# Verlinde Series for Hirzebruch Surfaces

Ian Cavey

###### Abstract

We give an explicit formula for Euler characteristics of line bundles on the
Hilbert scheme of points on $\mathbb{P}^{1}\times\mathbb{P}^{1}$. Combined
with structural results of Ellingsrud, Göttsche, and Lehn [5], this determines
the Euler characteristic of any line bundle on the Hilbert scheme of points on
any smooth, projective surface. We also give an enumerative description of the
dimensions of spaces of global sections of ample line bundles on Hilbert
schemes of points on Hirzebruch surfaces, extending the polytope-line bundle
correspondence on the underlying toric surface.

##  1 Introduction

Hilbert schemes of points on surfaces are fundamental examples of moduli
spaces in algebraic geometry with connections to a wide variety of topics in
math, as well as theoretical physics (for a brief survey see [7]). Verlinde
series are central objects of study in the enumerative geometry of these
Hilbert schemes. For a smooth, projective surface $X$, let $X^{[n]}$ denote
the Hilbert scheme of $n$ points on $X$. This Hilbert scheme is a smooth,
projective variety of dimension $2n$, and can be thought of as a
compactification of the set of unordered $n$-tuples of distinct points in $X$.
Given a line bundle $L$ on $X$, there is an induced line bundle $L_{n}$ on
$X^{[n]}$ pulled back from the symmetric power (see Section 2 for the precise
definitions). Verlinde series, introduced by Ellingsrud, Göttsche, and Lehn
[5], are the generating functions for holomorphic Euler characteristics of
line bundles,

$\mathbf{V}_{X,L,r}(z)=\sum_{n=0}^{\infty}z^{n}\cdot\chi(X^{[n]},L_{n}\otimes
E^{r})\in\mathbb{Z}[[z]],$

where $E$ is $-1/2$ times the exceptional divisor on $X^{[n]}$ and $r$ is a
fixed integer.

Verlinde series contain fundamental enumerative information about Hilbert
schemes of points. All line bundles on $X^{[n]}$ are of the form $L_{n}\otimes
E^{r}$ [6], so the coefficients of Verlinde series contain the holomorphic
Euler characteristics of all line bundles. In particular, sufficiently ample
line bundles on $X^{[n]}$ correspond to projective embeddings
$X^{[n]}\hookrightarrow\mathbb{P}(H^{0}(X^{[n]},L_{n}\otimes E^{r}))$, and in
this case the coefficient on the $z^{n}$ term of the Verlinde series
$\mathbf{V}_{X,L,r}(z)$ is the dimension of the vector space
$H^{0}(X^{[n]},L_{n}\otimes E^{r}).$

Verlinde series are also known to be related to other important enumerative
invariants of Hilbert schemes. Johnson [10] and Marian, Oprea, and
Pandharipande [14] conjectured that Verlinde series can be transformed into
the generating series for the top Segre classes of higher rank tautological
vector bundles on $X^{[n]}$ by an explicit change of variables. These higher
rank Segre series generalize those in Lehn's well-known conjecture [12], which
was proved by Voison [15] and Marian, Oprea, and Pandharipande [14]. Higher
rank Segre series are known completely for $K$-trivial surfaces [13], while
formulas for arbitrary surfaces are known only for tautological vector bundles
of certain ranks [13, 16]. The conjectured correspondance between Verlinde
series and higher rank Segre series was recently proved by Göttsche and Mellit
[8], although neither series was determined in general. Consequently, the
determination of general formulas for either the Verlinde series or higher
rank Segre series also determines the other.

An essential feature of Verlinde series established by Ellingsrud, Göttsche,
and Lehn [5] is the factorization into universal power series
$A_{r},B_{r},C_{r},D_{r}\in\mathbb{Q}[[z]]$ depending only on $r$,

$\mathbf{V}_{X,L,r}(z)=A_{r}(z)^{\chi(L)}\cdot
B_{r}(z)^{\chi(\mathcal{O}_{X})}\cdot C_{r}(z)^{c_{1}(L)\cdot
K_{X}-\frac{1}{2}K_{X}^{2}}\cdot D_{r}(z)^{K_{X}^{2}},$ (1)

where $K_{X}$ is the canonical divisor on $X$. In particular,
$\chi(X^{[n]},L_{n}\otimes E^{r})$ does not depend on the pair $(X,L)$ beyond
the four enumerative invariants $\chi(L),\chi(\mathcal{O}_{X}),c_{1}(L)\cdot
K_{X},$ and $K_{X}^{2}$. Computing Verlinde series can therefore be reduced to
finding formulas for the series $A_{r},B_{r},C_{r}$ and $D_{r}$ for each
integer $r.$ Using Serre duality one can show that these series satisfy the
symmetry relations $A_{-r}(z)=A_{r}(z),$ $B_{-r}(z)=B_{r}(z),$
$D_{-r}(z)=D_{r}(z),$ and $C_{-r}(z)=(C_{r}(z))^{-1}$ ([5] Theorem 5.3), so we
restrict to the case $r\geq 0$.

For $r=0,1$, the coefficients of the Verlinde series are given by the
combinatorially suggestive formulas,

$\chi(X^{[n]},L_{n})={\chi(L)+n-1\choose n},\hskip 28.45274pt\text{and}\hskip
28.45274pt\chi(X^{[n]},L_{n}\otimes E)={\chi(L)\choose n},$ (2)

valid for any smooth surface $X$ and line bundle $L$ ([5] Lemma 5.1). For
$\chi(L)>0$, these formulas count the number of ways to choose $n$ objects
from a set of $\chi(L)$ with and without repetitions respectively. In both
cases $r=0,1$ we have $B_{r}=C_{r}=D_{r}=1$, and one can give formulas for
$A_{r}$.

One approach to determine Verlinde series for $r>1$ is to focus on particular
surfaces with additional structure. For example, the Hilbert scheme of points
on a K3 surface is a symplectic manifold, and Ellingsrud, Göttsche, and Lehn
use this additional structure to show that $\chi(X^{[n]},L_{n}\otimes
E^{r})={\chi(L)-(r^{2}-1)(n-1)\choose n}$ for any K3 surface $X$, line bundle
$L$, and integer $r$ ([5] Theorem 5.3). The enumerative data of the underlying
K3 surface is $\chi(\mathcal{O}_{X})=2$ and $K_{X}=0$, so this result is a
formula for the coefficients of the Verlinde series
$\mathbf{V}(z)=A_{r}(z)^{\chi(L)}\cdot B_{r}(z)^{2}$. From this, Ellingsrud,
Göttsche, and Lehn extract formulas for $A_{r}$ and $B_{r}$ for any integer
$r$. A key consequence of the structural formula (1) is that these formulas
for $A_{r}$ and $B_{r}$ deduced from the K3 case determine the Verlinde series
for any surface $X$ for which $K_{X}=0$, since the unknown series $C_{r}$ and
$D_{r}$ do not contribute to their Verlinde series.

Formulas for Verlinde series involving $C_{r}$ and $D_{r}$ have proved more
difficult to find. Recently, using the theory of Macdonald polynomials,
Göttsche and Mellit [8] gave a substantially more complicated formula for
$C_{r}$ and a conjectural formula for $D_{r}$ for arbitrary $r$. In
particular, their formula for $C_{r}$ determines the Verlinde series for any
surface $X$ for which $K_{X}^{2}=0$, extending the known $K_{X}=0$ case.

Our first main result is a formula for the Euler characteristics of line
bundles on the Hilbert scheme of points on
$\mathbb{P}^{1}\times\mathbb{P}^{1}$. The relevant enumerative invariants of
the pair $(X,L)=(\mathbb{P}^{1}\times\mathbb{P}^{1},\mathcal{O}(d_{1},d_{2}))$
are $\chi(L)=(d_{1}+1)(d_{2}+1)$, $\chi(\mathcal{O}_{X})=1$, $c_{1}(L)\cdot
K_{X}=-2(d_{1}+d_{2})$, and $K_{X}^{2}=8$. This result is therefore an
explicit formula for the coefficients of the Verlinde series

$\mathbf{V}(z)=(A_{r}(z))^{(d_{1}+1)(d_{2}+1)}\cdot
B_{r}(z)\cdot(C_{r}(z))^{-2(d_{1}+d_{2})-4}\cdot(D_{r}(z))^{8}$ (3)

for any integers $d_{1},d_{2}$ and $r>0$.

To state the formula, we first set up some combinatorial notation. For any
vector $\delta=(\delta_{1},\dots,\delta_{n-1})\in\\{0,1,\dots,r\\}^{n-1}$,
define the statistics $|\delta|=\sum_{i=1}^{n-1}\delta_{i},$
$c(\delta)=1+\\#\\{i=1,\dots,n-1\,|\,\delta_{i}\neq 0\\},$ and
$\ell(\delta)=1+\\#\\{i=1,\dots,n-1\,|\,\delta_{i}=r\\}$. There are exactly
$c(\delta)$ distinct numbers in the list
$0,\delta_{1},\delta_{1}+\delta_{2},\dots,\delta_{1}+\cdots+\delta_{n-1}$
which we label in increasing order $a_{1},\dots,a_{c}$ and we write
$n_{k}=n_{k}(\delta)$ for the number of occurrences of $a_{k}$ in this list.
Finally, for each $k=1,\dots,c$ we define
$w_{k}(\delta)=\sum_{i=1}^{c}n_{i}\max\\{r-|a_{k}-a_{i}|,0\\}.$

######  Theorem 1.1.

For $X=\mathbb{P}^{1}\times\mathbb{P}^{1}$, any line bundle
$L=\mathcal{O}(d_{1},d_{2})$, and $r>0$,

$\chi(X^{[n]},L_{n}\otimes
E^{r})=\sum_{\delta\in\\{0,1,\dots,r\\}^{n-1}}{d_{1}-|\delta|+\ell(\delta)\choose\ell(\delta)}\prod_{k=1}^{c(\delta)}{d_{2}-w_{k}(\delta)+r+n_{k}(\delta)\choose
n_{k}(\delta)}.$

Theorem 1.1 is proved in Section 4. We note that the quantity
$\chi(X^{[n]},L_{n}\otimes E^{r})$ is clearly symmetric in $d_{1},d_{2}$,
whereas the symmetry of the formula given in Theorem 1.1 is not at all
apparent. The asymmetry of the expression comes from our study of a
(symmetric) collection of polynomials using a non-symmetric term order (see
Sections 3 and 5). It would be interesting to show directly that the formula
in Theorem 1.1 is symmetric in $d_{1},d_{2}$.

Specializing Theorem 1.1 allows one to determine the series $C_{r}$ and
$D_{r}$ explicitly. For example, when $(d_{1},d_{2})=(-1,-1)$ and
$(d_{1},d_{2})=(-1,-2)$ the previous formula gives the coefficients of
$\mathbf{V}(z)=B_{r}(z)\cdot D_{r}(z)^{8}$ and $\mathbf{V}(z)=B_{r}(z)\cdot
C_{r}(z)^{2}\cdot D_{r}(z)^{8}$ respectively. Since $B_{r}(z)$ is known, one
can solve for $D_{r}(z)$ and $C_{r}(z)$ by dividing and taking roots of these
power series. Theorem 1.1 therefore determines the Verlinde series
$\mathbf{V}_{X,L,r}(z)$ for any surface $X$, line bundle $L$, and integer $r$.

The expressions for $C_{r}$ and $D_{r}$ extracted from Theorem 1.1 appear
quite different from those given by Göttsche and Mellit [8]. In particular, it
remains an open problem to show that their formula for $D_{r}$ is correct, and
to show directly that the two different expressions for $C_{r}$ agree. Theorem
1.1 provides a new way to establish such conjectures: It suffices to show that
a conjectured expression for $D_{r}$ gives the correct value of any one of the
Verlinde series determined by Theorem 1.1, e.g. $\mathbf{V}(z)=B_{r}(z)\cdot
D_{r}(z)^{8}$.

Finally, we note that the determination of Verlinde series by Theorem 1.1 also
determines all higher rank Segre series by the Segre-Verlinde correspondence
established by Göttsche and Mellit [8]. However, it is not clear how to
extract closed form formulas for these Segre series from Theorem 1.1. A first
step in this direction might be to show that Theorem 1.1 is compatible under
this correspondence with the formulas for Segre series established by Marian,
Oprea, and Pandharipande [13] and Yao [16] in certain ranks.

We obtain Theorem 1.1 as a consequence of our second main result, a
combinatorial interpretation for global sections of ample line bundles on
Hilbert schemes of points on Hirzebruch surfaces. For now, we remain
restricted to the special case $X=\mathbb{P}^{1}\times\mathbb{P}^{1}$.
Consider the line bundle $L=\mathcal{O}(d_{1},d_{2})$ on
$X=\mathbb{P}^{1}\times\mathbb{P}^{1}$. By the toric geometry of $X$ [4],
there is a basis for the global sections $H^{0}(X,L)$ indexed by the integer
points in the rectangle
$P_{L}=[0,d_{1}]\times[0,d_{2}]\subseteq\mathbb{R}^{2}$. To generalize this to
$X^{[n]}$, we introduce the following terminology. Let
$(\mathbf{a},\mathbf{b})=(a_{1},b_{1}),\dots,(a_{n},b_{n})$ be an ordered
$n$-tuple of integer points in $P_{L}$. For any integer $r\geq 0$, we say that
$(\mathbf{a},\mathbf{b})$ is $r$-lexicographically increasing in $P_{L}$ if:

  1. 1.

$a_{i}\leq a_{i+1}$ for each $i=1,\dots,n-1$,

  2. 2.

if $a_{i}=a_{i+1}$ then $b_{i+1}\geq b_{i}+r$ for each $i=1,\dots,n-1$, and

  3. 3.

$\sum_{i=1}^{j-1}\max\\{r-(a_{j}-a_{i}),0\\}\leq b_{j}\leq
d_{2}-\sum_{k=j+1}^{n}\max\\{r-(a_{k}-a_{j}),0\\}$ for each $j=1,\dots,n$.

This final condition says that not only do we have $b_{j}\in[0,d_{2}]$
(because $(a_{j},b_{j})\in P_{L}$), but any other point in the tuple
$(\mathbf{a},\mathbf{b})$ whose $a$-coordinate is close to $a_{j}$ imposes an
additional constraint on $b_{j}$. One checks that an $n$-tuple of integer
points $(\mathbf{a},\mathbf{b})$ in $P_{L}$ is $0$-lexicographically
increasing in $P_{L}$ if and only if
$(a_{1},b_{1})\leq\cdots\leq(a_{n},b_{n})$ in lexicographic order, and
$1$-lexicographically increasing if and only if
$(a_{1},b_{1})<\cdots<(a_{n},b_{n})$ in lexicographic order. For larger $r$,
$r$-lexicographic increasingness in $P_{L}$ is a stronger notion of
separatedness for $n$-tuples.

######  Example 1.2.

Let $P=[0,2]\times[0,4]$, and $n=r=3$. There are ten $3$-lexicographically
increasing triples of points in $P$. These triples are depicted below where
the points in each triple are in increasing lexicographic order. For example,
the first diagram represents the triple $(a_{1},b_{1})=(0,0)$,
$(a_{2},b_{2})=(0,3)$, and $(a_{3},b_{3})=(2,2)$.

In Section 4 we extend the definition of $r$-lexicographically increasing
$n$-tuples of points in rectangles to trapezoids corresponding to line bundles
on Hirzebruch surfaces. Our second main result uses this notion to extend the
toric combinatorics of Hirzebruch surfaces to their Hilbert scheme of points.

######  Theorem 1.3.

Let $X$ be a Hirzebruch surface and $L_{n}\otimes E^{r}$ an ample line bundle
on $X^{[n]}$ (so in particular $r>0$). The set of $r$-lexicographically
increasing $n$-tuples of points in $P_{L}$ indexes a basis of
$H^{0}(X^{[n]},L_{n}\otimes E^{r}).$ In this case, $\chi(X^{[n]},L_{n}\otimes
E^{r})$ is equal to the number of $r$-lexicographically increasing $n$-tuples
of points in $P_{L}$.

The statement that $\chi(X^{[n]},L_{n}\otimes E^{r})$ coincides with the
number of $r$-lexicographically increasing $n$-tuples of points in $P_{L}$ is
proved in Theorem 4.2. The Frobenius splitting of $X^{[n]}$ implies that
$\chi(X^{[n]},L_{n}\otimes E^{r})=\dim H^{0}(X^{[n]},L_{n}\otimes E^{r})$ for
any ample $L_{n}\otimes E^{r}$ [11], so abstractly we know that any basis of
$H^{0}(X^{[n]},L_{n}\otimes E^{r})$ has as many elements as the number of such
$n$-tuples. In Section 5, however, we give a much more direct sense in which
these $n$-tuples correspond to global sections. The global sections can be
naturally identified with a certain $\mathbb{C}$-linear span of polynomials
$W\subseteq\mathbb{C}[x_{1},y_{1},\dots,x_{n},y_{n}]$, and the
$r$-lexicographically increasing $n$-tuples $(\mathbf{a},\mathbf{b})$ in
$P_{L}$ are precisely the leading term exponents of polynomials
$f=x_{1}^{a_{1}}y_{1}^{b_{1}}\cdots x_{n}^{a_{n}}y_{n}^{b_{n}}+\cdots$ in $W$
with respect to a certain term order.

The proof of Theorem 1.3 uses the (standard) fact that the Verlinde series for
toric surfaces can be expressed in terms of equivariant Verlinde series for
$\mathbb{C}^{2}$, which we review in Section 2. Equivariant Verlinde series
for $\mathbb{C}^{2}$ can then be computed using the equivariant localization
formula, a strategy that was used to compute several coefficients for $C_{r}$
and $D_{r}$ in [5]. However, the combinatorics of this expression are
unwieldy. It is not even clear from these expressions that the Euler
characteristics of line bundles on $X^{[n]}$ should be integers.

Our present results are based on a new combinatorial interpretation of the
equivariant Verlinde series for $\mathbb{C}^{2}$, given in Section 3. We
interpret the coefficients as generating functions of integer points in
certain convex sets, or equivalently of certain $n$-tuples of integer points
in the plane. This reduces Theorem 1.3 to an identity of generating functions
of integer points in certain convex sets, and we show in Section 4 that this
identity can be deduced from Brion's formula [2].

Acknowledgements: I thank Rahul Pandharipande and Anton Mellit for helpful
comments on a preliminary version of this paper, and Dave Anderson for
valuable discussions during the early stages of this project.

##  2 Background on Verlinde Series and Toric Surfaces

Let $X$ be a smooth, projective, toric surface equipped with an action of
$T\simeq(\mathbb{C}^{*})^{2}$. For background on the theory of toric varieties
we refer to [4]. The same torus $T$ acts on the Hilbert scheme $X^{[n]}$ by
pull-back of subschemes. For any line bundle $L$ on $X$, the symmetric line
bundle $L\boxtimes\cdots\boxtimes L$ on $X^{n}$ descends to the symmetric
power $X^{n}/S_{n}$. Let $L_{n}$ denote the pullback of this line bundle to
the Hilbert scheme $X^{[n]}$ via the Hilbert-Chow morphism $X^{[n]}\to
X^{n}/S_{n}$. When $L$ is a $T$-equivariant line bundle, $L_{n}$ inherits a
$T$-equivariant structure as well.

Let $\Sigma_{n}\subseteq X^{[n]}\times X$ denote the universal family, and
$\mathcal{O}_{n}$ its structure sheaf. The line bundle $E$ on $X^{[n]}$ is
defined as $E=\det(q_{*}(\mathcal{O}_{n}\otimes p^{*}\mathcal{O}_{X}))$ where
$q$ and $p$ are the natural projections of $X^{[n]}\times X$ onto $X^{[n]}$
and $X$ respectively. In terms of divisor classes, we have
$c_{1}(E)=-\frac{1}{2}[D]$ where $D\subseteq X^{[n]}$ is the exceptional
divisor of the Hilbert-Chow morphism parametrizing reduced length-$n$
subschemes of $X$. The line bundle $E$ is also equipped with a natural
$T$-equivariant structure. A fundamental theorem due to Fogarty [6] states
that
$\operatorname{Pic}(X^{[n]})\simeq\operatorname{Pic}(X)\times\mathbb{Z}E$, so
that every line bundle on $X^{[n]}$ is of the form $L_{n}\otimes E^{r}$ for
some line bundle $L$ on $X$ and $r\in\mathbb{Z}$.

Given a $T$-equivariant line bundle $L$ on $X$ and an integer $r$, the
equivariant Euler characteristic of the line bundle $L_{n}\otimes E^{r}$ is
defined as

$\chi^{T}\left(X^{[n]},L_{n}\otimes
E^{r}\right)=\sum_{i=0}^{2n}\sum_{(a,b)\in\mathbb{Z}^{2}}(-1)^{i}\,t^{a}q^{b}\,\dim_{\mathbb{C}}H^{i}\left(X^{[n]},L_{n}\otimes
E^{r}\right)_{(a,b)},$

where $V_{(a,b)}$ denotes the $(a,b)$-weight space of a vector space $V$ on
which $T$ acts. In other words, $V_{(a,b)}$ is the set of all $v\in V$ such
that $(t,q)\cdot v=t^{a}q^{b}v$ for all $(t,q)\in T$.

Since $X^{[n]}$ is projective, the cohomology groups of line bundles on
$X^{[n]}$ are finite-dimensional and so their equivariant Euler
characteristics are Laurent polynomials in $t$ and $q$. These can be assembled
into equivariant refinements of the Verlinde series introduced in the
introduction,

$\mathbf{V}^{T}_{X,L,r}(z)=\sum_{n=0}^{\infty}z^{n}\cdot\chi^{T}(X^{[n]},L_{n}\otimes
E^{r})\in\mathbb{Z}[t^{\pm 1},q^{\pm 1}][[z]].$

Unlike ordinary Velinde series, these equivariant series can be defined for
non-projective toric surfaces. In particular, consider the action of $T$ on
$\mathbb{C}^{2}$ defined on the coordinate ring $\mathbb{C}[x,y]$ by
$(t,q)\cdot x=tx$ and $(t,q)\cdot y=qy$. We equip $\mathbb{C}^{2}$ with a
$T$-linearized line bundle $L$. Although any such line bundle is trivial, it
may be equipped with a non-trivial torus action and it will be useful for us
to keep track of this data. In this case the equivariant Verlinde series is
defined as

$\mathbf{V}^{T}_{\mathbb{C}^{2},L,r}(z)=\sum_{n=0}^{\infty}z^{n}\cdot\chi^{T}((\mathbb{C}^{2})^{[n]},L_{n}\otimes
E^{r}),$

where the coefficients are now formal power series in $t,q$ rather than
Laurent polynomials. In fact, these coefficients can be represented by
rational functions in $t,q$. Suppose that $L$ is equipped with the $T$-action
by the character $t^{m_{1}}q^{m_{2}}$. The equivariant Euler characteristic of
$L_{n}\otimes E^{r}$ can be computed by the equivariant localization formula
[9],

$\chi^{T}((\mathbb{C}^{2})^{[n]},L_{n}\otimes
E^{r})=t^{nm_{1}}q^{nm_{2}}\sum_{\lambda\vdash
n}\frac{t^{rn(\lambda)}q^{rn(\lambda^{\prime})}}{\prod_{e\in\lambda}(1-t^{1+l(e)}q^{-a(e)})(1-t^{-l(e)}q^{1+a(e)})}.$
(4)

The above sum is over partitions $\lambda$ of $n$, $a(e)$ and $l(e)$ denote
the arm and leg lengths respectively of a cell $e$ in $\lambda$,
$\lambda^{\prime}$ denotes the conjugate partition to $\lambda$, and
$n(\mu)=\sum_{e\in\mu}l(e)$. More generally, for an action of $T$ on
$\mathbb{C}^{2}$ by distinct characters $t^{u}q^{v}$ and
$t^{u^{\prime}}q^{v^{\prime}}$ rather than $t$ and $q$, one replaces $t$ and
$q$ in the sum above with the corresponding characters.

The following standard result expresses equivariant Verlinde series for
projective toric surfaces in terms of those for $\mathbb{C}^{2}$.

######  Proposition 2.1.

For any smooth, projective, toric surface $X$ equipped with a $T$-linearized
line bundle $L$ and integer $r$,

$\mathbf{V}^{T}_{X,L,r}(z)=\prod_{i}\mathbf{V}^{T}_{U_{i},L|_{U_{i}},r}(z),$

where the product is over the open sets $\mathbb{C}^{2}\simeq U_{i}\subseteq
X$ in the standard $T$-invariant affine open cover of $X$.

###### Proof.

Let $U_{1},\dots,U_{k}$ denote the standard affine open cover of $X$, with
$p_{1},\dots,p_{k}$ the corresponding torus fixed points. The generating
function identity in the proposition is equivalent to the expression

$\chi^{T}(X^{[n]},L_{n}\otimes
E^{r})=\sum_{n_{1}+\cdots+n_{k}=n}\prod_{i=1}^{k}\chi^{T}(U_{i}^{[n_{i}]},(L|_{U_{i}})_{n_{i}}\otimes
E^{r}).$

To see that these coincide, expand each equivaraint Euler characteristic using
the localization formula as in [9]. The fixed points $\xi\in(X^{[n]})^{T}$ can
be decomposed as $\xi=\xi_{1}\sqcup\cdots\sqcup\xi_{k}$ where
$\xi_{i}\subseteq U_{i}$ is a $T$-fixed scheme supported at $p_{i}$. The term
corresponding to $\xi$ in the localization expression for
$\chi^{T}(X^{[n]},L_{n}\otimes E^{r})$ is the product of the terms
corresponding to $\xi_{1},\dots,\xi_{k}$ in
$\prod_{i=1}^{k}\chi^{T}(U_{i}^{[n_{i}]},(L|_{U_{i}})_{n_{i}}\otimes E^{r})$
where $n_{i}$ denotes the length of the component $\xi_{i}$. ∎

In Section 4 we study the case $X=\mathcal{H}_{s}$ a Hirzebruch surface,
defined as the projectivization of the split rank two vector bundle
$\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}(s)$ over
$\mathbb{P}^{1}$ for some fixed integer $s\geq 0$. Let $D_{1}$ be the class of
the projectivized $T$-fixed section with self-intersection $s$, and $D_{2}$ be
the class of a fiber. The line bundles associated to $D_{1}$ and $D_{2}$
freely generate the Picard group of $X$. We identify the Newton polygon of the
line bundle $L=\mathcal{O}_{X}(d_{1}D_{1}+d_{2}D_{2})$ with the polygon in
$\mathbb{R}^{2}$ defined by $0\leq x\leq d_{1}$ and $0\leq y\leq d_{2}+sd_{1}$
as depicted below, and denote this polygon by $P_{L}\subseteq\mathbb{R}^{2}$.

$t$$q$$t^{-1}$$q$$q^{-1}t^{-s}\hskip 12.91663pt$$q^{-1}$$qt^{s}\hskip
4.30554pt$$q^{-1}$$1$$t^{d_{1}}$$t^{d_{1}}q^{d_{2}+sd_{1}}$$q^{d_{2}}$ Figure
1: The polygon $P_{L}$ corresponding to
$L=\mathcal{O}_{X}(d_{1}D_{1}+d_{2}D_{2})$. Each vertex corresponds to a fixed
point $p_{i}$ and is labeled with the character of $L|_{U_{p_{i}}}$. The rays
extending from each vertex are labelled with the characters by which $T$ acts
on $U_{i}$.

There are four standard affine opens $U_{1},\dots,U_{4}$ with corresponding
fixed points $p_{1},\dots,p_{4}$ corresponding to the vertices
$(0,0),(d_{1},0),(d_{1},d_{2}+sd_{1}),$ and $(0,d_{2})$ of $P_{L}$
respectively. For readability, we sometimes use the notation
$U_{\,\leavevmode\hbox to3.61pt{\vbox
to3.61pt{\pgfpicture\makeatletter\hbox{\hskip 0.3pt\lower-0.3pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}}{{}}{} {}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}=U_{1}$,
$U_{\,\leavevmode\hbox to3.61pt{\vbox
to3.61pt{\pgfpicture\makeatletter\hbox{\hskip 3.31389pt\lower-0.3pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{}}{{}}{} {}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{3.01389pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{-3.01389pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}=U_{2},$
$U_{\,\leavevmode\hbox to3.61pt{\vbox
to3.61pt{\pgfpicture\makeatletter\hbox{\hskip 3.31389pt\lower-3.31389pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{}}{{}}{} {}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-3.01389pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}=U_{3}$, and
$U_{\,\leavevmode\hbox to3.61pt{\vbox
to3.61pt{\pgfpicture\makeatletter\hbox{\hskip 0.3pt\lower-3.31389pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{}}{{}}{} {}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-3.01389pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{3.01389pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}=U_{4}$.

We introduce notation for the rational function
$\sigma_{n,r}(t,q)=\chi^{T}((\mathbb{C}^{2})^{[n]},E^{r})$ given by (4). For
the Hirzebruch surface $X=\mathcal{H}_{s}$ and line bundle
$L=\mathcal{O}_{X}(d_{1}D_{1}+d_{2}D_{2})$, we will need four variants of this
rational function corresponding to the open sets $U_{1},\dots,U_{4}\subseteq
X$:

$\displaystyle\begin{split}\sigma^{(1)}_{n,r}(t,q)=\sigma^{\,\leavevmode\hbox
to3.61pt{\vbox to3.61pt{\pgfpicture\makeatletter\hbox{\hskip
0.3pt\lower-0.3pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}}{{}}{} {}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}_{n,r}(t,q)&:=\chi^{T}(U_{1}^{[n]},(L|_{U_{1}})_{n}\otimes
E^{r})=\sigma_{n,r}(t,q)\\\ \sigma^{(2)}_{n,r}(t,q)=\sigma^{\,\leavevmode\hbox
to3.61pt{\vbox to3.61pt{\pgfpicture\makeatletter\hbox{\hskip
3.31389pt\lower-0.3pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{}}{{}}{} {}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{3.01389pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{-3.01389pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}_{n,r}(t,q)&:=\chi^{T}(U_{2}^{[n]},(L|_{U_{2}})_{n}\otimes
E^{r})=t^{nd_{1}}\sigma_{n,r}(t^{-1},q)\\\
\sigma^{(3)}_{n,r}(t,q)=\sigma^{\,\leavevmode\hbox to3.61pt{\vbox
to3.61pt{\pgfpicture\makeatletter\hbox{\hskip 3.31389pt\lower-3.31389pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{}}{{}}{} {}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-3.01389pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}_{n,r}(t,q)&:=\chi^{T}(U_{3}^{[n]},(L|_{U_{3}})_{n}\otimes
E^{r})=t^{nd_{1}}q^{n(d_{2}+sd_{2})}\sigma_{n,r}(q^{-1}t^{-s},q^{-1})\\\
\sigma^{(4)}_{n,r}(t,q)=\sigma^{\,\leavevmode\hbox to3.61pt{\vbox
to3.61pt{\pgfpicture\makeatletter\hbox{\hskip 0.3pt\lower-3.31389pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{}}{{}}{} {}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-3.01389pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{3.01389pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}_{n,r}(t,q)&:=\chi^{T}(U_{4}^{[n]},(L|_{U_{4}})_{n}\otimes
E^{r})=q^{nd_{2}}\sigma_{n,r}(qt^{s},q^{-1}).\end{split}$ (5)

With this notation, the equality between coefficients on $z^{n}$ in
Proposition 2.1 gives the identity

$\chi^{T}(X^{[n]},L_{n}\otimes
E^{r})=\sum_{n_{1}+\cdots+n_{4}=n}\left(\prod_{i=1}^{4}\sigma^{(i)}_{n_{i},r}(t,q)\right).$
(6)

##  3 Combinatorial Verlinde Series for the Affine Plane

In this section, we give a new formula for the rational function
$\sigma_{n,r}(t,q)=\chi^{T}((\mathbb{C}^{2})^{[n]},E^{r})$ in the case $r>0$.
First, we recall an explicit construction of $(\mathbb{C}^{2})^{[n]}$
following Haiman [9].

Consider the action of the symmetric group $S_{n}$ on
$\mathbb{C}[\mathbf{x,y}]=\mathbb{C}[x_{1},y_{1},\dots,x_{n},y_{n}]$ defined
by $\sigma\cdot x_{i}=x_{\sigma(i)}$ and $\sigma\cdot y_{i}=y_{\sigma(i)}$. A
polynomial $f\in\mathbb{C}[\mathbf{x,y}]$ is said to be symmetric if
$\sigma\cdot f=f$ for all $\sigma\in S_{n}$, and alternating if $\sigma\cdot
f=\mathrm{sgn}(\sigma)f$ for all $\sigma\in S_{n}$.

Let $A^{0}\subseteq\mathbb{C}[\mathbf{x,y}]$ denote the space of all symmetric
polynomials, and $A^{1}\subseteq\mathbb{C}[\mathbf{x,y}]$ the space of all
alternating polynomials. Each $n$-tuple of not-necessarily distinct points
$(a_{1},b_{1}),\dots,(a_{n},b_{n})\in\mathbb{Z}^{2}_{\geq 0}$ corresponds to a
monomial symmetric polynomial $x_{1}^{a_{1}}y_{1}^{b_{1}}\cdots
x_{n}^{a_{n}}y_{n}^{b_{n}}+$ (symmetric terms). Similarly, each $n$-tuple of
distinct points $(a_{1},b_{1}),\dots,(a_{n},b_{n})\in\mathbb{Z}^{2}_{\geq 0}$
corresponds to an alternating polynomial
$\det(x_{i}^{a_{j}}y_{i}^{b_{j}})_{ij}.$ These polynomials form bases of
$A^{0}$ and $A^{1}$ as $(a_{1},b_{1}),\dots,(a_{n},b_{n})$ vary over all such
$n$-tuples up to reordering.

For $r>1$, let $A^{r}\subseteq\mathbb{C}[\mathbf{x,y}]$ denote the span of
products $f_{1}\cdots f_{r}$ where $f_{1},\dots,f_{r}\in A^{1}$, so that
$R=A^{0}\oplus A^{1}\oplus A^{2}\oplus\cdots$ forms a graded ring. Haiman
shows that the Hilbert scheme of $n$ points in $\mathbb{C}^{2}$ is isomorphic
to $\operatorname{Proj}(R)$ in such a way that the natural morphism
$\operatorname{Proj}(R)\to\operatorname{Spec}(A^{0})$ corresponds to the
Hilbert-Chow morphism ([9], Proposition 2.6).

In the notation of the previous section, the line bundle $E$ corresponds to
$\mathcal{O}(1)$ under the identification of $(\mathbb{C}^{2})^{[n]}$ with
$\operatorname{Proj}(R)$. Haiman's results can be used to show that the ring
$R$ is integrally closed, and therefore for each $r\geq 0$ there is an
isomorphism (see [3], Corollary 3.10)

$H^{0}((\mathbb{C}^{2})^{[n]},E^{r})\simeq A^{r}.$

In contrast to the spaces of symmetric and alternating polynomials, it is
unclear how to obtain a basis of $A^{r}$ when $r>1$. Such a basis can be
extracted from our earlier results [3], which we now summarize.

Let
$(\mathbf{a},\mathbf{b})=(a_{1},b_{1},\dots,a_{n},b_{n})\in\mathbb{R}^{2n}$,
and regard a point $(\mathbf{a},\mathbf{b})\in\mathbb{R}^{2n}$ as an ordered
$n$-tuple of points $(a_{1},b_{1}),\dots,(a_{n},b_{n})\in\mathbb{R}^{2}$.
Define $P_{n}\subseteq\mathbb{R}^{2n}$ to be the convex hull of the set of
nonnegative integer vectors $(\mathbf{a},\mathbf{b})\in\mathbb{Z}^{2n}_{\geq
0}$ such that $(a_{1},b_{1})<\cdots<(a_{n},b_{n})$ in lexicographic order. In
[3] we showed by induction on $n$ that $P_{n}$ is given explicitly by

$P_{n}=\left\\{\minipage{722.7pt} $(\mathbf{a},\mathbf{b})\in\mathbb{R}^{2n}$
\endminipage\hskip 4.30554pt\left|\hskip 4.30554pt\minipage{722.7pt} $0\leq
a_{1}\leq a_{2}\leq\cdots\leq a_{n}$, \\\ for each $j=1,\dots,n-1$, if
$a_{j}=a_{j+1}$ then $b_{j+1}\geq b_{j}+1$, and\\\
$b_{j}\geq\sum_{i=1}^{j-1}\max\\{1-(a_{j}-a_{i}),0\\}$ for all $1\leq j\leq n$
\endminipage\right.\right\\}.$

We equip $\mathbb{C}[\mathbf{x,y}]$ with the lexicographic term order where
the variables are ordered as $x_{1}>\cdots>x_{n}>y_{1}>\cdots>y_{n}$. The
following result allows us to extract a basis of $A^{r}$ for $r>1$.

######  Theorem 3.1.

[3] For each $r>0$, the integer points in the $r$-fold dilation
$(\mathbf{a},\mathbf{b})\in(rP_{n})\cap\mathbb{Z}^{2n}$ are precisely the
vectors that appear as exponents of trailing terms $x_{1}^{a_{1}}\cdots
x_{n}^{a_{n}}y_{1}^{b_{1}}\cdots y_{n}^{b_{n}}$ of polynomials in $A^{r}$.

The dilation $rP_{n}$ can be described explicitly by

$rP_{n}=\left\\{\minipage{722.7pt} $(\mathbf{a},\mathbf{b})\in\mathbb{R}^{2n}$
\endminipage\hskip 4.30554pt\left|\hskip 4.30554pt\minipage{722.7pt} $0\leq
a_{1}\leq a_{2}\leq\cdots\leq a_{n}$, \\\ for each $j=1,\dots,n-1$, if
$a_{j}=a_{j+1}$ then $b_{j+1}\geq b_{j}+r$, and\\\
$b_{j}\geq\sum_{i=1}^{j-1}\max\\{r-(a_{j}-a_{i}),0\\}$ for all $1\leq j\leq n$
\endminipage\right.\right\\}.$

We call any integer vector $(\mathbf{a},\mathbf{b})\in rP_{n}$ an
$r$-lexicographically increasing $n$-tuple in $\mathbb{R}^{2}_{\geq 0}$, as
these are precisely the $n$-tuples that can be written as coordinate-wise sums
of $r$ $n$-tuples of distinct points in $\mathbb{Z}^{2}_{\geq 0}$ written in
increasing lexicographical order.

######  Corollary 3.2.

For any $r>0$, the rational function
$\sigma_{n,r}(t,q)=\chi^{T}((\mathbb{C}^{2})^{[n]},E^{r})$ is given by

$\sigma_{n,r}(t,q)=\sum_{(\mathbf{a,b})\in(rP_{n})\cap\mathbb{Z}^{2n}}t^{a_{1}+\cdots+a_{n}}q^{b_{1}+\cdots+b_{n}}$

###### Proof.

By the Frobenius splitting of $(\mathbb{C}^{2})^{[n]}$ [11], the higher
cohomology groups of $E^{r}$ vanish for all $r>0$, and so the coefficient on
$t^{a}q^{b}$ in the series $\chi^{T}((\mathbb{C}^{2})^{[n]},E^{r})$ is equal
to the dimension of $H^{0}((\mathbb{C}^{2})^{[n]},E^{r})_{(a,b)}$. Under the
identification $H^{0}((\mathbb{C}^{2})^{[n]},E^{r})\simeq A^{r}$ described
above, the $T$-action on $H^{0}((\mathbb{C}^{2})^{[n]},E^{r})$ corresponds to
the action on $A^{r}$ given by $(t,q)\cdot x_{i}=tx_{i}$ and $(t,q)\cdot
y_{i}=qy_{i}$ for all $i=1,\dots,n$ and $(t,q)\in T$. The weight space
$A^{r}_{(a,b)}$ therefore consists of those polynomials $f\in A^{r}$ such that
$(a_{1}+\cdots+a_{n},b_{1}+\cdots+b_{n})=(a,b)$ for every term
$x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}y_{1}^{mb_{1}}\cdots y_{n}^{b_{n}}$ of $f$.

By Theorem 3.1, the integer points $(\mathbf{a,b})\in
rP_{n}\cap\mathbb{Z}^{2}$ are the lexicographic trailing term exponents of
polynomials in $A^{r}$. The integer points $(\mathbf{a,b})\in
rP_{n}\cap\mathbb{Z}^{2n}$ corresponding to trailing terms of polynomials in
the weight space $A^{r}_{(a,b)}$ are those with
$(a_{1}+\cdots+a_{n},b_{1}+\cdots+b_{n})=(a,b)$. Any collection of polynomials
in $A^{r}_{(a,b)}$ with pairwise distinct trailing terms is linearly
independent, and any additional polynomial not in their linear span can be
reduced modulo the collection to obtain a new trailing term. This implies that
the number of $(\mathbf{a},\mathbf{b})\in rP_{n}\cap\mathbb{Z}^{2n}$ with
$(a_{1}+\cdots+a_{n},b_{1}+\cdots+b_{n})=(a,b)$, corresponding to all the
trailing terms of polynomials in $A^{r}_{(a,b)}$, is equal to the dimension of
$A^{r}_{(a,b)}$, which completes the proof. ∎

######  Example 3.3.

Let $n=2$ and $r=2$. By formula (4), we have

$\chi^{T}((\mathbb{C}^{2})^{[2]},E^{2})=\frac{t^{2}+tq+q^{2}-t^{2}q^{2}}{(t^{2}-1)(t-1)(q^{2}-1)(q-1)}.$

One can compute, for example, that the coefficient on the $t^{3}q^{2}$ term of
the power series expansion at $t=q=0$ of this rational function is $5$.
Correspondingly, there are 5 integer pairs $((a_{1},b_{1}),(a_{2},b_{2}))$
corresponding to points $(a_{1},b_{1},a_{2},b_{2})\in 2P_{2}$ with
$a_{1}+a_{2}=3$ and $b_{1}+b_{2}=2$. They are the pairs $((0,0),(3,2))$,
$((0,1),(3,1))$, $((0,2),(3,0))$, $((1,0),(2,2))$, and $((1,1),(2,1))$.

##  4 Combinatorial Verlinde Series for Hirzebruch Surfaces

In this section, we use the combinatorial interpretation of
$\sigma_{n,r}(t,q)$ given in Corollary 3.2 to study line bundles on the
Hilbert scheme of points on a Hirzebruch surface. Fix the Hirzebruch surface
$X=\mathcal{H}_{s}$ and line bundle $L=\mathcal{O}(d_{1}D_{1}+d_{2}D_{2})$
with corresponding polytope $P_{L}$ as defined in Section 2, and fix an
integer $r>0$.

Our basic collection of $n$-tuples will be the integer points in the set

$P_{n,r}^{\circ}=\left\\{\minipage{722.7pt}
$(\mathbf{a},\mathbf{b})\in\mathbb{R}^{2n}$ \endminipage\hskip
4.30554pt\left|\hskip 4.30554pt\minipage{722.7pt} $a_{1}\leq
a_{2}\leq\cdots\leq a_{n}$, and \\\ for each $j=1,\dots,n-1$, if
$a_{j}=a_{j+1}$ then $b_{j+1}\geq b_{j}+r$ \endminipage\right.\right\\}.$

This is the $r$-fold dilation of the convex hull of the set of all $n$-tuples
of integer points $(\mathbf{a},\mathbf{b})$ such that
$(a_{1},b_{1})<\cdots<(a_{n},b_{n})$ in lexicographic order. Next, we
introduce four constraints on vectors $(\mathbf{a},\mathbf{b})\in
P_{n,r}^{\circ}$ depending on $r$ and corresponding to the left, right,
bottom, and top edges of $P_{L}$ respectively (as drawn in Figure 1):

(Left) $\displaystyle 0\leq a_{1},$ (Right) $\displaystyle a_{n}\leq d_{1},$
(Bottom) $\displaystyle
b_{j}\geq\sum_{i=1}^{j-1}\max\\{r-(a_{j}-a_{i}),0\\}\text{ for all }1\leq
j\leq n,\text{ and},$ (Top) $\displaystyle b_{j}\leq
d_{2}+sa_{j}-\sum_{k=j+1}^{n}\max\\{r-(a_{k}-a_{j}),0\\}\text{ for all }1\leq
j\leq n.$

We will need the following sets of $n$-tuples corresponding to the four vertex
cones of $P_{L}$ and the polygon $P_{L}$ itself:

$\displaystyle\begin{split}P^{(1)}_{n,r}&=P^{\,\leavevmode\hbox to3.61pt{\vbox
to3.61pt{\pgfpicture\makeatletter\hbox{\hskip 0.3pt\lower-0.3pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}}{{}}{} {}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{3.01389pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}_{n,r}:=\left\\{(\mathbf{a},\mathbf{b})\in
P_{n,r}^{\circ}\,|\,(\mathbf{a},\mathbf{b})\text{ satisfies the bottom and
left constraints}\right\\}\\\ P^{(2)}_{n,r}&=P^{\,\leavevmode\hbox
to3.61pt{\vbox to3.61pt{\pgfpicture\makeatletter\hbox{\hskip
3.31389pt\lower-0.3pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{}}{{}}{} {}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{3.01389pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{-3.01389pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}_{n,r}:=\left\\{(\mathbf{a},\mathbf{b})\in
P_{n,r}^{\circ}\,|\,(\mathbf{a},\mathbf{b})\text{ satisfies the bottom and
right constraints}\right\\}\\\ P^{(3)}_{n,r}&=P^{\,\leavevmode\hbox
to3.61pt{\vbox to3.61pt{\pgfpicture\makeatletter\hbox{\hskip
3.31389pt\lower-3.31389pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{}}{{}}{} {}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-3.01389pt}{0.0pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-3.01389pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}_{n,r}:=\left\\{(\mathbf{a},\mathbf{b})\in
P_{n,r}^{\circ}\,|\,(\mathbf{a},\mathbf{b})\text{ satisfies the top and right
constraints}\right\\}\\\ P^{(4)}_{n,r}&=P^{\,\leavevmode\hbox to3.61pt{\vbox
to3.61pt{\pgfpicture\makeatletter\hbox{\hskip 0.3pt\lower-3.31389pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}{}}{{}}{} {}{}
{{}{}}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{-3.01389pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@lineto{3.01389pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,}_{n,r}:=\left\\{(\mathbf{a},\mathbf{b})\in
P_{n,r}^{\circ}\,|\,(\mathbf{a},\mathbf{b})\text{ satisfies the top and left
constraints}\right\\}\\\ P^{\,\leavevmode\hbox to3.61pt{\vbox
to3.61pt{\pgfpicture\makeatletter\hbox{\hskip 0.3pt\lower-0.3pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{} {}{} {{}{}}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{3.01389pt}\pgfsys@lineto{3.01389pt}{3.01389pt}\pgfsys@lineto{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,{}}_{n,r}&:=\left\\{(\mathbf{a},\mathbf{b})\in
P_{n,r}^{\circ}\,|\,(\mathbf{a},\mathbf{b})\text{ satisfies the top, bottom,
left, and right constraints}\right\\}\end{split}$ (7)

By the inequalities of $P_{L}$ given in Section 2, any $n$-tuple
$(\mathbf{a},\mathbf{b})\in P^{\,\leavevmode\hbox to3.61pt{\vbox
to3.61pt{\pgfpicture\makeatletter\hbox{\hskip 0.3pt\lower-0.3pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{} {}{} {{}{}}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{3.01389pt}\pgfsys@lineto{3.01389pt}{3.01389pt}\pgfsys@lineto{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,{}}_{n,r}$ has
$(a_{1},b_{1}),\dots,(a_{n},b_{n})\in P_{L}$ since in particular $0\leq
a_{j}\leq d_{1}$ and $0\leq b_{j}\leq d_{2}+sa_{j}$ for all $j=1,\dots,n$. The
integer points $(\mathbf{a},\mathbf{b})\in P^{\,\leavevmode\hbox
to3.61pt{\vbox to3.61pt{\pgfpicture\makeatletter\hbox{\hskip
0.3pt\lower-0.3pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{} {}{} {{}{}}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{3.01389pt}\pgfsys@lineto{3.01389pt}{3.01389pt}\pgfsys@lineto{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,{}}_{n,r}\cap\mathbb{Z}^{2n}$
are the $r$-lexicographically increasing $n$-tuples in $P_{L}$, as described
in the introduction. As a consequence of Corollary 3.2, the integer points
$P^{(i)}_{n,r}$ give a combinatorial interpretation of the series
$\sigma_{n,r}^{(i)}(t,q)$.

######  Corollary 4.1.

For any $n,r>0$ and $i=1,\dots,4$ we have

$\chi^{T}(U_{i}^{[n]},(L|_{U_{i}})_{n}\otimes
E^{r})=\sigma^{(i)}_{n,r}(t,q)=\sum_{(\mathbf{a,b})\in
P^{(i)}_{n,r}\cap\mathbb{Z}^{2n}}t^{a_{1}+\cdots+a_{n}}q^{b_{1}+\cdots+b_{n}}.$

Our main object of study is the generating function of $r$-lexicographically
increasing $n$-tuples in $P_{L}$,

$\sigma^{\,\leavevmode\hbox to3.61pt{\vbox
to3.61pt{\pgfpicture\makeatletter\hbox{\hskip 0.3pt\lower-0.3pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{} {}{} {{}{}}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{3.01389pt}\pgfsys@lineto{3.01389pt}{3.01389pt}\pgfsys@lineto{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,{}}_{n,r}(t,q):=\sum_{(\mathbf{a,b})\in
P^{\,\leavevmode\hbox to2.75pt{\vbox
to2.75pt{\pgfpicture\makeatletter\hbox{\hskip 0.3pt\lower-0.3pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{} {}{} {{}{}}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{2.15277pt}\pgfsys@lineto{2.15277pt}{2.15277pt}\pgfsys@lineto{2.15277pt}{0.0pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,{}}_{n,r}\cap\mathbb{Z}^{2n}}t^{a_{1}+\cdots+a_{n}}q^{b_{1}+\cdots+b_{n}}.$

The goal of this section is to establish the following relationship between
this generating function and the Euler characteristic of $L_{n}\otimes E^{r}$.

######  Theorem 4.2.

Let $X=\mathcal{H}_{s}$, $L=\mathcal{O}(d_{1}D_{1}+d_{2}D_{2})$ and $r>0$ as
above. If $d_{1},d_{2}>r(n-1)$, then $\sigma^{\,\leavevmode\hbox
to3.61pt{\vbox to3.61pt{\pgfpicture\makeatletter\hbox{\hskip
0.3pt\lower-0.3pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{} {}{} {{}{}}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{3.01389pt}\pgfsys@lineto{3.01389pt}{3.01389pt}\pgfsys@lineto{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,{}}_{n,r}(t,q)=\chi^{T}(X^{[n]},L_{n}\otimes
E^{r}).$

The conditions $r>0$, and $d_{1},d_{2}>r(n-1)$ exactly describe the set of
ample line bundles on $X^{[n]}$ [1].

The idea of the proof is as follows: By (6) we have an expression for
$\chi^{T}(X^{[n]},L_{n}\otimes E^{r})$ in terms of the rational functions
$\sigma^{(i)}_{n,r}(t,q)$, and by Corollary 4.1 these rational functions are
the generating functions of integer points in the convex sets $P^{(i)}_{n,r}.$
We will show that the generating function $\sigma^{\,\leavevmode\hbox
to3.61pt{\vbox to3.61pt{\pgfpicture\makeatletter\hbox{\hskip
0.3pt\lower-0.3pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{} {}{} {{}{}}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{3.01389pt}\pgfsys@lineto{3.01389pt}{3.01389pt}\pgfsys@lineto{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,{}}_{n,r}(t,q)$ of
integer points in $P^{\,\leavevmode\hbox to3.61pt{\vbox
to3.61pt{\pgfpicture\makeatletter\hbox{\hskip 0.3pt\lower-0.3pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{} {}{} {{}{}}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{3.01389pt}\pgfsys@lineto{3.01389pt}{3.01389pt}\pgfsys@lineto{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,{}}_{n,r}$ satisfies
the same identity (6) in terms of the $\sigma^{(i)}_{n,r}(t,q)$'s by expanding
the sum using Brion's formula [2]. To carry this plan out, we first need a
relatively detailed study of the combinatorics of these $n$-tuples.

Given an integer point $(\mathbf{a},\mathbf{b})\in
P^{\circ}_{n,r}\cap\mathbb{Z}^{2n}$ which we think of as an $n$-tuple of
points $p_{j}=(a_{j},b_{j})$, let
$\delta_{i}=\delta_{i}(\mathbf{a},\mathbf{b})=\max\\{a_{i+1}-a_{i},r\\}$ for
each $i=1,\dots,n-1$. We refer to the vector
$\delta=(\delta_{1},\dots,\delta_{n-1})\in\\{0,1,\dots,r\\}^{n-1}$ as the type
of the $n$-tuple $(\mathbf{a},\mathbf{b}).$ For a fixed element
$\delta=(\delta_{1},\dots,\delta_{n-1})\in\\{0,1,\dots,r\\}^{n-1}$, we define
two partitions of each $n$-tuple $\\{p_{1},\dots,p_{n}\\}$ of type $\delta$. A
block is a maximal collection of successive points
$\\{p_{i},p_{i+1},\dots,p_{j}\\}$ such that $\delta_{i},\dots,\delta_{j-1}<r$.
A column is a maximal collection of successive points
$\\{p_{i},p_{i+1},\dots,p_{j}\\}$ such that $\delta_{i},\dots,\delta_{j-1}=0.$
Clearly, each block is made up of a union of consecutive columns. We define
$I=\\{i_{1},\dots,i_{\ell}\\}\subseteq\\{1,\dots,n\\}$ to be the indices of
the first points in each block for any $n$-tuple of type $\delta$. In terms of
$\delta$, this means $i_{1}=1$ and $i_{k+1}$ is the index at which the $k$th
entry equal to $r$ appears in $\delta$. Finally, for each column
$C=\\{p_{j},\dots,p_{j^{\prime}}\\}$ we define the statistics

$L_{C}=\sum_{i=1}^{j-1}\max\\{r-(\delta_{i}+\cdots+\delta_{j-1}),0\\},\hskip
14.22636pt\text{and}\hskip
14.22636ptR_{C}=\sum_{k=j^{\prime}+1}^{n}\max\\{r-(\delta_{j^{\prime}}+\cdots+\delta_{k-1}),0\\}.$

######  Lemma 4.3.

The set of integer points $(\mathbf{a},\mathbf{b})\in P^{\,\leavevmode\hbox
to3.61pt{\vbox to3.61pt{\pgfpicture\makeatletter\hbox{\hskip
0.3pt\lower-0.3pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{} {}{} {{}{}}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{3.01389pt}\pgfsys@lineto{3.01389pt}{3.01389pt}\pgfsys@lineto{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,{}}_{n,r}\cap\mathbb{Z}^{2n}$
of type $\delta$ are exactly the integer points satisfying the conditions

  1. 1.

$0\leq a_{i_{1}},\hskip
14.22636pta_{i_{k}}+\delta_{i_{k}}+\cdots+\delta_{i_{k+1}-1}\leq
a_{i_{k+1}}\text{ for each }k=1,\dots,\ell-1,\text{ and}\hskip
14.22636pta_{i_{\ell}}+\delta_{i_{\ell}}+\cdots+\delta_{n-1}\leq d_{1},$

  2. 2.

for each $j\notin I$ we have $a_{j}=a_{i}+\delta_{i}+\cdots+\delta_{j-1}$
where $i\in I$ is the largest index less than $j$, and

  3. 3.

$L_{C}\leq b_{j},\hskip 14.22636ptb_{k}+r\leq b_{k+1},\text{ for each
}k=j,\dots,j^{\prime}-1,\text{ and}\hskip 14.22636ptb_{j^{\prime}}\leq
d_{2}+sa_{j}-R_{C}$ for each column $C=\\{p_{j},\dots,p_{j^{\prime}}\\}$.

###### Proof.

The first two conditions describing the $a$-coordinates precisely say that
$(\mathbf{a},\mathbf{b})$ has type $\delta$ and $0\leq a_{1}\leq\cdots\leq
a_{n}\leq d_{1}.$ The constraint that for each column
$C=\\{p_{j},\dots,p_{j^{\prime}}\\}$ we have $b_{k}+r\leq b_{k+1}$ for each
$k=j,\dots,j^{\prime}-1$ are the remaining defining conditions for
$(\mathbf{a},\mathbf{b})\in P^{\circ}_{n,r}.$ Finally, the ``top" and
``bottom" conditions defining $P^{\,\leavevmode\hbox to3.61pt{\vbox
to3.61pt{\pgfpicture\makeatletter\hbox{\hskip 0.3pt\lower-0.3pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{} {}{} {{}{}}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{3.01389pt}\pgfsys@lineto{3.01389pt}{3.01389pt}\pgfsys@lineto{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,{}}_{n,r}$ are
redundant except at the top and bottom points of each column, which in this
case reduce to the remaining inequalities $L_{C}\leq b_{j}$ and
$b_{j^{\prime}}\leq d_{2}+sa_{j}-R_{C}$ respectively. ∎

Let $P^{\delta}_{n,r}\subseteq\mathbb{R}^{2n}$ be the polytope defined by the
equations and inequalities in Lemma 4.3. We study the combinatorics of
$P^{\delta}_{n,r}$ in the case $d_{1},d_{2}>r(n-1)$. By the second condition
in Lemma 4.3, we can write all the $a$-coordinates of points
$(\mathbf{a},\mathbf{b})\in P^{\delta}_{n,r}$ in terms of those with indices
in $I$, $a_{i_{1}},\dots,a_{i_{\ell}}.$ The first condition says that these
points are contained in $[0,d_{1}]$ and between each pair of points there is a
minimum increase determined by $\delta$. When $d_{1}$ is greater than the sum
of all these minimum gaps, $\delta_{1}+\cdots+\delta_{n-1}$, the set of all
such collections of $a$-coordinates is nonempty and forms a simplex of
dimension $I$. This is always the case when $d_{1}>r(n-1)$ since each entry of
$\delta$ is at most $r$.

For any collection of $a$-coordinates satisfying conditions $1$ and $2$ in the
lemma and any column $C=\\{p_{j},\dots,p_{j^{\prime}}\\}$, the coordinates
$b_{j},\dots,b_{j^{\prime}}$ are contained in $[L_{C},d_{2}+sa_{j}-R_{C}]$ and
between each pair of points there is an increase of at least $r$. When the
length of this interval, $d_{2}+sa_{j}-R_{C}-L_{C}$, is greater than the sum
of all these minimum gaps, $r(j^{\prime}-j)$, the set of all such coordinates
$b_{j},\dots,b_{j^{\prime}}$ is nonempty and forms a simplex of dimension
$j^{\prime}-j+1$, the number of points in the column. This is always the case
when $d_{2}>r(n-1)$ since it follows from the definitions of $L_{C}$ and
$R_{C}$ that $L_{C}\leq r(j-1)$ and $R_{C}\leq r(n-j^{\prime})$, and so

$d_{2}+sa_{j}-R_{C}-L_{C}-r(j^{\prime}-j)\geq d_{2}-r(n-1)+sa_{j}\geq 0.$

The size of the interval containing $b_{j},\dots,b_{j^{\prime}}$ varies
depending on $a_{j}=\cdots=a_{j^{\prime}}$ or equivalently on $a_{i}$ where
$i$ is the index of the first point in the block containing this column. This
analysis shows that when $d_{1},d_{2}>r(n-1)$, the polytope $P^{\delta}_{n,r}$
is combinatorially equivalent to a product of simplices. In particular, we can
describe its vertices and their tangent cones.

######  Lemma 4.4.

If $d_{1},d_{2}>r(n-1)$, $P^{\delta}_{n,r}$ is a lattice polytope with
$(|I|+1)\prod_{C}(|C|+1)$ vertices, where the product is over all columns $C$
for an $n$-tuple of type $\delta$. For each vertex, exactly one of the
inequalities in condition $1$ of Lemma 4.3 is strict, and for each column $C$
exactly one of the inequalities in condition $3$ is strict.

We can index the vertices of $P^{\delta}_{n,r}$ as follows: Choose a number of
blocks $k=0,1,\dots,|I|$ and move the points in the first $k$ blocks as far
left as possible, and the remaining points as far right as possible. For left
points this means $a_{j}=\delta_{1}+\cdots+\delta_{j-1}$ and for right points
it means $a_{j}=d_{1}-\delta_{j}-\cdots-\delta_{n-1}.$ Then, for each column
$C=\\{p_{j},\dots,p_{j^{\prime}}\\}$ choose a number of points
$k_{C}=0,1\dots,|C|$ and move the first $k_{C}$ points in the column as far
down as possible and the remaining points in the column as far up as possible.
For bottom points this means $b_{i}=L_{C}+r(i-j)$ and for top points this
means $b_{i}=d_{2}+sa_{j}-R_{C}-r(j^{\prime}-i).$

Using the above terminology, we partition each vertex of $P^{\delta}_{n,r}$
considered as an $n$-tuple $(\mathbf{a},\mathbf{b})$ into four separate
$n$-tuples of points: $(\mathbf{a}^{(1)},\mathbf{b}^{(1)})$ the points in a
left block and bottom of their column, $(\mathbf{a}^{(2)},\mathbf{b}^{(2)})$
the points in a right block and bottom of their column,
$(\mathbf{a}^{(3)},\mathbf{b}^{(3)})$ the points in a right block and top of
their column, and $(\mathbf{a}^{(4)},\mathbf{b}^{(4)})$ the points in a left
block and top of their column.

The equations defining a vertex cone of a polytope are obtained by removing
all the inequalities that are strict at a given vertex. Fixing a vertex $v$ of
$P^{\delta}_{n,r}$, we partition any integer point $(\mathbf{a},\mathbf{b})$
in the corresponding vertex cone $\mathcal{K}_{v}P^{\delta}_{n,r}$ into four
$n$-tuples the same way as the vertex $n$-tuple. In other words,
$(\mathbf{a}^{(i)},\mathbf{b}^{(i)})$ consists of the points in the $n$-tuple
$(\mathbf{a},\mathbf{b})$ with index in $J^{(i)}.$

###### Proof of Theorem 4.2.

By the localization formula (6), we have

$\chi^{T}(X^{[n]},L_{n}\otimes
E^{r})=\sum_{n_{1}+\cdots+n_{4}=n}\left(\prod_{i=1}^{4}\sigma^{(i)}_{n_{i},r}(t,q)\right),$

and by Corollary 4.1 the rational functions $\sigma^{(i)}_{n_{i},r}(t,q)$ are
generating functions summing over the integer points in the polyhedron
$P^{(i)}_{n_{i},r}$. In other words, $\chi^{T}(X^{[n]},L_{n}\otimes E^{r})$ is
equal to the sum of the generating functions of integer points in the products
$P^{(1)}_{n_{1},r}\times\cdots\times P^{(4)}_{n_{4},r}$ for all
$n_{1}+\cdots+n_{4}=n$.

On the other hand, $\sigma^{\,\leavevmode\hbox to3.61pt{\vbox
to3.61pt{\pgfpicture\makeatletter\hbox{\hskip 0.3pt\lower-0.3pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{} {}{} {{}{}}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{3.01389pt}\pgfsys@lineto{3.01389pt}{3.01389pt}\pgfsys@lineto{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,{}}_{n,r}(t,q)$ is
defined as a generating function summing over the integer points in
$P^{\,\leavevmode\hbox to3.61pt{\vbox
to3.61pt{\pgfpicture\makeatletter\hbox{\hskip 0.3pt\lower-0.3pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{} {}{} {{}{}}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{3.01389pt}\pgfsys@lineto{3.01389pt}{3.01389pt}\pgfsys@lineto{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,{}}_{n,r}.$ These
integer points are divided among the polytopes $P^{\delta}_{n,r}$, so we have

$\sigma^{\,\leavevmode\hbox to3.61pt{\vbox
to3.61pt{\pgfpicture\makeatletter\hbox{\hskip 0.3pt\lower-0.3pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{} {}{} {{}{}}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{3.01389pt}\pgfsys@lineto{3.01389pt}{3.01389pt}\pgfsys@lineto{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,{}}_{n,r}(t,q)=\sum_{\delta\in\\{0,1,\dots,r\\}^{n-1}}\sigma^{\delta}_{n,r}(t,q),$

where

$\sigma^{\delta}_{n,r}(t,q)=\sum_{(\mathbf{a},\mathbf{b})\in
P^{\delta}_{n,r}\cap\mathbb{Z}^{2n}}t^{a_{1}+\cdots+a_{n}}q^{b_{1}+\cdots+b_{n}}.$

By Brion's formula [2] the generating function of integer points in
$P^{\delta}_{n,r}$ is equal to the sum of those in its vertex cones. For a
vertex $v\in P^{\delta}_{n,r}$, the vertex cone $K_{v}P^{\delta}_{n,r}$ is the
polyhedron defined by all the conditions in Lemma 4.3 that are active at the
vertex $v$. Brion's formula gives the identity

$\sigma^{\delta}_{n,r}(t,q)=\sum_{\begin{subarray}{c}v\in P^{\delta}_{n,r}\\\
\text{a vertex}\end{subarray}}\sum_{(\mathbf{a},\mathbf{b})\in
K_{v}P^{\delta}_{n,r}\cap\mathbb{Z}^{2n}}t^{a_{1}+\cdots+a_{n}}q^{b_{1}+\cdots+b_{n}},$

and so $\sigma^{\,\leavevmode\hbox to3.61pt{\vbox
to3.61pt{\pgfpicture\makeatletter\hbox{\hskip 0.3pt\lower-0.3pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{} {}{} {{}{}}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{3.01389pt}\pgfsys@lineto{3.01389pt}{3.01389pt}\pgfsys@lineto{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,{}}_{n,r}(t,q)$ is
the sum of these over all $\delta$. To conclude the proof, we show that
choices of $\delta\in\\{0,1,\dots,r\\}^{n-1}$, vertex $v\in P^{\delta}_{n,r}$,
and integer point of $\mathcal{K}_{v}P^{\delta}_{n,r}$ are in bijection with
choices of $n_{1}+\cdots+n_{4}=n$ and integer point of
$P^{(1)}_{n_{1},r}\times\cdots\times P^{(4)}_{n_{4},r}$ in such a way that
preserves the weights of the terms in the respective generating functions
$\sigma^{\,\leavevmode\hbox to3.61pt{\vbox
to3.61pt{\pgfpicture\makeatletter\hbox{\hskip 0.3pt\lower-0.3pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{} {}{} {{}{}}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{3.01389pt}\pgfsys@lineto{3.01389pt}{3.01389pt}\pgfsys@lineto{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,{}}_{n,r}(t,q)$ and
$\chi^{T}(X^{[n]},L_{n}\otimes E^{r})$.

Fix $\delta$ and a vertex $v$ of $P^{\delta}_{n,r}$, and let
$(\mathbf{a},\mathbf{b})$ be an integer point in
$\mathcal{K}_{v}P^{\delta}_{n,r}.$ We use the description of the vertices of
$P^{\delta}_{n,r}$ given in Lemma 4.4. Subdivide the $n$-tuple
$(\mathbf{a},\mathbf{b})$ into left and right points depending on whether the
corresponding point in the vertex $n$-tuple is in a left or right block.
Further subdivide each column of $(\mathbf{a},\mathbf{b})$ depending on
whether the corresponding point in the vertex $n$-tuple is a top or bottom
point. Label in increasing lexicographic order four separate tuples of points
$(\mathbf{a}^{(1)},\mathbf{b}^{(1)}),\dots,(\mathbf{a}^{(4)},\mathbf{b}^{(4)})$
which are the bottom-left, bottom-right, top-right, and top-left points in
$(\mathbf{a},\mathbf{b})$ respectively. Let $n_{1},\dots,n_{4}$ be the number
of points in each of these tuples.

Consider a column of points $\\{p_{j},\dots,p_{j^{\prime}}\\}$ in some
$n$-tuple $(\mathbf{a},\mathbf{b})\in\mathcal{K}_{v}P^{\delta}_{n,r},$ and
suppose that the column lies in a left block. Let $j^{\prime\prime}$ be the
largest index corresponding to a left point. The equations defining
$P^{\delta}_{n,r}$ imply that heights of the bottom points in the column are
at least $\sum_{i=1}^{j-1}\max\\{r-(a_{j}-a_{i}),0\\}$, and similarly the
heights of the top points are at most
$\sum_{k=j^{\prime}+1}^{j^{\prime\prime}}\max\\{r-(a_{k}-a_{j}),0\\}.$
Comparing these conditions to those defining $P^{(1)}_{n_{i},r}$ and
$P^{(4)}_{n_{4},r}$, the only difference is that the sums for the lower and
upper bounds are restricted to the lower and upper points respectively. For
each $i<i^{\prime}$ where $(a_{i},b_{i})$ is a top-left point and
$(a_{i^{\prime}},b_{i^{\prime}})$ is a bottom-left point, we therefore shift
$b_{i}$ up by $\max\\{r-(a_{i^{\prime}}-a_{i}),0\\}$ and shift
$b_{i^{\prime}}$ down by $\max\\{r-(a_{i^{\prime}}-a_{i}),0\\}$. Similarly,
for each $i<i^{\prime}$ where $(a_{i},b_{i})$ is a top-right point and
$(a_{i^{\prime}},b_{i^{\prime}})$ is a bottom-right point, we shift $b_{i}$ up
by $\max\\{r-(a_{i^{\prime}}-a_{i}),0\\}$ and shift $b_{i^{\prime}}$ down by
$\max\\{r-(a_{i^{\prime}}-a_{i}),0\\}$. Call the new collection of points
after all the translations
$(\mathbf{a}^{(i)},\tilde{\mathbf{b}}^{(i)})_{i=1,\dots,4}.$

As discussed in the previous paragraph, the conditions defining the vertical
heights of points in the transformed tuples
$(\mathbf{a}^{(i)},\tilde{\mathbf{b}}^{(i)})_{i=1,\dots,4}$ for any fixed
$a$-coordinates are exactly the same as those defining
$P^{(1)}_{n_{1},r}\times\cdots\times P^{(4)}_{n_{4},r}$. As
$\delta\in\\{0,1,\dots,r\\}^{n-1}$ and $v$ vary, these transformed collections
of points are in bijection with the integer points each products
$P^{(1)}_{n_{1},r}\times\cdots\times P^{(4)}_{n_{4},r}$. Furthermore, the
transformation is defined by shifting certain pairs of points up and down by
the same amount so the sum of the $a$ and $b$-coordinates of all the points is
preserved. This shows that $\chi^{T}(X^{[n]},L_{n}\otimes E^{r})$, a sum of
generating functions of the products $P^{(1)}_{n_{1},r}\times\cdots\times
P^{(4)}_{n_{4},r}$, coincides with $\sigma^{\,\leavevmode\hbox to3.61pt{\vbox
to3.61pt{\pgfpicture\makeatletter\hbox{\hskip 0.3pt\lower-0.3pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{} {}{} {{}{}}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{3.01389pt}\pgfsys@lineto{3.01389pt}{3.01389pt}\pgfsys@lineto{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,{}}_{n,r}(t,q)$, a
sum of generating functions of cones $\mathcal{K}_{v}P^{\delta}_{n,r}$,
completing the proof. ∎

Next we extract an explicit formula for $\chi(X^{[n]},L_{n}\otimes E^{r})$ in
the case $X=\mathbb{P}^{1}\times\mathbb{P}^{1}$. We have $\ell=\ell(\delta)$
denoting the number of blocks in an $n$-tuple of type $\delta$, as well as
$L_{C}(\delta)$ and $R_{C}(\delta)$ for each column
$C=\\{p_{j},\dots,p_{j^{\prime}}\\}$. We also define the statistic
$|\delta|=\delta_{1}+\cdots+\delta_{n-1}$.

######  Corollary 4.5.

For $X=\mathbb{P}^{1}\times\mathbb{P}^{1}$, any line bundle
$L=\mathcal{O}(d_{1},d_{2})$, and $r>0$,

$\chi(X^{[n]},L_{n}\otimes
E^{r})=\sum_{\delta\in\\{0,1,\dots,r\\}^{n-1}}{d_{1}-|\delta|+\ell(\delta)\choose\ell(\delta)}\prod_{C}{d_{2}-R_{C}-L_{C}-r|C|+r+|C|\choose|C|},$

where the product is over all columns of points $C$ in an $n$-tuple of type
$\delta$.

The version of this formula given in Theorem 1.1 from the introduction
enumerates the columns $k=1,\dots,c(\delta)$. In the notation of the
introduction, $n_{k}(\delta)=|C|$ and $w_{k}(\delta)=R_{C}-L_{C}-r|C|$ where
$C$ is the $k$th column.

###### Proof.

Suppose first that $d_{1},d_{2}>r(n-1)$ so that by Theorem 4.2 we have

$\chi(X^{[n]},L_{n}\otimes E^{r})=\\#(P^{\,\leavevmode\hbox to3.61pt{\vbox
to3.61pt{\pgfpicture\makeatletter\hbox{\hskip 0.3pt\lower-0.3pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{} {}{} {{}{}}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{3.01389pt}\pgfsys@lineto{3.01389pt}{3.01389pt}\pgfsys@lineto{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,{}}_{n,r}\cap\mathbb{Z}^{2n})=\sum_{\delta\in\\{0,1,\dots,r\\}^{n-1}}\\#(P^{\delta}_{n,r}\cap\mathbb{Z}^{2n}).$

Plugging $s=0$ into the inequalities defining $P^{\delta}_{n,r}$ in Lemma 4.3,
we see that each $P^{\delta}_{n,r}$ is a product of simplices: one simplex for
the horizontal positions of the blocks, and one simplex for each column of
points controlling their heights.

The horizontal positions of the blocks are nonnegative integers
$a_{i_{1}},\dots,a_{i_{\ell}}$ such that $a_{i_{k+1}}\geq
a_{i_{k}}+\delta_{i_{k}}+\cdots+\delta_{i_{k+1}-1}$ and
$a_{i_{\ell}}+\delta_{i_{\ell}}+\cdots+\delta_{n-1}\leq d_{1}$. There are
$d_{1}-|\delta|+\ell(\delta)\choose\ell(\delta)$ such choices of block
positions. For each choice of block positions and column
$C=\\{p_{j},\dots,p_{j^{\prime}}\\}$, the heights $b_{j},\dots,b_{j^{\prime}}$
are integers in the interval $[L_{C}(\delta),d_{2}-R_{C}(\delta)]$ increasing
by at least $r$ in each step. There are
$d_{2}-R_{C}-L_{C}-r|C|+r+|C|\choose|C|$ choices of such integers.

This shows that for fixed $n,r>0$, the formula holds for all sufficiently
large $d_{1},d_{2}$. But both sides are polynomials in $d_{1},d_{2}$ so the
formula holds in general, completing the proof. ∎

##  5 Global Sections

In this final section, we explain the sense in which the $n$-tuples we have
studied correspond to global sections of line bundles on the Hilbert scheme.
Let $A^{r}\subseteq\mathbb{C}[\mathbf{x,y}]$ be the spaces defined in Section
3 which can be identified $A^{r}\simeq H^{0}((\mathbb{C}^{2})^{[n]},E^{r}).$
As before, we equip $\mathbb{C}[\mathbf{x,y}]$ with the lexicographic term
order with $x_{1}>\cdots>x_{n}>y_{1}>\cdots>y_{n}.$

Consider a smooth, projective, toric surface $X$ equipped with line bundle
$L$. The global sections $H^{0}(X,L)$ can be identified with Laurent
polynomials in $x$ and $y$ whose support is contained in $P_{L}$. The
following result generalizes this correspondence to Hilbert schemes.

######  Theorem 5.1 ([3] Proposition 4.2).

For any smooth, projective, toric surface $X$ with line bundle $L$ and integer
$r\geq 0$, the global sections $H^{0}(X^{[n]},L_{n}\otimes E^{r})$ can be
identified with the set of polynomials in
$A^{r}\subseteq\mathbb{C}[\mathbf{x,y}]$ whose support with respect to each
pair of variables $(x_{i},y_{i})$ is contained in $P_{L}$.

Here we assume without loss of generality (choosing an appropriate $T$-action
on $L$) that the corresponding polygon $P_{L}\subseteq\mathbb{R}^{2}$ is
contained in the first quadrant. In the $\mathbb{C}^{2}$ case, corresponding
to the entire collections of polynomials $A^{r}$, we were able to determine
the exact sets of trailing term exponents (see Theorem 3.1). For the support
restricted collections of polynomials appearing in the projective case, only
an upper bound for the sets of trailing terms was obtained in [3]. In the
Hirzebruch surface case studied in the previous section, Proposition 4.7 in
[3] states that the exponent $(\mathbf{a},\mathbf{b})$ appearing on the
trailing term $x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}y_{1}^{b_{1}}\cdots
y_{n}^{b_{n}}$ of any polynomial corresponding to a section of $L_{n}\otimes
E^{r}$ must be an $r$-separated $n$-tuple in $P_{L}$. We conjectured that
every such $n$-tuple appears one of these trailing term exponents, and thanks
to Theorem 4.2 we can prove this in the case $L_{n}\otimes E^{r}$ is ample.

######  Corollary 5.2.

Let $X$ be a Hirzebruch surface and $L=\mathcal{O}(d_{1}D_{1}+d_{2}D_{2})$
with polygon $P_{L}$ and $r>0$. If $L_{n}\otimes E^{r}$ is an ample line
bundle on $X^{[n]}$, then the set of exponents of trailing terms of
polynomials $f\in A^{r}$ corresponding to sections $H^{0}(X^{[n]},L_{n}\otimes
E^{r})$ is precisely the set of $r$-lexicographically increasing $n$-tuples of
points in $P_{L}$.

###### Proof.

By [3] Proposition 4.7, the trailing term exponent $(\mathbf{a},\mathbf{b})$
of any such polynomial is an $r$-lexicographically increasing $n$-tuple in
$P_{L}$. The number of trailing term exponents attained by polynomials
corresponding to sections of $L_{n}\otimes E^{r}$ is equal to the dimension of
$H^{0}(X^{[n]},L_{n}\otimes E^{r})$, so it suffices to show that the number of
such $n$-tuples coincides with $\dim H^{0}(X^{[n]},L_{n}\otimes E^{r}).$ By
Theorem 4.2, $\chi(X^{[n]},L_{n}\otimes E^{r})$ is equal to the number of such
$n$-tuples, and by the Frobenius splitting of $X^{[n]}$ [11], we have
$\chi(X^{[n]},L_{n}\otimes E^{r})=\dim H^{0}(X^{[n]},L_{n}\otimes E^{r})$
completing the proof. ∎

It would be interesting to give a more direct proof of the previous corollary
by constructing the polynomials with each given trailing term. Such an
approach would likely allow for more general results. For example, we expect
that similar results should hold for $X=\mathbb{P}^{2}$ and/or non-ample line
bundles $L_{n}\otimes E^{r}$ on $X^{[n]}$, but our methods relying on the
specific combinatorics of the trapezoid $P_{L}$ and corresponding sets
$P^{\,\leavevmode\hbox to3.61pt{\vbox
to3.61pt{\pgfpicture\makeatletter\hbox{\hskip 0.3pt\lower-0.3pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{} {}{} {{}{}}{}
{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.6pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{3.01389pt}\pgfsys@lineto{3.01389pt}{3.01389pt}\pgfsys@lineto{3.01389pt}{0.0pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,{}}_{n,r}$ and
passing through the Euler characteristic do not directly extend to these
cases.

## References

  * [1] Aaron Bertram and Izzet Coskun.  The birational geometry of the Hilbert scheme of points on surfaces.  Birational Geometry, Rational Curves, and Arithmetic, Simons Symposia, pages 15–55, 2013.
  * [2] Michel Brion.  Points entiers dans les polyèdres convexes.  Annales scientifiques de l'École Normale Supérieure, 21(4):653–663, 1988.
  * [3] Ian Cavey.  Effective divisors and Newton-Okounkov bodies of Hilbert schemes of points on toric surfaces.  Journal of Algebra, 632:602–640, 2023.
  * [4] David Cox, John Little, and Hal Schenck.  Toric Varieties.  American Mathematical Society, 2011.
  * [5] Geir Ellingsrud, Lothar Göttsche, and Manfred Lehn.  On the Cobordism Class of the Hilbert Scheme of a Surface.  Journal of Algebraic Geometry, 10, 05 1999.
  * [6] John Fogarty.  Algebraic families on an algebraic surface, II, the Picard scheme of the punctual Hilbert scheme.  American Journal of Mathematics, 95(3):660–687, 1973.
  * [7] Lothar Göttsche.  Hilbert schemes of points on surfaces.  Proceedings of the ICM, 2:483–494, 2002.
  * [8] Lothar Göttsche and Anton Mellit.  Refined Verlinde and Segre formula for Hilbert schemes.  arXiv:math/0304302, 2022.
  * [9] Mark D. Haiman.  t, q-Catalan numbers and the Hilbert scheme.  Discrete Math, 193:201–224, 1998.
  * [10] Drew Johnson.  Universal Series for Hilbert Schemes and Strange Duality.  International Mathematics Research Notices, 2020(10):3130–3152, 05 2018.
  * [11] Shrawan Kumar and Jesper Funch Thomsen.  Frobenius splitting of Hilbert schemes of points on surfaces.  Mathematische Annalen, 319(4):797–808, 2001.
  * [12] Manfred Lehn.  Chern classes of tautological sheaves on Hilbert schemes of points on surfaces.  Inventiones mathematicae, 136(1):157–207, 1999.
  * [13] Alina Marian, Dragos Oprea, and Rahul Pandharipande.  Higher rank segre integrals over the hilbert scheme of points.  Journal of the European Mathematical Society, 24, 12 2017.
  * [14] Alina Marian, Dragos Oprea, and Rahul Pandharipande.  The combinatorics of Lehn's conjecture.  Journal of the Mathematical Society of Japan, 71(1):299 – 308, 2019\.
  * [15] Claire Voisin.  Segre classes of tautological bundles on Hilbert schemes of surfaces.  Algebraic Geometry, 6(2):186–195, 2019.
  * [16] Yao Yuan.  Rank zero Segre integrals on Hilbert schemes of points on surfaces.  arXiv:2209.06600, 2022.