 3.8. A minimal set of generators for the ideal of relations is
given by $\alpha_{1,0}^{r,d},\alpha_{1,1}^{r,d}$ and the
$\alpha_{p,0}^{r,d}$’s where $p$ runs over all primes that divide $d$.

Regardless of the optimal generating set, we set forth to compute all
relations coming from the first component of the envelope or from the
pushforward of fundamental classes of any other component of the envelope. We
use two different techniques to compute the two types of relations. For the
relations coming from the first envelope, the embedding of $\mathbb{P}^{r}$ as
a coordinate hyperplane in $\mathbb{P}^{r+1}$ gives the relations a recursive
structure which allows to reconstruct them for all values of $r$ from the
degenerate $r=0$ case. Encoding the relations in generating function form, the
recursions become a linear system of functional equations which is easily
solved. Such recursive structure is present for the relations coming from
higher envelopes, but it becomes substantially more computationally intensive
to use this technique to extract the relations. Hence, for the fundamental
class relations, we instead use the Atyiah-Bott localization theorem on the
left factor of the envelope $Z_{i}$ to obtain an expression for the
fundamental class supported at the fixed points. Such an expression is then
readily pushed forward to obtain the formulas in the Main Theorem111Using the
Atyiah-Bott localization might in principle cause the loss of some torsion
classes, but we are applying the theorem to projective space, whose integral
Chow ring is known to not have torsion.. The drawback of this technique is
that the answer is not produced immediately as a polynomial in $c_{1},c_{2}$,
but rather as a (non obviously) symmetric polynomial in the Chern roots
$l_{1},l_{2}$. It would be interesting to symmetrize formula (43).

###  1.2. History, motivation, and considerations

One of the key conceptual leaps in modern enumerative geometry has been the
translation of enumerative questions into intersection theory on appropriate
moduli spaces of geometric objects. Thus, for example, the unique conic
through five points in the plane is obtained as the intersection of five
hyperplanes in the $\mathbb{P}^{5}$ of all conics, and twelve rational cubics
through eight points arise as the intersection of eight hyperplanes and the
discriminant hypersurface in the $\mathbb{P}^{9}$ of plane cubics.

The rich and rapid development of the field that followed in the late 1800’s
led to many exciting computations, some of which were alas incorrect; these
early mistakes brought awareness of many delicate issues and technical
difficulties in implementing the intuitive plan of counting geometric objects
by intersecting subvarieties in moduli spaces. On the one hand, this led to
Hilbert’s fifteenth problem [Hil30], requesting rigorous foundations for
Schubert calculus, which we now understand as intersection theory on
Grassmannians and flag varieties. On the other, new perspectives on geometric
objects and their moduli were developed to tackle even the most classical
problems. For example, the classical enumerative problem of counting the
number of plane rational curves of degree $d$ through $3d-1$ general points
was solved in [Kon95] by introducing the moduli spaces of stable maps, that
shifted the perspective by thinking of plane curves as the images of functions
from abstract curves.

The main object of study in algebraic intersection theory [Ful98, EH16] is the
Chow ring, a codimension-graded ring generated by equivalence classes of
closed subvarieties up to rational equivalence, where the product extends the
notion of transverse intersection of subvarieties. A full understanding of the
Chow ring of a moduli space gives in principle access to any enumerative
geometric problem involving the geometric objects described by the moduli
space. It should come as no surprise then that Chow rings of moduli spaces are
typically very sophisticated, and hard to compute objects. To further
complicate things, automorphisms of geometric objects cause most moduli spaces
to be represented only by stacks, rather than varieties or schemes.

Working with rational coefficients gives the significant advantage that the
Chow ring of a Deligne-Mumford stack agrees with that of its coarse moduli
space, see [Edi13, Theorem 4.40]. Still, the Chow ring of moduli spaces of
curves $\mathcal{M}_{g}$ are known only up to $g=9$ [Mum83, Fab90a, Fab90b,
Iza95, PV15, CL21], and for $\overline{\mathcal{M}}_{g}$ up to $g=3$ [Mum83,
Fab90a].

While rational coefficients are often sufficient for any application to
enumerative problems, they cause to lose all torsion classes, arguably
containing interesting information about the geometry of the moduli space. For
example, the stack of smooth hyperelliptic curves of genus $g$ has coarse
moduli space which is a finite quotient of $M_{0,2g+2}$ (an open subset of
$\mathbb{A}^{2g-1})$, and has therefore trivial Chow ring, i.e.
$A^{\ast}(\mathcal{H}_{g},\mathbb{Q})=\mathbb{Q}$.

Edidin and Graham ([EG98a]) generalize work of Totaro ([Tot99]) and approach
the study of Chow rings of moduli spaces with integer coefficients via
equivariant geometry: if a moduli space is presented as a global stack
quotient $[X/G]$, then its integral Chow ring is the equivariant Chow ring
$A^{\ast}_{G}(X).$

This perspective was used to unveil rich torsion structure in the integral
Chow ring of hyperelliptic loci [EF09, DL21], and in the locus of non-
hyperelliptic curves of genus $3$ [DLFV21]. The only moduli spaces of curves
for which the integral Chow ring has been computed are
$\overline{\mathcal{M}}_{2}$ ([Lar21]) and $\overline{\mathcal{M}}_{2,1}$
([dLPV21]).

Besides hyperelliptic loci, the only infinite family of moduli spaces for
which the integral Chow ring is known is given by Grassmannians $G(k,n)$. A
structurally satisfying presentation for all these Chow rings is given, for
example, in [EH16, Theorem 5.26] : $A^{\ast}(G(k,r+1),\mathbb{Z})$ is a
quotient of the polynomial ring $\mathbb{Z}[c_{1},\ldots,c_{k}]$, where
$c_{i}$ is a variable of degree $i$. The ideal of relations is generated by
the homogeneous terms of degrees between $r+2-k$ and $r+1$ in the power series
expansion of222For us $c_{i}$ denotes the $i$-th Chern class of the standard
representation of $GL_{n}$; in [EH16] they are choosing the dual of the
standard representation, hence the different signs in the formulas.

(1) $\frac{1}{1-c_{1}+c_{2}-\ldots+(-1)^{k}c_{k}}.$

We view (1) as a generating function for the ideal of relations of all
Grassmannians of $k$ planes as the ambient dimension $r+1$ varies.

When $k=2$, we have $Gr(2,r+1)\cong\mathcal{M}_{0}(\mathbb{P}^{r},1)$, and it
is easily seen that the terms of degree $r$ and $r+1$ of $1/(1-c_{1}+c_{2})$
agree with the generating functions $\mathcal{A}_{1,0}(1)$ and
$\mathcal{A}_{1,1}(1)$ from the Main Theorem.

In [Pan98], Pandharipande considers the function
$\Phi_{f}:Gr(2,r+1)=\mathcal{M}_{0}(\mathbb{P}^{r},1)\to\mathcal{M}_{0}(\mathbb{P}^{r},d)$
induced by post-composition with a fixed degree $d$ map
$f:\mathbb{P}^{r}\to\mathbb{P}^{r}$. He shows that when working with rational
coefficients, the pull-back

(2) $\Phi_{f}^{\ast}:A^{\ast}(\mathcal{M}_{0}(\mathbb{P}^{r},d),\mathbb{Q})\to
A^{\ast}(\mathcal{M}_{0}(\mathbb{P}^{r},1),\mathbb{Q})$

is an isomorphism. It is interesting to observe how this result relates to the
Main Theorem. First off, all relations $\alpha_{i,0}$ with $i>1$ are
$i$-torsion, and therefore vanish when tensoring coefficients with
$\mathbb{Q}$. In order to check that relations from the first envelope agree,
one must analyze the lift

$\widehat{\Phi}_{f}:\widehat{U}_{r,1}\to\widehat{U}_{r,d},$

where the $\text{GL}_{2}$-action on the two spaces is as in Proposition 2.4. A
non-trivial endomorphism $\varphi_{f}$ of $\text{GL}_{2}$ is required to make
the map $\widehat{\Phi}_{f}$ equivariant, and this induces the transformation
$\varphi_{f}^{\ast}(c_{1})=\frac{c_{1}}{d},\varphi_{f}^{\ast}(c_{2})=c_{2}-\frac{(d^{2}-1)c_{1}^{2}}{4d^{2}}$.
It is then immediate to check that the relations agree up to global factors
which are powers of $d$ (irrelevant after tensoring with $\mathbb{Q})$.

We make two important assumptions on the base field $k$. The first one is that
$k$ is algebraically closed. We use this condition to unambiguously determine
the degree of certain maps in Lemma 4.2. Moreover, we assume that the base
field has characteristic greater than $d$. This condition allows to work with
localization formulas and to construct formula (49).

###  1.3. Notation

There are four main discrete invariants that take different values in this
work:

$r$:

the dimension of the target projective space;

$d$:

the degree of the map $f:\mathbb{P}^{1}\to\mathbb{P}^{r}$ or equivalently of
the $(r+1)$ defining polynomials;

$i$:

the component of the envelope parameterizing polynomials that contain a common
factor of degree $i$;

$k$:

the power of the hyperplane class on the left factor of
$\mathbb{P}(W_{i})\times\mathbb{P}(W_{d-i}^{\oplus r+1})$.

As a consequence, the generators of the ideal of relations for the
presentation of $A^{\ast}(\mathcal{M}_{0}(\mathbb{P}^{r},d))$ are polynomials
$\alpha_{i,k}^{r,d}$ depending on four indices.

######  Convention 1.1.

To lighten this notation, we adopt the convention of suppressing indices
corresponding to discrete invariants that remain fixed throughout a section.
Thus for example in Section 3, where $r$ and $d$ are fixed, the relations are
denoted $\alpha_{i,k}$; whereas in Sections 5, where we consider maps
comparing different $\mathbb{P}^{r}$’s, we mantain the superscript $r$. A
similar convention is adopted for any other quantity depending on these four
discrete invariants.

The Chow ring $A^{\ast}(\mathcal{M}_{0}(\mathbb{P}^{r},d))$ is presented as a
quotient of the equivariant Chow ring of $\mathbb{P}(W_{d}^{\oplus r+1})$,
where one of the relations is $H-(d+1)c_{1}/2$. One may then eliminate the
hyperplane class $H$ and regard $A^{\ast}(\mathcal{M}_{0}(\mathbb{P}^{r},d))$
as a quotient of the polynomial ring $\mathbb{Z}[c_{1},c_{2}]$. We adopt the
following notation to deal with this phenomenon.

######  Convention 1.2.

We denote the class $\pi_{i_{\ast}}(h_{i}^{k})$ by $\alpha_{i,k}(H)$ when we
wish to regard it as a Chow class on $\mathbb{P}(W_{d}^{\oplus r+1})$, by
$\alpha_{i,k}$ when we wish to regard it as a Chow class on
$\mathcal{M}_{0}(\mathbb{P}^{r},d)$, so that
$\alpha_{i,k}=\alpha_{i,k}((d+1)c_{1}/2)$. This convention extends to any
other class with such double personality.

Some of the commonly recurring notation throughout the paper is collected in
Table 1.

$W_{d}$ | homogeneous polynomials of degree $d$ in two variables
---|---
$U_{r,d}\subset\mathbb{P}(W_{d}^{r+1})$ |  $(r+1)$-tuples of polynomials containing no common factor of positive degree
$\Delta_{r,d}\subset\mathbb{P}(W_{d}^{r+1})$ | the complement of $U_{r,d}$
$\widehat{U}_{r,d}$ | the affine cone over $U_{r,d}$
$l_{1},l_{2}$ | torus weights, Chern roots for the dual of the standard representation of $\text{GL}_{2}$
$c_{1},c_{2}$ | Chern classes of the standard representation of $\text{GL}_{2}$
$H$ | equivariant hyperplane class for $\mathbb{P}(W_{d}^{r+1})$
$P_{r,d}(H)$ | basic relation in equivariant Chow ring of $\mathbb{P}(W_{d}^{r+1})$, see (6)
$P_{d}$ | $P_{0,d}((d+1)c_{1}/2)$
$Z_{i}\subset\Delta_{r,d}$ | stratum of $(r+1)$ polynomials with largest common factor of degree $i$
$\widetilde{Z}_{i}$ |  $\mathbb{P}(W_{i})\times\mathbb{P}\left(W_{d-i}^{\oplus r+1}\right)$, with a natural map $\pi_{i}$, an isomorphism onto the closure of $Z_{i}$
$h_{i}\in A^{*}_{T}(\widetilde{Z}_{i})$ | equivariant hyperplane class for left factor
$\eta_{i}\in A^{*}_{T}(\widetilde{Z}_{i})$ | equivariant hyperplane class for right factor
$\alpha_{i,k}^{r,d}$ | $\pi_{i_{\ast}}(h_{i}^{k})$
Table 1. Recurring notation.

###  1.4. Acknowledgements

We would like to thank Seth Ireland for writing the code that gave extensive
verifications of Conjecture 7.2, and Chris Peterson for several conversations
related to the project. We are grateful to Rahul Pandharipande, Jim Bryan, Tom
Graber and the participants of the intercontinental algebraic geometry and
moduli zoominar as well as Ravi Vakil, Jesse Kass, Peter Petrov and the
participants of the Stanford virtual algebraic geometry seminar for the many
interesting questions and observations following the presentation of this
work. We gratefully acknowledge the following funding institutions: R.C. is
supported by NSF grant DMS-2100962, D.F. by Simons collaboration grant 360311.

##  2\. A Presentation of the Stack $\mathcal{M}_{0}(\mathbb{P}^{r},2s+1)$

Let $k$ be an algebraically closed field of characteristic 0 or larger than
$2s+1$. We refer the reader to [FP97] for a general definition of the stack of
stable maps $\mathcal{M}_{g,n}(X,\beta)$ where $X$ is a projective variety and
$\beta$ is a class in $A_{1}(X,\mathbb{Z})$. In this paper we focus on the
special case $\mathcal{M}_{0}(\mathbb{P}^{r},d)$ of maps from rational curves
to projective space. For completeness, we recall the definition.

######  Definition 2.1.

Let $\mathcal{M}_{0}(\mathbb{P}^{r},d)$ be the category whose objects are
diagrams

$\xymatrix{C\ar[d]_{\pi}\ar[r]^{f}&\mathbb{P}^{r}\\\ S}$

such that:

  * •

$S$ is a $k$-scheme,

  * •

the morphism $\pi:C\to S$ is a projective flat family of curves isomorphic to
$\mathbb{P}^{1}$,

  * •

the degree of $f^{*}\mathcal{O}_{\mathbb{P}^{r}}(1)$ on the geometric fibers
of $\pi:C\to S$ is $d$.

As a consequence of the third bullet point, the restriction of $f$ on the
geometric fibers of $\pi:C\to S$ is in particular not constant. Arrows are
cartesian diagrams:

$\xymatrix{C_{1}\ar<EMAIL_ADDRESS>S_{1}\ar[r]^{\varphi}S_{2}}$

The category $\mathcal{M}_{0}(\mathbb{P}^{r},d)$ is naturally a category
fibered in groupoids over the category $(\text{Sch}/k)$ of $k$-schemes. It is
a proper Deligne-Mumford stack, see, for example, [BM96, Theorem 3.14]. As an
example of a moduli point with non-trivial isotropy, let
$f_{d}:\mathbb{P}^{1}\to\mathbb{P}^{1}$ be defined as
$f_{d}(x:y)=(x^{d}:y^{d})$, and let
$\iota_{L}:\mathbb{P}^{1}\to\mathbb{P}^{r}$ be the inclusion of
$\mathbb{P}^{1}$ as a line in $\mathbb{P}^{r}$; then
$(\mathbb{P}^{1},\iota_{L}\circ f_{d})\in\mathcal{M}_{0}(\mathbb{P}^{r},d)$
has isotropy group $\mu_{d}$.

Observe that $\mathcal{M}_{0}(\mathbb{P}^{r},d)$ has a structure of a global
quotient by a linear algebraic group. Let $E$ be the standard representation
of $\text{GL}_{2}$. We identify the projective line as

$\mathbb{P}^{1}\cong\mathbb{P}(E).$

Consider the vector space of homogeneous forms of degree $d$ over
$\mathbb{P}^{1}$

$W_{d}:=H^{0}(\mathbb{P}^{1},\mathcal{O}_{\mathbb{P}^{1}}(d))=\text{Sym}^{d}(E^{\vee})$

where $E^{\vee}$ is the dual of the representation $E$.

The set of regular maps $\mathbb{P}^{1}\to\mathbb{P}^{r}$ of degree $d$ is an
open subset $U_{r,d}$ of $\mathbb{P}\left(W_{d}^{\oplus r+1}\right)$. More
precisely, a general element $f$ of $U_{r,d}$ can be written as

$f=\left[f_{1}(x,y),\dots,f_{r+1}(x,y)\right]$

where

  * •

for all $i=1,\dots,r+1$, the polynomial in two variables $f_{i}(x,y)$ is
homogeneous of degree $d$;

  * •

the map $f$ is free of base points, in other words, the polynomials
$f_{i}(x,y)$ have no common factors.

Call $\Delta_{r,d}$ the complement of $U_{r,d}$ in
$\mathbb{P}\left(W_{d}^{\oplus r+1}\right)$, and $\widehat{U}_{r,d}$ the
affine cone over $U_{r,d}$, that is the preimage of $U_{r,d}$ via the
tautological map

$W_{d}^{\oplus r+1}\backslash 0\to\mathbb{P}\left(W_{d}^{\oplus r+1}\right).$

One has the following isomorphism of stacks (see [FP97, Section 2] for a more
general result):

(3)
$\mathcal{M}_{0}(\mathbb{P}^{r},d)\cong\left[U_{r,d}/\text{PGL}_{2}\right]$

where the action of $\text{PGL}_{2}$ on $U_{r,d}$ is given by:

(4)
$(A\cdot[f_{1},f_{2},\dots,f_{r+1}])(x,y)=[f_{1}(A^{-1}(x,y)),f_{2}(A^{-1}(x,y)),\dots,f_{r+1}(A^{-1}(x,y))].$

The use of the inverse matrix is necessary to have a well defined left action.

######  Lemma 2.2.

Let $H$ be a normal subgroup of a linear group $G$ and let $X$ be a quasi-
projective scheme equipped with a $G$-action. Assume that $H$ acts freely on
$X$ so that the quotient $X/H$ is a quasi-projective scheme as well. Then we
have an isomorphism of quotient stacks:

$\left[X/G\right]\cong\left[(X/H)/(G/H)\right]$

###### Proof.

See, for example, [Hei05, Example 3.3]. ∎

######  Remark 2.3.

A consequence of Lemma 2.2 is the induced isomorphism of equivariant
intersection rings:

$A^{*}_{G}(X)\cong A^{*}_{G/H}(X/H).$

For a direct proof of this isomorphism see [MRV06, Lemma 2.1].

From now on, assume the degree $d$ to be odd.

######  Proposition 2.4.

Let $d=2s+1$ be a positive odd integer. The stack
$\mathcal{M}_{0}(\mathbb{P}^{r},d)$ is isomorphic to the quotient stack

$\mathcal{M}_{0}(\mathbb{P}^{r},d)\cong\left[\widehat{U}_{r,d}/\text{GL}_{2}\right]$

where the action of $\text{GL}_{2}$ is

$A\cdot(f_{0},f_{1},\dots,f_{r})(x,y)=\det{A}^{s+1}(f_{0}(A^{-1}(x,y)),f_{1}(A^{-1}(x,y)),\dots,f_{r}(A^{-1}(x,y))).$

###### Proof.

Apply Lemma 2.2 with the substitutions:

$X=\widehat{U}_{r,d};\quad G=\text{GL}_{2};\quad H=\mathbb{G}_{m}.$

$\mathbb{G}_{m}$ is normal in $\text{GL}_{2}$; one must describe the induced
action of $\mathbb{G}_{m}$ on $\widehat{U}_{r,d}\subset W_{d}^{\oplus r+1}$.

Refer to the exact sequence of groups:

$1\to\mathbb{G}_{m}\xrightarrow{\varphi}\text{GL}_{2}\xrightarrow{\pi}\text{PGL}_{2}\to
1$

Let $\lambda\in\mathbb{G}_{m}$ and
$\varphi(\lambda)=\left[\begin{array}[]{cc}\lambda&0\\\
0&\lambda\end{array}\right]=:A$, we have

$\displaystyle A\cdot(f_{1},f_{2},\dots,f_{r+1})(x,y)$ $\displaystyle=$
$\displaystyle\det{A}^{s+1}(f_{1}(A^{-1}(x,y)),f_{2}(A^{-1}(x,y)),\dots,f_{r+1}(A^{-1}(x,y)))=$
$\displaystyle=$
$\displaystyle\lambda^{2(s+1)}\frac{1}{\lambda^{d}}(f_{1},f_{2},\dots,f_{r+1})(x,y)=\lambda(f_{1},f_{2},\dots,f_{r+1})(x,y),$

therefore $\widehat{U}_{r,d}/\mathbb{G}_{m}=U_{r,d}$. Notice that the action
of $\mathbb{G}_{m}$ is free on $\widehat{U}_{r,d}$. Lemma 2.2 implies

$\left[\widehat{U}_{r,d}/\text{GL}_{2}\right]\cong\left[U_{r,d}/\text{PGL}_{2}\right],$

where the action of $\text{PGL}_{2}$ over $U_{r,d}$ is as in (3). We conclude
thanks to isomorphism (3). ∎

##  3\. Generators and Relations

By Proposition 2.4, we are reduced to computing the equivariant intersection
ring $A^{*}_{\text{GL}_{2}}\left(\widehat{U}_{r,d}\right)$. First, we recall
some facts about equivariant Chow rings we will use, and set notation. Let $T$
denote the maximal torus for $\text{GL}_{2}$ represented by diagonal matrices,
and consider the induced morphism $Bi:BT\to B\text{GL}_{2}$. We denote by $E$
the standard representation of $\text{GL}_{2}$, which we think of as (the
pull-back to the point via the quotient map $pt\to B\text{GL}_{2}$ of) a rank
two vector bundle over $B\text{GL}2$. Since the Chern classes of $E$ will be
frequently used, we denote $c_{i}(E)$ simply by $c_{i}$. The pull-back
$Bi^{\ast}(E^{\vee})$ splits as the direct sum of two line bundles on $BT$:
the characters $\lambda_{1},\lambda_{2}$ are given by the two coordinate
projections of $T$.

Denoting $A^{*}_{T}=A^{*}_{T}(pt.)$, it is known (see e.g. [HKK+03, Chapter
4]) that $A^{*}_{T}=\mathbb{Z}\left[l_{1},l_{2}\right]$, with
$l_{i}=c_{1}\left(\lambda_{i}\right)$. By a slight abuse of notation we also
denote by $l_{i}$ the Chern roots of the vector bundle $E^{\vee}$, since we
have

$\displaystyle Bi^{\ast}c_{1}$ $\displaystyle=$ $\displaystyle-(l_{1}+l_{2})$
$\displaystyle Bi^{\ast}c_{2}$ $\displaystyle=$ $\displaystyle l_{1}l_{2}.$

The Weyl group $S_{2}$ acts on $A^{*}_{T}$ by permuting the classes $l_{i}$
and $A^{*}_{\text{GL}_{2}}=\left(A^{*}_{T}\right)^{S_{2}}$ (see [EG98a,
Proposition 6]).

Consider the following commutative $\text{GL}_{2}$-equivariant diagram

$\xymatrix{\widehat{U}_{r,d}\ar[r]^{(}0.40){j}\ar[d]^{\pi}&W_{d}^{\oplus
r+1}\ar[d]^{\pi}\backslash 0\\\
U_{r,d}\ar[r]^{(}0.40){j}\mathbb{P}\left(W_{d}^{\oplus r+1}\right)}$

where the horizontal arrows are the natural open inclusions and the vertical
arrows are the natural quotient maps by the action of $\mathbb{G}_{m}$. The
vertical maps may be interpreted as principal $\mathbb{G}_{m}$-bundles
associated to the $\text{GL}_{2}$-equivariant line bundle
$\mathcal{D}^{\otimes s+1}\otimes\mathcal{O}(-1)$, where $\mathcal{O}(-1)$ is
the tautological bundle over $\mathbb{P}\left(W_{d}^{\oplus r+1}\right)$ and
$\mathcal{D}$ is the one dimensional representation of $\text{GL}_{2}$
associated to the determinant $\bigwedge^{2}E$. The first Chern class of
$\mathcal{D}$ is $c_{1}(\mathcal{D})=c_{1}$, therefore we have

$c_{1}\left(\mathcal{D}^{\otimes
s+1}\otimes\mathcal{O}(-1)\right)=(s+1)c_{1}-H$

where $H$ is the canonical equivariant lift of the hyperplane class of
$\mathbb{P}\left(W_{d}^{\oplus r+1}\right)$. By arguing as in Lemma 3.2 of
[EF09], the pull-back morphism

$\pi^{*}:A^{*}_{\text{GL}_{2}}\left(U_{r,d}\right)\to
A^{*}_{\text{GL}_{2}}\left(\widehat{U}_{r,d}\right)$

is surjective and its kernel is generated by $H-(s+1)c_{1}$. We may therefore
determine $A^{*}_{\text{GL}_{2}}\left(\widehat{U}_{r,d}\right)$ from a
presentation of the ring $A^{*}_{\text{GL}_{2}}\left(U_{r,d}\right)$ by
applying the substitution $H=(s+1)c_{1}$.

In order to compute $A^{*}_{\text{GL}_{2}}\left(U_{r,d}\right)$, we consider
the following exact sequence of $A^{*}_{\text{GL}_{2}}$-modules:

$A^{*}_{\text{GL}_{2}}\left(\Delta_{r,d}\right)\xrightarrow{i_{*}}A^{*}_{\text{GL}_{2}}\left(\mathbb{P}\left(W_{d}^{\oplus
r+1}\right)\right)\xrightarrow{j^{*}}A^{*}_{\text{GL}_{2}}\left(U_{r,d}\right)\to
0.$

Using standard techniques as in [FV18, Section 3.2], we have the following
isomorphism:

(5) $A^{*}_{\text{GL}_{2}}\left(\mathbb{P}\left(W_{d}^{\oplus
r+1}\right)\right)\cong\frac{\mathbb{Z}[c_{1},c_{2},H]}{(P_{r,d}(H))}$

where

(6) $P_{r,d}(H)=\prod_{k=0}^{d}(H+(d-k)l_{1}+kl_{2})^{r+1}.$

In conclusion, we have:

(7)
$A^{*}\left(\mathcal{M}_{0}(\mathbb{P}^{r},d)\right)\cong\frac{A^{*}_{\text{GL}_{2}}\left(\mathbb{P}\left(W_{d}^{\oplus
r+1}\right)\right)}{(\text{Im}(i_{*}),H-(s+1)c_{1})}\cong\frac{\mathbb{Z}[c_{1},c_{2},H]}{(\text{Im}(i_{*}),H-(s+1)c_{1},P_{r,d}((s+1)c_{1}))},$

where $\text{Im}(i_{*})$ is the image of the push-forward $i_{*}$. The goal is
now to compute generators for the ideal $\text{Im}(i_{*})$.

Let $X$ be a $G$-scheme and $\Delta\xrightarrow{i}X$ an equivariant closed
embedding. A consolidated method to determine the image of the group
homomorphism

$A_{G}^{*}(\Delta)\xrightarrow{i_{*}}A_{G}^{*}(X),$

is to use a so-called equivariant envelope of $\Delta$. Recall that an
envelope $\widetilde{\Delta}\xrightarrow{\widetilde{\pi}}\Delta$, see [Ful98,
Definition 18.3], is a proper map such that for every subvariety $V$ of
$\Delta$, there is a subvariety $\widetilde{V}$ of $\widetilde{\Delta}$ such
that the morphism $\widetilde{\pi}$ maps $\widetilde{\Delta}$ birationally
onto $V$. In the category of $G$-schemes, $\widetilde{\pi}$ is called an
equivariant envelope, see [EG98a, Section 2.6], if $\widetilde{\pi}$ is
$G$-equivariant and if one can choose $\widetilde{V}$ to be $G$-invariant
whenever $V$ is $G$-invariant. An equivariant envelope
$\widetilde{\Delta}\xrightarrow{\widetilde{\pi}}\Delta$ for a closed subscheme
$\Delta\xrightarrow{i}X$ is especially helpful when one can explicitly
describe the Chow group of $\widetilde{\Delta}$ and the image of the group
homomorphism

$A_{G}^{*}(\widetilde{\Delta})\xrightarrow{(i\circ\widetilde{\pi})_{*}}A_{G}^{*}(X).$

Since the group homomorphism $\widetilde{\pi}_{\ast}$ is surjective (see
[EG98a, Lemma 3] and [Ful98, Lemma 18.3(6)]), one has the following result.

######  Theorem 3.1.

Let $X$ be a $G$-scheme and $\Delta\xrightarrow{i}X$ an equivariant closed
embedding. If $\widetilde{\Delta}\xrightarrow{\widetilde{\pi}}\Delta$ is an
equivariant envelope of $\Delta$, then

$(i\circ\widetilde{\pi})_{*}\left(A_{G}^{*}(\widetilde{\Delta})\right)={i_{*}}\left(A_{G}^{*}\left(\Delta\right)\right).$

Returning to the computation, we construct an equivariant envelope for the
locus of degenerate maps $\Delta_{r,d}$.

######  Definition 3.2.

For every $i=1,\dots,d$, denote by $Z_{i}$ the subspace of
$\mathbb{P}\left(W_{d}^{\oplus r+1}\right)$ representing $(r+1)$-tuples of
polynomials with a common factor of degree $i$, but not $i+1$:

(8) $Z_{i}=\\{(f_{1},\ldots,f_{r+1})|\deg(\gcd(f_{1},\ldots,f_{r+1})=i\\}.$

The family $\\{Z_{i}\\}_{i=1,\dots,d}$ is an equivariant stratification of
$\Delta_{r,d}$, in the sense of [DLFV21, Definition 1.2].

Define

(9) $\widetilde{Z}_{i}:=\mathbb{P}(W_{i})\times\mathbb{P}\left(W_{d-i}^{\oplus
r+1}\right)$

and the morphisms

$\displaystyle\pi_{i}:\widetilde{Z}_{i}$ $\displaystyle\to$
$\displaystyle\mathbb{P}\left(W_{d}^{\oplus r+1}\right)$ (10)
$\displaystyle([c(x,y)],[g_{1}(x,y),g_{2}(x,y),\dots,g_{r+1}(x,y)])$
$\displaystyle\mapsto$ $\displaystyle[c(x,y)\cdot g_{1}(x,y),c(x,y)\cdot
g_{2}(x,y),\dots,c(x,y)\cdot g_{r+1}(x,y)].$

######  Proposition 3.3.

Denote by $\widetilde{Z}:=\bigsqcup_{i=1,\dots,d}\widetilde{Z}_{i}$ and by
$\pi:=\sqcup\pi_{i}$. Denoting by $\widetilde{\pi}$ the restriction of $\pi$
onto its image $\Delta_{r,d}$, we have that

$\widetilde{\pi}:\widetilde{Z}\to\Delta_{r,d}$

is an equivariant envelope on $\Delta_{r,d}$.

###### Proof.

We follow the argument in [Vis98, Lemma 3.2]. Observe that each
$\widetilde{Z}_{i}$ maps to the closure $\overline{Z}_{i}$ of a stratum of
$\Delta_{r,d}$. The morphism $\widetilde{\pi}$ is proper and
$\text{GL}_{2}$-invariant. To check the remaining equivariant envelope
hypothesis, we can restrict to each $Z_{i}$. Let $V$ be a
$\text{GL}_{2}$-invariant subvariety of $Z_{i}$ and $\omega$ the generic point
of $V$. The point $\omega$ is represented by an $(r+1)$-tuple
$[f]:=[f_{1},\dots,f_{r+1}]$ of forms of degree $d$ in $K[x,y]$ for some field
extension $k\subset K$. By definition of $Z_{i}$, the greatest common divisor
of the forms in $[f]$ has degree $i$. Therefore, there is a unique $K$-valued
point $\widetilde{\omega}$ in $\widetilde{Z}_{i}$ mapping to $\omega$. We
define $\widetilde{V}$ as the schematic closure of $\widetilde{\omega}$ in
$\widetilde{Z}_{i}$. Because of the uniqueness of the generic point,
$\widetilde{V}$ is in fact $\text{GL}_{2}$-invariant. ∎

######  Corollary 3.4.

We have the identity:

$\pi_{*}\left(A_{\text{GL}_{2}}^{*}(\widetilde{Z})\right)={i_{*}}\left(A_{\text{GL}_{2}}^{*}\left(\Delta_{r,d}\right)\right).$

###### Proof.

This is a consequence of Proposition 3.3 and Theorem 3.1. ∎

For the next definition, we adopt the notation in the following diagram:

(11)
$\xymatrix{&\widetilde{Z}_{i}\ar[rr]^{\pi_{i}}\ar[dl]_{p_{1}}\ar[dr]^{p_{2}}\mathbb{P}(W_{d}^{\oplus
r+1})\\\ \mathbb{P}(W_{i})\mathbb{P}(W_{d-i}^{\oplus r+1})}$

######  Definition 3.5.

For all $i=1,\dots,d$, denote by
$h_{i}=p_{1}^{\ast}c_{1}^{eq}(\mathcal{O}_{\mathbb{P}(W_{i})}(1))$ the (pull-
back via the first projection of the) canonical equivariant lift of the
hyperplane class on the first component of
$\widetilde{Z}_{i}=\mathbb{P}(W_{i})\times\mathbb{P}\left(W_{d-i}^{\oplus
r+1}\right)$. Also, denote by
$H=c_{1}^{eq}(\mathcal{O}_{\mathbb{P}(W_{d}^{\oplus r+1})}(1))$ and by
$\eta_{i}=p_{2}^{\ast}c_{1}^{eq}(\mathcal{O}_{\mathbb{P}\left(W_{d-i}^{\oplus
r+1}\right)}(1))$. Define the classes:

$\alpha_{i,k}(H):=\pi_{i_{*}}\left(h_{i}^{k}\right)\in
A_{\text{GL}_{2}}^{*}\left(\mathbb{P}(W_{d}^{\oplus r+1})\right).$

######  Proposition 3.6.

We have the ideal identity:

$i_{*}\left(A_{\text{GL}_{2}}^{*}\left(\Delta_{r,d}\right)\right)=\left(\alpha_{i,k}(H)\right)_{i=1,\dots,d;\;k=0,\dots,i}.$

###### Proof.

Using Corollary 3.4, it suffices to show that
$\pi_{*}\left(A_{\text{GL}_{2}}^{*}(\widetilde{Z})\right)$ is contained in the
ideal $I$ generated by the classes $\alpha_{i,k}(H)$. From (3),

(12) $\pi_{i}^{\ast}(H)=h_{i}+\eta_{i}.$

We prove that any class of the form $\pi_{i_{\ast}}(h_{i}^{k}\eta_{i}^{m})$ is
in $I$ by induction on $m$. For $m=0$, $k>i$, notice that the class
$\pi_{i_{\ast}}(h_{i}^{k})$ is a linear combination of the classes
$\alpha_{i_{k}}$, $k\leq i$, with coefficients in $\mathbb{Z}[l_{1},l_{2}]$;
hence the base case $m=0$ is established.

Using projection formula and (12) one obtains the following equalities.

$H\pi_{i_{\ast}}(h_{i}^{k}\eta_{i}^{m-1})=\pi_{i_{\ast}}(h_{i}^{k}\eta_{i}^{m-1}\pi_{i}^{\ast}H)=\pi_{i_{\ast}}(h_{i}^{k}\eta_{i}^{m-1}(h_{i}+\eta_{i}))=\pi_{i_{\ast}}(h_{i}^{k+1}\eta_{i}^{m-1})+\pi_{i_{\ast}}(h_{i}^{k}\eta_{i}^{m}).$

Solving

$\pi_{i_{\ast}}(h_{i}^{k}\eta_{i}^{m})=H\pi_{i_{\ast}}(h_{i}^{k}\eta_{i}^{m-1})-\pi_{i_{\ast}}(h_{i}^{k+1}\eta_{i}^{m-1})$

completes the inductive step. ∎

######  Lemma 3.7.

For every value of $r,d$, we have $P_{r,d}(H)\in Im(i_{\ast})$.

###### Proof.

The case $d=1$ is treated by direct inspection: it is known that
$\alpha_{1,0},\alpha_{1,1}$ generate the ideal of relations for the integral
Chow ring of Grassmannians ([EH16, Theorem 5.26]).

For $d\geq 2$, we consider the component $\widetilde{Z}_{2}$ of the
equivariant envelope and the point $P=(0:1:0)=[xy]\in\mathbb{P}(W_{2}).$ The
class $\pi_{2_{\ast}}(p_{1}^{\ast}([P]))$ is contained in the ideal of
relations. The lemma is proved by showing that such class divides
$P_{r,d}(H)$.

The map ${\pi_{2}}_{|p_{1}^{-1}([P])}$ has degree one onto the coordinate
linear subspace of $\mathbb{P}(W_{d}^{\oplus r+1})$ obtained by setting to
zero all homogeneous coordinates corresponding to monomials of the form
$x^{d}$ or $y^{d}$. It follows that

(13)
$\pi_{2_{\ast}}(p_{1}^{\ast}([P]))=\left((H+dl_{1})(H+dl_{2})\right)^{r+1},$

which indeed is a factor of $P_{r,d}(H)$. ∎

Using Proposition 3.6, Lemma 3.7 and recalling the notational convention 1.2,
we have obtained from (7) the following presentation for the Chow ring of
$\mathcal{M}_{0}(\mathbb{P}^{r},d)$.

######  Theorem 3.8.

For any odd positive integer $d$, and any integer $r\geq 0$ we have

(14)
$A^{*}\left(\mathcal{M}_{0}(\mathbb{P}^{r},d)\right)\cong\frac{\mathbb{Z}[c_{1},c_{2}]}{(\alpha_{i,k})_{i=1,\dots,d,k=0,\dots,i}}.$

######  Remark 3.9.

While for $r=0$ the constructions in this section do not yield a space of
stable maps (but rather a space of quasi maps as in [CFKM14]), we find
convenient to include this case in our study since we will be computing some
of the generating relations inductively on $r$, with $r=0$ being the base
case.

##  4\. Eliminating redundant relations

In this section we reduce the number of generators of the ideal of relations
in the quotient ring (14). More precisely, we prove that the ideal

$\left(\pi_{*}\left(A_{\text{GL}_{2}}^{*}(\widetilde{Z})\right)\right)=(\alpha_{i,k})_{i=1,\dots,d,k=0,\dots,i}$

is generated by $\alpha_{1,0},\alpha_{1,1},$ and $\alpha_{i,0}$ for all
positive integers $i$ that are powers of a prime number.

The goal is to prove that the image of the push-forward map

$\pi_{i*}:A_{\text{GL}_{2}}^{*}\left(\mathbb{P}(W_{i})\times\mathbb{P}\left(W_{d-i}^{\oplus
r+1}\right)\right)\to A_{\text{GL}_{2}}^{*}\left(\mathbb{P}\left(W_{d}^{\oplus
r+1}\right)\right)$

is contained in the image of the push-forward maps $\pi_{a*}$ with $a$ less
than $i$, whenever $i$ is not the power of a prime number. Consider the
commutative diagram

(15)
$\xymatrix{{\displaystyle\bigsqcup_{a=1}^{i-1}}\mathbb{P}(W_{a})\times\mathbb{P}(W_{i-a})\times\mathbb{P}(W_{d-i}^{\oplus
r+1})\ar[d]_{\psi_{i}:=\bigsqcup\pi_{a}\times\text{id}}\ar[rrr]^{\bigsqcup\text{id}\times\pi_{i-a}}&{\displaystyle\bigsqcup_{a=1}^{i-1}}\mathbb{P}(W_{a})\times\mathbb{P}(W_{d-a}^{\oplus
r+1})\ar[d]^{\bigsqcup\pi_{a}}\\\
\mathbb{P}(W_{i})\times\mathbb{P}(W_{d-i}^{\oplus
r+1})\ar[rrr]^{\pi_{i}}\mathbb{P}(W_{d}^{\oplus r+1})}$

in which the only morphism that has not been already defined is
$\psi_{i}:=\bigsqcup\pi_{a}\times\text{id}$. In this context, the morphism
$\pi_{a}:\mathbb{P}(W_{a})\times\mathbb{P}(W_{i-a})\to\mathbb{P}(W_{i})$ is
defined as

$\pi_{a}([c(x,y)],[g(x,y)])=[c(x,y)\cdot g(x,y)].$

In other words, on the left side of diagram (15), the morphisms $\pi_{a}$ are
considered in the case $r=0$, while in the rest of the diagram, the morphisms
$\pi_{k}$ are considered for a fixed value of $r$.

Since the above diagram is commutative, in order to show that the image of the
push-forward map $\pi_{i*}$ is contained in the image of the push-forward maps
$\pi_{a*}$ with $a$ less than $i$, it is enough to show that the push-forward
$\psi_{i*}$ is surjective. Moreover, it suffices to verify that $\psi_{i*}$ is
surjective when considering the torus-equivariant intersection groups.
Therefore, we focus on the push-forward map:

$\pi_{a*}:A^{*}_{T}\left(\mathbb{P}(W_{a})\times\mathbb{P}(W_{i-a})\right)\to
A^{*}_{T}\left(\mathbb{P}(W_{i})\right)$

######  Remark 4.1.

Arguing as in Proposition 3.6, the image of $\pi_{a*}$ is generated by the
classes $\alpha_{a,k}(h_{i}):=\pi_{a*}(h_{a}^{k})$ for $k=0,\dots,a$. Since
$\pi_{a*}$ is a homomorphism of $A^{*}_{\text{GL}_{2}}$-modules, one can also
view the image of $\pi_{a*}$ as generated by the classes

(16) $\widetilde{\alpha}_{a,k}:=\pi_{a*}(P_{a,k}(h_{a}))$

where $\\{P_{a,k}(h_{a})\\}_{k=0,\dots,a}$ is a suitable family of monic
polynomials of degree $k$ that we define as

$P_{a,k}(h_{a}):=\prod_{s=0}^{k-1}(h_{a}+(a-s)l_{1}+sl_{2}).$

In other words, $P_{a,k}(h_{a})$ is the equivariant Chow class of the locus of
polynomials in $W_{a}$ that are divisible by $y^{k}$. Note that
$P_{a,0}(h_{a})=1$. In the following Lemma we determine the classes
$\widetilde{\alpha}_{a,k}$.

######  Lemma 4.2.

Let $\widetilde{\alpha}_{a,k}$ be the class in $A^{T}_{*}(\mathbb{P}(W_{i}))$
defined in (16). Then we have the following identity:

$\widetilde{\alpha}_{a,k}=\binom{i-k}{a-k}P_{i,k}(h_{i})$

###### Proof.

Let $L_{a,k}\subset\mathbb{P}(W_{a})\times\mathbb{P}(W_{i-a})$ be the locus of
pairs of polynomials where the first one is a multiple of $y^{k}$. The locus
$L_{a,k}$ is invariant for the action of the torus $T$ and its equivariant
class is $P_{a,k}(h_{a})$. The restriction of the map

$\pi_{a}:\mathbb{P}(W_{a})\times\mathbb{P}(W_{i-a})$

to $L_{a,k}$ is finite of generic degree $\displaystyle\binom{i-k}{a-k}$ onto
its image. This is due to the following facts:

  * •

the image of $L_{a,k}$ is the locus of polynomials of degree $i$ which are
divisible by $y^{k}$;

  * •

let $R(x,y)=y^{k}S(x,y)$ be a generic homogeneous polynomial of degree $i$
which is divisible by $y^{k}$. Since we are in an algebraically closed field,
and assuming by generality that $S(x,y)$ has $i-k$ distinct roots, there are
exactly $\displaystyle\binom{i-k}{a-k}$ ways to choose a factor $F(x,y)$ of
$S(x,y)$ of degree $a-k$ so that $[y^{k}F(x,y),S(x,y)/F(x,y)]$ in
$\mathbb{P}(W_{a})\times\mathbb{P}(W_{i-a})$ maps to $R(x,y)$.

To conclude the proof, we notice that $\left[L_{a,k}\right]=P_{a,k}(h_{a})$
and $\left[\pi_{a}(L_{a,k})\right]=P_{i,k}(h_{i})$, therefore we have

$\widetilde{\alpha}_{a,k}=\binom{i-k}{a-k}P_{i,k}(h_{i}).$

∎

######  Theorem 4.3.

The ideal of relations $(\alpha_{i,k})_{i=1,\dots,d,k=0,\dots,i}$ is generated
by the classes $\alpha_{1,0}$, $\alpha_{1,1}$ from $\widetilde{Z}_{1}$ and the
pushforwards $\alpha_{i,0}$ of the fundamental classes of $\widetilde{Z}_{i}$
where $i$ is the power of a prime number.

###### Proof.

Assume that $i>1$ and fix a positive integer $1\leq k\leq i$. We want to show
that there exists a monic polynomial in $h_{i}$ of degree $k$ that is in the
image of $\psi_{*}$. For every integer $a<i$ and integer $j\leq\min(a,k)$, we
have that $h_{i}^{k-j}\widetilde{\alpha}_{a,j}$ is a polynomial in $h_{j}$
with leading coefficient equal to $\displaystyle\binom{i-j}{a-j}$. If $k\geq
2$ we can choose $a=j=k-1$, so that the polynomial
$\widetilde{\alpha}_{k-1,k-1}$ is monic in $h_{j}$. If $k=1$ then $k<i$ and
$\widetilde{\alpha}_{1,1}$ is monic.

For the case $k=0$, the only homogeneous monic polynomial in $h_{i}$ of degree
zero is the fundamental class 1. We know that the greatest common divisor of
the binomials $\displaystyle\binom{i}{a}$ for $a$ going from $1$ to $i-1$ is
in the image of $\psi_{*}$. It is a consequence of Lucas’s theorem [Luc78]
that such GCD is 1 when $i$ is not the power of a prime number (and the prime
number otherwise), and this concludes the proof. ∎

##  5\. Relations from $Z_{1}$

In this section we compute the relations coming from the first envelope
$Z_{1}$. We exhibit generating functions, whose coefficients are functions of
the degree $d$, encoding the classes $\alpha_{1,i}^{r}$ for all values of $r$.

######  Theorem 5.1.

For $k=0,1$, consider the
$A^{*}\left(\mathcal{M}_{0}(\mathbb{P}^{r},d)\right)$ valued generating
functions:

$\mathcal{A}_{1,k}(c_{1},c_{2},d)=\sum_{r=0}^{\infty}\pi_{1,\ast}[h_{1}^{k}]=\sum_{r=0}^{\infty}\alpha^{r}_{1,k},$

encoding the push-forwards of the fundamental class, and equivariant
hyperplane class from the first component $Z_{1}$ of the envelope. We have:

(17)
$\mathcal{A}_{1,0}(c_{1},c_{2},d)=\frac{d}{(1+\frac{d-1}{2}c_{1})(1-\frac{d+1}{2}c_{1})+d^{2}c_{2}}$

(18)
$\mathcal{A}_{1,1}(c_{1},c_{2},d)=\frac{1+\frac{d-1}{2}c_{1}}{(1+\frac{d-1}{2}c_{1})(1-\frac{d+1}{2}c_{1})+d^{2}c_{2}}-1.$

We begin by describing the strategy of proof of Theorem 5.1. The starting
point is the following commutative diagram, comparing the maps $\pi_{1}$ from
(11) for target dimensions $r$ and $r+1$:

(19) $\xymatrix{\mathbb{P}(W_{1})\times\mathbb{P}(W_{d-1}^{\oplus
r+1})\ar[rr]^{\pi_{1}}\ar[d]_{\text{id}\times i_{r}}&\mathbb{P}(W_{d}^{\oplus
r+1})\ar[d]^{i_{r}}\\\ \mathbb{P}(W_{1})\times\mathbb{P}(W_{d-1}^{\oplus
r+2})\ar[rr]^{\pi_{1}}\mathbb{P}(W_{d}^{\oplus r+2})},$

where the maps denoted $i_{r}$ are the linear inclusions into the subspace
where the homogeneous coordinates corresponding to the $(r+2)$-th direct
summand are equal to zero. The equality of the push-forwards along the two
different compositions in (19) gives rise to recursive relations that
determine the classes $\alpha_{1,k}^{r+1}$ from the classes
$\alpha_{1,k}^{r}$. Appropriately organizing the recursive relations gives
rise to a system of functional equations. The generating functions
$\mathcal{A}_{1,k}$ from (17) and (18) are then shown to satify the system of
functional equations and to have the correct initial conditions, corresponding
to the case $r=0$ (the target is a point).

###### Proof.

We first establish some conventions meant to simplify notation: the two
horizontal arrows of diagram (19) are both denoted by $\pi_{1}$, and we will
let the context determine which map any given equation is referring to.
Similarly, the equivariant hyperplane classes for the three projective spaces
in both rows of (19) are denoted $h_{1},\eta_{1}$ and $H$ (as in Section 3);
each class in the top row is the pull-back of the corresponding class in the
bottom row. So, for instance, any polynomial in $H,c_{1},c_{2}$ denotes both a
class for $\mathbb{P}(W_{d}^{\oplus r+2})$ and one for
$\mathbb{P}(W_{d}^{\oplus r+1})$, with the latter being the pullback via
$i_{r}$ of the former.

The commutativity of the diagram (19) implies that for $k=0,1$:

(20) $i_{r\ast}(\pi_{1_{\ast}}(h_{1}^{k}))=\pi_{1_{\ast}}((\text{id}\times
i_{r})_{\ast}(h_{1}^{k})).$

Starting with the left hand side of (20),
$\pi_{1_{\ast}}(h_{1}^{k})=\alpha^{r}_{1,k}(H)$ by definition, and by the
projection formula one has

(21) $i_{r_{\ast}}(\alpha^{r}_{1,k}(H))=\alpha^{r}_{1,k}(H)P_{d}(H),$

where $P_{d}(H):=P_{0,d}(H)$ from (6).

For the right hand side of (19), we apply the projection formula to obtain

(22) $\pi_{1_{\ast}}((\text{id}\times
i_{r})_{\ast}(h_{1}^{k}))=h_{1}^{k}P_{d-1}(\eta_{1}).$

Using that $H=h_{1}+\eta_{1}$ (where we omit $\pi_{1}^{\ast}$ from the
notation for the pullback of the class $H$), and the presentation of the
equivariant Chow ring of $\mathbb{P}(W_{1})$, we may write

(23)
$h_{1}^{k}P_{d-1}(\eta_{1})=R_{1,k,0}^{r,d}(H,c_{1},c_{2})+R_{1,k,1}^{r,d}(H,c_{1},c_{2})h_{1},$

where the right hand side of (23) is the remainder of the division of
$P_{d-1}(H-h_{1})$ by the polynomial
$P_{1}(h_{1})=(h_{1}^{2}-c_{1}h_{1}+c_{2})$.

Pushing forward and using the projection formula, we then obtain:

(24) $\pi_{1_{\ast}}((\text{id}\times
i_{r})_{\ast}(h_{1}^{k}))=R_{1,k,0}^{r,d}(H,c_{1},c_{2})\alpha^{r+1}_{1,0}(H)+R_{1,k,1}^{r,d}(H,c_{1},c_{2})\alpha^{r+1}_{1,1}(H).$

From equations (19),(21) and (23), by substituting $H=\frac{d+1}{2}c_{1}$ one
obtains the square size $2$ linear system

(25)
$P_{d}\alpha_{1,k}^{r}=R_{1,k,0}^{r,d}\alpha^{r+1}_{1,0}+R_{1,k,1}^{r,d}\alpha^{r+1}_{1,1},$

where $k=0,1$.

We observe that all coefficients in (25) are divisible by
$P_{d-2}:=P_{d-2}\left(\frac{d-1}{2}c_{1}\right).$

######  Claim 5.2.

We have the following identities:

(26) $\displaystyle P_{d}$ $\displaystyle=$
$\displaystyle\left(d^{2}c_{2}-\frac{d^{2}-1}{4}c_{1}^{2}\right)P_{d-2}$ (27)
$\displaystyle R_{1,0,0}^{r,d}$ $\displaystyle=$
$\displaystyle\left(\frac{d+1}{2}c_{1}\right)P_{d-2}$ (28) $\displaystyle
R_{1,0,1}^{r,d}$ $\displaystyle=$ $\displaystyle-dP_{d-2}$ (29) $\displaystyle
R_{1,1,0}^{r,d}$ $\displaystyle=$ $\displaystyle dc_{2}P_{d-2}$ (30)
$\displaystyle R_{1,1,1}^{r,d}$ $\displaystyle=$
$\displaystyle\left(\frac{1-d}{2}c_{1}\right)P_{d-2}$

Using the results from Claim 5.2, proved at the end of the section, the linear
system (25) simplifies to

(31)
$\left(d^{2}c_{2}-\frac{d^{2}-1}{4}c_{1}^{2}\right)\left[\begin{array}[]{c}\alpha^{r}_{1,0}\\\
\alpha^{r}_{1,1}\end{array}\right]=\left[\begin{array}[]{cc}\frac{d+1}{2}c_{1}&-d\\\
dc_{2}&\frac{1-d}{2}c_{1}\par\end{array}\right]\left[\begin{array}[]{c}\alpha^{r+1}_{1,0}\\\
\alpha^{r+1}_{1,1}\end{array}\right]$

The linear system (31) may be solved for the $\alpha_{1,k}^{r+1}$’s to obtain:

$\displaystyle\alpha^{r+1}_{1,0}$ $\displaystyle=$
$\displaystyle\frac{1-d}{2}c_{1}\alpha^{r}_{1,0}+d\alpha^{r}_{1,1}$ (32)
$\displaystyle\alpha^{r+1}_{1,1}$ $\displaystyle=$ $\displaystyle-
dc_{2}\alpha^{r}_{1,0}+\frac{d+1}{2}c_{1}\alpha^{r}_{1,1}.$

Together with the initial conditions

(33) $\alpha_{1,0}^{0}=d,\ \ \ \alpha_{1,1}^{0}=\frac{d+1}{2}c_{1},$

the equations in (5) for all $r\geq 0$ determine recursively all values of
$\alpha_{1,k}^{r}$. To conclude the proof, it suffices to observe that (5) and
(33) give rise to the system of functional equations:

(34)
$\left\\{\begin{array}[]{ccc}\mathcal{A}_{1,0}-d&=&\frac{1-d}{2}c_{1}\mathcal{A}_{1,0}+d\mathcal{A}_{1,1}\\\
\mathcal{A}_{1,1}-\frac{d+1}{2}c_{1}&=&-dc_{2}\mathcal{A}_{1,0}+\frac{d+1}{2}c_{1}\mathcal{A}_{1,1},\end{array}\right.$

together with the boundary conditions

(35) $\mathcal{A}_{1,0}(0,0,d)=d,\ \ \ \ \mathcal{A}_{1,1}(0,0,d)=0.$

It is then immediate to verify that (34),(35) are satisfied by the generating
functions (17),(18). Thus Theorem 5.1 is proved (as soon as the identities
from Claim 5.2 are established). ∎

###### Proof of Claim 5.2.

Equation (26) follows from the definition of the polynomial $P_{d}$ in terms
of the weights $l_{1}$ and $l_{2}$, as defined in (6); substituting
$H=\frac{d+1}{2}c_{1}$ in the polynomial $P_{0,d}(H)$ and
$H=\frac{d-1}{2}c_{1}$ in the polynomial $P_{0,d-2}(H)$ it is immediate to see
that the factors of $P_{d-2}$ are equal to the internal factors of $P_{d}$. It
follows that the quotient $P_{d}/P_{d-2}$ consists of the product of the first
and last terms of $P_{d}$:

(36)
$\frac{P_{d}}{P_{d-2}}=\left(\frac{d+1}{2}c_{1}+dl_{1}\right)\left(\frac{d+1}{2}c_{1}+dl_{2}\right).$

Equation (26) follows from (36) by expanding and substituting
$-l_{1}-l_{2}=c_{1},l_{1}l_{2}=c_{2}$.

Equations (27),(28),(29) and (30) are proved by induction on odd values of
$d$, with the base case $d=1$ easily established after noting $P_{-1}=1$.

We work in the quotient ring
$\mathbb{Z}[h_{1},c_{1},c_{2}]/(h_{1}^{2}-c_{1}h+c_{2})$, adopting the
convention that for any polynomial $q$, its image $[q]$ in the quotient ring
is identified with the remainder of division by $h_{1}^{2}-c_{1}h+c_{2}$. From
the definitions for the $(d+2)$-th case one may observe that

(37)
$P_{d+1}\left(\frac{d+3}{2}-h_{1}\right)=\left(\frac{d+3}{2}c_{1}+(d+1)l_{1}-h_{1}\right)\left(\frac{d+3}{2}c_{1}+(d+1)l_{2}-h_{1}\right)P_{d-1}\left(\frac{d+1}{2}-h_{1}\right).$

By (23)

(38)
$\left[P_{d+1}\left(\frac{d+3}{2}-h_{1}\right)\right]=\left[R^{r,d+2}_{1,0,0}+R^{r,d+2}_{1,0,1}h_{1}\right],$

so after symmetrizing the quadratic factor in (37) and replacing the last
factor using the inductive hypothesis, one has

(39)
$\left[R^{r,d+2}_{1,0,0}+R^{r,d+2}_{1,0,1}h_{1}\right]=\left[\left(h_{1}^{2}-2c_{1}h_{1}-\frac{(d+1)(d-3)}{4}c_{1}^{2}+(d+1)^{2}c_{2}\right)\left(-dh_{1}+\frac{d+1}{2}c_{1}\right)P_{d-2}\right]$

In order to obtain (27) and (28) one must check that:

(40)
$\left[R^{r,d+2}_{1,0,0}+R^{r,d+2}_{1,0,1}h_{1}\right]=\left[\left(-(d+2)h_{1}+\frac{d+3}{2}c_{1}\right)P_{d}\right]=\left[\left(-(d+2)h_{1}+\frac{d+3}{2}c_{1}\right)\left(d^{2}c_{2}-\frac{d^{2}-1}{4}c_{1}^{2}\right)P_{d-2}\right],$

where the last equality is obtained applying (26). It is then sufficient to
verify

(41)
$\left[\left(h_{1}^{2}-2c_{1}h_{1}-\frac{(d+1)(d-3)}{4}c_{1}^{2}+(d+1)^{2}c_{2}\right)\left(-dh_{1}+\frac{d+1}{2}c_{1}\right)\right]=\left[\left(-(d+2)h_{1}+\frac{d+3}{2}c_{1}\right)\left(d^{2}c_{2}-\frac{d^{2}-1}{4}c_{1}^{2}\right)\right],$

which is easily done.

In order to prove (29), (30), one has

(42)
$\left[R^{r,d}_{1,1,0}+R^{r,d}_{1,1,1}h_{1}\right]=\left[h_{1}P_{d-1}\left(\frac{d+1}{2}c_{1}-h_{1}\right)\right]=\left[R^{r,d}_{1,0,0}h_{1}+R^{r,d}_{1,0,1}h_{1}^{2}\right]=\left[-R^{r,d}_{1,0,1}c_{2}+\left(R^{r,d}_{1,0,0}+c_{1}R^{r,d}_{1,0,1}\right)h_{1}\right],$

from which the result follows by direct substitution. ∎

##  6\. Relations of type $\alpha_{i,0}$

In this section we exhibit formulas for the relations obtained by pushing-
forward the fundamental classes of the various components $Z_{i}$ of the
envelope. These classes exhibit the following interesting structure: for any
fixed values of $d,r$, the relations $\alpha_{i,0}$ are obtained for all $i$
via the action of a differential operator on a single monomial function.

######  Theorem 6.1.

For any $d>0,r\geq 0,0\leq i\leq d$, the relations $\alpha_{i,0}\in
A^{\ast}(\mathcal{M}_{0}(\mathbb{P}^{r},d))$ are given by the formula:

(43)
$\alpha_{i,0}=\displaystyle\sum_{j=0}^{i}\frac{(-1)^{j}}{i!(l_{2}-l_{1})^{i}}\binom{i}{j}\left(\frac{P_{d}}{\prod_{k=j}^{d-i+j}\left(\frac{c_{1}}{2}+\left(k-\frac{d}{2}\right)(l_{2}-l_{1})\right)}\right)^{r+1}.$

######  Remark 6.2.

We observe that in order to get from formula (43) to the generating function
$\mathcal{A}_{i,0}(d)$ from the Main Theorem, it is just a matter of noticing
the geometric series arising when adding (43) for all values of $r$.

We obtain formula (43) through an application of Atiyah-Bott localization,
which we now recall.

######  Theorem 6.3 ([EG98b, Theorem 2]).

Define the $A^{*}_{T}$-module $\mathcal{Q}:=((A^{*}_{T})^{+})^{-1}A^{*}_{T}$,
where $(A^{*}_{T})^{+}$ is the multiplicative system of homogeneous elements
of $A^{*}_{T}$ of positive degree.

Let $X$ be a smooth $T$-variety and consider the locus $F$ of fixed points for
the action of $T$. Let $F=\cup F_{j}$ be the decomposition of $F$ into
irreducible components. For every $\gamma$ in
$A^{*}_{T}(X)\otimes\mathcal{Q}$, we have the identity

$\gamma=\sum_{j}\frac{i_{F_{j}}^{*}(\gamma)}{c_{top}^{T}(N_{F_{j}/X})}$

where $i_{F_{j}}$ is the inclusion of $F_{j}$ in $X$ and $N_{F_{j}/X}$ is the
normal bundle of $F_{j}$ in $X$.

###### Proof of Theorem 6.1.

Referring to diagram (11) for notation, we compute

(44)
$\alpha_{i,0}=\pi_{1_{\ast}}([\widetilde{Z}_{i}])=\pi_{1_{\ast}}(p_{1}^{\ast}([\mathbb{P}(W_{i})]))$

by first localizing the fundamental class of $[\mathbb{P}(W_{i})]$ and then
pull-pushing the corresponding expression taking advantage of the fact that it
is supported on a torus invariant subvariety.

For $0\leq j\leq i$, denote by $q_{j}$ the torus invariant point where only
the $j$-th homogeneous coordinate is non-zero. From Theorem 6.3 we have

(45)
$[\mathbb{P}(W_{i})]=\sum_{j=0}^{i}\frac{[q_{j}]}{e(N_{q_{j}/\mathbb{P}(W_{i})})},$

where the equivariant Euler class of the normal bundle to $q_{j}$ is the
product of the $i$ tangent weights

(46)
$e(N_{q_{j}/\mathbb{P}(W_{i})})=\prod_{k\not=j}[((i-k)l_{1}+kl_{2})-((i-j)l_{1}+jl_{2})]=(-1)^{j}j!(i-j)!(l_{2}-l_{1})^{i}.$

Since both $p_{1}^{\ast}$ and $\pi_{1_{\ast}}$ are morphisms of $\mathcal{Q}$
modules, we have

(47)
$\pi_{1_{\ast}}(p_{1}^{\ast}([\mathbb{P}(W_{i})]))=\sum_{j=0}^{i}(-1)^{j}\frac{\pi_{1_{\ast}}[q_{j}\times\mathbb{P}(W_{d-i}^{\oplus
r+1})]}{j!(i-j)!(l_{2}-l_{1})^{i}}$

It remains to compute the equivariant Chow class of
$\pi_{1_{\ast}}[q_{j}\times\mathbb{P}(W_{d-i}^{\oplus r+1})]$. One may see
that $\pi_{1}$ maps the subspace $q_{j}\times\mathbb{P}(W_{d-i}^{\oplus r+1})$
isomorphically onto the linear subspace of $\mathbb{P}(W_{d}^{\oplus r+1})$
parameterizing $(r+1)$-tuples of polynomials which are divisible by $x^{j}$
and $y^{i-j}$. The class of this subspace corresponds to the product of the
linear factors of the polynomial $P_{r,d}(H)$ corresponding to the homogeneous
coordinates that vanish. By equivalently dividing $P_{r,d}(H)$ by the factors
corresponding to the coordinates that do not vanish, one may write:

(48) $\pi_{1_{\ast}}[q_{j}\times\mathbb{P}(W_{d-i}^{\oplus
r+1})]=\frac{P_{r,d}(H)}{\prod_{k=j}^{d-i+j}(H+(d-k)l_{1}+kl_{2})^{r+1}}$

Plugging (48) into (47) and substituting $H=\frac{d+1}{2}c_{1}$ one readily
obtains the expression in (43).

∎

######  Corollary 6.4.

For any fixed values of $r,d>0$, the relations $\alpha_{i,0}$ for all values
of $i$ are encoded in the generating function:

(49)
$\mathcal{A}_{0}(d)=\sum_{i=0}^{d}\alpha_{i,0}=\left[\left(\text{e}^{t_{1}\partial_{x}+t_{2}\partial_{y}}\left(x^{\frac{c_{1}}{2(l_{1}-l_{2})}+\frac{d}{2}}y^{\frac{c_{1}}{2(l_{2}-l_{1})}+\frac{d}{2}}\right)\right)^{\odot
r+1}_{|x=y=1}\right]_{\left|t_{1}=t_{2}=1,\;deg\leq dr\right.},$

where the symbol $\odot$ refers to a Hadamard power for a two-variable
exponential power series in variables
$t_{1}/(l_{1}-l_{2}),t_{2}/(l_{2}-l_{1})$, defined as follows:

(50)
$\left(\sum_{\mu,\nu}a_{\mu,\nu}\frac{\left(\frac{t_{1}}{l_{1}-l_{2}}\right)^{\mu}}{\mu!}\frac{\left(\frac{t_{2}}{l_{2}-l_{1}}\right)^{\nu}}{\nu!}\right)^{\odot
r+1}:=\sum_{\mu,\nu}a_{\mu,\nu}^{r+1}\frac{\left(\frac{t_{1}}{l_{1}-l_{2}}\right)^{\mu}}{\mu!}\frac{\left(\frac{t_{2}}{l_{2}-l_{1}}\right)^{\nu}}{\nu!}.$

###### Proof.

We show that the degree $ri$ summand of (49) coincides with (43).

The exponential differential operator is in commuting variables and hence it
admits the natural Taylor expansion:

(51)
$\text{e}^{t_{1}\partial_{x}+t_{2}\partial_{y}}=\sum_{\mu,\nu}\frac{t_{1}^{\mu}}{\mu!}\frac{t_{2}^{\nu}}{\nu!}\partial_{x}^{\mu}\partial_{y}^{\nu}$

Applying the general summand in (51) to the monomial
$\left(x^{\frac{c_{1}}{2(l_{1}-l_{2})}+\frac{d}{2}}y^{\frac{c_{1}}{2(l_{2}-l_{1})}+\frac{d}{2}}\right)$
and specializing $x=y=1$ one obtains:

(52)
$\frac{t_{1}^{\mu}}{\mu!}\frac{t_{2}^{\nu}}{\nu!}\frac{(-1)^{\mu}}{(l_{2}-l_{1})^{\mu+\nu}}\prod_{k=0}^{\mu-1}\left(\frac{c_{1}}{2}+\left(k-\frac{d}{2}\right)(l_{2}-l_{1})\right)\prod_{m=0}^{\nu-1}\left(\frac{c_{1}}{2}+\left(\frac{d}{2}-m\right)(l_{2}-l_{1})\right)$

Reindexing the second product in (52) by $m=d-k$ one may rewrite (52) to
obtain

(53)
$=\sum_{\mu,\nu}\frac{t_{1}^{\mu}}{\mu!}\frac{t_{2}^{\nu}}{\nu!}\frac{(-1)^{\mu}}{(l_{2}-l_{1})^{\mu+\nu}}\frac{P_{d}}{\prod_{k=\mu}^{d-\nu+1}\left(\frac{c_{1}}{2}+\left(k-\frac{d}{2}\right)(l_{2}-l_{1})\right)}$

By the definition of Hadamard power in (50), taking the $(r+1)$-th power of
the power series in 53 has the effect of raising the last rational function to
the power of $(r+1)$. The Chow degree of the coefficient of
$t_{1}^{\mu}t_{2}^{\nu}$ is $(r+1)(\mu+\nu)-(\mu+\nu)=r(\mu+\nu)$, from which
follows that the degree $ri$ coefficient of the series is obtained by summing
over all non-negative pairs of values of $\mu$ and $\nu$ adding to $i$.
Setting $\mu=j,\nu=i-j$ one may immediately recognize formula (43). ∎

Formula (43) allows us to also observe some structure of the relations
$\alpha_{i,0}$ as the degree $d$ varies.

######  Corollary 6.5.

For fixed values of $i$ and $r$, the coefficients of the classes
$\alpha_{i,0}\in\mathbb{Z}[c_{1},c_{2}]$ are polynomials in $d$ of degree
$i(r+1)$.

##  7\. Examples

###  7.1. The Case $d=1$.

Let us consider linear maps. The space $\mathcal{M}_{0}(\mathbb{P}^{r},1)$ is
the Grassmannian of lines in $\mathbb{P}^{r}$. The integral Chow ring of
Grassmannians is well known (see for example [EH16, Theorem 5.26]), therefore,
in this case, the result is not original but it allows us to show how to apply
the formulas in the simplest case. Thanks to Theorem 4.3, the integral
intersection ring of $\mathcal{M}_{0}(\mathbb{P}^{r},1)$ has the form

$\mathbb{Z}[c_{1},c_{2}]/I$

where $I$ is the ideal generated by the classes $\alpha_{1,0}$ and
$\alpha_{1,1}$. By replacing $d=1$ in equations (17) and (18), we get that the
classes $\alpha_{1,0}$ and $\alpha_{1,1}$ are the terms of total degree $r$
and $r+1$ in the power series expansion:

$\frac{1}{1-c_{1}+c_{2}}=1-(-c_{1}+c_{2})+(-c_{1}+c_{2})^{2}-\dots$

The generating functions $\mathcal{A}_{1,0}(c_{1},c_{2},1)$ and
$\mathcal{A}_{1,1}(c_{1},c_{2},1)$ from equations (17) and (18) are the same
up to the constant term. The above result is consistent with [EH16, Theorem
5.26] up to replacing our $c_{1}$ with $-c_{1}$.

###  7.2. The Case $r=2$ and $d=3$.

We explicitly determine the case of rational cubics in $\mathbb{P}^{2}$. By
Theorem 4.3, the integral intersection ring of
$\mathcal{M}_{0}(\mathbb{P}^{2},3)$ has the form $\mathbb{Z}[c_{1},c_{2}]/I,$
where $I$ is the ideal generated by the classes $\alpha_{1,0}$,
$\alpha_{1,1}$, $\alpha_{2,0}$, and $\alpha_{3,0}$.

The generating functions for $\alpha_{1,0}$ and $\alpha_{1,1}$, when $d=3$,
are

$\displaystyle\mathcal{A}_{1,0}(3)$ $\displaystyle=$
$\displaystyle\frac{3}{(1+c_{1})(1-2c_{1})+9c_{2}}=\frac{3}{1-c_{1}-2c_{1}^{2}+9c_{2}}=3\left(1+(c_{1}+2c_{1}^{2}-9c_{2})+(c_{1}+2c_{1}^{2}-9c_{2})^{2}+\dots\right);$
$\displaystyle\mathcal{A}_{1,1}(3)$ $\displaystyle=$
$\displaystyle\frac{2c_{1}+2c_{1}^{2}-9c_{2}}{1-c_{1}-2c_{1}^{2}+9c_{2}}=(2c_{1}+2c_{1}^{2}-9c_{2})\left(1+(c_{1}+2c_{1}^{2}-9c_{2})+(c_{1}+2c_{1}^{2}-9c_{2})^{2}+\dots\right).$

When $r=2$, the class $\alpha_{1,0}$ is the term of degree 2 of
$\mathcal{A}_{1,0}(c_{1},c_{2},3)$. More precisely:

$\alpha_{1,0}=3(2c_{1}^{2}-9c_{2}+c_{1}^{2})=9c_{1}^{2}-27c_{2}.$

On the other hand, the class $\alpha_{1,1}$ is the term of degree 3 of
$\mathcal{A}_{1,1}(c_{1},c_{2},3)$, that is to say:

$\alpha_{1,1}=2c_{1}(3c_{1}^{2}-9c_{2})+(2c_{1}^{2}-9c_{2})(c_{1})=8c_{1}^{3}-27c_{1}c_{2}.$

We apply formula (43) to determine $\alpha_{2,0}$ and $\alpha_{3,0}$. We use
the following script on Maple that works for any values of $d$ and $r$:

Hence we have:

$\displaystyle\alpha_{2,0}$ $\displaystyle=$ $\displaystyle
12c_{1}^{4}-90c_{1}^{2}c_{2}+189c_{2}^{2};$ $\displaystyle\alpha_{3,0}$
$\displaystyle=$ $\displaystyle
4c_{1}^{6}-42c_{1}^{4}c_{2}+129c_{1}^{2}c_{2}^{2}-90c_{2}^{3}.$

One can simplify the generators to write the final result in a more compact
form:

Classes | Reduction
---|---
$\alpha_{1,0}=9c_{1}^{2}-27c_{2}$ | $9c_{1}^{2}-27c_{2}$
$\alpha_{1,1}=8c_{1}^{3}-27c_{1}c_{2}$ | $\text{\framebox{$c_{1}^{3}$}}=c_{1}\cdot\alpha_{1,0}-\alpha_{1,1}$
| moreover, $27c_{1}c_{2}$ is generated by $c_{1}^{3}$ and $\alpha_{1,1}$
$\alpha_{2,0}=12c_{1}^{4}-90c_{1}^{2}c_{2}+189c_{2}^{2}$ | $\alpha_{2,0}=(4c_{1}^{2}-7c_{2})\cdot\alpha_{1,0}-3c_{1}\cdot\alpha_{1,1}$
$\alpha_{3,0}=4c_{1}^{6}-42c_{1}^{4}c_{2}+129c_{1}^{2}c_{2}^{2}-90c_{2}^{3}$ | $\framebox{$6c_{2}^{2}c_{1}^{2}+9c_{2}^{3}$}=(4c1^{4}-10c1^{2}c2+3c2^{2})\alpha_{1,0}-(4c1^{3}-6c1c2)\alpha_{1,1}-\alpha_{3,0}$

The class $\alpha_{2,0}$ is redundant since it belongs to the ideal generated
by $\alpha_{1,0}$ and $\alpha_{1,1}$. On the other hand, the class
$\alpha_{3,0}$ does not belong to the ideal generated by $\alpha_{1,0}$ and
$\alpha_{1,1}$. One may easily check this as follows: taking a further
quotient by the ideal generated by $c_{1}$ and $27$, we get
$\alpha_{1,0}\equiv\alpha_{1,1}\equiv 0$ but $\alpha_{3,0}\equiv
9c_{2}^{3}\not\equiv 0$. One can also see that

$3(6c_{2}^{2}c_{1}^{2}+9c_{2}^{3})=-c_{2}^{2}(9c_{1}^{2}-27c_{2})+c_{1}c_{2}(27c_{1}c_{2}),$

showing in particular that the class $3\alpha_{3,0}$ is in the ideal generated
by $\alpha_{1,0}$ and $\alpha_{1,1}$. In conclusion, we have shown the
following result.

######  Theorem 7.1.

We have the isomorphism:

$A^{*}\left(\mathcal{M}_{0}\left(\mathbb{P}^{2},3\right)\right)\cong\frac{\mathbb{Z}[c_{1},c_{2}]}{(9c_{1}^{2}-27c_{2},\;c_{1}^{3},\;6c_{2}^{2}c_{1}^{2}+9c_{2}^{3})}.$

As seen in Theorem 7.1, the set of generators described in Theorem 4.3 is not
necessarily minimal. We propose the following conjecture.

######  Conjecture 7.2.

Let $r$ be a positive integer and $d$ a positive odd number. Then

$A^{*}\left(\mathcal{M}_{0}\left(\mathbb{P}^{r},d\right)\right)\cong\frac{\mathbb{Z}[c_{1},c_{2}]}{(\alpha_{1,0},\alpha_{1,1},\\{\alpha_{p,0}|\;\text{$p$
is a prime that divides $d$}\\})}.$

Further, all the relations listed are necessary.

Seth Ireland programmed a Macaulay2 code that provided extensive verification
for this conjecture [Ire22]. At this point we know the conjecture to be true
for $r\leq 9$, $d\leq 49$. A weaker version of the conjecture, asserting
generation but not minimality, has been verified for $r\leq 5$, $d\leq 99$.

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