# Hyperquot schemes on curves:
virtual class and motivic invariants

Sergej Monavari and Andrea T. Ricolfi  École Polytechnique Fédérale de
Lausanne (EPFL), CH-1015 Lausanne (Switzerland) SISSA, Via Bonomea 265, 34136,
Trieste (Italy)

###### Abstract.

Let $C$ be a smooth projective curve, $E$ a locally free sheaf. Hyperquot
schemes on $C$ parametrise flags of coherent quotients of $E$ with fixed
Hilbert polynomial, and offer alternative compactifications to the spaces of
maps from $C$ to partial flag varieties. Motivated by enumerative geometry, in
this paper we construct a perfect obstruction theory (and hence a virtual
class and a virtual structure sheaf) on these moduli spaces, which we use to
provide criteria for smoothness and unobstructedness. Under these assumptions,
we determine their motivic partition function in the Grothendieck ring of
varieties, in terms of the motivic zeta function of $C$.

###### Key words and phrases:

Quot schemes, Deformation Theory, Grothendieck ring of varieties

###### Contents

  1. 1 Introduction
  2. 2 Hyperquot scheme
  3. 3 Perfect obstruction theory
  4. 4 Smoothness of hyperquot schemes
  5. 5 Torus fixed locus
  6. 6 Motives of hyperquot schemes
  7. A Functoriality of virtual classes
  8. B Virtual $\chi_{-y}$-genus

##  1\. Introduction

###  1.1. Overview

Let $C$ be a smooth projective curve over an algebraically closed field
$\mathbf{k}$, and let $E$ be a locally free sheaf of rank $r$ on $C$. Fix a
positive integer $l$, along with two $l$-tuples of integers
$\boldsymbol{s}=(s_{1},\ldots,s_{l})$ and
$\boldsymbol{d}=(d_{1},\ldots,d_{l})$, where $0\leq s_{1}\leq\cdots\leq
s_{l}\leq r$. In this paper we study a generalisation of the classical
Grothendieck’s Quot scheme [28], namely the _hyperquot scheme_

$\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d}),$

parametrising sequences

${K_{l}}$${\cdots}$${K_{1}}$${E}$${T_{l}}$${\cdots}$${T_{1}}$

in $\operatorname{Coh}(C)$, where $K_{i}=\ker(E\twoheadrightarrow T_{i})$ and
$(\operatorname{rk}T_{i},\deg T_{i})=(s_{i},d_{i})$ for all $i=1,\ldots,l$.

When $E=\mathscr{O}_{C}^{\oplus r}$, the hyperquot scheme provides a natural
compactification of the space of morphisms
$M_{\boldsymbol{d}}(C,\mathrm{Flag}(\boldsymbol{s},r))$, of multidegree
$\boldsymbol{d}$, from $C$ to the partial flag variety
$\mathrm{Flag}(\boldsymbol{s},r)$. This compactification is a natural
alternative to the moduli space of _stable maps_ of Kontsevich, and was
effectively exploited to compute the quantum cohomology ring of partial flag
varieties, see e.g. [13, 12, 14, 11, 4] for a non-exhaustive list of
references.

It is natural to approach the study of a moduli space by examining its
_invariants_. In this paper we pursue this direction in two different ways:

  * (1)

We construct a perfect obstruction theory in the sense of Behrend–Fantechi
[3], leading to a virtual fundamental class and a virtual structure sheaf, on
the space $\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d})$. This
paves the way to the study of the virtual intersection theory of the hyperquot
scheme.

  * (2)

We compute the motivic class of
$\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d})$ in the Grothendieck
ring of varieties $K_{0}(\operatorname{Var}_{\mathbf{k}})$, whenever $E$ is
split and the space is smooth and unobstructed.

The ring valued invariants that ‘behave like’ the topological Euler
characteristic are called _motivic_. By the well-known universal property of
$K_{0}(\operatorname{Var}_{\mathbf{k}})$, our results compute all motivic
invariants of the spaces
$\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d})$ under the smooth
and unobstructed assumption.

We note that (2) generalises Chen’s results [10] on the Poincaré polynomials
of the hyperquot scheme attached to
$(C,E)=({\mathbb{P}}^{1},\mathscr{O}^{\oplus r})$. In particular, we discover
new instances of smooth hyperquot schemes.

In the next subsection, we state our results in greater detail.

###  1.2. Statements of the main results

Our first main result is the existence of a perfect obstruction theory on the
hyperquot scheme of a curve.

######  Theorem A (Theorem 3.2, Corollary 3.7).

Let $C$ be a smooth projective curve of genus $g$, $E$ a locally free sheaf of
rank $r>0$, and $l$ a positive integer. Fix tuples
$\boldsymbol{s}=(s_{1}\leq\dots\leq s_{l})$ and
$\boldsymbol{d}=(d_{1},\dots,d_{l})$. The hyperquot scheme
$\operatorname{\mathrm{Q}}=\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d})$
admits a perfect obstruction theory

${{\mathbb{E}}_{\operatorname{\mathrm{Q}}}=\operatorname{\mathrm{Cone}}\Biggl{(}\displaystyle\bigoplus_{i=1}^{l}\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\pi}({\mathcal{K}}_{i},{\mathcal{K}}_{i})}$${\displaystyle\bigoplus_{i=0}^{l-1}\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\pi}({\mathcal{K}}_{i+1},{\mathcal{K}}_{i})\Biggr{)}^{\vee}}$${{\mathbb{L}}_{\operatorname{\mathrm{Q}}},}$

where we set ${\mathcal{K}}_{0}=E_{\operatorname{\mathrm{Q}}}$. In particular,
$\operatorname{\mathrm{Q}}$ admits a virtual fundamental class and a virtual
structure sheaf

$[\operatorname{\mathrm{Q}}]^{\operatorname{\mathrm{vir}}}\in
A_{\operatorname{vdim}}(\operatorname{\mathrm{Q}}),\qquad\mathscr{O}^{\operatorname{\mathrm{vir}}}_{\operatorname{\mathrm{Q}}}\in
K_{0}(\operatorname{\mathrm{Q}}).$

Setting $s_{l+1}=r$ and $s_{0}=0$, the virtual class lives in dimension

$\operatorname{vdim}=(1-g)\cdot\dim\mathrm{Flag}(\boldsymbol{s},r)+\sum_{i=1}^{l}d_{i}(s_{i+1}-s_{i-1})-\deg
E\cdot s_{l}.$

In the statement, $\mathrm{Flag}(\boldsymbol{s},r)$ refers to the partial flag
variety (cf. Section 2.1), whose dimension is
$\sum_{i=1}^{l}s_{i}(s_{i+1}-s_{i})$, as recalled in (2.1). It can be seen as
a hyperquot scheme over a point, cf. Example 2.1.

The perfect obstruction theory of A allows one to study the classical
deformation theory of points in the hyperquot scheme (cf. Corollary 3.5). We
give in 4.2 a sufficient condition for smoothness (and unobstructedness) in
the unnested case. In the nested case, we obtain the following result,
generalising the known case of the trivial bundle over ${\mathbb{P}}^{1}$.

######  Theorem B (Theorem 4.4).

Let $L$ be any line bundle on ${\mathbb{P}}^{1}$ and
$E=L\otimes\left(\mathscr{O}^{\oplus n}\oplus\mathscr{O}(1)^{\oplus m}\right)$
for some $n,m$. Then the hyperquot scheme
$\operatorname{Quot}_{{\mathbb{P}}^{1}}(E,\boldsymbol{s},\boldsymbol{d})$ is
smooth and unobstructed for all $\boldsymbol{d}$ and for all $\boldsymbol{s}$.

Assume that $E=\bigoplus_{\alpha=1}^{r}L_{\alpha}$ splits into a sum of line
bundles on $C$. Then
$\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d})$ admits the action
of the algebraic torus $\mathbf{T}={\mathbb{G}}_{m}^{r}$ as in [6]. We provide
in Section 5.2 a combinatorial description of the fixed locus, indexing it by
permutations $\sigma\in P_{\boldsymbol{r}}$ (cf. Section 5.2.2). We are then
in a position to apply the Białynicki-Birula decomposition to the hyperquot
scheme, provided it is smooth. This allows us to determine a closed formula
for the motivic generating function

$\mathsf{Z}_{C,E,\boldsymbol{s}}(\boldsymbol{q})=\sum_{\boldsymbol{d}}\,\bigl{[}\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d})\bigr{]}\boldsymbol{q}^{\boldsymbol{d}}\in
K_{0}(\operatorname{Var}_{\mathbf{k}})(\\!(q_{1},\dots,q_{l})\\!),$

in terms of the motivic zeta function

$\zeta_{C}(t)=\sum_{n\geq 0}\,[C^{(n)}]t^{n}$

of the curve and two combinatorial exponents $h^{\sigma}_{0}$ and
$h^{\sigma}_{2}$, defined in (6.1). The result is the following.

######  Theorem C (Theorem 6.3, Theorem 6.5).

Let $C$ be a smooth projective curve of genus $g$ and
$E=\bigoplus_{\alpha=1}^{r}L_{\alpha}$ a split locally free sheaf such that
the hyperquot scheme
$\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d})$ is smooth and
unobstructed. Set $r_{i}=r-s_{i}$ for $i=1,\ldots,l$ and $r_{l+1}=0$. There is
an identity

$\mathsf{Z}_{C,E,\boldsymbol{s}}(\boldsymbol{q})=\sum_{\sigma\in
P_{\boldsymbol{r}}}\prod_{j=1}^{l}\left({\mathbb{L}}^{h^{\sigma}_{2}(j,g,\deg
L_{1},\dots,\deg L_{r})}q_{j}^{\sum_{\alpha=r_{j}+1}^{r}\deg
L_{\sigma(\alpha)}}\right)\\\ \cdot\prod_{1\leq i\leq j\leq
l}\prod_{\alpha=r_{j+1}+1}^{r_{j}}\zeta_{C}\left({\mathbb{L}}^{h^{\sigma}_{0}(i,\alpha)+r_{i}-r_{j}+\alpha-
r_{j+1}-1}q_{i}\cdots q_{j}\right).$

In particular, $\mathsf{Z}_{C,E,\boldsymbol{s}}(\boldsymbol{q})$ is a rational
function.

For the pair $(C,E)=({\mathbb{P}}^{1},\mathscr{O}^{\oplus r})$, one has the
identity

$\mathsf{Z}_{{\mathbb{P}}^{1},\mathscr{O}^{\oplus
r},\boldsymbol{s}}(\boldsymbol{q})=[\mathrm{Flag}(\boldsymbol{s},r)]\prod_{1\leq
i\leq j\leq
l}\prod_{\alpha=r_{j+1}+1}^{r_{j}}\frac{1}{\left(1-{\mathbb{L}}^{r_{i}-\alpha}q_{i}\cdots
q_{j}\right)\left(1-{\mathbb{L}}^{r_{i-1}-\alpha+1}q_{i}\cdots q_{j}\right)}.$

Our formulas fit nicely in the philosophical idea that _rank $r$ theories are
determined by the rank $1$ theory_. Up to now, this phenomenon has been
observed in several places, see for instance the motivic computations [43, 9,
8] and the recent results in Donaldson–Thomas theory [21, 20, 22, 23] and
String Theory [45, 17, 16].

We also notice that the computation of the motivic partition functions in C
allows one to spot the connectedness of the moduli space, by first
specialising to Poincaré polynomials
$\mathsf{P}(\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d}),q)$ and
then extracting the constant coefficient, which coincides with
$b_{0}(\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d}))$. This is
carried out in Corollary 6.7 for $(C,E)=({\mathbb{P}}^{1},\mathscr{O}^{\oplus
r})$.

At the level of Euler characteristics (over the complex field
$\mathbf{k}={\mathbb{C}}$), we have a complete description of the generating
function

$\mathsf{Z}^{\mathrm{top}}_{C,E,\boldsymbol{s}}(\boldsymbol{q})=\sum_{\boldsymbol{d}}\,e_{\mathrm{top}}(\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d}))\boldsymbol{q}^{\boldsymbol{d}}\in{\mathbb{Z}}(\\!(q_{1},\dots,q_{l})\\!),$

without any smoothness assumption.

######  Theorem D (Theorem 5.2).

Let $C$ be a smooth projective complex curve of genus $g$ and
$E=\bigoplus_{\alpha=1}^{r}L_{\alpha}$ a split locally free sheaf. There is an
identity

$\displaystyle\mathsf{Z}^{\mathrm{top}}_{C,E,\boldsymbol{s}}(\boldsymbol{q})$
$\displaystyle=\left(\sum_{\sigma\in
P_{\boldsymbol{r}}}\prod_{j=1}^{l}q_{j}^{\sum_{\alpha=r_{j}+1}^{r}\deg
L_{\sigma(\alpha)}}\right)\cdot\prod_{1\leq i\leq j\leq l}(1-q_{i}\cdots
q_{j})^{(2g-2)(r_{j}-r_{j+1})}.$

###  1.3. Relation with Enumerative Geometry

We conclude the introduction by pointing out some interesting applications of
our results, and how they should be considered in the more general framework
of enumerative geometry.

####  1.3.1. Virtual invariants

There has been a recent interest in defining and studying virtual invariants
of Quot schemes on varieties of various dimensions, usually motivated by
parallel computations in topological string theory, see e.g. [21, 20, 1, 37,
7]. It would be of particular interest to extend to the hyperquot setting the
_Vafa–Intriligator formulas_ proved in [40] — see also [57] for an isotropic
version of the latter — and the tautological integrals, studied e.g. in [48,
49]. As a first step of this program, we prove in Appendix A some functorial
properties of the virtual fundamental class, and in Appendix B we discuss some
structural properties of the virtual $\chi_{-y}$-genus, leaving more general
computations for future work. Finally, another compactification of the space
of maps to partial flag varieties is provided by moduli spaces of stable
_quasimaps_ , see e.g. [15] where it is shown that the moduli stack of
quasimaps has a perfect obstruction theory in good circumstances.

####  1.3.2. Higher dimensional varieties

Moduli spaces of sheaves with nesting conditions should not be regarded as
mere variations of classical moduli spaces, as they appear rather naturally
while studying moduli spaces of sheaves on higher dimensional _local_
varieties, e.g. vector bundles on curves and surfaces. The prototypical
situation is the following: let $Y$ be a projective variety, and $X$ be the
total space of a (split) locally free sheaf on $Y$. Then $X$ admits a natural
torus action scaling the fibers, which lifts to the moduli spaces of sheaves
on $X$ [53]. The fixed locus of this torus action can be usually identified
with another moduli space of sheaves on $Y$, satisfying suitable nesting
conditions. Therefore, by localisation techniques, enumerative invariants of
moduli spaces of sheaves on $X$ are reduced to enumerative invariants on
moduli spaces of _nested_ sheaves on $Y$. See for instance applications of
this idea for virtual invariants in Donaldson–Thomas and Vafa–Witten theory
[42, 35, 25, 36] and for motivic invariants of moduli spaces of Higgs bundles
[24, 30, 29].

####  1.3.3. Cohomology

Recently Marian–Neguţ [39] described an action of the shifted Yangian of
$\mathfrak{sl}_{2}$ on the cohomology groups of the Quot schemes of
0-dimensional quotients on a smooth projective curve. It would be interesting
to understand how to generalise this action to our setting with nested and
possibly 1-dimensional quotients, see also [41].

###  1.4. Conventions

All _schemes_ are of finite type over a fixed algebraically closed field
$\mathbf{k}$. A _variety_ is a reduced separated $\mathbf{k}$-scheme. If $Y$
is a scheme and $Y_{1},\ldots,Y_{s}$ are locally closed subschemes of $Y$, we
say that they form a (locally closed) _stratification_ , denoted
‘$Y=Y_{1}\amalg\cdots\amalg Y_{s}$’ with a slight abuse of notation, if the
natural morphism of schemes $Y_{1}\amalg\cdots\amalg Y_{s}\rightarrow Y$ is
bijective. We denote by $\mathbf{D}^{[a,b]}(Y)$ the full subcategory of the
derived category of coherent sheaves $\mathbf{D}(Y)$, consisting of complexes
having (possibly) nonvanishing cohomology in the interval $[a,b]$. We use the
same notation for the full subcategory $\operatorname{Perf}^{[a,b]}(Y)$ of the
triangulated category $\operatorname{Perf}(Y)$ of perfect complexes on $Y$. We
denote the $\mathscr{O}_{Y}$-linear dual
$\operatorname{\mathscr{H}\kern-3.00003pt\mathit{om}}_{\mathscr{O}_{Y}}(F,\mathscr{O}_{Y})$
of a coherent sheaf $F\in\operatorname{Coh}(Y)$ by $F^{\ast}$, reserving the
notation $(-)^{\vee}$ for the derived dual
$\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\mathscr{O}_{Y}}(-,\mathscr{O}_{Y})$.
Given a morphism $\pi\colon Y\rightarrow B$, we set
$\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\pi}(-,-)=\operatorname{\mathbf{R}}\pi_{\ast}\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}(-,-)$.
We sometimes write derived pullbacks $\mathbf{L}\pi^{\ast}$ simply by
$\pi^{\ast}$.

### Acknowledgements

We thank Alina Marian for bringing to our attention Chen’s work [10]. We are
grateful to Shubham Sinha for pointing out Lemma 3.1 to us. We thank Barbara
Fantechi, Alina Marian and Shubham Sinha for very helpful discussions. S. M.
thanks Eric Chen and Denis Nesterov for useful discussions. S.M. is supported
by the Chair or Arithmetic Geometry, EPFL. Both authors are members of GNSAGA
of INDAM.

##  2\. Hyperquot scheme

###  2.1. Partial flag varieties

Fix integers $r,l>0$ and a nondecreasing $l$-tuple of nonnegative integers
$\boldsymbol{s}=(s_{1}\leq\dots\leq s_{l})$ such that $s_{l}\leq r$. To ease
the notation, we set $s_{0}=0$, $s_{l+1}=r$ and $r_{i}=r-s_{i}$ for all
$i=0,\dots,l+1$.

We denote by $\mathrm{Flag}(\boldsymbol{s},r)$ the _partial flag variety_ ,
namely the scheme parametrising flags of vector spaces

$V_{l}\subset\dots\subset V_{1}\subset V,$

where $V$ is a fixed $r$-dimensional $\mathbf{k}$-vector space and $\dim
V_{i}=r_{i}$. It is well-known that $\mathrm{Flag}(\boldsymbol{s},r)$ is a
smooth projective variety of dimension

(2.1)
$\dim\mathrm{Flag}(\boldsymbol{s},r)=\sum_{i=1}^{l}s_{i}(s_{i+1}-s_{i}).$

The case $l=1$ yields the Grassmannian $\mathrm{Flag}(s_{1},r)=G(r_{1},V)$.
The _hyperquot scheme_ , introduced in the next subsection, can be seen as a
vaste generalisation of the partial flag variety.

###  2.2. The hyperquot scheme

Let $Y$ be a projective variety, $E$ a coherent sheaf on $Y$. Fix an integer
$l>0$ and a vector

$\boldsymbol{v}=(v_{1},\ldots,v_{l})\in\mathrm{H}^{\ast}(Y,{\mathbb{Q}})^{\oplus
l}$

of Chern characters $v_{i}\in\mathrm{H}^{\ast}(Y,{\mathbb{Q}})$. We define the
_hyperquot functor_ associated to $(Y,E,\boldsymbol{v})$ to be the functor
$\mathsf{Quot}_{Y}(E,\boldsymbol{v})\colon\operatorname{Sch}_{\mathbf{k}}^{\mathrm{op}}\rightarrow\operatorname{Sets}$
sending a $\mathbf{k}$-scheme $B$ to the set of isomorphism classes of
subsequent quotients

${E_{B}}$${{\mathcal{T}}_{l}}$${\cdots}$${{\mathcal{T}}_{1}}$

where $E_{B}$ is the pullback of $E$ along $Y\times B\rightarrow Y$ and
${\mathcal{T}}_{i}\in\operatorname{Coh}(Y\times B)$ is a $B$-flat family of
coherent sheaves such that ${\mathcal{T}}_{i}|_{Y\times
b}\in\operatorname{Coh}(Y)$ has Chern character $v_{i}$ for all $b\in B$ and
$i=1,\ldots,l$. Two ‘nested quotients’

${E_{B}}$${{\mathcal{T}}_{l}}$${\cdots}$${{\mathcal{T}}_{1},}$${E_{B}}$${{\mathcal{T}}_{l}^{\prime}}$${\cdots}$${{\mathcal{T}}_{1}^{\prime}}$

are considered isomorphic when
$\ker(E_{B}\twoheadrightarrow{\mathcal{T}}_{i})=\ker(E_{B}\twoheadrightarrow{\mathcal{T}}_{i}^{\prime})$
for all $i=1,\ldots,l$.

The representability of the functor $\mathsf{Quot}_{Y}(E,\boldsymbol{v})$ can
be proved adapting the proof of [55, Thm. 4.5.1] or by an explicit induction
on $l$ as in [31, Sec. 2.A.1]. We define

$\operatorname{Quot}_{Y}(E,\boldsymbol{v})$

to be the moduli scheme representing the hyperquot functor. We call it the
_hyperquot scheme_. Its closed points are then in bijection with the set of
isomorphism classes of subsequent quotients

(2.2) ${E}$${T_{l}}$${\cdots}$${T_{1}}$

where each $T_{i}\in\operatorname{Coh}(Y)$ is a quotient of $E$ with
$\mathsf{ch}(T_{i})=v_{i}$. Equivalently, it parametrises sequences

(2.3) ${K_{l}}$${\cdots}$${K_{1}}$${E}$${T_{l}}$${\cdots}$${T_{1}}$

where we have set $K_{i}=\ker(E\twoheadrightarrow T_{i})$. The hyperquot
scheme comes with a closed immersion, inside a product of ordinary Quot
schemes

(2.4)
${\operatorname{Quot}_{Y}(E,\boldsymbol{v})}$${\displaystyle\prod_{i=1}^{l}\operatorname{Quot}_{Y}(E,v_{i}),}$

cut out by the nesting condition $K_{l}\hookrightarrow\cdots\hookrightarrow
K_{1}$. In particular, it is a projective scheme.

######  Example 2.1.

The following examples recover well-known moduli spaces.

  1. (1)

Fix $r\in{\mathbb{Z}}_{>0}$. If
$Y=\operatorname{Spec}\mathbf{k}={\mathsf{pt}}$ and
$\boldsymbol{v}=(s_{1},\ldots,s_{l})$ is a nondecreasing tuple of nonnegative
integers with $s_{l}\leq r$, we have
$\operatorname{Quot}_{{\mathsf{pt}}}(\mathbf{k}^{\oplus
r},\boldsymbol{v})=\mathrm{Flag}(\boldsymbol{s},r)$, the partial flag variety
of Section 2.1.

  2. (2)

If $E=L$ is locally free of rank $1$ and $v_{i}=(0,0,\ldots,n_{i})$ with
$n_{l}\geq\cdots\geq n_{1}\geq 0$, then
$\operatorname{Quot}_{Y}(L,\boldsymbol{v})$ is the nested Hilbert scheme
$Y^{[\boldsymbol{n}]}$. If $E$ is an arbitrary locally free sheaf, one obtains
the nested Quot scheme of points.

###  2.3. Tangent space to hyperquot scheme

Let $z\in\operatorname{Quot}_{Y}(E,\boldsymbol{v})$ be a closed point,
representing a sequence as in (2.3). There is an inclusion of
$\mathbf{k}$-vector spaces

${T_{z}\operatorname{Quot}_{Y}(E,\boldsymbol{v})}$${\displaystyle\bigoplus_{i=1}^{l}\operatorname{Hom}_{Y}(K_{i},T_{i}),}$

which is just the tangent map to (2.4) at $z$. An $l$-tuple
$(\delta_{1},\delta_{2},\ldots,\delta_{l})$ of tangent vectors
$\delta_{i}\in\operatorname{Hom}_{Y}(K_{i},T_{i})$ belongs to
$T_{z}\operatorname{Quot}_{Y}(E,\boldsymbol{v})$ if and only if the diagram

${K_{l}}$${K_{l-1}}$${\cdots}$${K_{2}}$${K_{1}}$${T_{l}}$${T_{l-1}}$${\cdots}$${T_{2}}$${T_{1}}$$\scriptstyle{\iota_{l-1}}$$\scriptstyle{\delta_{l}}$$\scriptstyle{\iota_{l-2}}$$\scriptstyle{\delta_{l-1}}$$\scriptstyle{\iota_{2}}$$\scriptstyle{\iota_{1}}$$\scriptstyle{\delta_{2}}$$\scriptstyle{\delta_{1}}$$\scriptstyle{p_{l-1}}$$\scriptstyle{p_{l-2}}$$\scriptstyle{p_{2}}$$\scriptstyle{p_{1}}$

commutes (see also [43, Sec. 2]). Such commutativity relations,

$\delta_{i}\circ\iota_{i}-p_{i}\circ\delta_{i+1}=0,\quad i=1,\ldots,l-1,$

can be encoded in an exact sequence of $\mathbf{k}$-linear maps

(2.5)
${0}$${T_{z}\operatorname{Quot}_{Y}(E,\boldsymbol{v})}$${\displaystyle\bigoplus_{i=1}^{l}\operatorname{Hom}_{Y}(K_{i},T_{i})}$${\displaystyle\bigoplus_{i=1}^{l-1}\operatorname{Hom}_{Y}(K_{i+1},T_{i}),}$$\scriptstyle{\Delta_{z}}$

where

$\Delta_{z}(\delta_{1},\ldots,\delta_{l})=(\delta_{i}\circ\iota_{i}-p_{i}\circ\delta_{i+1})_{1\leq
i\leq l-1}.$

###  2.4. Our setup: the curve case

From now on, we work with the hyperquot scheme of a fixed smooth, projective,
connected curve $C$ of genus $g$, i.e. we set $Y=C$. The following data are
fixed throughout:

  * $\circ$

two positive integers $r,l$,

  * $\circ$

a locally free sheaf $E$ of rank $r$,

  * $\circ$

a nondecreasing $l$-tuple $\boldsymbol{s}=(s_{1}\leq\dots\leq s_{l})$ of
integers $s_{i}\in{\mathbb{Z}}_{\geq 0}$ such that $s_{l}\leq r$,

  * $\circ$

a tuple $\boldsymbol{d}=(d_{1},\ldots,d_{l})$ of integers
$d_{i}\in{\mathbb{Z}}$.

With this data, we form the vector of Chern characters
$\boldsymbol{v}=((s_{1},d_{1}),\ldots,(s_{l},d_{l}))\in\mathrm{H}^{\ast}(C,{\mathbb{Z}})^{\oplus
l}$ and we set

$\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d})=\operatorname{Quot}_{C}(E,\boldsymbol{v}).$

This hyperquot scheme parametrises sequences (2.3) where $T_{i}$ is a coherent
quotient of $E$ of rank $s_{i}$ and degree $d_{i}$. In particular, the kernels
$K_{i}=\ker(E\twoheadrightarrow T_{i})$ are locally free of rank
$r_{i}=r-s_{i}$.

##  3\. Perfect obstruction theory

###  3.1. Perfect obstruction theories

Let $X$, $B$ be algebraic stacks over $\mathbf{k}$, and assume $B$ is smooth.
Let $h\colon X\rightarrow B$ be a morphism of Deligne–Mumford type (i.e. such
that the diagonal $\Delta_{h}\colon X\rightarrow X\times_{B}X$ is unramified).
In this situation, the relative cotangent complex $L_{h}$ defined by Illusie
[32] has cohomology concentrated in nonpositive degrees, i.e. it belongs to
$\mathbf{D}^{\leq 0}(X)$. Let ${\mathbb{L}}_{h}\in\mathbf{D}^{[-1,0]}(X)$
denote its cut-off at $-1$.

A _relative perfect obstruction theory_ [3, Sec. 7] on $h\colon X\rightarrow
B$ is the datum of a morphism

${{\mathbb{E}}}$${{\mathbb{L}}_{h}}$$\scriptstyle{\phi}$

in the derived category $\mathbf{D}^{[-1,0]}(X)$, where ${\mathbb{E}}$ is a
perfect complex of perfect amplitude contained in $[-1,0]$, such that

  * $\circ$

$h^{0}(\phi)$ is an isomorphism,

  * $\circ$

$h^{-1}(\phi)$ is surjective.

If $B=\operatorname{Spec}\mathbf{k}$, we say $\phi$ is an _absolute_ perfect
obstruction theory.

The dual ${\mathbb{E}}^{\vee}\in\mathbf{D}^{[0,1]}(X)$ is called the _virtual
tangent bundle_ of the obstruction theory, and its first cohomology
$\operatorname{Ob}_{\phi}=h^{1}({\mathbb{E}}^{\vee})$ is called the
_obstruction sheaf_ attached to $({\mathbb{E}},\phi)$. The _virtual dimension_
of $X$ with respect to $({\mathbb{E}},\phi)$ is the integer
$\operatorname{vdim}=\operatorname{rk}{\mathbb{E}}+\dim B$. By [3], a perfect
obstruction theory induces a _virtual fundamental class_

$[X]^{\operatorname{\mathrm{vir}}}\in A_{\operatorname{vdim}}(X)$

and, if $B=\operatorname{Spec}\mathbf{k}$, a _virtual structure sheaf_

$\mathscr{O}_{X}^{\operatorname{\mathrm{vir}}}\in K_{0}(X).$

By a result of Siebert [56] (resp. Thomas [59]) the virtual fundamental class
in the absolute case $B=\operatorname{Spec}\mathbf{k}$ (resp. the virtual
structure sheaf) depends only on the $K$-theory class of ${\mathbb{E}}$.
Moreover, Behrend–Fantechi prove [3, Prop. 7.3] that if
$h^{-1}({\mathbb{E}})=0$ then $h\colon X\rightarrow B$ is smooth, and in this
case $[X]^{\operatorname{\mathrm{vir}}}=[X]$. Furthermore, if $h\colon
X\rightarrow B$ is smooth, then $\operatorname{Ob}_{\phi}$ is locally free and
$[X]^{\operatorname{\mathrm{vir}}}=e(\operatorname{Ob}_{\phi})\cap[X]$, where
$e(-)$ denotes the Euler class.

We now recall two basic facts that will be needed in the next subsection.

First of all, given a smooth algebraic stack $B$ and a morphism $h\colon
X\rightarrow B$ with a relative perfect obstruction theory
$\phi_{h}\colon{\mathbb{E}}_{h}\rightarrow{\mathbb{L}}_{h}$, the shifted cone

${\mathbb{E}}=\operatorname{\mathrm{Cone}}({\mathbb{E}}_{h}\rightarrow
h^{\ast}\Omega_{B}[1])[-1]$

is an absolute perfect obstruction theory for $X$. This can be shown via the
diagram

(3.1)
${h^{\ast}\Omega_{B}}$${{\mathbb{E}}}$${{\mathbb{E}}_{h}}$${h^{\ast}\Omega_{B}[1]}$${h^{\ast}\Omega_{B}}$${{\mathbb{L}}_{X}}$${{\mathbb{L}}_{h}}$${h^{\ast}\Omega_{B}[1]}$$\scriptstyle{\phi_{h}}$$\scriptstyle{\varepsilon\circ\phi_{h}}$$\scriptstyle{\varepsilon}$

where the dotted arrows can be completed to solid ones, to form a morphism of
exact triangles, by the axioms of the derived category, and
${\mathbb{E}}\rightarrow{\mathbb{L}}_{X}$ turns out to be a perfect
obstruction theory thanks to a simple computation in the long exact sequence
in cohomology for the two triangles involved. See also [38, Constr. 3.13] for
a more general setup.

The second fact we need is the following lemma.

######  Lemma 3.1 ([57, Lemma 2.8] and [54]).

Let $p\colon B^{\prime}\rightarrow B$ be a flat, projective, $1$-dimensional
family of algebraic stacks. Let $F\in\operatorname{Coh}(B^{\prime})$ be a
$B$-flat coherent sheaf. Consider the abelian cone

${\mathbf{S}=\operatorname{Spec}_{\mathscr{O}_{B}}\operatorname{Sym}(\operatorname{\mathbf{R}}^{1}p_{\ast}F)}$${B.}$$\scriptstyle{\eta}$

Let $\overline{p}\colon\mathbf{S}\times_{B}B^{\prime}\rightarrow\mathbf{S}$
and $\overline{\eta}\colon\mathbf{S}\times_{B}B^{\prime}\rightarrow
B^{\prime}$ be the projections. Then there is a relative perfect obstruction
theory

${\operatorname{\mathbf{R}}\overline{p}_{\ast}\overline{F}[1]}$${{\mathbb{L}}_{\eta},}$

where $\overline{F}=\overline{\eta}^{\ast}F$.

###  3.2. Perfect obstruction theory on hyperquot

Set
$\operatorname{\mathrm{Q}}=\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d})$,
where we have fixed $\boldsymbol{s}=(s_{1}\leq\dots\leq s_{l})$ and
$\boldsymbol{d}=(d_{1},\dots,d_{l})$ as in Section 2.4. Let
$\pi\colon\operatorname{\mathrm{Q}}\times
C\rightarrow\operatorname{\mathrm{Q}}$ denote the projection. We prove in this
section that $\operatorname{\mathrm{Q}}$ admits a perfect obstruction theory.

Over $\operatorname{\mathrm{Q}}\times C$, the universal kernels realise a
filtration

(3.2)
${{\mathcal{K}}_{l}}$${{\mathcal{K}}_{l-1}}$${\cdots}$${{\mathcal{K}}_{2}}$${{\mathcal{K}}_{1}}$${{\mathcal{K}}_{0}=E_{\operatorname{\mathrm{Q}}}.}$$\scriptstyle{\iota_{l-1}}$$\scriptstyle{\iota_{l-2}}$$\scriptstyle{\iota_{2}}$$\scriptstyle{\iota_{1}}$$\scriptstyle{\iota_{0}}$

######  Theorem 3.2.

The hyperquot scheme
$\operatorname{\mathrm{Q}}=\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d})$
admits a perfect obstruction theory

${{\mathbb{E}}_{\operatorname{\mathrm{Q}}}=\operatorname{\mathrm{Cone}}\Biggl{(}\displaystyle\bigoplus_{i=1}^{l}\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\pi}({\mathcal{K}}_{i},{\mathcal{K}}_{i})}$${\displaystyle\bigoplus_{i=0}^{l-1}\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\pi}({\mathcal{K}}_{i+1},{\mathcal{K}}_{i})\Biggr{)}^{\vee}}$${{\mathbb{L}}_{\operatorname{\mathrm{Q}}}}$

###### Proof.

We prove the case $l=2$. The case $l>2$ is identical and the case $l=1$ is
well-known, but also a special case of the case $l=2$ (cf. Remark 3.3).

So set $l=2$. We need to fix some notation. For $i=1,2$, set
$\operatorname{\mathrm{Q}}_{i}=\operatorname{Quot}_{C}(E,s_{i},d_{i})$. Set
$B=B_{1}\times B_{2}$ where $B_{i}=\operatorname{Bun}_{C}(r-s_{i},\deg
E-d_{i})$ is the (smooth) algebraic stack of vector bundles on $C$, with the
indicated Chern character. For $i=1,2$, we have universal bundles
${\mathcal{U}}_{i}\in\operatorname{Coh}(B_{i}\times C)$ and we denote by

${B_{i}\times C}$${B_{i},}$${B\times C}$${B_{i},}$${B\times
C}$${B}$$\scriptstyle{q_{i}}$$\scriptstyle{\rho_{i}}$$\scriptstyle{p}$

the projections. Consider the map

${\operatorname{\mathrm{Q}}}$${B}$$\scriptstyle{h}$

sending $[K_{2}\hookrightarrow K_{1}\hookrightarrow E\twoheadrightarrow
T_{2}\twoheadrightarrow T_{1}]\mapstochar\rightarrow([K_{1}],[K_{2}])$. We
have

$\Omega_{B}=\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{q_{1}}({\mathcal{U}}_{1},{\mathcal{U}}_{1})^{\vee}[-1]\boxplus\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{q_{2}}({\mathcal{U}}_{2},{\mathcal{U}}_{2})^{\vee}[-1]$

and therefore

$h^{\ast}\Omega_{B}[1]=\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\pi}({\mathcal{K}}_{1},{\mathcal{K}}_{1})^{\vee}\oplus\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\pi}({\mathcal{K}}_{2},{\mathcal{K}}_{2})^{\vee}.$

Define the sheaf

$\displaystyle F$ $\displaystyle=\rho_{1}^{\ast}({\mathcal{U}}_{1}\otimes
E_{B_{1}}^{\ast}\otimes\omega_{q_{1}})\oplus\rho_{2}^{\ast}{\mathcal{U}}_{2}\otimes\rho_{1}^{\ast}{\mathcal{U}}_{1}^{\ast}\otimes\rho_{2}^{\ast}\omega_{q_{2}}$
$\displaystyle=\rho_{1}^{\ast}{\mathcal{U}}_{1}\otimes
E_{B}^{\ast}\otimes\omega_{p}\oplus\rho_{2}^{\ast}{\mathcal{U}}_{2}\otimes\rho_{1}^{\ast}{\mathcal{U}}_{1}^{\ast}\otimes\omega_{p}$

in $\operatorname{Coh}(B\times C)$. Then $\operatorname{\mathrm{Q}}$ is
open111This is because the locus in
$\operatorname{Hom}(K_{1},E)\oplus\operatorname{Hom}(K_{2},K_{1})=\mathrm{H}^{0}(C,K_{1}^{\ast}\otimes
E)\oplus\mathrm{H}^{0}(C,K_{2}^{\ast}\otimes
K_{1})=\mathrm{H}^{1}(C,K_{1}\otimes
E^{\ast}\otimes\omega_{C})^{\ast}\oplus\mathrm{H}^{1}(K_{2}\otimes
K_{1}^{\ast}\otimes\omega_{C})^{\ast}$ of pairs of injective maps is open. in
the total space of the abelian cone

${\mathbf{S}=\operatorname{Spec}_{\mathscr{O}_{B}}\operatorname{Sym}\operatorname{\mathbf{R}}^{1}p_{\ast}F}$${B,}$$\scriptstyle{\eta}$

and the map $h\colon\operatorname{\mathrm{Q}}\rightarrow B$ agrees with
$\eta|_{\operatorname{\mathrm{Q}}}\colon\operatorname{\mathrm{Q}}\hookrightarrow\mathbf{S}\rightarrow
B$.

We can thus form the diagram

(3.3) ${\operatorname{\mathrm{Q}}\times C}$${\mathbf{S}\times C}$${B\times
C}$${\operatorname{\mathrm{Q}}}$${\mathbf{S}}$${B}$$\scriptstyle{\square}$$\scriptstyle{\pi}$$\scriptstyle{h_{C}}$$\scriptstyle{\square}$$\scriptstyle{\overline{\eta}}$$\scriptstyle{\overline{p}}$$\scriptstyle{p}$$\scriptstyle{h}$$\scriptstyle{\mathrm{open}}$$\scriptstyle{\eta}$

and apply Lemma 3.1 to get a perfect obstruction
$\phi_{h}\colon{\mathbb{E}}_{h}\rightarrow{\mathbb{L}}_{h}$ theory relative to
$h\colon\operatorname{\mathrm{Q}}\rightarrow B$. Using the transitivity
triangle for cotangent complexes relative to
$h\colon\operatorname{\mathrm{Q}}\rightarrow B$, we find a morphism of exact
triangles

${{\mathbb{E}}_{\operatorname{\mathrm{Q}}}}$${{\mathbb{E}}_{h}}$${\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\pi}({\mathcal{K}}_{1},{\mathcal{K}}_{1})^{\vee}\oplus\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\pi}({\mathcal{K}}_{2},{\mathcal{K}}_{2})^{\vee}}$${{\mathbb{L}}_{\operatorname{\mathrm{Q}}}}$${{\mathbb{L}}_{h}}$${h^{\ast}\Omega_{B}[1]}$$\scriptstyle{\phi_{h}}$$\scriptstyle{\alpha}$

where we have set

${\mathbb{E}}_{\operatorname{\mathrm{Q}}}=\operatorname{\mathrm{Cone}}(\alpha)[-1]=\operatorname{\mathrm{Cone}}(\alpha^{\vee})^{\vee}.$

Note that

$\displaystyle{\mathbb{E}}_{h}$
$\displaystyle=\operatorname{\mathbf{R}}\pi_{\ast}(h_{C}^{\ast}F)[1]$
$\displaystyle=\operatorname{\mathbf{R}}\pi_{\ast}({\mathcal{K}}_{1}\otimes
E_{\operatorname{\mathrm{Q}}}^{\ast}\otimes\omega_{\pi})[1]\oplus\operatorname{\mathbf{R}}\pi_{\ast}({\mathcal{K}}_{2}\otimes{\mathcal{K}}_{1}^{\ast}\otimes\omega_{\pi})[1]$
$\displaystyle=\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\pi}({\mathcal{K}}_{1},E_{\operatorname{\mathrm{Q}}})^{\vee}\oplus\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\pi}({\mathcal{K}}_{2},{\mathcal{K}}_{1})^{\vee}.$

We used Grothendieck–Verdier duality for the last identity. This finishes the
proof. ∎

######  Remark 3.3.

The case $l=2$ also covers the case $l=1$, which corresponds to setting
$s_{1}=d_{1}=0$. In this case the hyperquot scheme is the classical Quot
scheme $\operatorname{Quot}_{C}(E,s_{2},d_{2})$ and its obstruction theory is

${\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\pi}({\mathcal{K}}_{2},{\mathcal{T}}_{2})^{\vee}}$${{\mathbb{L}}_{\operatorname{Quot}_{C}(E,s_{2},d_{2})}.}$

This has been already constructed by Marian–Oprea [40] for
$E=\mathscr{O}_{C}^{\oplus r}$ (but the proof generalises to arbitrary locally
free sheaves), Oprea [47], and Gillam [26, Lemma 4.7].

######  Remark 3.4.

The universal quotients
${\mathcal{T}}_{i}\in\operatorname{Coh}(\operatorname{\mathrm{Q}}\times C)$
realise subsequent quotients

${E_{\operatorname{\mathrm{Q}}}}$${{\mathcal{T}}_{l}}$${{\mathcal{T}}_{l-1}}$${\cdots}$${{\mathcal{T}}_{2}}$${{\mathcal{T}}_{1}.}$$\scriptstyle{p_{l}}$$\scriptstyle{p_{l-1}}$$\scriptstyle{p_{l-2}}$$\scriptstyle{p_{2}}$$\scriptstyle{p_{1}}$

By functoriality, for each $i=1,\ldots,l-1$ we obtain maps

${\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\pi}({\mathcal{K}}_{i},{\mathcal{T}}_{i})}$${\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\pi}({\mathcal{K}}_{i+1},{\mathcal{T}}_{i})}$${\delta\mapstochar\rightarrow\delta\circ\iota_{i}}$${\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\pi}({\mathcal{K}}_{i+1},{\mathcal{T}}_{i+1})}$${\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\pi}({\mathcal{K}}_{i+1},{\mathcal{T}}_{i})}$${\delta\mapstochar\rightarrow
p_{i}\circ\delta}$$\scriptstyle{u_{i}}$$\scriptstyle{v_{i}}$

in $\operatorname{Perf}(\operatorname{\mathrm{Q}})$, where
$\iota_{i}\colon{\mathcal{K}}_{i}\hookrightarrow{\mathcal{K}}_{i-1}$ are the
universal inclusions of (3.2). We obtain an exact triangle

(3.4)
${\Gamma[-1]}$${\displaystyle\bigoplus_{i=1}^{l}\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\pi}({\mathcal{K}}_{i},{\mathcal{T}}_{i})}$${\displaystyle\bigoplus_{i=1}^{l-1}\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\pi}({\mathcal{K}}_{i+1},{\mathcal{T}}_{i}).}$$\scriptstyle{\operatorname{\mathbf{R}}_{\pi}\Delta}$

where
$\Gamma=\operatorname{\mathrm{Cone}}(\operatorname{\mathbf{R}}_{\pi}\Delta)$.
The map $\operatorname{\mathbf{R}}_{\pi}\Delta$ can be seen as a universal
analogue of the map $\Delta_{z}$ in (2.5). Indeed, the sequence (2.5) is
obtained from $\operatorname{\mathbf{R}}_{\pi}\Delta$ by restricting (3.4) to
$z\in\operatorname{\mathrm{Q}}$ (see (3.5) below) and taking cohomology. The
complex

$\Gamma[-1]\in\mathbf{D}(\operatorname{\mathrm{Q}})$

is perfect in $[0,1]$, i.e. it belongs to
$\operatorname{Perf}^{[0,1]}(\operatorname{\mathrm{Q}})$, and has the
following two properties:

  * (i)

It has the same $K$-theory class as
${\mathbb{E}}_{\operatorname{\mathrm{Q}}}^{\vee}$, and

  * (ii)

$h^{0}(\Gamma[-1])=T_{\operatorname{\mathrm{Q}}}=h^{0}({\mathbb{E}}_{\operatorname{\mathrm{Q}}}^{\vee})$,
by the exact sequence (2.5), and because
${\mathbb{E}}_{\operatorname{\mathrm{Q}}}$ is a perfect obstruction theory.

It follows that
$h^{1}(\Gamma[-1])=h^{1}({\mathbb{E}}_{\operatorname{\mathrm{Q}}}^{\vee})$ as
well.

Let $z\in\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d})$ be a closed
point representing (2.3), with inclusion morphism $i_{z}$. Applying
$\operatorname{\mathbf{L}}i_{z}^{\ast}$ to (3.4) we obtain an exact triangle

(3.5)
${\Gamma_{z}[-1]}$${\displaystyle\bigoplus_{i=1}^{l}\operatorname{\mathbf{R}}\operatorname{Hom}(K_{i},T_{i})}$${\displaystyle\bigoplus_{i=1}^{l-1}\operatorname{\mathbf{R}}\operatorname{Hom}(K_{i+1},T_{i}),}$$\scriptstyle{\operatorname{\mathbf{R}}_{z}\Delta}$

where
$\Gamma_{z}=\operatorname{\mathrm{Cone}}(\operatorname{\mathbf{R}}_{z}\Delta)$.
On the other hand, recall the definition of the deformation functor
$\mathsf{Def}_{z}\subset\mathsf{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d})|_{\operatorname{Art}_{\mathbf{k}}}$,
sending a local artinian $\mathbf{k}$-algebra $A$ (with residue field
$\mathbf{k}$) to the set of $A$-valued points in
$\mathsf{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d})(\operatorname{Spec}A)$
restricting to $z$ over the closed point of $\operatorname{Spec}A$.

We then obtain the following corollary of Theorem 3.2, which we will use to
detect smoothness of hyperquot schemes in Section 4.

######  Corollary 3.5.

Let $z\in\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d})$ be a closed
point. Then the pair of $\mathbf{k}$-vector spaces

$T^{1}_{z}=h^{0}(\Gamma_{z}[-1]),\qquad T^{2}_{z}=h^{1}(\Gamma_{z}[-1])$

defines a tangent-obstruction theory on the deformation functor
$\mathsf{Def}_{z}$.

###### Proof.

Combine Theorem 3.2 and Remark 3.4 with one another. ∎

######  Remark 3.6.

Define

(3.6)
${\mathbb{F}}_{\operatorname{\mathrm{Q}}}=\Gamma^{\vee}[1]=\operatorname{\mathrm{Cone}}((\operatorname{\mathbf{R}}_{\pi}\Delta)^{\vee}).$

Given the evidence provided in Remark 3.4, we expect an equality of
_complexes_

${\mathbb{E}}_{\operatorname{\mathrm{Q}}}={\mathbb{F}}_{\operatorname{\mathrm{Q}}}.$

We will confirm functoriality properties of the virtual class
$[\operatorname{\mathrm{Q}}]^{\operatorname{\mathrm{vir}}}$, granting this
identity, in Appendix A.

###  3.3. Virtual structure

By the general machinery recalled in Section 3.1, hyperquot schemes on curves
admit natural virtual fundamental classes. In this subsection we give a
formula for the _virtual dimension_ of the perfect obstruction theory, namely
the rank of the complex ${\mathbb{E}}_{\operatorname{\mathrm{Q}}}$.

Recall that, with the conventions of Section 2.1, the dimension of the partial
flag variety is

$\displaystyle\dim\mathrm{Flag}(\boldsymbol{s},r)=\sum_{i=1}^{l}s_{i}(s_{i+1}-s_{i}).$

The following is a direct consequence of Theorem 3.2.

######  Corollary 3.7.

Let $C$ be a smooth projective curve of genus $g$, $E$ a locally free sheaf of
rank $r$, and fix tuples $\boldsymbol{s}=(s_{1}\leq\dots\leq s_{l})$ and
$\boldsymbol{d}=(d_{1},\dots,d_{l})$. The hyperquot scheme
$\operatorname{\mathrm{Q}}=\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d})$
admits a virtual fundamental class and a virtual structure sheaf

$\displaystyle[\operatorname{\mathrm{Q}}]^{\operatorname{\mathrm{vir}}}$
$\displaystyle\in\,A_{\operatorname{vdim}}(\operatorname{\mathrm{Q}}),$
$\displaystyle\mathscr{O}_{\operatorname{\mathrm{Q}}}^{\operatorname{\mathrm{vir}}}$
$\displaystyle\in\,K_{0}(\operatorname{\mathrm{Q}}),$

where the virtual dimension is

$\operatorname{vdim}=(1-g)\cdot\dim\mathrm{Flag}(\boldsymbol{s},r)+\sum_{i=1}^{l}d_{i}(s_{i+1}-s_{i-1})-\deg
E\cdot s_{l},$

where we set $s_{l+1}=r$ and $s_{0}=0$.

###### Proof.

Let

$[K_{l}\hookrightarrow\dots\hookrightarrow K_{1}\hookrightarrow
E\twoheadrightarrow T_{l}\twoheadrightarrow\cdots\twoheadrightarrow
T_{1}]\in\operatorname{\mathrm{Q}}$

be any closed point. Then, thanks to Remark 3.4, the virtual dimension is
computed by

$\operatorname{vdim}=\operatorname{rk}{\mathbb{E}}_{\operatorname{\mathrm{Q}}}=\sum_{i=1}^{l}\operatorname{rk}\operatorname{\mathbf{R}}\operatorname{Hom}(K_{i},T_{i})-\sum_{i=1}^{l-1}\operatorname{rk}\operatorname{\mathbf{R}}\operatorname{Hom}(K_{i+1},T_{i}).$

By Riemann–Roch, for each $i,j=1,\dots,l$ we have

$\operatorname{rk}\operatorname{\mathbf{R}}\operatorname{Hom}(K_{j},T_{i})=r_{j}s_{i}(1-g)+r_{j}d_{i}+s_{i}(d_{j}-\deg
E),$

which implies

$\displaystyle\operatorname{vdim}$
$\displaystyle=\sum_{i=1}^{l}\left(r_{i}s_{i}(1-g)+r_{i}d_{i}+s_{i}(d_{i}-\deg
E)\right)-\sum_{i=1}^{l-1}\left(r_{i+1}s_{i}(1-g)+r_{i+1}d_{i}+s_{i}(d_{i+1}-\deg
E)\right)$
$\displaystyle=(1-g)\sum_{i=1}^{l}s_{i}(s_{i+1}-s_{i})-s_{l}\cdot\deg
E+\sum_{i=1}^{l}s_{i+1}d_{i}-\sum_{i=1}^{l-1}s_{i}d_{i+1}$
$\displaystyle=(1-g)\cdot\dim\mathrm{Flag}(\boldsymbol{s},r)-s_{l}\cdot\deg
E+\sum_{i=1}^{l}d_{i}(s_{i+1}-s_{i-1}),$

as required. ∎

######  Remark 3.8.

Assume $g>1$, $\deg E=0$ and $d_{i}=0$ for all $i=1,\ldots,l$. Then
$\operatorname{\mathrm{Q}}=\operatorname{Quot}_{C}(E,\boldsymbol{s},\mathbf{0})$
has negative virtual dimension, namely

$\operatorname{vdim}=(1-g)\cdot\dim\mathrm{Flag}(\boldsymbol{s},r)<0.$

######  Remark 3.9.

It was recently proved in [51] that, for $l=2$ and the multidegree
$\boldsymbol{d}$ large enough, the hyperquot scheme
$\operatorname{\mathrm{Q}}=\operatorname{Quot}_{C}(E,\boldsymbol{s},\mathbf{\boldsymbol{d}})$
is irreducible, local complete intersection and that its dimension coincides
with our formula for the virtual dimension, which implies that the virtual
fundamental class coincides with the ordinary fundamental class. See also [50]
for the case of ordinary Quot schemes.

##  4\. Smoothness of hyperquot schemes

We exploit the tangent-obstruction theory of Corollary 3.5 to discuss the
smoothness of the hyperquot scheme
$\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d})$, where $E$ is a
locally free sheaf over a smooth projective curve $C$, as in Section 2.4.
Notice that if $\operatorname{rk}E=1$, then
$\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d})$ is isomorphic to a
product of symmetric products of $C$, and therefore it is always smooth and
irreducible.

###  4.1. The case of 0-dimensional quotients

The smoothness in the case of $0$-dimensional quotients was obtained in [43,
44] by a direct computation. We provide here an alternative proof.

######  Proposition 4.1.

Let $C$ be a smooth projective curve and $E$ a locally free sheaf on $C$. Then
for all $\boldsymbol{d}$ the hyperquot scheme
$\operatorname{Quot}_{C}(E,\mathbf{0},\boldsymbol{d})$ is smooth and
unobstructed.

###### Proof.

Let

$z=[K_{l}\hookrightarrow\dots\hookrightarrow K_{1}\hookrightarrow
E\twoheadrightarrow T_{l}\twoheadrightarrow\cdots\twoheadrightarrow
T_{1}]\in\operatorname{Quot}_{C}(E,\mathbf{0},\boldsymbol{d})$

be any point and consider the exact sequences

${0}$${K_{i}}$${K_{i-1}}$${Q_{i}}$${0}$${0}$${Q_{i}}$${T_{i}}$${T_{i-1}}$${0}$

for every $i=1,\dots,l$. Since $Q_{i},T_{i}$ are $0$-dimensional sheaves and
$K_{i}$ is locally free, we obtain the vanishings

$\displaystyle\operatorname{Ext}^{j}_{C}(K_{i},T_{i})=\operatorname{Ext}^{j}_{C}(K_{i+1},T_{i})=\operatorname{Ext}^{j}_{C}(K_{i},Q_{i})=0,\quad
j>0,$

which implies that $\Delta_{z}$, the map introduced in (2.5), is surjective.
Therefore, taking the long exact sequence in cohomology attached to (3.4) we
obtain

${h^{1}(\Gamma_{z}[-1])}$${\displaystyle\bigoplus_{i=1}^{l}\operatorname{Ext}^{1}_{C}(K_{i},T_{i})=0,}$

by which we conclude that the obstruction space $h^{1}(\Gamma_{z}[-1])$
vanishes as well. ∎

###  4.2. Smoothness in the unnested case

We start discussing criteria for smoothness of the ordinary Quot scheme, i.e.
without nesting conditions.

######  Proposition 4.2.

Fix $r\geq 2$ and $0<s\leq r$. Let $C$ be a smooth projective curve of genus
$g$, and let $E=\bigoplus_{\alpha=1}^{r}L_{\alpha}$ be a split locally free
sheaf on $C$. Assume that

(4.1) $\displaystyle\mathrm{H}^{0}(C,L_{\alpha}\otimes
L_{\beta}^{-1}\otimes\omega_{C})=0,\qquad 1\leq\alpha\neq\beta\leq r.$

Then, for all $d\in{\mathbb{Z}}$, the Quot scheme
$\operatorname{Quot}_{C}(E,s,d)$ is smooth and unobstructed, and $g\leq 1$.

###### Proof.

In Section 5.1 we describe a $\mathbf{T}$-action on
$\operatorname{Quot}_{C}(E,s,d)$, where $\mathbf{T}={\mathbb{G}}_{m}^{r}$,
such that all the torus fixed point are of the form

$\bigoplus_{\alpha=1}^{r}\,[K_{\alpha}\hookrightarrow
L_{\alpha}\twoheadrightarrow
T_{\alpha}]\in\operatorname{Quot}_{C}(E,s,d)^{\mathbf{T}}.$

For such a torus fixed point, by Serre duality its obstruction space satisfies

$\operatorname{Ext}^{1}_{C}\left(\bigoplus_{\alpha=1}^{r}K_{\alpha},\bigoplus_{\beta=1}^{r}T_{\beta}\right)=\bigoplus_{\alpha,\beta=1}^{r}\mathrm{H}^{0}(C,K_{\alpha}\otimes
T_{\beta}^{\mathrm{fr},*}\otimes\omega_{C})^{*},$

where $T_{\beta}^{\mathrm{fr}}$ denotes the free part of $T_{\beta}$. Now, for
each $\alpha=1,\dots,r$ we have that $(K_{\alpha},T_{\alpha})$ is either
$(0,L_{\alpha})$ or
$(L_{\alpha}\otimes\mathscr{O}_{C}(-Z_{\alpha}),\mathscr{O}_{Z_{\alpha}})$,
for a suitable closed subscheme $Z_{\alpha}\subset C$ (see (5.2)). Denote by
$J$ the collection of indices $\alpha$ such that
$(K_{\alpha},T_{\alpha})=(0,L_{\alpha})$. Therefore we have

$\displaystyle\bigoplus_{\alpha,\beta=1}^{r}\mathrm{H}^{0}(C,K_{\alpha}\otimes
T_{\beta}^{\mathrm{fr},*}\otimes\omega_{C})$
$\displaystyle=\bigoplus_{\begin{subarray}{c}\alpha\notin J\\\ \beta\in
J\end{subarray}}\mathrm{H}^{0}(C,L_{\alpha}\otimes
L_{\beta}^{-1}\otimes\mathscr{O}_{C}(-Z_{\alpha})\otimes\omega_{C})$
$\displaystyle\hookrightarrow\bigoplus_{\begin{subarray}{c}\alpha\notin J\\\
\beta\in J\end{subarray}}\mathrm{H}^{0}(C,L_{\alpha}\otimes
L_{\beta}^{-1}\otimes\omega_{C})=0.$

The vanishing of the obstruction space implies that all torus fixed points are
smooth and unobstructed, which implies that all points are smooth and
unobstructed as well222This follows by the fact that, in our case, the
dimension of the obstruction space is upper semi-continuous, and that the
closure of each torus orbit contains a torus fixed point..

Assume now by contradiction that $g\geq 2$. Then (4.1) implies that for all
$\alpha\neq\beta$ we have

$\displaystyle\mathrm{H}^{1}(L_{\alpha}\otimes
L_{\beta}^{-1})=\mathrm{H}^{0}(L_{\alpha}^{-1}\otimes
L_{\beta}\otimes\omega_{C})^{*}=0,$
$\displaystyle\mathrm{H}^{0}(L_{\alpha}\otimes
L_{\beta}^{-1})\hookrightarrow\mathrm{H}^{0}(L_{\alpha}\otimes
L_{\beta}^{-1}\otimes\omega_{C})=0,$

which by Riemann–Roch implies that

$\deg L_{\alpha}-\deg L_{\beta}=g-1,$

which is satisfied for all $\alpha\neq\beta$ only if $g=1$. ∎

We give some nontrivial instances of the vanishing (4.1).

######  Example 4.3.

Let $C\cong{\mathbb{P}}^{1}$, then all locally free sheaves admit a splitting
$E=\bigoplus_{\alpha=1}^{r}L_{\alpha}$, and (4.1) translates into

$\deg L_{\alpha}-\deg L_{\beta}\leq 1,$

for all $\alpha\neq\beta$, i.e. $E=L\otimes\left(\mathscr{O}^{\oplus
n}\oplus\mathscr{O}(1)^{\oplus m}\right)$ where $L$ is any line bundle.

Let $C$ be an elliptic curve and
$E=\bigoplus_{\alpha=1}^{r}\mathscr{O}_{C}(p_{\alpha})$, where $p_{\alpha}\in
C$ are distinct points. Then for each $\alpha\neq\beta$ the line bundles
$\mathscr{O}_{C}(p_{\alpha}-p_{\beta})$ have no global sections, which implies
that $(C,E)$ satifies (4.1) and the corresponding Quot scheme is smooth.

###  4.3. Smoothness in the nested case

In the nested case, we show that there are non-trivial examples of smooth and
unobstructed hyperquot schemes at least in genus 0.

######  Theorem 4.4.

Let $L$ be any line bundle on ${\mathbb{P}}^{1}$ and
$E=L\otimes\left(\mathscr{O}^{\oplus n}\oplus\mathscr{O}(1)^{\oplus m}\right)$
for some $n,m$. Then the hyperquot scheme
$\operatorname{Quot}_{{\mathbb{P}}^{1}}(E,\boldsymbol{s},\boldsymbol{d})$ is
smooth and unobstructed for all $\boldsymbol{d}$ and for all $\boldsymbol{s}$.

###### Proof.

The proof is similar to the ones of Propositions 4.1, 4.2. Without loss of
generality, we just need to prove our claim for a torus fixed point

$z=\bigoplus_{\alpha=1}^{r}\left(K_{l,\alpha}\hookrightarrow\cdots\hookrightarrow
K_{1,\alpha}\hookrightarrow L_{\alpha}\twoheadrightarrow
T_{l,\alpha}\twoheadrightarrow\dots\twoheadrightarrow
T_{1,\alpha}\right)\,\in\,\operatorname{Quot}_{{\mathbb{P}}^{1}}(E,\boldsymbol{s},\boldsymbol{d})^{\mathbf{T}}.$

Set $E=\bigoplus_{\alpha}L_{\alpha}$. Consider the exact sequences

${0}$${K_{i,\alpha}}$${K_{i-1,\alpha}}$${Q_{i,\alpha}}$${0}$${0}$${Q_{i,\alpha}}$${T_{i,\alpha}}$${T_{i-1,\alpha}}$${0}$

for every $i=1,\dots,l$ and $\alpha=1,\dots,r$. Each $Q_{i,\alpha}$ either
satisfies $Q_{i,\alpha}\cong L_{\alpha}$ or is $0$-dimensional. Since
$K_{i,\alpha}$ is locally free, by Serre duality we obtain the vanishings

$\displaystyle\operatorname{Ext}^{p}_{C}(K_{i,\alpha},T_{j,\beta})=\operatorname{Ext}^{p}_{C}(K_{i+1,\alpha},T_{j,\beta})=\operatorname{Ext}^{p}_{C}(K_{i,\alpha},Q_{j,\beta})=0,\quad
p>0,$

which implies that $\Delta_{z}$, the map introduced in (2.5) is surjective.
Therefore the obstruction space satisfies

${h^{1}(\Gamma_{z}[-1])}$${\displaystyle\bigoplus_{i=1}^{l}\operatorname{Ext}^{1}_{{\mathbb{P}}^{1}}(K_{i},T_{i})=0,}$

by which we conclude that the obstruction space $h^{1}(\Gamma_{z}[-1])$
vanishes as well. ∎

In general, there are examples of smooth yet obstructed hyperquot schemes.

######  Example 4.5.

Set $E=\mathscr{O}_{C}^{\oplus r}$ and $\boldsymbol{d}=\mathbf{0}$. Then for
all $\boldsymbol{s}$, the hyperquot scheme is isomorphic to the partial flag
variety $\operatorname{Quot}_{C}(\mathscr{O}_{C}^{\oplus
r},\boldsymbol{s},\mathbf{0})\cong\mathrm{Flag}(\boldsymbol{s},r)$, but by
Theorem 3.2 it is endowed with a perfect obstruction theory of virtual
dimension $(1-g(C))\cdot\dim\mathrm{Flag}(\boldsymbol{s},r)$, which implies
that it is smooth but obstructed as soon as $g(C)\geq 1$.

##  5\. Torus fixed locus

###  5.1. Torus action

We fix $\boldsymbol{s}=(s_{1}\leq\dots\leq s_{l})$ an $l$-tuple of non-
decreasing non-negative integers and $\boldsymbol{d}=(d_{1},\dots,d_{l})$ an
$l$-tuple of integers. Let $E$ be a locally free sheaf of rank $r>0$ on a
smooth projective curve $C$. Assume that
$E=\bigoplus_{\alpha=1}^{r}L_{\alpha}$ splits into a sum of line bundles on
$C$. Then $\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d})$ admits
the action of the algebraic torus $\mathbf{T}={\mathbb{G}}_{m}^{r}$ as in [6].
Indeed, $\mathbf{T}$ acts diagonally on the product
$\prod_{i=1}^{l}\operatorname{Quot}_{C}(E,s_{i},d_{i})$ and the closed
subscheme $\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d})$ is
$\mathbf{T}$-invariant. Its fixed locus is determined by a straightforward
generalisation of the main result of [6], whose proof is analogous to [43,
Prop. 3.1] (see also [10, Sec. 3.1]).

######  Lemma 5.1.

If $E=\bigoplus_{\alpha=1}^{r}L_{\alpha}$, there is a scheme-theoretic
identity

$\displaystyle\operatorname{Quot}_{C}\left(E,\boldsymbol{s},\boldsymbol{d}\right)^{\mathbf{T}}=\coprod_{\boldsymbol{s}=\boldsymbol{s}_{1}+\dots+\boldsymbol{s}_{r}}\coprod_{\boldsymbol{d}=\boldsymbol{d}_{1}+\dots+\boldsymbol{d}_{r}}\prod_{\alpha=1}^{r}\operatorname{Quot}_{C}(L_{\alpha},\boldsymbol{s}_{\alpha},\boldsymbol{d}_{\alpha}).$

In other words, Lemma 5.1 says that all torus fixed points

$\displaystyle[K_{l}\hookrightarrow\cdots\hookrightarrow K_{1}\hookrightarrow
E\twoheadrightarrow T_{l}\twoheadrightarrow\dots\twoheadrightarrow
T_{1}]\,\in\,\operatorname{Quot}_{C}\left(E,\boldsymbol{s},\boldsymbol{d}\right)^{\mathbf{T}}$

are of the form

(5.1)
$\displaystyle\bigoplus_{\alpha=1}^{r}\left(K_{l,\alpha}\hookrightarrow\cdots\hookrightarrow
K_{1,\alpha}\hookrightarrow L_{\alpha}\twoheadrightarrow
T_{l,\alpha}\twoheadrightarrow\dots\twoheadrightarrow
T_{1,\alpha}\right)\,\in\,\prod_{\alpha=1}^{r}\operatorname{Quot}_{C}\left(L_{\alpha},\boldsymbol{s}_{\alpha},\boldsymbol{d}_{\alpha}\right),$

for some suitable $\boldsymbol{s}_{\alpha},\boldsymbol{d}_{\alpha}$ for
$\alpha=1,\dots,r$. Notice that for each pair $(K_{i,\alpha},T_{i,\alpha})$
there are two possibilities. Namely, we have

(5.2)
$\displaystyle\begin{cases}K_{i,\alpha}=L_{\alpha}\otimes\mathscr{O}_{C}(-Z_{i,\alpha})\\\
T_{i,\alpha}=\mathscr{O}_{Z_{i,\alpha}}\end{cases}\mbox{ or
}\quad\begin{cases}K_{i,\alpha}=0\\\ T_{i,\alpha}=L_{\alpha}\end{cases}$

for some 0-dimensional closed subschemes $Z_{i,\alpha}\subset C$, satisfying
the required nesting conditions. In particular, this implies that not all
possible choices of splittings
$\boldsymbol{s}=\sum_{\alpha=1}^{r}\boldsymbol{s}_{\alpha}$ and
$\boldsymbol{d}=\sum_{\alpha=1}^{r}\boldsymbol{d}_{\alpha}$ contribute to the
connected components of
$\operatorname{Quot}_{C}\left(E,\boldsymbol{s},\boldsymbol{d}\right)^{\mathbf{T}}$.
In the next section we offer a convenient combinatorial description of the
actual connected components appearing in
$\operatorname{Quot}_{C}\left(E,\boldsymbol{s},\boldsymbol{d}\right)^{\mathbf{T}}$.

###  5.2. Combinatorial description of the fixed locus

We keep the notation as in Section 5.1. We give a combinatorial
characterisation of the connected components appearing in the fixed locus of
Lemma 5.1, following a description inspired by [10, Sec. 3.1]. Given the non-
decreasing tuple $\boldsymbol{s}=(s_{1}\leq\dots\leq s_{l})$, define a new
tuple $\boldsymbol{r}=(r_{1}\geq\dots\geq r_{l})$ by $r_{i}=r-s_{i}$ for
$i=1,\dots,l$. For convenience, we also set $r_{0}=r$ and $r_{l+1}=0$.

####  5.2.1. The unnested case

We start with the easier case of $l=1$, i.e. no nesting conditions. By Lemma
5.1 and (5.2), a fixed point as in (5.1) decomposes as

${\displaystyle\bigoplus_{\alpha=1}^{r_{1}}L_{c_{1,\alpha}}(-Z_{1,\alpha})}$${\displaystyle\bigoplus_{\alpha=1}^{r}L_{\alpha}}$${\displaystyle\bigoplus_{\alpha=1}^{r_{1}}\mathscr{O}_{Z_{1,\alpha}}\oplus\bigoplus_{\begin{subarray}{c}\beta\neq
c_{1,\alpha}\\\
\alpha=1,\dots,r_{1}\end{subarray}}L_{\beta}}$${K_{1}}$${E}$${T_{1}}$

for some closed 0-dimensional subschemes $Z_{1,\alpha}\subset C$ and some
tuple of integers

(5.3) $\displaystyle 1\leq c_{1,1}<\dots<c_{1,r_{1}}\leq r,$

such that the inclusion of $K_{1}\hookrightarrow E$ is “component-wise”, i.e.
each summand $\mathscr{O}_{Z_{1},\alpha}$ is a quotient of $L_{c_{1,\alpha}}$.
In other words, the tuple (5.3) selects exactly $r_{1}$ nonzero components
(out of the $r$ possible ones) of $K_{1}$.

Clearly, all tuples of the form (5.3) are in bijection with elements333By
convention, we represent elements of $S_{r}$ as bijections
$\sigma\colon\set{1,\dots,r}\rightarrow\set{1,\dots,r}.$ $\sigma\in S_{r}$,
i.e. permutations of $r$ elements, with the condition that

$\displaystyle\sigma(\alpha)<\sigma(\beta)\quad\mbox{ if }\quad
1\leq\alpha<\beta\leq r_{1}\quad\mbox{ or }\quad r_{1}+1\leq\alpha<\beta\leq
r_{0}=r,$

by declaring $c_{1,\alpha}=\sigma(\alpha)$. We denote the collection of such
permutations by $P_{r_{1}}$.

Therefore, the fixed locus decomposes as

(5.4)
$\displaystyle\begin{split}\operatorname{Quot}_{C}\left(E,s_{1},d_{1}\right)^{\mathbf{T}}&=\coprod_{\sigma\in
P_{r_{1}}}\coprod_{\begin{subarray}{c}d_{1}=\sum_{\alpha=1}^{r_{1}}n_{1,\alpha}+\\\
\sum_{\alpha=r_{1}+1}^{r}\deg
L_{\sigma(\alpha)}\end{subarray}}\prod_{\alpha=1}^{r_{1}}\operatorname{Quot}_{C}\left(L_{\sigma(\alpha)},0,n_{1,\alpha}\right)\\\
&\cong\coprod_{\sigma\in
P_{r_{1}}}\coprod_{\begin{subarray}{c}d_{1}=\sum_{\alpha=1}^{r_{1}}n_{1,\alpha}+\\\
\sum_{\alpha=r_{1}+1}^{r}\deg
L_{\sigma(\alpha)}\end{subarray}}\prod_{\alpha=1}^{r_{1}}C^{(n_{1,\alpha})},\end{split}$

where in the last line we used the isomorphism

$\operatorname{Quot}_{C}\left(L_{\sigma(\alpha)},0,n_{1,\alpha}\right)\cong
C^{(n_{1,\alpha})}.$

####  5.2.2. The nested case

If the lenght of the nestings is $l\geq 1$, the flag of the kernels of a fixed
point as in (5.1) is of the form

(5.5) ${\displaystyle\bigoplus_{1\leq\alpha\leq
r_{l}}L_{c_{l,\alpha}}(-Z_{l,\alpha})}$${\displaystyle\bigoplus_{\begin{subarray}{c}l-1\leq
j\leq l\\\ r_{j+1}+1\leq\alpha\leq
r_{j}\end{subarray}}L_{c_{j,\alpha}}(-Z_{l-1,\alpha})}$${\cdots}$${\displaystyle\bigoplus_{\begin{subarray}{c}1\leq
j\leq l\\\ r_{j+1}+1\leq\alpha\leq
r_{j}\end{subarray}}L_{c_{j,\alpha}}(-Z_{1,\alpha})}$${E}$${K_{l}}$${K_{l-1}}$${K_{1}}$

for some closed 0-dimensional subschemes $Z_{j,\alpha}\subset C$ and some
tuples of integers

(5.6) $\displaystyle\begin{split}&1\leq c_{l,1}<\dots<c_{l,r_{l}}\leq r,\\\
&1\leq c_{l-1,r_{l}+1}<\dots<c_{l-1,r_{l-1}}\leq r,\\\ &\vdots\\\ &1\leq
c_{1,r_{2}+1}<\dots<c_{1,r_{1}}\leq r,\end{split}$

where the collections of indices $\set{c_{j,r_{j+1}+1},\dots,c_{j,r_{j}}}$ are
pairwise disjoint, for all $j=1,\dots,l$.

As for the case of no nestings, each inclusion $K_{j+1}\hookrightarrow K_{j}$
is “component-wise”, i.e. each $\mathscr{O}_{Z_{j},\alpha}$ is a quotient of
$L_{c_{j,\alpha}}$ (as a summand of $E$). In other words, the tuples in (5.6)
select which components of $K_{j}$ are nonzero.

Again, as for the case $l=1$, all tuples of the form (5.6) are in bijection
with elements $\sigma\in S_{r}$, with the condition that

$\displaystyle\sigma(\alpha)<\sigma(\beta)\quad\mbox{ if }\quad
r_{j+1}+1\leq\alpha<\beta\leq r_{j}\quad\mbox{ for }\quad j=0,\dots,l,$

by declaring $c_{j,\alpha}=\sigma(\alpha)$ whenever $r_{j+1}+1\leq\alpha\leq
r_{j}$ for $j=1,\dots,l$. We denote the collection of such permutations by
$P_{\boldsymbol{r}}$. Therefore (5.5) can be rewritten more compactly as

${\displaystyle\bigoplus_{\alpha=1}^{r_{l}}L_{\sigma(\alpha)}(-Z_{l,\alpha})}$${\displaystyle\bigoplus_{\alpha=1}^{r_{l-1}}L_{\sigma(\alpha)}(-Z_{l-1,\alpha})}$${\cdots}$${\displaystyle\bigoplus_{\alpha=1}^{r_{1}}L_{\sigma(\alpha)}(-Z_{1,\alpha})}$${E}$${K_{l}}$${K_{l-1}}$${K_{1}}$

and the fixed locus decomposes as

(5.7)
$\displaystyle\begin{split}\operatorname{Quot}_{C}\left(E,\boldsymbol{s},\boldsymbol{d}\right)^{\mathbf{T}}&=\coprod_{\sigma\in
P_{\boldsymbol{r}}}\coprod_{\begin{subarray}{c}(\boldsymbol{n}_{\alpha})_{\alpha},\\\
d_{j}=\sum_{\alpha=1}^{r_{j}}n_{j,\alpha}+\\\ \sum_{\alpha=r_{j}+1}^{r}\deg
L_{\sigma(\alpha)}\end{subarray}}\prod_{\alpha=1}^{r_{1}}\operatorname{Quot}_{C}\left(L_{\sigma(\alpha)},\mathbf{0},\boldsymbol{n}_{\alpha}\right)\\\
&\cong\coprod_{\sigma\in
P_{\boldsymbol{r}}}\coprod_{\begin{subarray}{c}(\boldsymbol{n}_{\alpha})_{\alpha},\\\
d_{j}=\sum_{\alpha=1}^{r_{j}}n_{j,\alpha}+\\\ \sum_{\alpha=r_{j}+1}^{r}\deg
L_{\sigma(\alpha)}\end{subarray}}\prod_{\alpha=1}^{r_{1}}C^{[\boldsymbol{n}_{\alpha}]}.\end{split}$

Here, the second disjoint union is over all collection of tuples
$(\boldsymbol{n}_{\alpha})_{\alpha=1,\dots,r_{1}}$ where for each
$r_{j+1}+1\leq\alpha\leq r_{j}$, we have
$\boldsymbol{n}_{\alpha}=(n_{1,\alpha}\leq\dots\leq n_{j,\alpha})$, and in the
last equality we used that

$\operatorname{Quot}_{C}\left(L_{\sigma(\alpha)},\mathbf{0},\boldsymbol{n}_{\alpha}\right)\cong
C^{[\boldsymbol{n}_{\alpha}]}.$

Notice also that, if $r_{j+1}+1\leq\alpha\leq r_{j}$, then

$\displaystyle
C^{[\boldsymbol{n}_{\alpha}]}\cong\prod_{i=1}^{j}C^{(n_{i,\alpha}-n_{i-1,\alpha})},$

where we set $n_{0,\alpha}=0$ for convenience.

###  5.3. Topological Euler characteristic

Consider $\mathbf{k}={\mathbb{C}}$ for this section. Let
$\boldsymbol{s}=(s_{1}\leq\dots\leq s_{l})$ be a non-decreasing tuple. Let $C$
be a smooth projective curve of genus $g$ and
$E=\bigoplus_{\alpha=1}^{r}L_{\alpha}$ a split locally free sheaf. We compute
the generating series of topological Euler characteristics

$\mathsf{Z}^{\mathrm{top}}_{C,E,\boldsymbol{s}}(\boldsymbol{q})=\sum_{\boldsymbol{d}}\,e_{\mathrm{top}}(\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d}))\boldsymbol{q}^{\boldsymbol{d}}\in{\mathbb{Z}}(\\!(q_{1},\dots,q_{l})\\!),$

without any smoothness assumption.

######  Theorem 5.2.

There is an identity

$\displaystyle\mathsf{Z}^{\mathrm{top}}_{C,E,\boldsymbol{s}}(\boldsymbol{q})$
$\displaystyle=\left(\sum_{\sigma\in
P_{\boldsymbol{r}}}\prod_{j=1}^{l}q_{j}^{\sum_{\alpha=r_{j}+1}^{r}\deg
L_{\sigma(\alpha)}}\right)\cdot\prod_{1\leq i\leq j\leq l}(1-q_{i}\cdots
q_{j})^{(2g-2)(r_{j}-r_{j+1})}.$

###### Proof.

The topological Euler characteristic of a space with a torus action coincides
with the topological Euler characteristic of its torus fixed locus. Therefore
by (5.7) we have

$\displaystyle\mathsf{Z}^{\mathrm{top}}_{C,E,\boldsymbol{s}}(\boldsymbol{q})$
$\displaystyle=\sum_{\boldsymbol{d}}\boldsymbol{q}^{\boldsymbol{d}}\sum_{\sigma\in
P_{\boldsymbol{r}}}\sum_{\begin{subarray}{c}\boldsymbol{n}=(\boldsymbol{n}_{\alpha})_{\alpha},\\\
d_{j}=\sum_{\alpha=1}^{r_{j}}n_{j,\alpha}+\\\ \sum_{\alpha=r_{j}+1}^{r}\deg
L_{\sigma(\alpha)}\end{subarray}}\prod_{\alpha=1}^{r_{1}}e_{\mathrm{top}}(C^{[\boldsymbol{n}_{\alpha}]})$
$\displaystyle=\sum_{\sigma\in
P_{\boldsymbol{r}}}\prod_{j=1}^{l}q_{j}^{\sum_{\alpha=r_{j}+1}^{r}\deg
L_{\sigma(\alpha)}}\prod_{j=1}^{l}\prod_{\alpha=r_{j+1}+1}^{r_{j}}\sum_{\boldsymbol{n}_{\alpha}}q_{1}^{n_{1,\alpha}}\cdots
q_{j}^{n_{j,\alpha}}e_{\mathrm{top}}(C^{[\boldsymbol{n}_{\alpha}]}).$

By [43, Cor. 4.5], we have

$\displaystyle\sum_{\boldsymbol{n}_{\alpha}}q_{1}^{n_{1,\alpha}}\cdots
q_{j}^{n_{j,\alpha}}e_{\mathrm{top}}(C^{[\boldsymbol{n}_{\alpha}]})=\prod_{i=1}^{j}\,(1-q_{i}\cdots
q_{j})^{2g-2},$

by which we conclude that

$\displaystyle\mathsf{Z}^{\mathrm{top}}_{C,E,\boldsymbol{s}}(\boldsymbol{q})$
$\displaystyle=\sum_{\sigma\in
P_{\boldsymbol{r}}}\prod_{j=1}^{l}q_{j}^{\sum_{\alpha=r_{j}+1}^{r}\deg
L_{\sigma(\alpha)}}\prod_{1\leq i\leq j\leq l}(1-q_{i}\cdots
q_{j})^{(2g-2)(r_{j}-r_{j+1})}.\qed$

Setting $C={\mathbb{P}}^{1}$ and $E=\mathscr{O}_{{\mathbb{P}}^{1}}^{\oplus r}$
in the previous theorem one recovers Chen’s result [10, Thm 3].

##  6\. Motives of hyperquot schemes

###  6.1. Grothendieck ring of varieties

The _Grothendieck ring of $\mathbf{k}$-varieties_, denoted
$K_{0}(\operatorname{Var}_{\mathbf{k}})$, is the free abelian group generated
by isomorphism classes $[X]$ of $\mathbf{k}$-varieties, modulo the scissor
relations, namely the identities $[X]=[Z]+[X\setminus Z]$ whenever
$Z\hookrightarrow X$ is a closed $\mathbf{k}$-subvariety of $X$. The ring
structure is defined by fibre product. The ring identity is
$[\operatorname{Spec}\mathbf{k}]$. The main rules for calculations in
$K_{0}(\operatorname{Var}_{\mathbf{k}})$ are the following:

  1. (1)

If $X\rightarrow Y$ is a geometric bijection, i.e. a bijective morphism, then
$[X]=[Y]$.

  2. (2)

If $X\rightarrow Y$ is Zariski locally trivial with fibre $F$, then
$[X]=[Y]\cdot[F]$.

These are, indeed, the only properties that we will use. Recall that the
_Lefschetz motive_ is defined to be the class
${\mathbb{L}}=[{\mathbb{A}}^{1}_{\mathbf{k}}]\in
K_{0}(\operatorname{Var}_{\mathbf{k}})$.

###  6.2. Białynicki-Birula decomposition for hyperquot schemes

We recall the classical result of Białynicki-Birula.

######  Theorem 6.1 (Białynicki-Birula [5, Sec. 4]).

Let $X$ be a smooth projective scheme with a ${\mathbb{G}}_{m}$-action and let
$\set{X_{i}}_{i}$ be the connected components of the ${\mathbb{G}}_{m}$-fixed
locus $X^{{\mathbb{G}}_{m}}\subset X$. Then there exists a locally closed
stratification $X=\coprod_{i}X^{+}_{i}$, such that each $X^{+}_{i}\rightarrow
X_{i}$ is an affine fibre bundle. Moreover, for every closed point $x\in
X_{i}$, the tangent space is given by
$T_{x}(X^{+}_{i})=T_{x}(X)^{\mathsf{fix}}\oplus T_{x}(X)^{+}$, where
$T_{x}(X)^{\mathsf{fix}}$ (resp. $T_{x}(X)^{+}$) denotes the
${\mathbb{G}}_{m}$-fixed (resp. positive) part of $T_{x}(X)$. In particular,
the relative dimension of $X^{+}_{i}\rightarrow X_{i}$ is equal to $\dim
T_{x}(X)^{+}$ for $x\in X_{i}$.

We now determine a Białynicki-Birula decomposition for
$\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d})$ whenever it is
smooth and unobstructed.

Let ${\mathbb{G}}_{m}\hookrightarrow\mathbf{T}$ be the generic 1-parameter
subtorus given by $w\mapstochar\rightarrow(w,w^{2},\dots,w^{r})$; it is clear
that
$\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d})^{\mathbf{T}}=\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d})^{{\mathbb{G}}_{m}}$.
Let

$Q_{\sigma,\boldsymbol{n}}=\prod_{\alpha=1}^{r_{1}}C^{[\boldsymbol{n}_{\alpha}]}\subset\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d})^{{\mathbb{G}}_{m}}$

be the connected component of the fixed locus corresponding to the data
$\sigma,\boldsymbol{n}=(\boldsymbol{n}_{\alpha})_{\alpha}$ as in (5.4).

Recall that we set $\boldsymbol{r}=(r_{1}\geq\dots\geq r_{l})$, where
$r_{i}=r-s_{i}$ and we set $r_{0}=r$ and $r_{l+1}=0$ for convenience. For a
fixed permutation $\sigma\in P_{\boldsymbol{r}}$, define for all
$\alpha,\beta=1,\dots,r$

$\epsilon^{\sigma}_{\alpha,\beta}=\begin{cases}1&\sigma(\beta)>\sigma(\alpha),\\\
0&\sigma(\beta)\leq\sigma(\alpha).\end{cases}$

Define moreover, for all $i=1,\dots,l+1$ and $\alpha=1,\dots,r$

(6.1)
$\begin{split}h^{\sigma}_{0}(i,\alpha)&=\sum_{\beta=r_{i}+1}^{r_{i-1}}\epsilon^{\sigma}_{\alpha,\beta}\\\
h^{\sigma}_{1}(i,\alpha)&=\sum_{\beta=r_{i+1}+1}^{r_{i}}\epsilon^{\sigma}_{\beta,\alpha}+\sum_{\beta=r_{i}+1}^{r_{i-1}}\epsilon^{\sigma}_{\alpha,\beta},\\\
h^{\sigma}_{2}(i,g,v_{1},\dots,v_{r})&=\sum_{\alpha=r_{i+1}+1}^{r_{i}}\sum_{\beta=r_{i}+1}^{r}(v_{\sigma(\beta)}-v_{\sigma(\alpha)}+1-g)\cdot\epsilon^{\sigma}_{\alpha,\beta},\end{split}$

where $v_{1},\ldots,v_{r}$ are integers. Notice that
$h^{\sigma}_{0},h^{\sigma}_{1},h^{\sigma}_{2}$ are algorithmically determined
simply by the initial data of $\boldsymbol{r},\sigma$.

######  Proposition 6.2.

Let $C$ be a smooth projective curve and
$E=\bigoplus_{\alpha=1}^{r}L_{\alpha}$ such that the hyperquot scheme
$\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d})$ is smooth and
unobstructed. Then there is a locally closed stratification

$\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d})=\coprod_{\sigma\in
P_{\boldsymbol{r}}}\coprod_{\begin{subarray}{c}\boldsymbol{n}=(\boldsymbol{n}_{\alpha})_{\alpha},\\\
d_{j}=\sum_{\alpha=1}^{r_{j}}n_{j,\alpha}+\\\ \sum_{\alpha=r_{j}+1}^{r}\deg
L_{\sigma(\alpha)}\end{subarray}}Q^{+}_{{\sigma,\boldsymbol{n}}},$

where the second disjoint union is over all collections of tuples
$(\boldsymbol{n}_{\alpha})_{\alpha=1,\dots,r_{1}}$ such that, for each
$r_{j+1}+1\leq\alpha\leq r_{j}$, we have
$\boldsymbol{n}_{\alpha}=(n_{1,\alpha}\leq\dots\leq n_{j,\alpha})$, and
$Q^{+}_{{\sigma,\boldsymbol{n}}}\rightarrow Q_{{\sigma,\boldsymbol{n}}}$ is an
affine fibre bundle of relative dimension

$\displaystyle\sum_{i=1}^{l}\sum_{\alpha=1}^{r_{i}}n_{i,\alpha}\cdot
h^{\sigma}_{1}(i,\alpha)+\sum_{i=1}^{l}h^{\sigma}_{2}(i,g,\deg
L_{1},\dots,\deg L_{r}).$

###### Proof.

By our assumption the hyperquot scheme
$\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d})$ is smooth (and
projective), therefore we can apply Theorem 6.1. Fix $\sigma,\boldsymbol{n}$.
To compute the relative dimension of the affine fibre bundle
$Q^{+}_{{\sigma,\boldsymbol{n}}}\rightarrow Q_{{\sigma,\boldsymbol{n}}}$, we
consider a $\mathbf{T}$-fixed point $z\in Q_{{\sigma,\boldsymbol{n}}}$ and
compute the positive part of the tangent space at $z$. Say that the flag

${\displaystyle\bigoplus_{\alpha=1}^{r_{l}}L_{\sigma(\alpha)}(-Z_{l,\alpha})}$${\displaystyle\bigoplus_{\alpha=1}^{r_{l-1}}L_{\sigma(\alpha)}(-Z_{l-1,\alpha})}$${\cdots}$${\displaystyle\bigoplus_{\alpha=1}^{r_{1}}L_{\sigma(\alpha)}(-Z_{1,\alpha})}$${E}$${K_{l}}$${K_{l-1}}$${K_{1}}$

corresponds to the point $z$. Let ${\mathbb{G}}_{m}\hookrightarrow\mathbf{T}$
be the generic 1-parameter subtorus given by
$w\mapstochar\rightarrow(w,w^{2},\dots,w^{r})$. Since by our assumption the
hyperquot scheme is unobstructed, the class in $K$-theory of the tangent
bundle at $z$ coincides with the class of the dual
${\mathbb{E}}_{\operatorname{\mathrm{Q}}}^{\vee}$ of the perfect obstruction
theory, which by Remark 3.4 can be computed as
$\operatorname{\mathrm{Cone}}(\operatorname{\mathbf{R}}_{\pi}\Delta)|_{z}$. In
particular, the tangent space decomposes ${\mathbb{G}}_{m}$-equivariantly as

$\displaystyle
T_{z}=\sum_{\alpha,\beta=1}^{r}\left(\sum_{i=1}^{l}\operatorname{\mathbf{R}}\operatorname{Hom}(K_{i,\alpha},T_{i,\beta})-\sum_{i=1}^{l-1}\operatorname{\mathbf{R}}\operatorname{Hom}(K_{i+1,\alpha},T_{i,\beta})\right)w^{\beta-\alpha}\in
K^{0}_{{\mathbb{G}}_{m}}(z),$

where, to set notation, we decompose the kernels and the quotients as

$\displaystyle K_{i}=\bigoplus_{\alpha=1}^{r}K_{i,\alpha},\quad
T_{i}=\bigoplus_{\alpha=1}^{r}T_{i,\alpha},\quad i=1,\dots,l,$

and we use the same notation as in Section 5.2. Therefore, the relative
dimension of the affine fibre bundle coincides with

(6.2)
$\displaystyle\begin{split}\operatorname{rk}T_{z}^{+}&=\sum_{1\leq\alpha<\beta\leq
r}\left(\sum_{i=1}^{l}\operatorname{rk}\operatorname{\mathbf{R}}\operatorname{Hom}(K_{i,\alpha},T_{i,\beta})-\sum_{i=1}^{l-1}\operatorname{rk}\operatorname{\mathbf{R}}\operatorname{Hom}(K_{i+1,\alpha},T_{i,\beta})\right)\\\
&=\sum_{\alpha,\beta=1}^{r}\left(\sum_{i=1}^{l}\operatorname{rk}\operatorname{\mathbf{R}}\operatorname{Hom}(K_{i,\sigma(\alpha)},T_{i,\sigma(\beta)})-\sum_{i=1}^{l-1}\operatorname{rk}\operatorname{\mathbf{R}}\operatorname{Hom}(K_{i+1,\sigma(\alpha)},T_{i,\sigma(\beta)})\right)\cdot\epsilon^{\sigma}_{\alpha,\beta}\\\
&=\sum_{i=1}^{l}\sum_{\alpha,\beta=1}^{r}\left(\operatorname{rk}\operatorname{\mathbf{R}}\operatorname{Hom}(K_{i,\sigma(\alpha)},T_{i,\sigma(\beta)})-\operatorname{rk}\operatorname{\mathbf{R}}\operatorname{Hom}(K_{i+1,\sigma(\alpha)},T_{i,\sigma(\beta)})\right)\cdot\epsilon^{\sigma}_{\alpha,\beta}\end{split}$

where we set for convenience $K_{l+1,\sigma(\alpha)}=0$. We consider now all
the possible values of
$\operatorname{rk}\operatorname{\mathbf{R}}\operatorname{Hom}(K_{i,\sigma(\alpha)},T_{j,\sigma(\beta)})$.

Case I. If $\alpha>r_{i}$, then $K_{i,\sigma(\alpha)}=0$ and

$\operatorname{rk}\operatorname{\mathbf{R}}\operatorname{Hom}(K_{i,\sigma(\alpha)},T_{j,\sigma(\beta)})=0.$

Case II. If $\alpha\leq r_{i}$ and $\beta\leq r_{j}$, we have

$\displaystyle\operatorname{rk}\operatorname{\mathbf{R}}\operatorname{Hom}(K_{i,\sigma(\alpha)},T_{j,\sigma(\beta)})$
$\displaystyle=\operatorname{rk}\operatorname{\mathbf{R}}\operatorname{Hom}(L_{\sigma(\alpha)}(-Z_{i,\alpha}),\mathscr{O}_{Z_{j,\beta}})$
$\displaystyle=n_{j,\beta}.$

Case III. If $\alpha\leq r_{i}$ and $\beta>r_{j}$, by Riemann-Roch we have

$\displaystyle\operatorname{rk}\operatorname{\mathbf{R}}\operatorname{Hom}(K_{i,\sigma(\alpha)},T_{j,\sigma(\beta)})$
$\displaystyle=\operatorname{rk}\operatorname{\mathbf{R}}\operatorname{Hom}(L_{\sigma(\alpha)}(-Z_{i,\alpha}),L_{\sigma(\beta)})$
$\displaystyle=\deg L_{\sigma(\beta)}-\deg
L_{\sigma(\alpha)}+n_{i,\alpha}+1-g.$

We compute now all the contributions to the rank in (6.2). Fix $i=1,\dots,l$.

Case I. If $\alpha>r_{i}$, then
$K_{i,\sigma(\alpha)}=K_{i+1,\sigma(\alpha)}=0$ and therefore

$\left(\operatorname{rk}\operatorname{\mathbf{R}}\operatorname{Hom}(K_{i,\sigma(\alpha)},T_{i,\sigma(\beta)})-\operatorname{rk}\operatorname{\mathbf{R}}\operatorname{Hom}(K_{i+1,\sigma(\alpha)},T_{i,\sigma(\beta)})\right)\cdot\epsilon^{\sigma}_{\alpha,\beta}=0.$

Case II. If $\alpha\leq r_{i+1}$ and $\beta\leq r_{i}$, we have

$\displaystyle\left(\operatorname{rk}\operatorname{\mathbf{R}}\operatorname{Hom}(K_{i,\sigma(\alpha)},T_{i,\sigma(\beta)})-\operatorname{rk}\operatorname{\mathbf{R}}\operatorname{Hom}(K_{i+1,\sigma(\alpha)},T_{i,\sigma(\beta)})\right)\cdot\epsilon^{\sigma}_{\alpha,\beta}$
$\displaystyle=(n_{i,\beta}-n_{i,\beta})\cdot\epsilon^{\sigma}_{\alpha,\beta}$
$\displaystyle=0.$

Case III. If $r_{i+1}+1\leq\alpha\leq r_{i}$ and $\beta\leq r_{i}$, we have

$\displaystyle\left(\operatorname{rk}\operatorname{\mathbf{R}}\operatorname{Hom}(K_{i,\sigma(\alpha)},T_{i,\sigma(\beta)})-\operatorname{rk}\operatorname{\mathbf{R}}\operatorname{Hom}(K_{i+1,\sigma(\alpha)},T_{i,\sigma(\beta)})\right)\cdot\epsilon^{\sigma}_{\alpha,\beta}=n_{i,\beta}\cdot\epsilon^{\sigma}_{\alpha,\beta}.$

Case IV. If $\alpha\leq r_{i+1}$ and $\beta>r_{i}$, we have

$\displaystyle\left(\operatorname{rk}\operatorname{\mathbf{R}}\operatorname{Hom}(K_{i,\sigma(\alpha)},T_{i,\sigma(\beta)})-\operatorname{rk}\operatorname{\mathbf{R}}\operatorname{Hom}(K_{i+1,\sigma(\alpha)},T_{i,\sigma(\beta)})\right)\cdot\epsilon^{\sigma}_{\alpha,\beta}=(n_{i,\alpha}-n_{i+1,\alpha})\cdot\epsilon^{\sigma}_{\alpha,\beta}.$

Case V. If $r_{i+1}+1\leq\alpha\leq r_{i}$ and $\beta>r_{i}$, we have

$\left(\operatorname{rk}\operatorname{\mathbf{R}}\operatorname{Hom}(K_{i,\sigma(\alpha)},T_{i,\sigma(\beta)})-\operatorname{rk}\operatorname{\mathbf{R}}\operatorname{Hom}(K_{i+1,\sigma(\alpha)},T_{i,\sigma(\beta)})\right)\cdot\epsilon^{\sigma}_{\alpha,\beta}\\\
=(\deg L_{\sigma(\beta)}-\deg
L_{\sigma(\alpha)}+n_{i,\alpha}+1-g)\cdot\epsilon^{\sigma}_{\alpha,\beta}.$

Patching everything together, (6.2) equals

(6.3)
$\operatorname{rk}T_{z}^{+}=\sum_{i=1}^{l}\left(\sum_{\alpha=r_{i+1}+1}^{r_{i}}\sum_{\beta=1}^{r_{i}}n_{i,\beta}\cdot\epsilon^{\sigma}_{\alpha,\beta}+\sum_{\alpha=1}^{r_{i+1}}\sum_{\beta=r_{i}+1}^{r}(n_{i,\alpha}-n_{i+1,\alpha})\cdot\epsilon^{\sigma}_{\alpha,\beta}\right.\\\
\left.+\sum_{\alpha=r_{i+1}+1}^{r_{i}}\sum_{\beta=r_{i}+1}^{r}(\deg
L_{\sigma(\beta)}-\deg
L_{\sigma(\alpha)}+n_{i,\alpha}+1-g)\cdot\epsilon^{\sigma}_{\alpha,\beta}\right).$

We simplify the above expression. We have that

(6.4)
$\begin{split}\sum_{i=1}^{l}&\left(\sum_{\alpha=1}^{r_{i+1}}\sum_{\beta=r_{i}+1}^{r}(n_{i,\alpha}-n_{i+1,\alpha})\cdot\epsilon^{\sigma}_{\alpha,\beta}+\sum_{\alpha=r_{i+1}+1}^{r_{i}}\sum_{\beta=r_{i}+1}^{r}n_{i,\alpha}\cdot\epsilon^{\sigma}_{\alpha,\beta}\right)\\\
&=\sum_{i=1}^{l}\left(\sum_{\alpha=1}^{r_{i}}\sum_{\beta=r_{i}+1}^{r}n_{i,\alpha}\cdot\epsilon^{\sigma}_{\alpha,\beta}-\sum_{\alpha=1}^{r_{i+1}}\sum_{\beta=r_{i}+1}^{r}n_{i+1,\alpha}\cdot\epsilon^{\sigma}_{\alpha,\beta}\right)\\\
&=\sum_{i=1}^{l}\left(\sum_{\alpha=1}^{r_{i}}\sum_{\beta=r_{i}+1}^{r_{i-1}}n_{i,\alpha}\cdot\epsilon^{\sigma}_{\alpha,\beta}+\sum_{\alpha=1}^{r_{i}}\sum_{\beta=r_{i-1}+1}^{r}n_{i,\alpha}\cdot\epsilon^{\sigma}_{\alpha,\beta}-\sum_{\alpha=1}^{r_{i+1}}\sum_{\beta=r_{i}+1}^{r}n_{i+1,\alpha}\cdot\epsilon^{\sigma}_{\alpha,\beta}\right)\\\
&=\sum_{i=1}^{l}\left(\sum_{\alpha=1}^{r_{i}}\sum_{\beta=r_{i}+1}^{r_{i-1}}n_{i,\alpha}\cdot\epsilon^{\sigma}_{\alpha,\beta}\right).\end{split}$

Therefore by plugging (6.4) into (6.3) and suitably relabelling $\alpha$ and
$\beta$ in the first summand, we obtain

$\operatorname{rk}T_{z}^{+}=\sum_{i=1}^{l}\left(\sum_{\alpha=1}^{r_{i}}n_{i,\alpha}\cdot\left(\sum_{\beta=r_{i+1}+1}^{r_{i}}\epsilon^{\sigma}_{\beta,\alpha}+\sum_{\beta=r_{i}+1}^{r_{i-1}}\epsilon^{\sigma}_{\alpha,\beta}\right)\right.\\\
\left.+\sum_{\alpha=r_{i+1}+1}^{r_{i}}\sum_{\beta=r_{i}+1}^{r}(\deg
L_{\sigma(\beta)}-\deg
L_{\sigma(\alpha)}+1-g)\cdot\epsilon^{\sigma}_{\alpha,\beta}\vphantom{}\right),$

which concludes the argument. ∎

Specialising in Proposition 6.2 the rank of all the quotients to be $0$, i.e.
$r_{1}=\dots=r_{l}=r$ and $r_{l+1}=0$, we recover the decomposition computed
in [43, Prop. 3.4].

###  6.3. Motivic partition function

Let $C$ be a smooth projective curve, $E$ a locally free sheaf of rank $r$ and
fix $\boldsymbol{s}=(s_{1}\leq\dots\leq s_{l})$ . Define

$\mathsf{Z}_{C,E,\boldsymbol{s}}(\boldsymbol{q})=\sum_{\boldsymbol{d}}\,\bigl{[}\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d})\bigr{]}\boldsymbol{q}^{\boldsymbol{d}}\in
K_{0}(\operatorname{Var}_{\mathbf{k}})(\\!(q_{1},\dots,q_{l})\\!),$

where $\boldsymbol{d}=(d_{1},\dots,d_{l})$ and we use the multivariable
notation $\boldsymbol{q}=(q_{1},\ldots,q_{l})$ and
$\boldsymbol{q}^{\boldsymbol{d}}=\prod_{i=1}^{l}q^{d_{i}}$. If $l=1$, $E=L$ is
a line bundle and $\boldsymbol{s}=0$, then $\mathsf{Z}_{C,L,0}(q)$ is simply
Kapranov’s motivic zeta function [33]

(6.5) $\mathsf{Z}_{C,L,0}(q)=\zeta_{C}(q)=\sum_{d\geq
0}\,\bigl{[}C^{(d)}\bigr{]}q^{d}.$

We prove an expression for the motive of the hyperquot scheme, in terms of
Kapranov motivic zeta functions.

######  Theorem 6.3.

Let $C$ be a smooth projective curve of genus $g$ and
$E=\bigoplus_{\alpha=1}^{r}L_{\alpha}$ such that the hyperquot scheme
$\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d})$ is smooth and
unobstructed. There is an identity

$\mathsf{Z}_{C,E,\boldsymbol{s}}(\boldsymbol{q})=\sum_{\sigma\in
P_{\boldsymbol{r}}}\prod_{j=1}^{l}\left({\mathbb{L}}^{h^{\sigma}_{2}(j,g,\deg
L_{1},\dots,\deg L_{r})}q_{j}^{\sum_{\alpha=r_{j}+1}^{r}\deg
L_{\sigma(\alpha)}}\right)\\\ \cdot\prod_{1\leq i\leq j\leq
l}\prod_{\alpha=r_{j+1}+1}^{r_{j}}\zeta_{C}\left({\mathbb{L}}^{h^{\sigma}_{0}(i,\alpha)+r_{i}-r_{j}+\alpha-
r_{j+1}-1}q_{i}\cdots q_{j}\right).$

In particular, $\mathsf{Z}_{C,E,\boldsymbol{s}}(\boldsymbol{q})$ is a rational
function.

###### Proof.

By the assumptions, to compute the motive of
$\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d})$ we can exploit the
stratification of Proposition 6.2. Every stratum is a Zariski locally trivial
fibration over a connected component of the fixed locus, with fibre an affine
space whose dimension we computed in 6.2. Therefore we have

$\displaystyle\mathsf{Z}_{C,E,\boldsymbol{s}}(\boldsymbol{q})$
$\displaystyle=\sum_{\boldsymbol{d}}\boldsymbol{q}^{\boldsymbol{d}}\sum_{\sigma\in
P_{\boldsymbol{r}}}\sum_{\begin{subarray}{c}\boldsymbol{n}=(\boldsymbol{n}_{\alpha})_{\alpha},\\\
d_{j}=\sum_{\alpha=1}^{r_{j}}n_{j,\alpha}+\\\ \sum_{\alpha=r_{j}+1}^{r}\deg
L_{\sigma(\alpha)}\end{subarray}}{\mathbb{L}}^{\sum_{j=1}^{l}\sum_{\alpha=1}^{r_{j}}n_{j,\alpha}\cdot
h^{\sigma}_{1}(j,\alpha)+\sum_{j=1}^{l}h^{\sigma}_{2}(j,g,\deg
L_{1},\dots,\deg
L_{r})}\cdot\prod_{\alpha=1}^{r_{1}}\left[C^{[\boldsymbol{n}_{\alpha}]}\right]$
$\displaystyle=\sum_{\boldsymbol{d}}\sum_{\sigma\in
P_{\boldsymbol{r}}}\prod_{j=1}^{l}\left({\mathbb{L}}^{h^{\sigma}_{2}(j,g,\deg
L_{1},\dots,\deg L_{r})}q_{j}^{\sum_{\alpha=r_{j}+1}^{r}\deg
L_{\sigma(\alpha)}}\right)$
$\displaystyle\qquad\qquad\qquad\cdot\sum_{\begin{subarray}{c}\boldsymbol{n}=(\boldsymbol{n}_{\alpha})_{\alpha},\\\
d_{j}=\sum_{\alpha=1}^{r_{j}}n_{j,\alpha}+\\\ \sum_{\alpha=r_{j}+1}^{r}\deg
L_{\sigma(\alpha)}\end{subarray}}\prod_{j=1}^{l}\left({\mathbb{L}}^{\sum_{\alpha=1}^{r_{j}}n_{j,\alpha}\cdot
h^{\sigma}_{1}(j,\alpha)}q_{j}^{\sum_{\alpha=1}^{r_{j}}n_{j,\alpha}}\right)\prod_{\alpha=1}^{r_{1}}\left[C^{[\boldsymbol{n}_{\alpha}]}\right]$
$\displaystyle=\sum_{\sigma\in
P_{\boldsymbol{r}}}\prod_{j=1}^{l}\left({\mathbb{L}}^{h^{\sigma}_{2}(j,g,\deg
L_{1},\dots,\deg L_{r})}q_{j}^{\sum_{\alpha=r_{j}+1}^{r}\deg
L_{\sigma(\alpha)}}\right)$
$\displaystyle\qquad\qquad\qquad\cdot\sum_{\begin{subarray}{c}\boldsymbol{n}=(\boldsymbol{n}_{\alpha})_{\alpha}\end{subarray}}\prod_{j=1}^{l}\left({\mathbb{L}}^{\sum_{\alpha=1}^{r_{j}}n_{j,\alpha}\cdot
h^{\sigma}_{1}(j,\alpha)}q_{j}^{\sum_{\alpha=1}^{r_{j}}n_{j,\alpha}}\right)\prod_{\alpha=1}^{r_{1}}\left[C^{[\boldsymbol{n}_{\alpha}]}\right].$

We redistribute the variables to simplify the above expression. We have

$\prod_{j=1}^{l}q_{j}^{\sum_{\alpha=1}^{r_{j}}n_{j,\alpha}}=\prod_{j=1}^{l}\prod_{\alpha=r_{j+1}+1}^{r_{j}}q_{1}^{n_{1,\alpha}}\cdots
q_{j}^{n_{j,\alpha}}.$

Similarly, we have

$\prod_{j=1}^{l}{\mathbb{L}}^{\sum_{\alpha=1}^{r_{j}}n_{j,\alpha}\cdot
h^{\sigma}_{1}(j,\alpha)}=\prod_{j=1}^{l}\prod_{\alpha=r_{j+1}+1}^{r_{j}}{\mathbb{L}}^{n_{1,\alpha}\cdot
h^{\sigma}_{1}(1,\alpha)}\cdots{\mathbb{L}}^{n_{j,\alpha}\cdot
h^{\sigma}_{1}(j,\alpha)}$

and

$\prod_{\alpha=1}^{r_{1}}\left[C^{[\boldsymbol{n}_{\alpha}]}\right]=\prod_{j=1}^{l}\prod_{\alpha=r_{j+1}+1}^{r_{j}}\left[C^{[\boldsymbol{n}_{\alpha}]}\right].$

Therefore we have

$\displaystyle\mathsf{Z}_{C,E,\boldsymbol{s}}(\boldsymbol{q})$
$\displaystyle=\sum_{\sigma\in
P_{\boldsymbol{r}}}\prod_{j=1}^{l}\left({\mathbb{L}}^{h^{\sigma}_{2}(j,g,\deg
L_{1},\dots,\deg L_{r})}q_{j}^{\sum_{\alpha=r_{j}+1}^{r}\deg
L_{\sigma(\alpha)}}\right)$
$\displaystyle\qquad\qquad\cdot\sum_{\begin{subarray}{c}\boldsymbol{n}=(\boldsymbol{n}_{\alpha})_{\alpha}\end{subarray}}\prod_{j=1}^{l}\prod_{\alpha=r_{j+1}+1}^{r_{j}}\left(({\mathbb{L}}^{h^{\sigma}_{1}(1,\alpha)}q_{1})^{n_{1,\alpha}}\cdots({\mathbb{L}}^{h^{\sigma}_{1}(j,\alpha)}q_{j})^{n_{j,\alpha}}\left[C^{[\boldsymbol{n}_{\alpha}]}\right]\right)$
$\displaystyle=\sum_{\sigma\in
P_{\boldsymbol{r}}}\prod_{j=1}^{l}\left({\mathbb{L}}^{h^{\sigma}_{2}(j,g,\deg
L_{1},\dots,\deg L_{r})}q_{j}^{\sum_{\alpha=r_{j}+1}^{r}\deg
L_{\sigma(\alpha)}}\right)$
$\displaystyle\qquad\qquad\cdot\prod_{j=1}^{l}\prod_{\alpha=r_{j+1}+1}^{r_{j}}\sum_{\boldsymbol{n}_{\alpha}}({\mathbb{L}}^{h^{\sigma}_{1}(1,\alpha)}q_{1})^{n_{1,\alpha}}\cdots({\mathbb{L}}^{h^{\sigma}_{1}(j,\alpha)}q_{j})^{n_{j,\alpha}}\left[C^{[\boldsymbol{n}_{\alpha}]}\right].$

Notice now that the last sum is a generating series of motives of hyperquot
schemes
$\operatorname{Quot}_{C}(\mathscr{O}_{C},\mathbf{0},\boldsymbol{n}_{\alpha})$,
where the rank of all the quotients is $0$. The associated generating series
was computed in [43, Thm. 4.2]444Alternatively one could directly see the
factorisation exploiting the fact that the nested Hilbert scheme on a smooth
projective curve is a product of symmetric products. and equals

$\displaystyle\sum_{\boldsymbol{n}_{\alpha}}\,({\mathbb{L}}^{h^{\sigma}_{1}(1,\alpha)}q_{1})^{n_{1,\alpha}}\cdots({\mathbb{L}}^{h^{\sigma}_{1}(j,\alpha)}q_{j})^{n_{j,\alpha}}\left[C^{[\boldsymbol{n}_{\alpha}]}\right]=\prod_{i=1}^{j}\zeta_{C}\left({\mathbb{L}}^{\sum_{k=i}^{j}h^{\sigma}_{1}(k,\alpha)}q_{i}\cdots
q_{j}\right).$

We conclude that

$\displaystyle\mathsf{Z}_{C,E,\boldsymbol{s}}(\boldsymbol{q})$
$\displaystyle=\sum_{\sigma\in
P_{\boldsymbol{r}}}\prod_{j=1}^{l}\left({\mathbb{L}}^{h^{\sigma}_{2}(j,g,\deg
L_{1},\dots,\deg L_{r})}q_{j}^{\sum_{\alpha=r_{j}+1}^{r}\deg
L_{\sigma(\alpha)}}\right)$
$\displaystyle\qquad\qquad\qquad\cdot\prod_{j=1}^{l}\prod_{\alpha=r_{j+1}+1}^{r_{j}}\prod_{i=1}^{j}\zeta_{C}\left({\mathbb{L}}^{\sum_{k=i}^{j}h^{\sigma}_{1}(k,\alpha)}q_{i}\cdots
q_{j}\right)$ $\displaystyle=\sum_{\sigma\in
P_{\boldsymbol{r}}}\prod_{j=1}^{l}\left({\mathbb{L}}^{h^{\sigma}_{2}(j,g,\deg
L_{1},\dots,\deg L_{r})}q_{j}^{\sum_{\alpha=r_{j}+1}^{r}\deg
L_{\sigma(\alpha)}}\right)$ $\displaystyle\qquad\qquad\qquad\cdot\prod_{1\leq
i\leq j\leq
l}\prod_{\alpha=r_{j+1}+1}^{r_{j}}\zeta_{C}\left({\mathbb{L}}^{h^{\sigma}_{0}(i,\alpha)+r_{i}-r_{j}+\alpha-
r_{j+1}-1}q_{i}\cdots q_{j}\right),$

where in the last equality we used Lemma 6.4. Rationality of
$\mathsf{Z}_{C,E,\boldsymbol{s}}(\boldsymbol{q})$ follows from the rationality
of $\zeta_{C}$ proved in [33]. ∎

Once more, specialising in Theorem 6.3 the ranks of all the quotients to be
$0$, i.e. setting $r_{1}=\dots=r_{l}=r$ and $r_{l+1}=0$, one recovers the
product formula in [43, Thm. 4.2] which, in turn, generalises the
corresponding computation in the unnested case [2, 52].

We prove now the combinatorial result we used in the proof of Theorem 6.3.

######  Lemma 6.4.

Let $1\leq i\leq j\leq r$ and $r_{j+1}+1\leq\alpha\leq r_{j}$. We have

$\sum_{k=i}^{j}h^{\sigma}_{1}(k,\alpha)=h^{\sigma}_{0}(i,\alpha)+r_{i}-r_{j}+\alpha-
r_{j+1}-1.$

###### Proof.

By definition, we have

$\displaystyle\sum_{k=i}^{j}h^{\sigma}_{1}(k,\alpha)$
$\displaystyle=\sum_{k=i}^{j}\left(\sum_{\beta=r_{k+1}+1}^{r_{k}}\epsilon^{\sigma}_{\beta,\alpha}+\sum_{\beta=r_{k}+1}^{r_{k-1}}\epsilon^{\sigma}_{\alpha,\beta}\right)$
$\displaystyle=\sum_{\beta=r_{i+1}+1}^{r_{i}}\epsilon^{\sigma}_{\beta,\alpha}+\sum_{\beta=r_{i}+1}^{r_{i-1}}\epsilon^{\sigma}_{\alpha,\beta}+\sum_{\beta=r_{i+2}+1}^{r_{i+1}}\epsilon^{\sigma}_{\beta,\alpha}$
$\displaystyle\qquad\qquad\qquad+\sum_{\beta=r_{i+1}+1}^{r_{i}}\epsilon^{\sigma}_{\alpha,\beta}+\dots+\sum_{\beta=r_{j+1}+1}^{r_{j}}\epsilon^{\sigma}_{\beta,\alpha}+\sum_{\beta=r_{j}+1}^{r_{j-1}}\epsilon^{\sigma}_{\alpha,\beta}$
$\displaystyle=\sum_{k=i}^{j-1}\sum_{\beta=r_{k+1}+1}^{r_{k}}(\epsilon^{\sigma}_{\alpha,\beta}+\epsilon^{\sigma}_{\beta,\alpha})+\sum_{\beta=r_{i}+1}^{r_{i}-1}\epsilon^{\sigma}_{\alpha,\beta}+\sum_{\beta=r_{j+1}+1}^{r_{j}}\epsilon^{\sigma}_{\beta,\alpha}.$

Notice that if $\alpha\neq\beta$, we have
$\epsilon^{\sigma}_{\alpha,\beta}+\epsilon^{\sigma}_{\beta,\alpha}=1$.
Therefore

$\displaystyle\sum_{k=i}^{j-1}\sum_{\beta=r_{k+1}+1}^{r_{k}}(\epsilon^{\sigma}_{\alpha,\beta}+\epsilon^{\sigma}_{\beta,\alpha})=\sum_{k=i}^{j-1}\sum_{\beta=r_{k+1}+1}^{r_{k}}1=r_{i}-r_{j}.$

Secondly, we have that

$\displaystyle\sum_{\beta=r_{j+1}+1}^{r_{j}}\epsilon^{\sigma}_{\beta,\alpha}=\alpha-
r_{j+1}-1.$

We conclude that

$\sum_{k=i}^{j}h^{\sigma}_{1}(k,\alpha)=h^{\sigma}_{0}(i,\alpha)+r_{i}-r_{j}+\alpha-
r_{j+1}-1.\qed$

###  6.4. Genus 0

In the case where the curve $C\cong{\mathbb{P}}^{1}$ has genus 0 and
$E=\mathscr{O}_{{\mathbb{P}}^{1}}^{\oplus r}$, we prove a different closed
expression for the motive of the hyperquot scheme, where the role of the
partial flag variety becomes more evident. We denote by
$\mathrm{Flag}(\boldsymbol{s},r)$ the partial flag variety parametrising flags
of vector subspaces

$V_{l}\subset\dots\subset V_{1}\subset V,$

of a fixed $r$-dimensional vector space $V$, where we set $\dim
V_{i}=r_{i}=r-s_{i}$. Setting $r_{0}=r$ and $r_{l+1}=0$ as ever, an easy
inductive argument yields the relation

$\displaystyle[\mathrm{Flag}(\boldsymbol{s},r)]=\frac{\prod_{k=1}^{r}\,\left({\mathbb{L}}^{k}-1\right)}{\prod_{j=0}^{l}\prod_{k=1}^{r_{j}-r_{j+1}}\,\left({\mathbb{L}}^{k}-1\right)}\in
K_{0}(\operatorname{Var}_{\mathbf{k}}).$

The inductive argument uses the base case

$[G(d,r)]=\frac{\prod_{k=1}^{r}\,\left({\mathbb{L}}^{k}-1\right)}{\prod_{k=1}^{d}\,\left({\mathbb{L}}^{k}-1\right)\prod_{k=1}^{r-d}\,\left({\mathbb{L}}^{k}-1\right)}$

and the fibration

${\mathrm{Flag}(r,s_{1},\ldots,s_{l-1},s_{l})}$${\mathrm{Flag}(r,s_{1},\ldots,s_{l-1})}$

forgetting $V_{l}$, which has fibre the Grassmannian $G(r_{l},r_{l-1})$.

######  Theorem 6.5.

There is an identity

$\mathsf{Z}_{{\mathbb{P}}^{1},\mathscr{O}^{\oplus
r},\boldsymbol{s}}(\boldsymbol{q})=[\mathrm{Flag}(\boldsymbol{s},r)]\prod_{1\leq
i\leq j\leq
l}\prod_{\alpha=r_{j+1}+1}^{r_{j}}\frac{1}{\left(1-{\mathbb{L}}^{r_{i}-\alpha}q_{i}\cdots
q_{j}\right)\left(1-{\mathbb{L}}^{r_{i-1}-\alpha+1}q_{i}\cdots q_{j}\right)}.$

###### Proof.

In the case ${\mathbb{P}}^{1}$ and $E=\mathscr{O}^{\oplus r}$ the hyperquot
scheme is smooth and unobstructed by Theorem 4.4. Therefore by Theorem 6.3 we
have

$\displaystyle\mathsf{Z}_{{\mathbb{P}}^{1},\mathscr{O}^{\oplus
r},\boldsymbol{s}}(\boldsymbol{q})=\sum_{\sigma\in
P_{\boldsymbol{r}}}{\mathbb{L}}^{\sum_{j=1}^{l}h^{\sigma}_{2}(j,0,0,\dots,0)}\prod_{1\leq
i\leq j\leq
l}\prod_{\alpha=r_{j+1}+1}^{r_{j}}\zeta_{C}\left({\mathbb{L}}^{h^{\sigma}_{0}(i,\alpha)+r_{i}-r_{j}+\alpha-
r_{j+1}-1}q_{i}\cdots q_{j}\right).$

Recall that the motivic zeta function of ${\mathbb{P}}^{1}$ is (see e.g. [33])

$\displaystyle\zeta_{{\mathbb{P}}^{1}}(q)=\frac{1}{(1-q)(1-{\mathbb{L}}q)}.$

Thus we conclude that

$\mathsf{Z}_{{\mathbb{P}}^{1},\mathscr{O}^{\oplus
r},\boldsymbol{s}}(\boldsymbol{q})=\sum_{\sigma\in
P_{\boldsymbol{r}}}{\mathbb{L}}^{\sum_{j=1}^{l}\sum_{\alpha=r_{j+1}+1}^{r_{j}}\sum_{\beta=r_{j}+1}^{r}\epsilon^{\sigma}_{\alpha,\beta}}\\\
\cdot\prod_{1\leq i\leq j\leq
l}\prod_{\alpha=r_{j+1}+1}^{r_{j}}\frac{1}{\left(1-{\mathbb{L}}^{h^{\sigma}_{0}(i,\alpha)+r_{i}-r_{j}+\alpha-
r_{j+1}-1}q_{i}\cdots
q_{j}\right)\left(1-{\mathbb{L}}^{h^{\sigma}_{0}(i,\alpha)+r_{i}-r_{j}+\alpha-
r_{j+1}}q_{i}\cdots q_{j}\right)}\\\
=\frac{\prod_{k=1}^{r}\,\left(1-{\mathbb{L}}^{k}\right)}{\prod_{j=0}^{l}\prod_{k=1}^{r_{j}-r_{j+1}}\,\left(1-{\mathbb{L}}^{k}\right)}\prod_{1\leq
i\leq j\leq
l}\prod_{\alpha=r_{j+1}+1}^{r_{j}}\frac{1}{\left(1-{\mathbb{L}}^{r_{i}-\alpha}q_{i}\cdots
q_{j}\right)\left(1-{\mathbb{L}}^{r_{i-1}-\alpha+1}q_{i}\cdots q_{j}\right)},$

where in the last expression we used the purely combinatorial identity [10,
Prop. 5], suitably adapted to our conventions on the indices. ∎

######  Remark 6.6.

It would be interesting to understand if the product formula in Theorem 6.5
could be geometrically explained by the existence of a fibration

${\operatorname{Quot}_{{\mathbb{P}}^{1}}(\mathscr{O}^{\oplus
r},\boldsymbol{s},\boldsymbol{d})}$${\mathrm{Flag}(\boldsymbol{s},r).}$

###  6.5. Irreducibility

Assume $\mathbf{k}$ has characteristic $0$ in this subsection. Specialising
the motivic class from Theorem 6.3 to the Poincaré polynomial and extracting
the $0$-th Betti number gives a simple criterion to count the number of
connected components of the hyperquot scheme, in the smooth and unobstructed
case. We do this explicitly in the genus 0 case.

######  Corollary 6.7.

The hyperquot scheme
$\operatorname{Quot}_{{\mathbb{P}}^{1}}(\mathscr{O}^{\oplus
r},\boldsymbol{s},\boldsymbol{d})$ is irreducible for all
$\boldsymbol{s},\boldsymbol{d}$.

###### Proof.

Without loss of generality, we can suppose that $r_{i}>r_{j}$ for all $0\leq
i<j\leq l+1$. Denote by $b_{0}$ the $0$-th Betti number. Its generating series
is obtained by the specialisation ${\mathbb{L}}=0$

$\displaystyle\sum_{\boldsymbol{d}}\,b_{0}\left(\operatorname{Quot}_{{\mathbb{P}}^{1}}(\mathscr{O}^{\oplus
r},\boldsymbol{s},\boldsymbol{d})\right)\boldsymbol{q}^{\boldsymbol{d}}$
$\displaystyle=\mathsf{Z}_{{\mathbb{P}}^{1},\mathscr{O}^{\oplus
r},\boldsymbol{s}}(\boldsymbol{q})|_{{\mathbb{L}}=0}$
$\displaystyle=\prod_{i=1}^{l}\frac{1}{1-q_{i}}=\sum_{\boldsymbol{d}}\boldsymbol{q}^{\boldsymbol{d}},$

which implies that
$b_{0}(\operatorname{Quot}_{{\mathbb{P}}^{1}}(\mathscr{O}^{\oplus
r},\boldsymbol{s},\boldsymbol{d}))=1$ and therefore
$\operatorname{Quot}_{{\mathbb{P}}^{1}}(\mathscr{O}^{\oplus
r},\boldsymbol{s},\boldsymbol{d})$ has only one connected component. ∎

###  6.6. Ordinary Quot scheme

As a special case, we obtain new formulas for the generating series of motives
of the classical Quot scheme.

######  Corollary 6.8.

Let $C$ be a smooth projective curve of genus $g$ and
$E=\bigoplus_{\alpha=1}^{r}L_{\alpha}$ such that the Quot scheme
$\operatorname{Quot}_{C}(E,s,d)$ is smooth and unobstructed. Set
$\tilde{r}=r-s$. There is an identity

$\displaystyle\mathsf{Z}_{C,E,s}(q)=\sum_{\sigma\in
P_{\tilde{r}}}{\mathbb{L}}^{h^{\sigma}_{2}(1,g,\deg L_{1},\dots,\deg
L_{r})}q^{\sum_{\alpha=\tilde{r}+1}^{r}\deg
L_{\sigma(\alpha)}}\cdot\prod_{\alpha=1}^{\tilde{r}}\zeta_{C}\left({\mathbb{L}}^{h^{\sigma}_{0}(i,\alpha)+\alpha-1}q\right).$

In particular, $\mathsf{Z}_{C,E,\boldsymbol{s}}(\boldsymbol{q})$ is a rational
function. Moreover there is an identity

$\displaystyle\mathsf{Z}_{{\mathbb{P}}^{1},\mathscr{O}^{\oplus
r},s}(q)=[G(s,r)]\cdot\prod_{\alpha=1}^{\tilde{r}}\frac{1}{\left(1-{\mathbb{L}}^{\tilde{r}-\alpha}q\right)\left(1-{\mathbb{L}}^{r-\alpha+1}q\right)}.$

##  Appendix A Functoriality of virtual classes

Fix a smooth projective curve $C$, a locally free sheaf $E$, two $l$-tuples
$\boldsymbol{s}$ and $\boldsymbol{d}$, and form the Quot scheme
$\operatorname{\mathrm{Q}}=\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d})$.

Given a locally free sheaf $F$ on $C$, define for each $i=1,\ldots,l$ the
associated _tautological complex_ on $\operatorname{\mathrm{Q}}$ by

$\displaystyle
F^{[\boldsymbol{d},i]}=\operatorname{\mathbf{R}}\pi_{*}\left(p^{*}F\otimes{\mathcal{T}}_{i}\right),$

where ${\mathcal{T}}_{i}$ is the $i$-th universal quotient over
$\operatorname{\mathrm{Q}}\times C$ and $\pi,p$ are the natural projections to
$\operatorname{\mathrm{Q}}$ and $C$ respectively. By cohomology and base
change, the fibre over a point $z=[E\twoheadrightarrow
T_{l}\twoheadrightarrow\cdots\twoheadrightarrow
T_{1}]\in\operatorname{\mathrm{Q}}$ is the complex

$F^{[\boldsymbol{d},i]}\big{|}_{z}=\operatorname{\mathbf{R}}\Gamma(C,F\otimes
T_{i}),$

which implies that
$F^{[\boldsymbol{d},i]}\in\operatorname{Perf}^{[0,1]}(\operatorname{\mathrm{Q}})$.
Note that

(A.1) $\mathrm{H}^{1}(C,F\otimes
T_{i})=h^{1}(F^{[\boldsymbol{d},i]}\big{|}_{z})=h^{1}(F^{[\boldsymbol{d},i]})\big{|}_{z}$

since there is no higher cohomology.

Now assume that $E$ sits in a short exact sequence

$0\rightarrow E_{1}\rightarrow E\rightarrow E_{2}\rightarrow 0$

of locally free sheaves. Set
$\operatorname{\mathrm{Q}}_{2}=\operatorname{Quot}_{C}(E_{2},\boldsymbol{s},\boldsymbol{d})$.
Precomposing with $E\twoheadrightarrow E_{2}$ yields a natural closed
immersion

${\operatorname{\mathrm{Q}}_{2}}$${\operatorname{\mathrm{Q}},}$$\scriptstyle{\iota}$

which realises $\operatorname{\mathrm{Q}}_{2}$ as the zero locus of a natural
section

$\sigma\in\mathrm{H}^{0}(\operatorname{\mathrm{Q}},h^{0}(F_{1}^{[\boldsymbol{d},l]})),\qquad
F_{1}=E_{1}^{\ast}.$

The proof of this assertion goes along the same lines of the one of [42, Thm.
2.8]. We remark that a similar situation was studied in [58, Sec. 4.1] for the
Quot scheme of $0$-dimensional quotients on a smooth projective surface.

Our next goal is to construct a morphism of exact triangles (which we will
call a ‘compatibility diagram’)

(A.2)
${\iota^{\ast}{\mathbb{F}}_{\operatorname{\mathrm{Q}}}}$${{\mathbb{F}}_{\operatorname{\mathrm{Q}}_{2}}}$${{\mathbb{F}}_{\operatorname{\mathrm{Q}}_{2}/\operatorname{\mathrm{Q}}}}$${\iota^{\ast}{\mathbb{L}}_{\operatorname{\mathrm{Q}}}}$${{\mathbb{L}}_{\operatorname{\mathrm{Q}}_{2}}}$${{\mathbb{L}}_{\operatorname{\mathrm{Q}}_{2}/\operatorname{\mathrm{Q}}}}$$\scriptstyle{\psi}$

where ${\mathbb{F}}_{\operatorname{\mathrm{Q}}}$ and
${\mathbb{F}}_{\operatorname{\mathrm{Q}}_{2}}$ are the complexes (3.6), and
the rightmost square is induced by the leftmost one. Once we have this, and
assuming
${\mathbb{F}}_{\operatorname{\mathrm{Q}}}={\mathbb{E}}_{\operatorname{\mathrm{Q}}}$
(cf. Remark 3.6), by Theorem 3.2 the map

${{\mathbb{F}}_{\operatorname{\mathrm{Q}}_{2}/\operatorname{\mathrm{Q}}}}$${{\mathbb{L}}_{\operatorname{\mathrm{Q}}_{2}/\operatorname{\mathrm{Q}}}}$

is automatically a relative obstruction theory for
$\iota\colon\operatorname{\mathrm{Q}}_{2}\hookrightarrow\operatorname{\mathrm{Q}}$
by [38, Constr. 3.13]. Under suitable additional assumptions, we can, and
will, make sure
${\mathbb{F}}_{\operatorname{\mathrm{Q}}_{2}/\operatorname{\mathrm{Q}}}$ is
concentrated in degree $-1$, equal to the shifted locally free sheaf of the
form $F^{\ast}\big{|}_{\operatorname{\mathrm{Q}}_{2}}[1]$, where
$F=h^{0}(F_{1}^{[\boldsymbol{d},l]})=\pi_{\ast}(p^{\ast}F_{1}\otimes{\mathcal{T}}_{l})$.
Then, in this situation, [34] will give us the compatibility relation

$\iota_{\ast}[\operatorname{\mathrm{Q}}_{2}]^{\operatorname{\mathrm{vir}}}=e(F)\cap[\operatorname{\mathrm{Q}}]^{\operatorname{\mathrm{vir}}}\,\in\,A_{\ast}(\operatorname{\mathrm{Q}})$

between virtual classes. See A.1 for the precise statement.

Let us go back to the contruction of $\psi$. Let us denote by
${\mathcal{K}}_{i}$ and ${\mathcal{T}}_{i}$ the universal kernels and the
universal quotients, respectively, on $\operatorname{\mathrm{Q}}\times C$.
Similarly, ${\mathcal{K}}_{i}^{2}$ and ${\mathcal{T}}_{i}^{2}$ are the
universal kernels and universal quotients on
$\operatorname{\mathrm{Q}}_{2}\times C$. Consider, to fix the notation, the
diagram

${\operatorname{\mathrm{Q}}_{2}\times C}$${\operatorname{\mathrm{Q}}\times
C}$${\operatorname{\mathrm{Q}}_{2}}$${C}$${\operatorname{\mathrm{Q}}}$$\scriptstyle{p_{2}}$$\scriptstyle{\iota_{C}}$$\scriptstyle{\pi_{2}}$$\scriptstyle{p}$$\scriptstyle{\pi}$

and note that one has
$\iota_{C}^{\ast}E_{\operatorname{\mathrm{Q}}}=p_{2}^{\ast}E$. Consider, for
each $i=1,\ldots,l$, the morphism of exact sequences

${0}$${\iota_{C}^{\ast}{\mathcal{K}}_{i}}$${\iota_{C}^{\ast}E_{\operatorname{\mathrm{Q}}}}$${\iota_{C}^{\ast}{\mathcal{T}}_{i}}$${0}$${0}$${{\mathcal{K}}_{i}^{2}}$${p_{2}^{\ast}E_{2}}$${{\mathcal{T}}_{i}^{2}}$${0}$

$\scriptstyle\sim$

in $\operatorname{Coh}(\operatorname{\mathrm{Q}}_{2}\times C)$, where, to
confirm exactness on the left for the top sequence, we use the vanishings of
the higher left derived restriction $\mathbf{L}_{>0}\iota_{C}^{\ast}$ on all
$\operatorname{\mathrm{Q}}$-flat sheaves such as ${\mathcal{T}}_{i}$.

The kernel of the middle vertical surjection is $p_{2}^{\ast}E_{1}$, therefore
we obtain an exact triangle

${p_{2}^{\ast}E_{1}}$${\iota_{C}^{\ast}{\mathcal{K}}_{i}}$${{\mathcal{K}}_{i}^{2}}$${p_{2}^{\ast}E_{1}[1]}$

in the derived category of $\operatorname{\mathrm{Q}}_{2}\times C$. Applying
$\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\pi_{2}}(-,{\mathcal{T}}_{i}^{2})$
yields an exact triangle

${\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\pi_{2}}({\mathcal{K}}_{i}^{2},{\mathcal{T}}_{i}^{2})}$${\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\pi_{2}}(\iota_{C}^{\ast}{\mathcal{K}}_{i},{\mathcal{T}}_{i}^{2})}$${\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\pi_{2}}(p_{2}^{\ast}E_{1},{\mathcal{T}}_{i}^{2})}$$\scriptstyle{a_{i}}$

in the derived category of $\operatorname{\mathrm{Q}}_{2}$. The middle term is

$\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\pi_{2}}(\iota_{C}^{\ast}{\mathcal{K}}_{i},\iota_{C}^{\ast}{\mathcal{T}}_{i})=\operatorname{\mathbf{R}}\pi_{2\ast}\iota_{C}^{\ast}\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}({\mathcal{K}}_{i},{\mathcal{T}}_{i})=\iota^{\ast}\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\pi}({\mathcal{K}}_{i},{\mathcal{T}}_{i}).$

The third term is, after setting $F_{1}=E_{1}^{\ast}$,

$\operatorname{\mathbf{R}}\pi_{2\ast}\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}(p_{2}^{\ast}E_{1},{\mathcal{T}}_{i}^{2})=\operatorname{\mathbf{R}}\pi_{2\ast}(p_{2}^{\ast}F_{1}\otimes{\mathcal{T}}_{i}^{2})=F_{1}^{[\boldsymbol{d},i]}\in\operatorname{Perf}^{[0,1]}(\operatorname{\mathrm{Q}}_{2}).$

We have thus obtained natural exact triangles

${\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\pi_{2}}({\mathcal{K}}_{i}^{2},{\mathcal{T}}_{i}^{2})}$${\iota^{\ast}\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\pi}({\mathcal{K}}_{i},{\mathcal{T}}_{i})}$${F_{1}^{[\boldsymbol{d},i]},\quad
i=1,\ldots,l,}$$\scriptstyle{a_{i}}$

in the derived category of $\operatorname{\mathrm{Q}}_{2}$. The same
construction, namely applying
$\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\pi_{2}}(-,{\mathcal{T}}_{i}^{2})$
to the exact triangle

${p_{2}^{\ast}E_{1}}$${\iota_{C}^{\ast}{\mathcal{K}}_{i+1}}$${{\mathcal{K}}_{i+1}^{2}}$${p_{2}^{\ast}E_{1}[1]}$

gives rise to natural exact triangles

${\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\pi_{2}}({\mathcal{K}}_{i+1}^{2},{\mathcal{T}}_{i}^{2})}$${\iota^{\ast}\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\pi}({\mathcal{K}}_{i+1},{\mathcal{T}}_{i})}$${F_{1}^{[\boldsymbol{d},i]},\quad
i=1,\ldots,l-1,}$$\scriptstyle{b_{i}}$

in the derived category of $\operatorname{\mathrm{Q}}_{2}$. Taking their
direct sums we obtain a diagram of (horizontal and vertical) exact triangles

${\displaystyle\bigoplus_{i=1}^{l}\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\pi_{2}}({\mathcal{K}}_{i}^{2},{\mathcal{T}}_{i}^{2})}$${\displaystyle\bigoplus_{i=1}^{l}\iota^{\ast}\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\pi}({\mathcal{K}}_{i},{\mathcal{T}}_{i})}$${\displaystyle\bigoplus_{i=1}^{l}F_{1}^{[\boldsymbol{d},i]}}$${\displaystyle\bigoplus_{i=1}^{l-1}\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\pi_{2}}({\mathcal{K}}_{i+1}^{2},{\mathcal{T}}_{i}^{2})}$${\displaystyle\bigoplus_{i=1}^{l-1}\iota^{\ast}\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\pi}({\mathcal{K}}_{i+1},{\mathcal{T}}_{i})}$${\displaystyle\bigoplus_{i=1}^{l-1}F_{1}^{[\boldsymbol{d},i]}}$${\Gamma_{2}}$${\iota^{\ast}\Gamma}$${\Gamma_{\operatorname{\mathrm{Q}}_{2}/\operatorname{\mathrm{Q}}}}$$\scriptstyle{\operatorname{\mathbf{R}}_{\pi_{2}}\Delta}$$\scriptstyle{\mathrm{diag}(a_{i})}$$\scriptstyle{\iota^{\ast}\operatorname{\mathbf{R}}_{\pi}\Delta}$$\scriptstyle{\mathrm{diag}(b_{i})}$

andapplying $(-)^{\vee}[1]$ to the bottom horizontal triangle gives the
triangle

${\Gamma_{\operatorname{\mathrm{Q}}_{2}/\operatorname{\mathrm{Q}}}^{\vee}[1]}$${(\iota^{\ast}\Gamma)^{\vee}[1]=\iota^{\ast}(\Gamma^{\vee}[1])=\iota^{\ast}{\mathbb{F}}_{\operatorname{\mathrm{Q}}}}$${\Gamma_{2}^{\vee}[1]={\mathbb{F}}_{\operatorname{\mathrm{Q}}_{2}}}$${\Gamma_{\operatorname{\mathrm{Q}}_{2}/\operatorname{\mathrm{Q}}}^{\vee}[2].}$$\scriptstyle{\psi}$

Note that

$\Gamma_{\operatorname{\mathrm{Q}}_{2}/\operatorname{\mathrm{Q}}}^{\vee}[1]=(F_{1}^{[\boldsymbol{d},l]})^{\vee}\,\in\,\operatorname{Perf}^{[-1,0]}(\operatorname{\mathrm{Q}}_{2}),$

the dual of the $l$-th tautological complex associated to $F_{1}=E_{1}^{\ast}$
(calculated on $\operatorname{\mathrm{Q}}_{2}$). Now set

${\mathbb{F}}_{\operatorname{\mathrm{Q}}_{2}/\operatorname{\mathrm{Q}}}=\operatorname{\mathrm{Cone}}(\psi)=\Gamma_{\operatorname{\mathrm{Q}}_{2}/\operatorname{\mathrm{Q}}}^{\vee}[2]=(F_{1}^{[\boldsymbol{d},l]})^{\vee}[1]\,\in\,\operatorname{Perf}^{[-2,-1]}(\operatorname{\mathrm{Q}}_{2}).$

We have obtained the sought after compatible triple (A.2), and in particular a
morphism of complexes

(A.3)
${{\mathbb{F}}_{\operatorname{\mathrm{Q}}_{2}/\operatorname{\mathrm{Q}}}}$${{\mathbb{L}}_{\operatorname{\mathrm{Q}}_{2}/\operatorname{\mathrm{Q}}}.}$

######  Proposition A.1.

Assume that
${\mathbb{E}}_{\operatorname{\mathrm{Q}}}={\mathbb{F}}_{\operatorname{\mathrm{Q}}}$
and
${\mathbb{E}}_{\operatorname{\mathrm{Q}}_{2}}={\mathbb{F}}_{\operatorname{\mathrm{Q}}_{2}}$
(cf. Remark 3.6), so that (A.3) is a relative perfect obstruction theory for
$\iota\colon\operatorname{\mathrm{Q}}_{2}\hookrightarrow\operatorname{\mathrm{Q}}$.
Assume, in addition, that for all closed points $z=[E\twoheadrightarrow
T_{l}\twoheadrightarrow\cdots\twoheadrightarrow
T_{1}]\in\operatorname{\mathrm{Q}}$ we have

$\mathrm{H}^{0}(C,E_{1}\otimes T_{l}^{\mathrm{fr},\ast}\otimes\omega_{C})=0,$

where $T_{l}^{\mathrm{fr}}$ denotes the free part of $T_{l}$. Then
$F_{1}^{[\boldsymbol{d},l]}\cong
h^{0}(F_{1}^{[\boldsymbol{d},l]})\in\operatorname{Coh}(\operatorname{\mathrm{Q}})$
is a locally free sheaf of rank $(1-g)s_{l}\operatorname{rk}E_{1}-s_{l}\deg
E_{1}+d_{l}\operatorname{rk}E_{1}$. Denote this sheaf by $F$. Then we also
have identities

$\displaystyle\iota_{*}\mathscr{O}^{\operatorname{\mathrm{vir}}}_{\operatorname{\mathrm{Q}}_{2}}$
$\displaystyle=\Lambda^{\bullet}\left(F^{\ast}\right)\otimes\mathscr{O}^{\operatorname{\mathrm{vir}}}_{\operatorname{\mathrm{Q}}}\in
K_{0}(\operatorname{\mathrm{Q}}),$
$\displaystyle\iota_{*}[\operatorname{\mathrm{Q}}_{2}]^{\operatorname{\mathrm{vir}}}$
$\displaystyle=e\left(F\right)\cap[\operatorname{\mathrm{Q}}]^{\operatorname{\mathrm{vir}}}\in
A_{*}(\operatorname{\mathrm{Q}})$

in $K$-theory and Chow, respectively.

###### Proof.

Our assumption, combined with Serre duality and Equation A.1, says that

$h^{1}(F_{1}^{[\boldsymbol{d},l]})\big{|}_{z}=0,$

from which we deduce that

$F_{1}^{[\boldsymbol{d},l]}=h^{0}(F_{1}^{[\boldsymbol{d},l]})=\pi_{\ast}(p^{\ast}F_{1}\otimes{\mathcal{T}}_{l})\,\in\,\operatorname{Coh}(\operatorname{\mathrm{Q}})$

is locally free. Let us set

$F=F_{1}^{[\boldsymbol{d},l]},\qquad\mathbf{V}(F)=\operatorname{Spec}_{\mathscr{O}_{\operatorname{\mathrm{Q}}}}\operatorname{Sym}F^{\ast}.$

We know that a canonical section
$\sigma\in\mathrm{H}^{0}(\operatorname{\mathrm{Q}},\mathbf{V}(F))=\operatorname{Hom}_{\mathscr{O}_{\operatorname{\mathrm{Q}}}}(\mathscr{O}_{\operatorname{\mathrm{Q}}},F)$
cuts out $\operatorname{\mathrm{Q}}_{2}$. In other words, we have a cartesian
diagram

${\operatorname{\mathrm{Q}}_{2}}$${\operatorname{\mathrm{Q}}}$${\operatorname{\mathrm{Q}}}$${\mathbf{V}(F)}$$\scriptstyle{\square}$$\scriptstyle{\iota}$$\scriptstyle{\iota}$$\scriptstyle{\sigma}$$\scriptstyle{0}$

where ‘$0$’ denotes the zero section. We claim that

(A.4)
$0^{!}[\operatorname{\mathrm{Q}}]^{\operatorname{\mathrm{vir}}}=[\operatorname{\mathrm{Q}}_{2}]^{\operatorname{\mathrm{vir}}}$

in $A_{\ast}(\operatorname{\mathrm{Q}}_{2})$. Note that, since the zero
section is a regular immersion, its cotangent complex is

${\mathbb{L}}_{\operatorname{\mathrm{Q}}/\mathbf{V}(F)}=(\mathscr{I}/\mathscr{I}^{2})[1]=F^{\ast}[1],$

where $\mathscr{I}$ is the ideal sheaf of the zero section. Its restriction to
$\operatorname{\mathrm{Q}}_{2}$ is

$\iota^{\ast}{\mathbb{L}}_{\operatorname{\mathrm{Q}}/\mathbf{V}(F)}=F^{\ast}\big{|}_{\operatorname{\mathrm{Q}}_{2}}[1]={\mathbb{F}}_{\operatorname{\mathrm{Q}}_{2}/\operatorname{\mathrm{Q}}}.$

We are then in the situation of a compatible diagram

${\iota^{\ast}{\mathbb{F}}_{\operatorname{\mathrm{Q}}}}$${{\mathbb{F}}_{\operatorname{\mathrm{Q}}_{2}}}$${\iota^{\ast}{\mathbb{L}}_{\operatorname{\mathrm{Q}}/\mathbf{V}(F)}}$${\iota^{\ast}{\mathbb{L}}_{\operatorname{\mathrm{Q}}}}$${{\mathbb{L}}_{\operatorname{\mathrm{Q}}_{2}}}$${{\mathbb{L}}_{\operatorname{\mathrm{Q}}_{2}/\operatorname{\mathrm{Q}}}}$$\scriptstyle{\psi}$

and [34, Thm. 1] gives the identity (A.4). Pushing (A.4) forward in Chow and
in $K$-theory gives the required identities. ∎

######  Remark A.2.

We expect to be possible to drop the assumption that
$F_{1}^{[\boldsymbol{d},l]}$ is a locally free sheaf, but only a perfect
2-term complex, by using a recent result of Oh–Thomas [46], in order to obtain
analogous pushforward formulae.

##  Appendix B Virtual $\chi_{-y}$-genus

Consider $\mathbf{k}={\mathbb{C}}$ for this section. Let $C$ be a smooth
projective curve, $E$ a locally free sheaf of rank $r$ and fix tuples
$\boldsymbol{s}=(s_{1}\leq\dots\leq s_{l})$ and
$\boldsymbol{d}=(d_{1},\dots,d_{l})$. Notice that by deformation invariance,
we can always assume that there exists a splitting
$E=\bigoplus_{\alpha=1}^{r}L_{\alpha}$. We proved in Corollary 3.7 that the
hyperquot scheme
$\operatorname{\mathrm{Q}}=\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d})$
admits a virtual structure sheaf

$\displaystyle\mathscr{O}_{\operatorname{\mathrm{Q}}}^{\operatorname{\mathrm{vir}}}\in
K_{0}(\operatorname{\mathrm{Q}}).$

The _virtual tangent bundle_ is the class in $K$-theory of the dual of the
perfect obstruction theory of Theorem 3.2, which by Remark 3.4 can be computed
as

$\displaystyle
T_{\operatorname{\mathrm{Q}}}^{\operatorname{\mathrm{vir}}}=\sum_{i=1}^{l}\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\pi}({\mathcal{K}}_{i},{\mathcal{T}}_{i})-\sum_{i=1}^{l-1}\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\pi}({\mathcal{K}}_{i+1},{\mathcal{T}}_{i})\in
K^{0}(\operatorname{\mathrm{Q}}).$

For any locally free sheaf $V$ on a projective scheme $X$, define the class in
$K$-theory

$\Lambda_{t}V=\sum_{i=0}^{\operatorname{rk}V}t^{i}\cdot\Lambda^{i}V\in
K^{0}(X)[t],$

and extend it by linearity, for any $V\in K^{0}(X)$, to a class
$\Lambda_{t}V\in K^{0}(X)[\\![t]\\!]$ (see e.g. [19, Def. 5.1]). Following
[19], we define the _virtual $\chi_{-y}$-genus_ of
$\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d})$ as555The
$\chi_{-y}$-genus is polynomial in $y$ by [19, Prop. 5.3].

$\displaystyle\chi^{\operatorname{\mathrm{vir}}}_{-y}(\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d}))=\chi\left(\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d}),\mathscr{O}^{\operatorname{\mathrm{vir}}}\otimes\Lambda_{-y}(T^{\operatorname{\mathrm{vir}}})^{\vee}\right)\in{\mathbb{Z}}[y].$

We define the generating series of $\chi_{-y}$-genera

$\displaystyle
Z_{C,E,\boldsymbol{s}}(\boldsymbol{q},y)=\sum_{\boldsymbol{d}}\boldsymbol{q}^{\boldsymbol{d}}\cdot\chi^{\operatorname{\mathrm{vir}}}_{-y}(\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d}))\in{\mathbb{Z}}[y](\\!(\boldsymbol{q})\\!),$

where we used the multinotation
$\boldsymbol{q}^{\boldsymbol{d}}=q_{1}^{d_{1}}\cdots q_{l}^{d_{l}}$. This
generating series is well defined as a Laurent power series in
$\boldsymbol{q}$, since the hyperquot scheme
$\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d})$ is empty for
$\boldsymbol{d}$ sufficiently negative.

As an application, we prove that the $\chi_{-y}$-genus of the hyperquot
schemes only depends on the genus of the curve and the degrees of the line
bundles $L_{\alpha}$. We remark that, due to the presence of nesting
conditions, we cannot apply verbatim the standard techniques of [18].

######  Proposition B.1.

The virtual $\chi_{-y}$-genus
$\chi_{-y}^{\operatorname{\mathrm{vir}}}(\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d}))$
depends on $(C,L_{1},\dots,L_{r})$ only on the genus $g=g(C)$ and the degrees
$\deg L_{\alpha}$, for $\alpha=1,\dots,r$.

###### Proof.

Consider the action of the torus $\mathbf{T}$ on
$\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d})$ as in Section 5.1.
To simplify the notation, we write

$\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d})^{\mathbf{T}}=\coprod_{\underline{\boldsymbol{s}},\underline{\boldsymbol{d}}}Q_{\underline{\boldsymbol{s}},\underline{\boldsymbol{d}}}$

the decomposition of the fixed locus of Lemma 5.1, where

$Q_{\underline{\boldsymbol{s}},\underline{\boldsymbol{d}}}=\prod_{\alpha=1}^{r}\operatorname{Quot}_{C}(L_{\alpha},\boldsymbol{s}_{\alpha},\boldsymbol{d}_{\alpha}).$

Denote by $w_{1},\ldots,w_{r}$ the irreducible representations of the torus
$\mathbf{T}$.

By the $K$-theoretic virtual localisation [19], each fixed component of
$\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d})$ inherits a virtual
structure sheaf and

(B.1)
$\displaystyle\chi^{\operatorname{\mathrm{vir}}}_{-y}(\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d}))=\left.\left(\sum_{\underline{\boldsymbol{s}},\underline{\boldsymbol{d}}}\chi\left(Q_{\underline{\boldsymbol{s}},\underline{\boldsymbol{d}}},\mathscr{O}^{\operatorname{\mathrm{vir}}}\otimes\frac{\Lambda_{-y}(T^{\operatorname{\mathrm{vir}}}|_{Q_{\underline{\boldsymbol{s}},\underline{\boldsymbol{d}}}})^{\vee}}{\Lambda_{-1}(N^{\operatorname{\mathrm{vir}}}_{Q_{\underline{\boldsymbol{s}},\underline{\boldsymbol{d}}}})^{\vee}}\right)\right)\right|_{w_{\alpha}=1}.$

Here, for each component of the fixed locus $\chi(\cdot)$ is performed
$\mathbf{T}$-equivariantly, and the (restriction of the) virtual tangent
bundle is

$T^{\operatorname{\mathrm{vir}}}|_{Q_{\underline{\boldsymbol{s}},\underline{\boldsymbol{d}}}}=T^{\operatorname{\mathrm{vir}}}_{Q_{\underline{\boldsymbol{s}},\underline{\boldsymbol{d}}}}+N^{\operatorname{\mathrm{vir}}}_{Q_{\underline{\boldsymbol{s}},\underline{\boldsymbol{d}}}}\in
K^{0}(Q_{\underline{\boldsymbol{s}},\underline{\boldsymbol{d}}}),$

where the _virtual normal bundle_
$N^{\operatorname{\mathrm{vir}}}_{Q_{\underline{\boldsymbol{s}},\underline{\boldsymbol{d}}}}$
is the $\mathbf{T}$-movable part and the virtual tangent bundle
$T^{\operatorname{\mathrm{vir}}}_{Q_{\underline{\boldsymbol{s}},\underline{\boldsymbol{d}}}}$
of $Q_{\underline{\boldsymbol{s}},\underline{\boldsymbol{d}}}$ is the
$\mathbf{T}$-fixed part by [27].

Fix now $\underline{\boldsymbol{s}},\underline{\boldsymbol{d}}$ and consider
the cartesian diagram

${Q_{\underline{\boldsymbol{s}},\underline{\boldsymbol{d}}}\times
C}$${\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d})\times
C}$${Q_{\underline{\boldsymbol{s}},\underline{\boldsymbol{d}}}}$${\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d}).}$$\scriptstyle{\iota}$$\scriptstyle{\pi}$$\scriptstyle{\pi}$$\scriptstyle{\iota}$

The universal sheaves ${\mathcal{K}}_{i},{\mathcal{T}}_{i}$ on
$\operatorname{Quot}_{C}(E,\boldsymbol{s},\boldsymbol{d})\times C$ restrict
via $\iota$ as

$\displaystyle{\mathcal{K}}_{i}$
$\displaystyle=\bigoplus_{\alpha=1}^{r}{\mathcal{K}}_{i,\alpha}$
$\displaystyle{\mathcal{T}}_{i}$
$\displaystyle=\bigoplus_{\alpha=1}^{r}{\mathcal{T}}_{i,\alpha},$

where ${\mathcal{K}}_{i,\alpha},{\mathcal{T}}_{i,\alpha}$ are the universal
sheaves on each factor
$\operatorname{Quot}_{C}(L_{\alpha},\boldsymbol{s}_{\alpha},\boldsymbol{d}_{\alpha})\times
C$ of $Q_{\underline{\boldsymbol{s}},\underline{\boldsymbol{d}}}\times C$.

The restriction of the virtual tangent bundle equivariantly decomposes as

$\displaystyle
T^{\operatorname{\mathrm{vir}}}|_{Q_{\underline{\boldsymbol{s}},\underline{\boldsymbol{d}}}}=\sum_{\alpha,\beta=1}^{r}\left(\sum_{i=1}^{l}\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\pi}({\mathcal{K}}_{i,\alpha},{\mathcal{T}}_{i,\beta})-\sum_{i=1}^{l-1}\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\pi}({\mathcal{K}}_{i+1,\alpha},{\mathcal{T}}_{i,\beta})\right)w_{\beta}w_{\alpha}^{-1}\in
K^{0}(Q_{\underline{\boldsymbol{s}},\underline{\boldsymbol{d}}}),$

and its $\mathbf{T}$-fixed part is

$\displaystyle
T^{\operatorname{\mathrm{vir}}}_{Q_{\underline{\boldsymbol{s}},\underline{\boldsymbol{d}}}}=\sum_{\alpha=1}^{r}\left(\sum_{i=1}^{l}\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\pi}({\mathcal{K}}_{i,\alpha},{\mathcal{T}}_{i,\alpha})-\sum_{i=1}^{l-1}\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\pi}({\mathcal{K}}_{i+1,\alpha},{\mathcal{T}}_{i,\alpha})\right).$

By the discussions of Section 5.2 it is easy to see that, whenever
$Q_{\underline{\boldsymbol{s}},\underline{\boldsymbol{d}}}$ is non-empty, it
is isomorphic to a product

(B.2) $\displaystyle
Q_{\underline{\boldsymbol{s}},\underline{\boldsymbol{d}}}\cong\prod_{\alpha}C^{[\boldsymbol{n}_{\alpha}]},$

where each $\boldsymbol{n}_{\alpha}=(n_{1,\alpha}\leq\dots\leq
n_{k_{\alpha},\alpha})$ for some $k_{\alpha}\geq 1$. Therefore
$Q_{\underline{\boldsymbol{s}},\underline{\boldsymbol{d}}}$ is smooth,
unobstructed and the virtual tangent bundle coincides with the actual tangent
bundle (as classes in $K$-theory)

$T^{\operatorname{\mathrm{vir}}}_{Q_{\underline{\boldsymbol{s}},\underline{\boldsymbol{d}}}}=T_{Q_{\underline{\boldsymbol{s}},\underline{\boldsymbol{d}}}}\in
K^{0}(Q_{\underline{\boldsymbol{s}},\underline{\boldsymbol{d}}}),$

which implies that the virtual structure sheaf coincides with the structure
sheaf by [59, Cor. 4.5]. On the product
$\prod_{\alpha}C^{[\boldsymbol{n}_{\alpha}]}\times C$, there are universal
exact sequences of sheaves

${0}$${{\mathcal{I}}_{i,\alpha}}$${\mathscr{O}}$${\mathscr{O}_{{\mathcal{Z}}_{i,\alpha}}}$${0}$

where we suppressed some obvious pullback maps. Under the identification
(B.2), the universal sequences of sheaves on
$Q_{\underline{\boldsymbol{s}},\underline{\boldsymbol{d}}}\times C$

${0}$${{\mathcal{K}}_{i,\alpha}}$${L_{\alpha}}$${{\mathcal{T}}_{i,\alpha}}$${0}$

are of one of the following two forms

(B.3)
${0}$${L_{\alpha}\otimes{\mathcal{I}}_{i,\alpha}}$${L_{\alpha}}$${L_{\alpha}\otimes\mathscr{O}_{{\mathcal{Z}}_{i,\alpha}}}$${0}$${0}$${L_{\alpha}}$${L_{\alpha}}$${0.}$$\scriptstyle{\sim}$

Using (B.3), we have that under the identification (B.2), for each
$\alpha,\beta=1,\dots,r$ and $i,j=1,\dots,l$ we can represent the classes

$\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\pi}({\mathcal{K}}_{i,\alpha},{\mathcal{T}}_{j,\beta})\in
K^{0}\left(Q_{\underline{\boldsymbol{s}},\underline{\boldsymbol{d}}}\times
C\right)$

as a linear combination of classes

$\displaystyle\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\pi}({\mathcal{I}}_{i,\alpha},{\mathcal{I}}_{j,\beta}\otimes
L_{\alpha}^{-1}L_{\beta}),\,\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\pi}({\mathcal{I}}_{i,\alpha},\mathscr{O}\otimes
L_{\alpha}^{-1}L_{\beta})\in
K^{0}\left(\prod_{\alpha}C^{[\boldsymbol{n}_{\alpha}]}\times C\right).$

Consider now a tuple $\boldsymbol{n}_{\alpha}=(n_{1,\alpha}\leq\dots\leq
n_{k_{\alpha},\alpha})$. We have the isomorphism

(B.4) $\displaystyle C^{[\boldsymbol{n}_{\alpha}]}\cong
C^{(n_{1,\alpha})}\times C^{(n_{2,\alpha}-n_{1,\alpha})}\times\dots\times
C^{(n_{k_{\alpha},\alpha}-n_{k_{\alpha}-1,\alpha})}.$

Again, on each factor $C^{(n_{i,\alpha}-n_{i-1,\alpha})}\times C$, there is a
universal sequence

${0}$${{\mathcal{J}}_{i,\alpha}}$${\mathscr{O}}$${\mathscr{O}_{{\mathcal{W}}_{i,\alpha}}}$${0,}$

and under the isomorphism (B.4), we have the identification

${\mathcal{I}}_{i,\alpha}\cong{\mathcal{J}}_{i,\alpha}\otimes{\mathcal{J}}_{i-1,\alpha}\otimes\dots\otimes{\mathcal{J}}_{1,\alpha}.$

To sum up, we showed that

$Q_{\underline{\boldsymbol{s}},\underline{\boldsymbol{d}}}\cong\prod_{k\in
I}C^{(n_{k})},$

for a suitable family of indices $I$ and suitable $(n_{k})_{k\in I}$, and that
under this identification, both
$T^{\operatorname{\mathrm{vir}}}|_{Q_{\underline{\boldsymbol{s}},\underline{\boldsymbol{d}}}},N^{\operatorname{\mathrm{vir}}}_{Q_{\underline{\boldsymbol{s}},\underline{\boldsymbol{d}}}}$
are linear combination of classes of the form

(B.5)
$\displaystyle\operatorname{\mathbf{R}\kern-0.24994pt\mathscr{H}\kern-3.00003pt\mathit{om}}_{\pi}\left(\bigotimes_{k\in
K}{\mathcal{J}}_{k},\bigotimes_{m\in M}{\mathcal{J}}_{m}\otimes L_{a}\right),$

for some families of indices $K,M$ and where $L_{a}$ is a line bundle on $C$
obtained as a linear combination of the $\set{L_{\alpha}}_{\alpha}$.

We are now ready to show that each localized contribution of (B.1) can be
expressed as a polynomial on $g=g(C)$ and the degrees $\deg L_{\alpha}$. By
the fact that $\mathscr{O}^{\operatorname{\mathrm{vir}}}=\mathscr{O}$ and
($\mathbf{T}$-equivariant) Grothendieck–Riemann–Roch, we have

$\displaystyle\chi\left.\left(Q_{\underline{\boldsymbol{s}},\underline{\boldsymbol{d}}},\mathscr{O}^{\operatorname{\mathrm{vir}}}\otimes\frac{\Lambda_{-y}(T^{\operatorname{\mathrm{vir}}}|_{Q_{\underline{\boldsymbol{s}},\underline{\boldsymbol{d}}}})^{\vee}}{\Lambda_{-1}(N^{\operatorname{\mathrm{vir}}}_{Q_{\underline{\boldsymbol{s}},\underline{\boldsymbol{d}}}})^{\vee}}\right)\right)$
$\displaystyle=\int_{Q_{\underline{\boldsymbol{s}},\underline{\boldsymbol{d}}}}\frac{\mathrm{ch}\left(\Lambda_{-y}(T^{\operatorname{\mathrm{vir}}}|_{Q_{\underline{\boldsymbol{s}},\underline{\boldsymbol{d}}}})^{\vee}\right)}{\mathrm{ch}\left(\Lambda_{-1}(N^{\operatorname{\mathrm{vir}}}_{Q_{\underline{\boldsymbol{s}},\underline{\boldsymbol{d}}}})^{\vee}\right)}\cdot\mathrm{td}(T_{Q_{\underline{\boldsymbol{s}},\underline{\boldsymbol{d}}}})$
$\displaystyle=\int_{\prod_{k\in I}C^{(n_{k})}}f,$

where $f$ is a polynomial in the Chern classes of $K$-theory classes of the
form (B.5). We conclude by [42, Prop. 5.3] that the dependence of the last
integral is only on the genus $g=g(C)$ and the degrees of the line bundles
$L_{\alpha}$. ∎

## References

  * [1] Noah Arbesfeld, Drew Johnson, Woonam Lim, Dragos Oprea, and Rahul Pandharipande, The virtual $K$-theory of Quot schemes of surfaces, J. Geom. Phys. 164 (2021), Paper No. 104154, 36.
  * [2] Massimo Bagnarol, Barbara Fantechi, and Fabio Perroni, On the motive of Quot schemes of zero-dimensional quotients on a curve, New York J. Math. 26 (2020), 138–148.
  * [3] Kai Behrend and Barbara Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997), no. 1, 45–88.
  * [4] Aaron Bertram, Quantum Schubert calculus, Adv. Math. 128 (1997), no. 2, 289–305.
  * [5] Andrzej Białynicki-Birula, Some theorems on actions of algebraic groups, Ann. of Math. (2) 98 (1973), 480–497.
  * [6] Emili Bifet, Sur les points fixes du schéma ${\rm Quot}_{{\mathcal{O}}^{r}_{X}/X/k}$ sous l’action du tore ${\bf G}^{r}_{m,k}$, C. R. Acad. Sci. Paris Sér. I Math. 309 (1989), no. 9, 609–612.
  * [7] Arkadij Bojko, Wall-crossing for punctual Quot-schemes, ArXiv: 2111.11102, 2021.
  * [8] Alberto Cazzaniga, Dimbinaina Ralaivaosaona, and Andrea T. Ricolfi, Higher rank motivic Donaldson-Thomas invariants of $\mathbb{A}^{3}$ via wall-crossing, and asymptotics, Math. Proc. Cambridge Philos. Soc. 174 (2023), no. 1, 97–122.
  * [9] Alberto Cazzaniga and Andrea T. Ricolfi, Framed motivic Donaldson-Thomas invariants of small crepant resolutions, Math. Nachr. 295 (2022), no. 6, 1096–1112.
  * [10] Linda Chen, Poincaré polynomials of hyperquot schemes, Math. Ann. 321 (2001), no. 2, 235–251.
  * [11] Linda Chen, Quantum cohomology of flag manifolds, Adv. Math. 174 (2003), no. 1, 1–34.
  * [12] Ionuţ Ciocan-Fontanine, Quantum cohomology of flag varieties, Int. Math. Res. Not. IMRN (1995), no. 6, 263–277.
  * [13] Ionuţ Ciocan-Fontanine, On quantum cohomology rings of partial flag varieties, Duke Math. J. 98 (1999), no. 3, 485–524.
  * [14] Ionuţ Ciocan-Fontanine, The quantum cohomology ring of flag varieties, Trans. Amer. Math. Soc. 351 (1999), no. 7, 2695–2729.
  * [15] Ionuţ Ciocan-Fontanine, Bumsig Kim, and Davesh Maulik, Stable quasimaps to GIT quotients, J. Geom. Phys. 75 (2014), 17–47.
  * [16] Michele Cirafici, On the M2-brane index on noncommutative crepant resolutions, Lett. Math. Phys. 112 (2022), no. 5, Paper No. 88, 43.
  * [17] Michele Del Zotto, Nikita Nekrasov, Nicolò Piazzalunga, and Maxim Zabzine, Playing with the index of M-theory, Comm. Math. Phys. 396 (2022), no. 2, 817–865.
  * [18] Geir Ellingsrud, Lothar Göttsche, and Manfred Lehn, On the cobordism class of the Hilbert scheme of a surface, J. Algebraic Geom. 10 (2001), no. 1, 81–100.
  * [19] Barbara Fantechi and Lothar Göttsche, Riemann–Roch theorems and elliptic genus for virtually smooth schemes, Geom. Topol. 14 (2010), no. 1, 83–115.
  * [20] Nadir Fasola and Sergej Monavari, Tetrahedron instantons in Donaldson-Thomas theory, ArXiv:2306.07145, 2023.
  * [21] Nadir Fasola, Sergej Monavari, and Andrea T. Ricolfi, Higher rank $K$-theoretic Donaldson-Thomas theory of points, Forum Math. Sigma 9 (2021), 1–51, e15.
  * [22] Soheyla Feyzbakhsh and Richard P. Thomas, Rank $r$ DT theory from rank $0$, To appear in Duke Math. J., ArXiv: 2103.02915, 2021.
  * [23] Soheyla Feyzbakhsh and Richard P. Thomas, Rank $r$ DT theory from rank 1, J. Am. Math. Soc. 36 (2023), no. 3, 795–826.
  * [24] Oscar García-Prada, Jochen Heinloth, and Alexander Schmitt, On the motives of moduli of chains and Higgs bundles, J. Eur. Math. Soc. (JEMS) 16 (2014), no. 12, 2617–2668.
  * [25] Amin Gholampour, Artan Sheshmani, and Shing-Tung Yau, Localized Donaldson–Thomas theory of surfaces, Amer. J. Math. 142 (2020), no. 2, 405–442.
  * [26] William D. Gillam, Deformation of quotients on a product, ArXiv:1103.5482, 2011.
  * [27] Tom Graber and Rahul Pandharipande, Localization of virtual classes, Invent. Math. 135 (1999), no. 2, 487–518.
  * [28] Alexander Grothendieck, Techniques de construction et théorèmes d’existence en géométrie algébrique. IV. Les schémas de Hilbert, Séminaire Bourbaki, Vol. 6, Soc. Math. France, Paris, 1961, pp. Exp. No. 221, 249–276.
  * [29] Victoria Hoskins and Simon Pepin Lehalleur, A formula for the Voevodsky motive of the moduli stack of vector bundles on a curve, Geom. Topol. 25 (2021), no. 7, 3555–3589.
  * [30] Victoria Hoskins and Simon Pepin Lehalleur, On the Voevodsky motive of the moduli space of Higgs bundles on a curve, Selecta Math. (N.S.) 27 (2021), no. 1, Paper No. 11, 37.
  * [31] Daniel Huybrechts and Manfred Lehn, The geometry of moduli spaces of sheaves, Second Edition, Cambridge University Press, 2010, pp. xviii+325.
  * [32] Luc Illusie, Complexe cotangent et déformations. I, Lecture Notes in Mathematics, vol. 239, Springer-Verlag, Berlin-New York, 1971.
  * [33] Mikhail Kapranov, The elliptic curve in the S-duality theory and Eisenstein series for Kac-Moody groups, ArXiv:0001005, 2000.
  * [34] Bumsig Kim, Andrew Kresch, and Tony Pantev, Functoriality in intersection theory and a conjecture of Cox, Katz, and Lee, J. Pure Appl. Algebra 179 (2003), no. 1-2, 127–136.
  * [35] Martijn Kool and Richard P. Thomas, Stable pairs with descendents on local surfaces I: the vertical component, Pure Appl. Math. Q. 13 (2017), no. 4, 581–638.
  * [36] Ties Laarakker, Monopole contributions to refined Vafa-Witten invariants, Geom. Topol. 24 (2020), no. 6, 2781–2828.
  * [37] Woonam Lim, Virtual $\chi_{-y}$-genera of Quot schemes on surfaces, J. Lond. Math. Soc. (2) 104 (2021), no. 3, 1300–1341.
  * [38] Cristina Manolache, Virtual pull-backs, J. Algebr. Geom. 21 (2012), no. 2, 201–245.
  * [39] Alina Marian and Andrei Neguţ, The cohomology of the Quot scheme on a smooth curve as a Yangian representation, ArXiv:2307.13671, 2023.
  * [40] Alina Marian and Dragos Oprea, Virtual intersections on the Quot scheme and Vafa-Intriligator formulas, Duke Math. J. 136 (2007), no. 1, 81–113.
  * [41] Takuro Mochizuki, The structure of the cohomology ring of the filt schemes, ArXiv:0301184, 2003.
  * [42] Sergej Monavari, Double nested Hilbert schemes and the local stable pairs theory of curves, Compos. Math. 158 (2022), no. 9, 1799–1849.
  * [43] Sergej Monavari and Andrea T. Ricolfi, On the motive of the nested Quot scheme of points on a curve, J. Algebra 610 (2022), 99–118.
  * [44] Sergej Monavari and Andrea T. Ricolfi, Sur la lissité du schéma Quot ponctuel emboîté, Can. Math. Bull. 66 (2023), no. 1, 178–184.
  * [45] Nikita Nekrasov and Nicolò Piazzalunga, Magnificent four with colors, Comm. Math. Phys. 372 (2019), no. 2, 573–597.
  * [46] Jeongseok Oh and Richard P. Thomas, Quantum Lefschetz without curves, ArXiv:2403.01011, 2024.
  * [47] Dragos Oprea, Notes on the moduli space of stable quotients, In: Compactifying moduli spaces, Adv. Courses Math. CRM Barcelona, Birkhäuser/Springer, Basel, 2016, pp. 69–135.
  * [48] Dragos Oprea and Rahul Pandharipande, Quot schemes of curves and surfaces: virtual classes, integrals, Euler characteristics, Geom. Topol. 25 (2021), no. 7, 3425–3505.
  * [49] Dragos Oprea and Shubham Sinha, Euler characteristics of tautological bundles over Quot schemes of curves, Adv. Math. 418 (2023), Paper No. 108943, 45.
  * [50] Mihnea Popa and Mike Roth, Stable maps and Quot schemes, Invent. Math. 152 (2003), no. 3, 625–663.
  * [51] Parvez Rasul and Ronnie Sebastian, Irreducibility of some nested Quot schemes, ArXiv:2404.05532, 2024.
  * [52] Andrea T. Ricolfi, On the motive of the Quot scheme of finite quotients of a locally free sheaf, J. Math. Pures Appl. (9) 144 (2020), 50–68.
  * [53] Andrea T. Ricolfi, The equivariant Atiyah class, C. R., Math., Acad. Sci. Paris 359 (2021), no. 3, 257–282.
  * [54] Giorgio Scattareggia, A perfect obstruction theory for moduli of coherent systems, ArXiv:1803.00869, 2018.
  * [55] Edoardo Sernesi, Deformations of algebraic schemes, Grundlehren der Mathematischen Wissenschaften, vol. 334, Springer-Verlag, Berlin, 2006.
  * [56] Bernd Siebert, Virtual fundamental classes, global normal cones and Fulton’s canonical classes, Frobenius manifolds, Aspects Math., vol. 36, Friedr. Vieweg, Wiesbaden, 2004, pp. 341–358.
  * [57] Shubham Sinha, The virtual intersection theory of isotropic Quot schemes, J. Geom. Phys. 198 (2024), 105128.
  * [58] Samuel Stark, On the Quot scheme $\mathrm{Quot}^{l}_{S}(\mathcal{E})$, ArXiv:2107.03991, 2021.
  * [59] Richard P. Thomas, A K-theoretic Fulton class, Facets of algebraic geometry. Vol. II, London Math. Soc. Lecture Note Ser., vol. 473, Cambridge Univ. Press, Cambridge, 2022, pp. 367–379.

Sergej Monavari

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Andrea T. Ricolfi

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