# Non-reductive geometric invariant theory and Thom polynomials

Gergely Bérczi  Department of Mathematics, Aarhus University
<EMAIL_ADDRESS>To Eszter, Dani, Flora and Epsilon

###### Abstract.

We combine recently developed intersection theory for non-reductive geometric
invariant theoretic quotients with equivariant localisation to prove a formula
for Thom polynomials of Morin singularities. These formulas use only toric
combinatorics of certain partition polyhedra, and our new approach circumvents
the poorly understood Borel geometry of existing models.

This work was partially supported by the Aarhus University Research Foundation
grant AUFF 29289.

###### Contents

  1. 1 Introduction
  2. 2 The strategy and the result
  3. 3 Hilbert schemes and tautological integrals
  4. 4 The test curve model of Morin singularities
  5. 5 Non-reductive geometric invariant theory
  6. 6 Moment maps and cohomology of non-reductive quotients
  7. 7 Equivariant localisation and multidegrees
  8. 8 NRGIT quotients versus curvilinear Hilbert schemes
  9. 9 Equivariant localisation and the Residue Vanishing Theorem
  10. 10 Examples: Thom polynomials for $k\leq 5$
  11. 11 The blowing up process : polynomial vs monomial maps

##  1\. Introduction

We combine the results of [4] with a blow-up process to construct a new non-
reductive GIT type model for the moduli space of $k$-jets, which parametrises
holomorphic $k$-jets of map germs $({\mathbb{C}},0)\to({\mathbb{C}}^{n},0)$
modulo polynomial reparametrisations of ${\mathbb{C}}$ at the origin.
Intersection theory developed in [8] for non-reductive GIT quotients
accompanied with equivariant localisation and a residue vanishing theorem
leads us to a closed iterated residue formula for Thom polynomials of Morin
singularities. Our formula builds only on the toric geometry of partition
polyhedra, and we avoid the use of unknown Borel geometry appeared in the
earlier works [26, 11, 36].

Global singularity theory studies the topology of holomorphic maps between
complex manifolds. A fundamental question is how to describe the topological
locus in the source space where a generic map has a given type of singularity.
A more precise formulation of this problem goes back to Thom [42] and reads as
follows. Let $f:N\to M$ be a map between two complex manifolds, $N$ and $M$,
of dimensions $n\leq m$. We say that $p\in N$ is a singular point of $f$ if
the rank of the differential $df_{p}:T_{p}N\to T_{f(p)}M$ is less than $n$. In
order to introduce a finer classification of singular points, choose local
coordinates near $p\in N$ and $f(p)\in M$, and consider the resulting map-germ
$f_{p}:({\mathbb{C}}^{n},0)\to({\mathbb{C}}^{m},0)$, which may be thought of
as a sequence of $m$ power series in $n$ variables without constant terms. Let
$J(n,m)$ denote the space of such map-germs endowed with the action of the
group $\mathrm{Diff}_{n}\times\mathrm{Diff}_{m}$ of infinitesimal local
coordinate changes. The $\mathrm{Diff}_{n}\times\mathrm{Diff}_{m}$-orbits or,
more generally, $\mathrm{Diff}_{n}\times\mathrm{Diff}_{m}$-invariant subsets
$O\subset J(n,m)$ are called singularities. For a singularity $O$ and
holomorphic $f:N\to M$, we can define the set

$\Sigma_{O}(f)=\\{p\in N;\;f_{p}\in O\\},$

which is independent of any coordinate choices. $N$. Assuming $N$ is compact
and $f$ is sufficiently generic, $\Sigma_{O}(f)$ is an analytic subvariety of
$N$ giving a Poincaré dual class $[\Sigma_{O}(f)]\in H^{*}(N,\mathbb{Z})$.
This problem was first studied by René Thom (cf. [42, 23]) in the category of
smooth varieties and smooth maps; in this case cohomology with
${\mathbb{Z}}/2{\mathbb{Z}}$-coefficients is used. Thom discovered that to
every singularity $O$ one can associate a bivariant characteristic class
$\tau_{O}$, which, when evaluated on the pair $(TN,f^{*}TM)$ produces the
Poincaré dual class $[\Sigma_{O}(f)]$. One of the consequences of this result
is that the class $[\Sigma_{O}(f)]$ depends only on the homotopy class of $f$.

The structure of the $\mathrm{Diff}_{n}\times\mathrm{Diff}_{m}$-action on
$J(n,m)$ is rather complicated; even the parametrization of the orbits is
difficult. There is, however, a simple invariant on the space of orbits: to
each map-germ
$f=(f_{1},\ldots,f_{m}):(\mathbb{C}^{n},0)\to(\mathbb{C}^{m},0)$, we can
associate the finite-dimensional nilpotent algebra
$A_{f}=(x_{1},\ldots,x_{n})/(f_{1},\ldots,f_{m})$ defined as the quotient of
the algebra of power series with no constant term by the ideal generated by
the pull-back subalgebra $f^{*}(y_{1}\ldots,y_{m})$. This algebra $A_{f}$ is
trivial if the map-germ $f$ is nonsingular, and it does not change along a
$\mathrm{Diff}_{n}\times\mathrm{Diff}_{m}$-orbit. Hence, for a fixed nilpotent
algebra $A$ with $n$ generators one can associate the invariant subset
$\Sigma_{A}=\\{f\in J(n,m):A_{f}\simeq A\\}$, the set of jets with local
algebra isomorphic to $A$. The Thom principle in this holomorphic setting (cf
[27, 20, 11]) for $\mathscr{O}=\Sigma_{A}$ tells that there is a jet order $k$
depending only on $A$ such that

$[\Sigma_{A}(f)]=\mathrm{eP}[\Sigma_{A},J_{k}(n,m)](f^{*}TN,TM),$

is obtained by substituting the Chern roots of $TN,f^{*}TM$ into the
$\mathrm{GL}(n)\times\mathrm{GL}(m)$-equivariant dual of $\Sigma_{A}$ in
$J_{k}(n,m)$. This equivariant dual sits in the ring
${\mathbb{C}}[x_{1},\ldots,x_{n},y_{1},\ldots,y_{m}]^{S_{n}\times S_{m}}$ of
bi-symmetric polynomials on $n+m$ variables. Damon and Ronga [14, 37] proved
that $[\Sigma_{A}(f)]$ depends on $TN,f^{*}TM$ only through the Chern classes
of the difference bundle $TN-f^{*}TM$, and hence
$[\Sigma_{A}(f)]=\mathrm{Tp}_{A}(c_{1},c_{2},\ldots)$ is a polynomial in these
classes, which we call the Thom polynomial of $A$. Calculation of these
polynomials remains a major open problem ever since then, see e.g [1, 35, 27,
19].

In this paper we study the Morin case, i.e when
$A_{k}={\mathbb{C}}[t]/t^{k+1}$ for some $k$. The problem of calculating
$\mathrm{Tp}_{k}=\mathrm{Tp}_{A_{k}}$ goes back to Thom [42, 40]. The case
$k=1$ is the classical formula of Porteous [32]: $\mathrm{Tp}_{1}=c_{m-n+1}$.
The Thom polynomial in the $k=2$ case was computed by Ronga in [37]. An
explicit formula for $\mathrm{Tp}_{3}$ was proposed in [7], and P. Pragacz has
given a sketch of its proof [33]. Finally, using his method of restriction
equations, Rimányi [35] was able to treat the $n=k$ case, and computed
$\mathrm{Tp}_{k}$ for $k\leq 8$ in the equidimensional $n=m$ case (cf. [22]
for the case $k=4$).

In [11] we introduced a new approach and used equivariant localisation to
prove iterated residue formulas for Morin singularities for any $n,k$. Let
$J_{k}(1,n)$ denote the $k$-jets of holomorphic map germs
$(\mathbb{C},0)\to(\mathbb{C}^{n},0)$. These jets can be reparametrised by the
polynomial reparametrisation group $\mathrm{Diff}_{k}=J_{k}^{reg}(1,1)$ formed
by jets with nonzero first derivative. The quotient
$J_{k}(1,n)/\mathrm{Diff}_{k}$ is quasi-projective, it is the moduli of
invariant jets of order $k$ in ${\mathbb{C}}^{n}$. A key observation of [11]
is that (some open part of) $\Sigma_{k}(n,m)$ fibers over the non-reductive
quotient $J_{k}(1,n)/\mathrm{Diff}_{k}$ with affine fibers. We show in [11]
that the curvilinear component of the Hilbert scheme of $k$ points on
${\mathbb{C}}^{n}$ provides a natural compactification of this non-reductive
quotient, and the main message of [11] is that the Thom polynomials are
certain equivariant intersection numbers on the curvilinear Hilbert scheme.

This formula, however, contains an unknown ingredient, the $Q_{k}$ polynomial,
which is the equivariant dual of a certain Borel orbit in a
$\mathrm{GL}(n)$-module. Our knowledge about $Q_{k}$ is very limited. In [26],
Kazarian reinterprets the defining equations of $Q_{k}$ as the ideal of the
Morin algebra sitting in a smooth ambient space what he calls the
nonassociative Hilbert scheme. These equations fully determine $Q_{k}$—however
the description of them is out of reach, and a major problem. The problem is
that besides the associativity equations (which do not cut out even the right
dimensional variety–there there are other exotic equations, which are
determined by the Borel geometry of the nonassociative Hilbert scheme.

In this paper we develop an approach which avoids working with $Q_{k}$ and
Borel orbits, and reduces the problem to toric geometry. We replace the
curvilinear Hilbert scheme with a birational model, namely, the non-reductive
GIT quotient $\overline{J_{k}(1,n)}/\mathrm{Diff}_{k}$ for some proper
projective completion $\overline{J_{k}(1,n)}$ of $J_{k}(1,n)$. This
compactification will be a result of a blow-up procedure fully determined by
the combinatorics of certain partition polyhedra. We combine intersection
theory developed in [8] for non-reductive GIT quotients with equivariant
localisation on the blown-up space, and in particular, we use the non-
reductive Jeffrey-Kirwan type integration formula proved in [8]. We prove a
residue vanishing theorem, which leads us to closed iterated residue formulas
for Thom polynomials of Morin singularities, detailed in the next section.

In particular, this paper provides a combinatorial (toric) approach to a
partial fraction decomposition of the rational expression
$\frac{Q_{k}}{\prod_{1\leq i+j\leq k}(z_{i}+z_{j}-z_{k})}$ in the formula of
Bérczi-Szenes [11], which might serve as a starting point towards Rimányi’s
conjecture on Chern positivity of Thom polynomials [35, 34].

As a final remark, we note that although this paper focuses only on Morin
singularities, our approach extends to compute Thom polynomials for other
contact singularity classes (see [36, 35]).

##  2\. The strategy and the result

Fix the parameters $k\leq n\leq m$. Let $J_{k}(n,m)$ denote the $k$-jets of
holomorphic map germs $(\mathbb{C}^{n},0)\to(\mathbb{C}^{m},0)$, and
$\Sigma_{k}(n,m)$ be the set of jets with local algebra
$t\mathbb{C}[t]/t^{k+1}$. Its closure, $\overline{\Sigma}_{k}(n,m)$ is a
$\mathrm{GL}(n)\times\mathrm{GL}(m)$-invariant, singular subvariety of the
affine space $J_{k}(n,m)$. The Thom principle (see e.g (see [35, 26, 11] says
that the polynomial of $A_{k}$ singularities is given by the
$\mathrm{GL}(n)\times\mathrm{GL}(m)$-equivariant dual of $\Sigma_{k}(n,m)$ in
$J_{k}(n,m)$ after substituting the Chern roots of $TN$ and $f^{*}TM$ into the
weights:

$\mathrm{Tp}_{k}^{n,m}=\mathrm{eP}_{\mathrm{GL}(n)\times\mathrm{GL}(m)}[\Sigma_{k}(n,m),J_{k}(n,m)](TN,f^{*}TM)$

Gaffney [22] observed that a generic germ $\Psi\in\Sigma_{k}$ has a test curve
$\gamma\in J_{k}^{reg}(1,n)$ (reg refers to nonvanishing linear part) such
that $\Psi\circ\gamma=0$ in $J_{k}(n,m)$. Here $\Psi=\Psi^{1}+\ldots+\Psi^{k}$
is generic if its linear part is generic, that is, the rank of $\Psi^{1}$ is
$1$. We denote the set of these jets by $\Sigma_{k}^{0}(n,m)$. The test curve
is unique up to polynomial reparametrizations of $(\mathbb{C},0)$; that is, up
to the action of the reparametrisation group
$\mathrm{Diff}_{k}=J_{k}^{reg}(1,1)$. Therefore $\Sigma_{k}^{0}(n,m)$ fibers
over the quasi-projective quotient $J_{k}^{reg}(1,n)/\mathrm{Diff}_{k}$, which
we call the moduli space of $k$-jets in ${\mathbb{C}}^{n}$. The fibers are
linear subspaces in $J_{k}(n,m)$ of codimension $km$.

In [11] we construct an embedding
$\phi:J_{k}^{reg}(1,n)/\mathrm{Diff}_{k}\hookrightarrow\mathrm{Grass}(k,J_{k}(n,1)^{*})$
and we work with the the closure $\overline{\mathrm{im}(\phi)}$ as a natural
projective completion of the quotient. The key property of this
compactification is that the linear fibration $\Sigma_{k}^{0}(n,m)$ over
$J_{k}^{reg}(1,n)/\mathrm{Diff}_{k}$ extends to a bundle $V$ over
$\overline{\mathrm{im}(\phi)}$ which sits in the trivial bundle $V\subset
J_{k}(n,m)\times\overline{\mathrm{im}(\phi)}$. We develop equivariant
localisation on $\overline{\mathrm{im}(\phi)}$ to come up with an iterated
residue formula for integrals over $\overline{\mathrm{im}(\phi)}$, and we
present Thom polynomials of $A_{k}$ singularities as an equivariant
intersection number, namely,

$\mathrm{eP}[\Sigma_{k}(n,m),J_{k}(n,m)]=\int_{\overline{\mathrm{im}(\phi)}}\mathrm{Thom}(V\subset
J_{k}(n,m))$

is the integral of the Thom form of $V$ in
$J_{k}(n,m)\times\overline{\mathrm{im}(\phi)}$, whose restriction to each
linear fiber $V_{f}$ is the Euler class of the normal bundle of $V_{f}$ in
$J_{k}(n,m)$.

In [3] we observed that $k$-jets of smooth curves at the origin on
${\mathbb{C}}^{n}$ correspond to so-called curvilinear subschemes on
${\mathbb{C}}^{n}$. These are points of the Hilbert scheme
$\mathrm{Hilb}^{k+1}({\mathbb{C}}^{n})$ which are supported at the origin and
isomorphic to the algebra $t{\mathbb{C}}[t]/t^{k+1}$. The closure of this
locus in the Hilbert scheme is a punctual component, called the curvilinear
component, denoted by $\mathrm{CHilb}^{k+1}_{0}({\mathbb{C}}^{n})$. Hence this
component is a natural projective compactification of the moduli space of
jets, and in [3] we showed that
$\mathrm{CHilb}^{k+1}_{0}({\mathbb{C}}^{n})=\overline{\mathrm{im}(\phi)}\subset\mathrm{Grass}_{k}(J_{k}(n,1)^{*})$.
Let $\mathscr{E}$ denote the tautological rank $k$ bundle, and
${\mathbb{C}}^{m}$ the trivial $\mathrm{GL}(m)$-equivariant bundle over
$\overline{\mathrm{im}(\phi)}$. Then $\mathscr{E}\otimes{\mathbb{C}}^{m}$ is
$\mathrm{GL}(n)\times\mathrm{GL}(m)$-equivariant, and in [11] we show that
$\mathrm{Thom}(V\subset
J_{k}(n,m))=\mathrm{Euler}(\mathscr{E}\otimes{\mathbb{C}}^{m})$ is the
equivariant Euler class, hence

$\mathrm{eP}[\Sigma_{k}(n,m),J_{k}(n,m)]=\int_{\mathrm{CHilb}^{k+1}_{0}({\mathbb{C}}^{n})}\mathrm{Euler}(\mathscr{E}\otimes{\mathbb{C}}^{m}).$

In this paper we substitute the curvilinear Hilbert scheme with a birational
model, namely, the non-reductive geometric invariant theoretic (NRGIT)
quotient $\mathrm{Jet}_{k}/\\!/\mathrm{Diff}_{k}$ for some suitable projective
completion $\mathrm{Jet}_{k}=\overline{J_{k}(1,n)}$ of $J_{k}(1,n)$. For our
purposes ’well-behaved’ means a completion $\overline{J_{k}(1,n)}$ such that
our $\mathrm{Diff}_{k}$-invariant morphism
$\phi:J_{k}^{reg}(1,n)\to\mathrm{CHilb}^{k+1}({\mathbb{C}}^{n})$ extends to
the NRGIT semistable locus in $\overline{J_{k}(1,n)}$. This morphism
guarantees that integration over the curvilinear Hilbert scheme can be pulled
back to the NRGIT quotient, and we can work with the much better behaved NRGIT
model. We start with the simple compactification
${\mathbb{P}}({\mathbb{C}}\oplus J_{k}(n,m))$, and we construct a
$\mathrm{Diff}_{k}$-equivariant blow-up $\mathrm{Jet}_{k}$ such that the
$\mathrm{Diff}_{k}$-semistable locus $\mathrm{Jet}_{k}^{ss}$ admits a
$\mathrm{Diff}_{k}$-invariant morphism
$\mathrm{Jet}_{k}^{ss}\to\mathrm{CHilb}^{k+1}({\mathbb{C}}^{n})$:

(1)
$\textstyle{\mathrm{Jet}_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\mathrm{Jet}_{k}^{ss}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\phi}}$$\textstyle{{\mathbb{P}}({\mathbb{C}}\oplus
J_{k}(1,n))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{\mathrm{CHilb}^{k+1}_{0}({\mathbb{C}}^{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Grass}_{k}(J_{k}(1,n)^{*})}$

With such a diagram in hand we will be in good position: the NRGIT quotient is
categorical, hence $\tilde{\phi}$ induces a morphism
$\varphi:\mathrm{Jet}_{k}/\\!/\mathrm{Diff}_{k}\to\mathrm{CHilb}^{k+1}_{0}({\mathbb{C}}^{n})$.
The Thom polynomials and multisingularity classes will be expressed as an
integral on the NRGIT quotient $\mathrm{Jet}_{k}/\\!/\mathrm{Diff}_{k}$:

$\mathrm{eP}[\Sigma_{k}(n,m),J_{k}(n,m)]=\int_{\mathrm{Jet}_{k}/\\!/\mathrm{Diff}_{k}}\varphi^{*}\mathrm{Euler}(\mathscr{E}\otimes{\mathbb{C}}^{m}).$

Our equivariant localisation strategy can be summarised as follows:

1) The non-reductive Jeffrey-Kirwan residue formula developed in [8] can be
applied to get

$\mathrm{eP}[\Sigma_{k}(n,m),J_{k}(n,m)]=\operatornamewithlimits{Res}_{z=\infty}\sum_{F\in(\mathrm{Jet}_{k}^{{\mathbb{C}}^{*}})_{\min}}\int_{F}\frac{(k-1)!z^{k-1}i_{F}^{*}\varphi^{*}\mathrm{Euler}(\mathscr{E}\otimes{\mathbb{C}}^{m})}{\mathrm{Euler}(\mathscr{N}_{F})(z)}dz$

Here the sum is taken over those
${\mathbb{C}}^{*}\subset\mathrm{Diff}_{k}$-fixed point components on the blown
up $\mathrm{Jet}_{k}$ (these will be isolated fixed points in our
construction) which have minimal ${\mathbb{C}}^{*}$-weight.

2) We will perform blow-ups at smooth centers, invariant under the
$\mathrm{Diff}_{k}$ action. Starting from the projective space
${\mathbb{P}}({\mathbb{C}}\oplus J_{k}(n,m))$, this way we will end up in
having a smooth iterated blow-up $\mathrm{Jet}_{k}$ with an induced
$\mathrm{Diff}_{k}$ action. In order to follow this plan, we need to start
with an initial master blow-up: instead of ${\mathbb{P}}({\mathbb{C}}\oplus
J_{k}(n,m))$, we start the blow-up process with a fiberwise compactification
$\tilde{{\mathbb{P}}}$ of the fibration over the complete flag
$J_{k}(1,n)^{nondeg}\to
J_{k}(1,n)^{nondeg}/B=\mathrm{Flag}_{k}({\mathbb{C}}^{n})$. This space is a
partial blow-up of ${\mathbb{P}}({\mathbb{C}}\oplus J_{k}(n,m))$, and the
crucial advantage of using it is that we can apply equivariant localisation on
$\mathrm{Flag}_{k}({\mathbb{C}}^{n})$ with respect to the $\mathrm{GL}(n)$
action (which commutes with the ${\mathbb{C}}^{*}$ action coming from
$\mathrm{Diff}_{k}$) and rewrite the localisation formula as an iterated
residue w.r.t $k$ variables $z_{1},\ldots,z_{k}$.

3) Let $X_{0}=\tilde{{\mathbb{P}}}$ and let $X_{i}$ denote the space we get
after performing $i$ blow-ups. The center of each blow-up will be given by an
ideal of the form $(\beta_{1},\ldots,\beta_{r})$ where the $\beta_{j}$’s are
some local affine coordinates on some $X_{i}$. Hence the blow-up process will
be described by a rooted tree, whose nodes are labeled by clusters of affine
coordinates on the intermediate spaces, and the edges are labeled by variables
indicating the affine chart we pick. Fixed points on $\mathrm{Jet}_{k}$
correspond to paths on the tree from the root to some leave (leave=node of
valency $1$).

4) Due to a residue vansihing theorem proved in §9.2, if a leaf (i.e torus
fixed point on $\mathrm{Jet}_{k}$) has nonzero contribution to the residue
formula, then it must be mapped by $\tilde{\phi}$ to the distinguished subset
$\Lambda_{k}=\\{\xi\in\mathrm{CHilb}^{k+1}({\mathbb{C}}^{n}):\mathscr{O}_{\xi}\simeq(x_{1},\ldots,x_{k})/(x_{1},\ldots,x_{k})^{2}\\}$.
This is a very strong condition, which allows us to drop most of the leaves
from the blow-up tree. The yellow labels in the blow-up diagrams in our
examples in §10 have zero contribution, and only the red leaves have nonzero
contribution.

Before stating the main result, we fix some terminology. We will refer to
fixed points on $\mathrm{Jet}_{k}$ as leaves of the blow-up tree. There is a
unique path from the root to the leaf $L$, and such a path has the form

$\textstyle{\mathscr{C}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{1}}$$\textstyle{\mathscr{C}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{2}}$$\textstyle{\ldots\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta_{r-1}}$$\textstyle{\mathscr{C}_{\max}}$

where $\mathscr{C}_{i}$ are clusters of affine coordinates, that is,
$\mathscr{C}_{i}\subset\mathbf{B}$ for a fixed set of coordinates
$\mathbf{B}$. The edges are labeled by variables such that
$\beta_{i}\in\mathscr{C}_{i}$ for all $i$. The set of associated paths is
denoted my $\Gamma_{k}$. Each variable of $\mathbf{B}$ at each level is
endowed with a $T^{k}\times{\mathbb{C}}^{*}$ weight in this path; the weight
of $\beta\in\mathbf{B}$ after $i$ blow-ups is denoted by
$\gamma^{L}_{i}(\beta)$. The simple rule of forming these weights from some
initial weights $\gamma_{1}(\beta)$ is the following:

(2) $\gamma^{L}_{i+1}(\beta)=\begin{cases}\gamma^{L}_{i}(\beta)&\text{ if
}\beta=\beta_{i}\text{ or }\beta\notin\mathscr{C}_{i+1}\\\
\gamma^{L}_{i}(\beta)-\gamma^{L}_{i}(\beta_{i})&\text{ if
}\beta\in\mathscr{C}_{i+1}\setminus\\{\beta_{i}\\}\\\ \end{cases}$

We will use the shorthand notation
$\gamma^{L}(\beta)=\gamma^{L}_{\max}(\beta)$. We are ready to formulate the
main theorem, which we will restate as Theorem Main Theorem at the end of this
paper.

######  Main Theorem.

For arbitrary integers $k\ll n\leq m$ the Thom polynomial for the
$A_{k}$-singularity with $n$-dimensional source space and $m$-dimensional
target space is given by the following iterated residue formula:

$\mathrm{Tp}_{k}^{n,m}=\operatornamewithlimits{Res}_{z_{1}=\infty}\ldots\operatornamewithlimits{Res}_{z_{k}=\infty}\cdot\sum_{L\in\tilde{\phi}^{-1}({\mathbf{g}})}\frac{(k-1)!z^{k-1}(z_{1}\ldots
z_{k})^{m-n}\prod_{i<j}(z_{i}-z_{j})}{\prod_{\beta\in\mathbf{B}}\gamma^{L}(\beta)}\prod_{i=1}^{k}c_{TM-
TN}(1/z_{i})d\mathbf{z}$

where we sum over those leaves of the blow-up tree, which are mapped to the
distinguished point ${\mathbf{g}}=[e_{1}\wedge\ldots\wedge
e_{k}]\in{\mathbb{P}}(\wedge^{k}\mathrm{Sym}^{\leq k}{\mathbb{C}}^{n})$ under
the morphism $\tilde{\phi}$.

Examples For $k=3$ we get back the formula of [11]:

$\mathrm{Tp}_{3}=\operatornamewithlimits{Res}_{z_{1},z_{2},z_{3}=\infty}\frac{(z_{1}-z_{2})(z_{1}-z_{3})(z_{2}-z_{3})}{(2z_{1}-z_{2})(2z_{1}-z_{3})(z_{1}+z_{2}-z_{3})}\prod_{i=1}^{3}c_{TM-
TN}(1/z_{i})$

For the blow-up tree and the details of the computation see §10.1. However,
for $k=4$ we already get something different:

$\displaystyle\mathrm{Tp}_{4}=\operatornamewithlimits{Res}_{\mathbf{z}}\frac{(z_{1}z_{2}z_{3}z_{4})^{N-n}\prod_{i<j}(z_{i}-z_{j})\prod_{i=1}^{4}c_{TM-
TN}(1/z_{i})d\mathbf{z}}{(z_{2}+z_{3}-z_{1}-z_{4})(z_{1}+z_{2}-z_{4})(2z_{1}-z_{3})(2z_{1}-z_{4})}\cdot\left(\frac{1}{(2z_{1}-z_{2})(z_{1}+z_{2}-z_{3})}+\frac{1}{(z_{4}-z_{1}-z_{3})(2z_{2}-z_{4})}\right)$

The details of the computation including the blow-up tree and the weight table
is given in §10.2. Recall formula from [11]:

$\displaystyle\mathrm{Tp}_{4}^{BSZ}$
$\displaystyle=\operatornamewithlimits{Res}_{z_{1},\ldots,z_{4}=\infty}\frac{(2z_{1}+z_{2}-z_{4})(z_{1}z_{2}z_{3}z_{4})^{N-n}\prod_{i<j}(z_{i}-z_{j})d\mathbf{z}}{(2z_{1}-z_{2})(2z_{1}-z_{3})(z_{1}+z_{2}-z_{3})(z_{1}+z_{2}-z_{4})(2z_{1}-z_{4})(z_{1}+z_{3}-z_{4})(2z_{2}-z_{4})}\cdot\prod_{i=1}^{4}c_{TM-
TN}(1/z_{i})$

The two formula give the same result, and in particular the new formula
provides a nontrivial partial fraction decomposition

(3)
$\frac{1}{(z_{2}+z_{3}-z_{1}-z_{4})(2z_{1}-z_{2})(z_{1}+z_{2}-z_{3})}+\frac{1}{(z_{2}+z_{3}-z_{1}-z_{4})(z_{4}-z_{1}-z_{3})(2z_{2}-z_{4})}=\\\
=\frac{2z_{1}+z_{2}-z_{4}}{(2z_{1}-z_{2})(z_{1}+z_{2}-z_{3})(z_{1}+z_{3}-z_{4})(2z_{2}-z_{4})}$

For $k=5$ the blow-up tree has $15$ leaves mapped to $\Xi_{5}$, and hence in
the residue formula we have the sum of $15$ rational expression. Two of these
have 0 contribution, and in short, our formula replaces the rational
expression

$\frac{Q_{5}(z_{1},\ldots,z_{5})}{\prod_{1\leq i+j\leq l\leq
k}(z_{i}+z_{j}-z_{l})}$

where

${\mathbb{Q}}_{5}(z_{1},z_{2},z_{3},z_{4},z_{5})=(2z_{1}+z_{2}-z_{5})(2z_{1}^{2}+3z_{1}z_{2}-2z_{1}z_{5}+2z_{2}z_{3}-z_{2}z_{4}-z_{2}z_{5}-z_{3}z_{4}+z_{4}z_{5}).$

which appears in [11] with the sum of $13$ rational expressions. This might
look as a bad deal, but the real benefit comes for $k\geq 6$; for $k=6$ we
know that $Q_{6}$ is a is a degree-$7$ polynomial in $6$ variables, with more
than $1000$ terms, but our blow-up tree has $\sim 60$ leaves. And for $k\geq
7$ there is no computing capacity to cope with the relations and hence $Q_{k}$
is unknown, whereas our algorithm is purely toric, working entirely with
monomial ideals.

##  3\. Hilbert schemes and tautological integrals

###  3.1. Tautological integrals

Let $X$ be a smooth projective variety of dimension $n$ and let $F$ be a rank
$r$ bundle (loc. free sheaf) on $X$. Let

$\mathrm{Hilb}^{k}(X)=\\{\xi\subset X:\dim(\xi)=0,\mathrm{length}(\xi)=\dim
H^{0}(\xi,\mathscr{O}_{\xi})=k\\}$

denote the Hilbert scheme of $k$ points on $X$ parametrizing length $k$
subschemes of $X$ and $F^{[k]}$ the corresponding rank $rk$ bundle on
$\mathrm{Hilb}^{k}(X)$ whose fibre over $\xi\in\mathrm{Hilb}^{k}(X)$ is
$F\otimes\mathscr{O}_{\xi}=H^{0}(\xi,F|_{\xi})$.

Equivalently, $F^{[k]}=q_{*}p^{*}(F)$ where $p,q$ denote the projections from
the universal family of subschemes $\mathscr{U}$ to $X$ and
$\mathrm{Hilb}^{k}(X)$ respectively:

$\mathrm{Hilb}^{k}(X)\times X\supset\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
7.53471pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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14.45435pt\raise 5.1875pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8264pt\hbox{$\scriptstyle{q}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 31.53471pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
0.0pt\raise-16.13889pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise-0.8264pt\hbox{$\scriptstyle{p}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-29.27779pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
31.53471pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathrm{Hilb}^{k}(X)}$}}}}}}}{\hbox{\kern-7.53471pt\raise-39.1111pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{X}$}}}}}}}{\hbox{\kern
50.80974pt\raise-39.1111pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

In this paper we work with singular homology and cohomology with rational
coefficients. For a smooth manifold $X$ the degree of a class $\eta\in
H_{*}(X)$ means its pushforward to $H_{*}(pt)={\mathbb{Q}}$. By choosing
$\alpha_{\eta}\in\Omega^{top}(X)$, a closed compactly supported differential
form representing the cohomology class $\eta$ this degree is equal to the
integral

$\eta\cap[X]=\int_{X}\alpha_{\eta}.$

Let $\mathscr{Z}\subset\mathrm{Hilb}^{k}(X)$ be a geometric subset with
closure $\overline{\mathscr{Z}}$ and $M(c_{1},\ldots,c_{rk})$ be a monomial in
the Chern classes $c_{i}=c_{i}(F^{[k]})$ of weighted degree equal to
$\dim\overline{\mathscr{Z}}$ where the weight of $c_{i}$ is $i$. By choosing
$\alpha_{M}\in\Omega^{*}(\mathrm{Hilb}^{k}(X))$, a closed compactly supported
differential form representing the cohomology class of
$M(c_{1},\ldots,c_{rk})$, the degree

(4) $[\overline{\mathscr{Z}}]\cap
M(c_{1},\ldots,c_{rk})=\int_{\overline{\mathscr{Z}}}\alpha_{M}$

is called a tautological integral of $F^{[k]}$.

###  3.2. Curvilinear Hilbert schemes

Let $X$ be a smooth projective variety of dimension $n$ and for $p\in X$ let

$\mathrm{Hilb}^{k}_{p}(X)=\\{\xi\in\mathrm{Hilb}^{k}(X):\mathrm{supp}(\xi)=p\\}$

denote the punctual Hilbert scheme consisiting of subschemes supported at $p$.
If $\rho:\mathrm{Hilb}^{k}(X)\to S^{k}X$, $\xi\mapsto\sum_{p\in
X}\mathrm{length}(\mathscr{O}_{\xi,p})p$ denotes the Hilbert-Chow morphism
then $\mathrm{Hilb}^{k}_{p}(X)=\rho^{-1}(kp)$.

######  Definition 3.1.

A subscheme $\xi\in\mathrm{Hilb}^{k}_{p}(X)$ is called curvilinear if $\xi$ is
contained in some smooth curve $C\subset X$. Equivalently, $\xi$ is
curvilinear if $\mathscr{O}_{\xi}$ is isomorphic to the ${\mathbb{C}}$-algebra
${\mathbb{C}}[z]/z^{k}$. The punctual curvilinear locus at $p\in X$ is the set
of curvilinear subschemes supported at $p$:

$\mathrm{Curv}^{k}(X)_{p}=\\{\xi\in\mathrm{Hilb}^{k}_{p}(X):\xi\subset\mathscr{C}_{p}\text{
for some smooth curve }\mathscr{C}\subset
X\\}=\\{\xi:\mathscr{O}_{\xi}\simeq{\mathbb{C}}[z]/z^{k}\\}$

For surfaces ($n=2$) $\mathrm{Curv}^{k}_{p}(X)$ is an irreducible quasi-
projective variety of dimension $k-1$ which is an open dense subset in
$\mathrm{Hilb}^{k}_{p}(X)$ and therefore its closure is the full punctual
Hilbert scheme at $p$, that is,
$\overline{\mathrm{Curv}}^{[k]}_{p}(X)=\mathrm{Hilb}^{k}_{p}(X)$. When $n\geq
3$ the punctual Hilbert scheme $\mathrm{Hilb}^{k}_{p}(X)$ is not necessarily
irreducible or reduced, but the closure of the curvilinear locus is one of its
irreducible components:

######  Lemma 3.2.

$\overline{\mathrm{Curv}}^{[k]}_{p}$ is an irreducible component of the
punctual Hilbert scheme $\mathrm{Hilb}^{k}_{p}(X)$ of dimension $(n-1)(k-1)$.
We call this punctual component the curvilinear Hilbert scheme (supported at
$p$) and denote it by $\mathrm{CHilb}^{k}_{p}(X)$.

###### Proof.

Note that $\xi\in\mathrm{Hilb}^{[k]}_{0}({\mathbb{C}}^{n})$ is not curvilinear
if and only if $\mathscr{O}_{\xi}$ does not contain elements of degree $k-1$,
that is, after fixing some local coordinates $x_{1},\ldots,x_{n}$ of
${\mathbb{C}}^{n}$ at the origin we have

$\mathscr{O}_{\xi}\simeq{\mathbb{C}}[x_{1},\ldots,x_{n}]/I\text{ for some
}I\supseteq(x_{1},\ldots,x_{n})^{k-1}.$

This is a closed condition and therefore curvilinear subschemes can’t be
approximated by non-curvilinear subschemes in
$\mathrm{Hilb}^{[k]}_{0}({\mathbb{C}}^{n})$. The dimension of
$\mathrm{CHilb}^{k}_{p}(X)$ comes from the description of it as a non-
reductive quotient in the previous section. ∎

Note that any curvilinear subscheme contains only one subscheme for any given
smaller length and any small deformation of a curvilinear subscheme is again
locally curvilinear.

######  Remark 1.

Fix coordinates $x_{1},\ldots,x_{n}$ on ${\mathbb{C}}^{n}$. Recall that the
defining ideal $I_{\xi}$ of any subscheme
$\xi\in\mathrm{Hilb}^{k+1}_{0}({\mathbb{C}}^{n})$ is a codimension $k$
subspace in the maximal ideal $\mathfrak{m}=(x_{1},\ldots,x_{n})$. The dual of
this is a $k$-dimensional subspace $S_{\xi}$ in
$\mathfrak{m}^{*}\simeq\mathrm{Sym}^{\leq k}{\mathbb{C}}^{n}$ giving us a
natural embedding
$\rho:X^{[k+1]}_{p}\hookrightarrow\mathrm{Grass}_{k}(\mathrm{Sym}^{\leq
k}{\mathbb{C}}^{n})$. The test curve model of Theorem 4.4 gives an explicit
parametrization of this embedding.

###  3.3. Tautological bundles over punctual components

Over the punctual Hilbert scheme $\mathrm{Hilb}^{k}_{0}(X)$ the tautological
integral can be described as follows. Let $F$ be a rank $r$ vector bundle over
$X$. The fibre of the corresponding rank $rk$ tautological bundle $F^{[k]}$ on
$\mathrm{CHilb}^{k}(X)$ at the point $\xi$ is

$F^{[k]}_{\xi}=H^{0}(\xi,F|_{\xi})=H^{0}(\mathscr{O}_{\xi}\otimes F).$

On the level of bundles we have the following.

######  Lemma 3.3 ([3] Lemma 3.15).

There is an isomorphism of topological vector bundles

$F^{[k]}|_{\mathrm{CHilb}^{k}_{0}(X)}\simeq\mathscr{O}_{X}^{[k]}\otimes\pi^{*}F$

where $\pi:\mathrm{CHilb}^{k}_{0}(X)\to X$ is the projection.

The natural embedding
$\rho:\mathrm{CHilb}^{k+1}(X)\hookrightarrow\mathrm{Grass}_{k}(J_{k}(n,1)^{*})$
in Remark 1 identifies the fibres of $\mathscr{O}_{X}^{[k+1]}$ over
$\xi\in\mathrm{CHilb}^{k+1}_{p}(X)$ with
$H^{0}(\mathscr{O}_{\xi})\simeq\mathscr{O}_{X,p}\oplus\mathscr{E}_{\rho(\xi)}$
where $\mathscr{E}$ is the tautological rank $k$ bundle over
$\mathrm{Grass}_{k}(J_{k}(n,1)^{*})$. Hence the total Chern class of
$F^{[k+1]}$ can be written as

(5)
$c(F^{[k+1]})=\prod_{j=1}^{r}(1+\theta_{j})\prod_{i=1}^{k}\prod_{j=1}^{r}(1+\eta_{i}+\theta_{j})$

where $c(F)=\prod_{j=1}^{r}(1+\theta_{j})$ and
$c(\mathscr{E})=\prod_{i=1}^{k}(1+\eta_{i})$ are the Chern classes for the
corresponding bundles. In particular the Chern class

(6) $c_{i}(F^{[k+1]})=\mathscr{C}_{i}(c_{1}(\mathscr{E}),\ldots
c_{k}(\mathscr{E}),c_{1}(F),\ldots,c_{r}(F))$

can be expressed as a polynomial function $\mathscr{C}_{i}$ in Chern classes
of $\mathscr{E}$ and $F$.

##  4\. The test curve model of Morin singularities

In this section we describe the so-called test jet model of the curvilinear
locus of the punctual Hilbert scheme
$\mathrm{Hilb}_{0}^{k}({\mathbb{C}}^{n})$. This open locus can be described as
the moduli of holomorphic map jets ${\mathbb{C}}\to{\mathbb{C}}^{n}$ of order
$k$, which has a canonical embedding into a Grassmannian. The compactification
in this Grassmannian is the curvilinear component.

###  4.1. The test jet model of Berczi-Szenes [11]

This model was developed as a compactification of the Morin contact
singularity class of type $A_{k}$, and localisation on this model resulted in
new formulas for Thom polynomials in [11].

####  4.1.1. Jets of holomorphic maps

If $u,v$ are positive integers let $J_{k}(u,v)$ denote the vector space of
$k$-jets of holomorphic maps $({\mathbb{C}}^{u},0)\to({\mathbb{C}}^{v},0)$ at
the origin, that is, the set of equivalence classes of maps
$f:({\mathbb{C}}^{u},0)\to({\mathbb{C}}^{v},0)$, where $f\sim g$ if and only
if $f^{(j)}(0)=g^{(j)}(0)$ for all $j=1,\ldots,k$. This is a finite-
dimensional complex vector space, which one can identify with
$J_{k}(u,1)\otimes{\mathbb{C}}^{v}$; hence $\dim
J_{k}(u,v)=v\binom{u+k}{k}-v$. We will call the elements of $J_{k}(u,v)$ map-
jets of order $k$, or simply map-jets.

Eliminating the terms of degree $k+1$ results in an algebra homomorphism
$J_{k}(u,1)\twoheadrightarrow J_{k-1}(u,1)$, and the chain
$J_{k}(u,1)\twoheadrightarrow
J_{k-1}(u,1)\twoheadrightarrow\ldots\twoheadrightarrow J_{1}(u,1)$ induces an
increasing filtration on $J_{k}(u,1)^{*}$:

(7) $J_{1}(u,1)^{*}\subset J_{2}(u,1)^{*}\subset\ldots\subset J_{k}(u,1)^{*}$

######  Remark 2.

The space $J_{i}(u,1)^{*}$ may be interpreted as set of differential operators
on ${\mathbb{C}}^{u}$ of degree at most $i$, and in particular, by taking
symbols, we have

(8) $J_{k}(u,1)^{*}\cong\,{\rm Sym}\,^{\leq
k}{\mathbb{C}}^{u}\overset{\mathrm{def}}{=}\oplus_{l=1}^{k}\mathrm{Sym}^{l}{\mathbb{C}}^{u},$

where $\mathrm{Sym}^{l}$ stands for the symmetric tensor product and the
isomorphism is that of filtered $\mathrm{GL}(n)$-modules. Given a regular
$k$-jet $f:({\mathbb{C}},0)\to({\mathbb{C}}^{n},0)$ in
$J_{k}^{\mathrm{reg}}(1,n)$ we may push forward the differential operators of
order $k$ on ${\mathbb{C}}$ (with constant coefficients) to ${\mathbb{C}}^{n}$
along $f$ which gives us a map

$\tilde{f}:J_{k}(1,1)^{*}\to\mathrm{Grass}(k,J_{k}(n,1)^{*}).$

Choosing coordinates on ${\mathbb{C}}^{u}$ and ${\mathbb{C}}^{v}$ a $k$-jet
$f\in J_{k}(u,v)$ can be identified with the set of derivatives at the origin,
that is the vector $(f^{\prime}(0),f^{\prime\prime}(0),\ldots,f^{(k)}(0))$,
where
$f^{(j)}(0)\in\mathrm{Hom}(\mathrm{Sym}^{j}{\mathbb{C}}^{u},{\mathbb{C}}^{v})$.
This way we get the equality

(9) $J_{k}(u,v)\simeq
J_{k}(u,1)\otimes{\mathbb{C}}^{v}\simeq\oplus_{j=1}^{k}\mathrm{Hom}(\mathrm{Sym}^{j}{\mathbb{C}}^{u},{\mathbb{C}}^{v}).$

One can compose map-jets via substitution and elimination of terms of degree
greater than $k$; this leads to the composition map

(10) $J_{k}({u},{v})\times J_{k}({v},{w})\to
J_{k}({u},{w}),\;\;(\Psi_{1},\Psi_{2})\mapsto\Psi_{2}\circ\Psi_{1}\mbox{
modulo terms of degree $>k$ }.$

When $k=1$, $J_{1}(u,v)$ may be identified with $u$-by-$v$ matrices, and (10)
reduces to multiplication of matrices.

The $k$-jet of a curve $({\mathbb{C}},0)\to({\mathbb{C}}^{n},0)$ is simply an
element of $J_{k}(1,n)$. We call such a curve $\gamma$ regular, if
$\gamma^{\prime}(0)\neq 0$; introduce the notation
$J_{k}^{\mathrm{reg}}({1},{n})$ for the set of regular curves:

$J_{k}^{\mathrm{reg}}({1},{n})=\left\\{\gamma\in
J_{k}({1},{n});\gamma^{\prime}(0)\neq 0\right\\}$

Note that $J_{k}^{\mathrm{reg}}({u},{u})$ with the composition map (10) has a
natural group structure and we will often use the notation

$\mathrm{Diff}_{k}(u)=J_{k}^{\mathrm{reg}}({u},{u})$

and refer to this set as the $k$-jet diffeomorphism group to underline this
property.

####  4.1.2. Test curves for curvilinear subschemes

Let $\xi\in\mathrm{CHilb}^{k+1}_{0}({\mathbb{C}}^{n})$ be a curvilinear
subscheme supported at the origin. Then $\xi$ is (scheme theoretically)
contained in a smooth curve germ $\mathscr{C}_{p}$ in ${\mathbb{C}}^{n}$:

$\xi\subset\mathscr{C}_{p}\subset{\mathbb{C}}^{n}.$

Let $f_{\xi}:({\mathbb{C}},0)\to({\mathbb{C}}^{n},0)$ be a $k$-jet of germ
parametrising $\mathscr{C}_{p}$. Then $f_{\xi}\in J^{\mathrm{reg}}_{k}(1,n)$
is determined only up to polynomial reparametrisation germs
$\phi:({\mathbb{C}},0)\to({\mathbb{C}},0)$ and therefore we get the following
lemma.

######  Lemma 4.1.

The punctual curvilinear locus $\mathrm{CHilb}^{[k+1]}_{0}({\mathbb{C}}^{n})$
is equal (as a set) to the set of $k$-jet of regular germs at the origin,
modulo polynomial reparametrisations:

$\mathrm{CHilb}^{k+1}_{0}({\mathbb{C}}^{n})=\\{\text{regular }k\text{-jets
}({\mathbb{C}},0)\to({\mathbb{C}}^{n},0)\\}/\\{\text{regular }k\text{-jets
}({\mathbb{C}},0)\to({\mathbb{C}},0)\\}=J^{\mathrm{reg}}_{k}(1,n)/\mathrm{Diff}_{k}(1).$

We can explicitely write out the reparametrisation action (defined in (10)) of
$\mathrm{Diff}_{k}(1)$ on $J_{k}^{\mathrm{reg}}({1},{n})$ as follows. Let
$f_{\xi}(z)=zf^{\prime}(0)+\frac{z^{2}}{2!}f^{\prime\prime}(0)+\ldots+\frac{z^{k}}{k!}f^{(k)}(0)\in
J_{k}^{\mathrm{reg}}({1},{n})$ the $k$-jet of a germ at the origin (i.e no
constant term) in ${\mathbb{C}}^{n}$ with $f^{(i)}\in{\mathbb{C}}^{n}$ such
that $f^{\prime}\neq 0$ and let
$\varphi(z)=\alpha_{1}z+\alpha_{2}z^{2}+\ldots+\alpha_{k}z^{k}\in
J_{k}^{\mathrm{reg}}({1},{1})$ with $\alpha_{i}\in{\mathbb{C}},\alpha_{1}\neq
0$. Then

$f\circ\varphi(z)=(f^{\prime}(0)\alpha_{1})z+(f^{\prime}(0)\alpha_{2}+\frac{f^{\prime\prime}(0)}{2!}\alpha_{1}^{2})z^{2}+\ldots+\left(\sum_{i_{1}+\ldots+i_{l}=k}\frac{f^{(l)}(0)}{l!}\alpha_{i_{1}}\ldots\alpha_{i_{l}}\right)z^{k}=$
(11)
$=(f^{\prime}(0),\ldots,f^{(k)}(0)/k!)\cdot\left(\begin{array}[]{ccccc}\alpha_{1}&\alpha_{2}&\alpha_{3}&\ldots&\alpha_{k}\\\
0&\alpha_{1}^{2}&2\alpha_{1}\alpha_{2}&\ldots&2\alpha_{1}\alpha_{k-1}+\ldots\\\
0&0&\alpha_{1}^{3}&\ldots&3\alpha_{1}^{2}\alpha_{k-2}+\ldots\\\
0&0&0&\ldots&\cdot\\\
\cdot&\cdot&\cdot&\ldots&\alpha_{1}^{k}\end{array}\right)$

where the $(i,j)$ entry is
$p_{i,j}(\bar{\alpha})=\sum_{a_{1}+a_{2}+\ldots+a_{i}=j}\alpha_{a_{1}}\alpha_{a_{2}}\ldots\alpha_{a_{i}}.$

######  Remark 3.

The linearisation of the action of $\mathrm{Diff}_{k}(1)$ on
$J_{k}^{\mathrm{reg}}({1},{n})$ given as the matrix multiplication in (11)
represents $\mathrm{Diff}_{k}(1)$ as a upper triangular matrix group in
$\mathrm{GL}(n)$. This is a non-reductive group so Mumford’s reductive GIT is
not applicable to study the geometry of the quotient
$J_{k}^{\mathrm{reg}}({1},{n})/\mathrm{Diff}_{k}(1)$, see Doran-Kirwan for
details. Note that our matrix group is parametrised along its first row with
the free parameters $\alpha_{1},\ldots,\alpha_{k}$ and the other entries are
certain (weighted homogeneous) polynomials in these free parameters. It is a
${\mathbb{C}}^{*}$ extension of its maximal unipotent radical

$\mathrm{Diff}_{k}(1)=U\rtimes{\mathbb{C}}^{*}$

where $U$ is the subgroup we get via substituting $\alpha_{1}=1$ and the
diagonal ${\mathbb{C}}^{*}$ acts with weights $0,1\ldots,n-1$ on the Lie
algebra $\mathrm{Lie}(U)$. In Bérczi and Kirwan [9] and Bérczi, Doran, Hawes
and Kirwan [5, 6] we study actions of groups of this type in a more general
context.

Fix an integer $N\geq 1$ and define

$\Theta_{k}=\left\\{\Psi\in J_{k}(n,N):\exists\gamma\in
J_{k}^{\mathrm{reg}}({1},{n}):\Psi\circ\gamma=0\right\\},$

that is, $\Theta_{k}$ is the set of those $k$-jets of germs on
${\mathbb{C}}^{n}$ at the origin which vanish on some regular curve. By
definition, $\Theta_{k}$ is the image of the closed subvariety of
$J_{k}({n},{N})\times J_{k}^{\mathrm{reg}}({1},{n})$ defined by the algebraic
equations $\Psi\circ\gamma=0$, under the projection to the first factor. If
$\Psi\circ\gamma=0$, we call $\gamma$ a test curve of $\Theta$.

######  Remark 4.

The subset $\Theta_{k}$ is the closure of an important singularity class in
the jet space $J_{k}(n,N)$. These are called Morin singularities and the
equivariant dual of $\Theta_{k}$ in $J_{k}(n,N)$ is called the Thom polynomial
of Morin singularities, see Bérczi and Szenes [11] and Fehér and Rimányi [18]
for details.

Test curves of germs are generally not unique. A basic but crucial observation
is the following. If $\gamma$ is a test curve of $\Psi\in\Theta_{k}$, and
$\varphi\in\mathrm{Diff}_{k}(1)$ is a holomorphic reparametrisation of
${\mathbb{C}}$, then $\gamma\circ\varphi$ is, again, a test curve of $\Psi$:

$\textstyle{{\mathbb{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{{\mathbb{C}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{{\mathbb{C}}^{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi}$$\textstyle{{\mathbb{C}}^{N}}$
$\Psi\circ\gamma=0\ \ \Rightarrow\ \ \ \Psi\circ(\gamma\circ\varphi)=0$

In fact, we get all test curves of $\Psi$ in this way if the following
property, open and dense in $\theta_{k}$, holds: the linear part of $\Psi$ has
$1$-dimensional kernel. Before stating this in Theorem 4.3 below, let us write
down the equation $\Psi\circ\gamma=0$ in coordinates in an illustrative case.
Let $\gamma=(\gamma^{\prime},\gamma^{\prime\prime},\ldots,\gamma^{(k)})\in
J_{k}^{\mathrm{reg}}({1},{n})$ and
$\Psi=(\Psi^{\prime},\Psi^{\prime\prime},\ldots,\Psi^{(k)})\in J_{k}({n},{N})$
be the $k$-jets of the test curve $\gamma$ and the map $\Psi$ respectively.
Using the chain rule and the notation $v_{i}=\gamma^{(i)}/i!$, the equation
$\Psi\circ\gamma=0$ reads as follows for $k=4$:

(12) $\displaystyle\Psi^{\prime}(v_{1})=0$
$\displaystyle\Psi^{\prime}(v_{2})+\Psi^{\prime\prime}(v_{1},v_{1})=0$
$\displaystyle\Psi^{\prime}(v_{3})+2\Psi^{\prime\prime}(v_{1},v_{2})+\Psi^{\prime\prime\prime}(v_{1},v_{1},v_{1})=0$
$\displaystyle\Psi^{\prime}(v_{4})+2\Psi^{\prime\prime}(v_{1},v_{3})+\Psi^{\prime\prime}(v_{2},v_{2})+3\Psi^{\prime\prime\prime}(v_{1},v_{1},v_{2})+\Psi^{\prime\prime\prime\prime}(v_{1},v_{1},v_{1},v_{1})=0$

######  Lemma 4.2 (Gaffney [22], Bérczi-Szenes [11]).

Let $\gamma=(\gamma^{\prime},\gamma^{\prime\prime},\ldots,\gamma^{(k)})\in
J_{k}^{\mathrm{reg}}({1},{n})$ and
$\Psi=(\Psi^{\prime},\Psi^{\prime\prime},\ldots,\Psi^{(k)})\in J_{k}({n},{N})$
be $k$-jets. Then substituting $v_{i}=\gamma^{(i)}/i!$, the equation
$\Psi\circ\gamma=0$ is equivalent to the following system of $k$ linear
equations with values in ${\mathbb{C}}^{N}$:

(13) $\sum_{\ell\in\mathscr{P}(m)}\Psi(\mathbf{v}_{\ell})=0,\quad
m=1,2,\dots,k$

Here $\mathscr{P}(m)$ denotes the set of partitions $\ell=1^{\ell_{1}}\ldots
m^{\ell_{m}}$ of $m$ into nonnegative integers and
$\mathbf{v}_{\ell}=v_{1}^{\ell_{1}}\cdots v_{m}^{\ell_{m}}$.

For a given $\gamma\in J_{k}^{\mathrm{reg}}({1},{n})$ and $1\leq i\leq k$ let
$\mathscr{S}^{i,N}_{\gamma}$ denote the set of solutions of the first $i$
equations in (13), that is,

(14) $\mathscr{S}^{i,N}_{\gamma}=\left\\{\Psi\in
J_{k}({n},{N}),\Psi\circ\gamma=0\text{ up to order }i\right\\}.$

The equations (13) are linear in $\Psi$, hence

$\mathscr{S}^{i,N}_{\gamma}\subset J_{k}({n},{N})$

is a linear subspace of codimension $iN$, i.e a point of
$\mathrm{Grass}_{\mathrm{codim}=iN}(J_{k}(n,N))$, whose orthogonal,
$(\mathscr{S}^{i,N}_{\gamma})^{\perp}$, is an $iN$-dimensional subspace of
$J_{k}(n,N)^{*}$. These subspaces are invariant under the reparametrization of
$\gamma$. In fact, $\Psi\circ\gamma$ has $N$ vanishing coordinates and
therefore

$(\mathscr{S}^{i,N}_{\gamma})^{\perp}=(\mathscr{S}^{i,1}_{\gamma})^{\perp}\otimes{\mathbb{C}}^{N}$

holds.

For $\Psi\in J_{k}(n,N)$ let $\Psi^{1}\in\,{\rm
Hom}\,({\mathbb{C}}^{n},{\mathbb{C}}^{N})$ denote the linear part. When $N\geq
n$ then the subset

$\tilde{\mathscr{S}}^{i,N}_{\gamma}=\\{\Psi\in\mathscr{S}^{i,N}_{\gamma}:\dim\ker\Psi^{1}=1\\}$

is an open dense subset of the subspace $\mathscr{S}^{i,N}_{\gamma}$. In fact
it is not hard to see that the complement
$\tilde{\mathscr{S}}^{i,N}_{\gamma}\setminus\mathscr{S}^{i,N}_{\gamma}$where
the kernel of $\Psi^{1}$ has dimension at least two is a closed subvariety of
codimension $N-n+2$.

######  Theorem 4.3.

The map

$\phi:J_{k}^{\mathrm{reg}}({1},{n})\rightarrow\mathrm{Grass}_{k}(J_{k}(n,1)^{*})$

defined as $\gamma\mapsto(\mathscr{S}^{k,1}_{\gamma})^{\perp}$ is
$\mathrm{Diff}_{k}(1)$-invariant and induces an injective map on the
$\mathrm{Diff}_{k}(1)$-orbits into the Grassmannian

$\phi^{\mathrm{Grass}}:J_{k}^{\mathrm{reg}}({1},{n})/\mathrm{Diff}_{k}(1)\hookrightarrow\mathrm{Grass}_{k}(J_{k}(n,1)^{*}).$

Moreover, $\phi$ and $\phi^{\mathrm{Grass}}$ are $\mathrm{GL}(n)$-equivariant
with respect to the standard action of $\mathrm{GL}(n)$ on
$J_{k}^{\mathrm{reg}}({1},{n})\subset\,{\rm
Hom}\,({\mathbb{C}}^{k},{\mathbb{C}}^{n})$ and the induced action on
$\mathrm{Grass}_{k}(J_{k}(n,1)^{*})$.

###### Proof.

For the first part it is enough to prove that for $\Psi\in\Theta_{k}$ with
$\dim\ker\Psi^{1}=1$ and $\gamma,\delta\in J_{k}^{\mathrm{reg}}({1},{n})$

$\Psi\circ\gamma=\Psi\circ\delta=0\Leftrightarrow\exists\Delta\in
J_{k}^{\mathrm{reg}}({1},{1})\text{ such that }\gamma=\delta\circ\Delta.$

We prove this statement by induction. Let $\gamma=v_{1}t+\dots+v_{k}t^{k}$ and
$\delta=w_{1}t+\dots+w_{k}t^{k}$. Since $\dim\ker\Psi^{1}=1$, $v_{1}=\lambda
w_{1}$, for some $\lambda\neq 0$. This proves the $k=1$ case.

Suppose the statement is true for $k-1$. Then, using the appropriate
order-($k-1$) diffeomorphism, we can assume that $v_{m}=w_{m}$, $m=1\ldots
k-1.$ It is clear then from the explicit form (13) (cf. (12)) of the equation
$\Psi\circ\gamma=0$, that $\Psi^{1}(v_{k})=\Psi^{1}(w_{k})$, and hence
$w_{k}=v_{k}-\lambda v_{1}$ for some $\lambda\in{\mathbb{C}}$. Then
$\gamma=\Delta\circ\delta$ for $\Delta=t+\lambda t^{k}$, and the proof is
complete. ∎

######  Remark 5.

For a point $\gamma\in J_{k}^{\mathrm{reg}}(1,n)$ let
$v_{i}=\frac{\gamma^{(i)}}{i!}\in{\mathbb{C}}^{n}$ denote the normed $i$th
derivative. Then from Lemma 4.2 immediately follows that for $1\leq i\leq k$
(see [11]):

(15)
$(\mathscr{S}^{i,1}_{\gamma})^{\perp}=\mathrm{Span}_{\mathbb{C}}(v_{1},v_{2}+v_{1}^{2},\ldots,\sum_{i_{1}+\ldots+i_{r}=k}v_{i_{1}}\ldots
v_{i_{r}})\subset\mathrm{Sym}^{\leq k}{\mathbb{C}}^{n}$

Recall from Remark 2 that $J_{k}(n,1)^{*}=\mathrm{Sym}^{\leq
k}{\mathbb{C}}^{n}$. Note that the image of $\phi$ and the image of $\rho$
defined in Remark 1 of the previous section coincide in
$\mathrm{Grass}_{k}(\mathrm{Sym}^{\leq k}{\mathbb{C}}^{n})$:

$\mathrm{im}(\phi)=\mathrm{im}(\rho)\subset\mathrm{Grass}_{k}(J_{k}(n,1)^{*}).$

Hence their closure is the same, which is the first part of the next theorem.
The second part follows from Remark 5 because $\phi$ is
$\mathrm{GL}(n)$-equivariant.

######  Theorem 4.4 (Bérczi-Szenes model for
$\mathrm{CHilb}^{k}_{0}({\mathbb{C}}^{n})$, [11]).

  1. (1)

For any $k,n$ we have

$\mathrm{CHilb}^{k+1}_{0}({\mathbb{C}}^{n})=\overline{\mathrm{im}(\phi)}\subset\mathrm{Grass}_{k}(J_{k}(n,1)^{*}).$

  2. (2)

Let $\\{e_{1},\ldots,e_{n}\\}$ be a basis of ${\mathbb{C}}^{n}$. For $k\leq n$
the $\mathrm{GL}(n)$-orbit of

$p_{k,n}=\phi(e_{1},\ldots,e_{k})=\mathrm{Span}_{\mathbb{C}}(e_{1},e_{2}+e_{1}^{2},\ldots,\sum_{i_{1}+\ldots+i_{r}=k}e_{i_{1}}\ldots
e_{i_{r}})$

forms a dense subset of the image $J_{k}^{\mathrm{reg}}({1},{n})$ and
therefore (see e.g Remark 1)

$\mathrm{CHilb}^{k+1}_{0}({\mathbb{C}}^{n})=\overline{\mathrm{GL_{n}}\cdot
p_{k,n}}.$

###  4.2. Tautological integrals and Thom polynomials

In this subsection we recall from [11] how the test curve model explained in
the previous section reinterprets Thom polynomials as certain equivariant
tautological integrals over the punctual curvilinear Hilbert scheme
$\mathrm{CHilb}^{k+1}_{0}({\mathbb{C}}^{n})$. Recall from Theorem 4.4 that

$\mathrm{CHilb}^{k+1}_{0}({\mathbb{C}}^{n})=\overline{\mathrm{im}(\phi)}\subset\mathrm{Grass}_{k}(\wedge^{k}\mathrm{Sym}^{\leq
k}{\mathbb{C}}^{n})$

and therefore there are two canonically defined bundles on
$\mathrm{CHilb}^{k+1}_{0}({\mathbb{C}}^{n})$:

  1. (1)

The tautological rank $k+1$ bundle $\mathscr{O}_{{\mathbb{C}}^{n}}^{[k+1]}$
associated to the trivial bundle on ${\mathbb{C}}^{n}$.

  2. (2)

The tautological rank $k$ bundle $\mathscr{E}$, which is the restriction of
the tautological bundle over $\mathrm{Grass}_{k}(\wedge^{k}\mathrm{Sym}^{\leq
k}{\mathbb{C}}^{n})$.

These fit into the short exact sequence

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
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29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathscr{O}_{\mathrm{Grass}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
81.08334pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
81.08334pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathscr{O}_{{\mathbb{C}}^{n}}^{[k+1]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
124.82632pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
124.82632pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathscr{E}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
161.6319pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
161.6319pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{0}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

Let ${\mathbb{C}}^{m}$ be the trivial $\mathrm{GL}(m)$-equivariant bundle over
$\mathrm{Grass}_{k}(J_{k}(n,1)^{*})$. By Lemma 3.3, by tensoring with
${\mathbb{C}}^{m}$ we obtain the short exact sequence over
$\mathrm{CHilb}^{k+1}_{0}({\mathbb{C}}^{n})$

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathbb{C}}^{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
71.6391pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
71.6391pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{({\mathbb{C}}^{m})^{[k+1]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
134.91434pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
134.91434pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathscr{E}\otimes{\mathbb{C}}^{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
191.63681pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
191.63681pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{0}$}}}}}}}\ignorespaces}}}}\ignorespaces.$

The equivariant Euler class of $\mathscr{E}\otimes{\mathbb{C}}^{m}$ can be
written as

(16)
$\mathrm{Euler}(\mathscr{E}\otimes{\mathbb{C}}^{m})=\prod_{i=1}^{k}\prod_{j=1}^{m}(\eta_{i}+\theta_{j})$

where $c^{T}(\mathscr{E})=\prod_{j=1}^{m}(1+\theta_{j})$ and
$c^{T}(\mathscr{E})=\prod_{i=1}^{k}(1+\eta_{i})$ are the equivariant Chern
classes for the corresponding bundles.

######  Theorem 4.5 ([11]).

Thom polynomials of Morin singularities can be expressed as equivariant
tautological integrals over curvilinear Hilbert schemes as follows:

$\mathrm{Tp}_{k}^{n,m}=\int_{\mathrm{CHilb}^{k+1}_{0}({\mathbb{C}}^{n})}\mathrm{Euler}(\mathscr{E}\otimes{\mathbb{C}}^{m})(TN,f^{*}(TM)),$

obtained by substituting the Chern roots of $TN,TM$ into the equivariant
integral, which sits in the ring
${\mathbb{C}}[x_{1},\ldots,x_{n},y_{1},\ldots,y_{m}]^{S_{n}\times S_{m}}$ of
bi-symmetric polynomials on $n+m$ variables.

##  5\. Non-reductive geometric invariant theory

In [6] an extension of Mumford’s classical GIT is developed for linear actions
of a non-reductive linear algebraic group with internally graded unipotent
radical over an algebraically closed field ${\mathbf{k}}$ of characteristic 0.

######  Definition 5.1.

We say that a linear algebraic group $H=U\rtimes R$ has internally graded
unipotent radical $U$ if there is a central one-parameter subgroup
$\lambda:{\mathbb{G}}_{m}\to Z(R)$ of the Levi subgroup $R$ of $H$ such that
the adjoint action of ${\mathbb{G}}_{m}$ on the Lie algebra of $U$ has all its
weights strictly positive. Then $\hat{U}=U\rtimes\lambda({\mathbb{G}}_{m})$ is
a normal subgroup of $H$ and $H/\hat{U}\cong R/\lambda({\mathbb{G}}_{m})$ is
reductive.

Let $H=U\rtimes R$ be a linear algebraic group with internally graded
unipotent radical $U$ acting linearly with respect to an ample line bundle $L$
on a projective variety $X$; that is, the action of $H$ on $X$ lifts to an
action on $L$ via automorphisms of the line bundle. When $H=R$ is reductive,
using Mumford’s classical geometric invariant theory (GIT) [31], we can define
$H$-invariant open subsets $X^{s}\subseteq X^{ss}$ of $X$ (the stable and
semistable loci for the linearisation) with a geometric quotient $X^{s}/H$ and
projective completion $X/\\!/H\supseteq X^{s}/H$ which is the projective
variety associated to the algebra of invariants $\bigoplus_{k\geq
0}H^{0}(X,L^{\otimes k})^{H}$. The variety $X/\\!/H$ is the image of a
surjective morphism $\phi$ from the open subset $X^{ss}$ of $X$ such that if
$x,y\in X^{ss}$ then $\phi(x)=\phi(y)$ if and only if the closures of the
$H$-orbits of $x$ and $y$ meet in $X^{ss}$. Furthermore the subsets $X^{s}$
and $X^{ss}$ can be described using the Hilbert–Mumford criteria for stability
and semistability.

Mumford’s GIT does not have an immediate extension to actions of non-reductive
linear algebraic groups $H$, since the algebra of invariants $\bigoplus_{k\geq
0}H^{0}(X,L^{\otimes k})^{H}$ is not necessarily finitely generated as a
graded algebra when $H$ is not reductive. It is still possible to define
semistable and stable subsets $X^{ss}$ and $X^{s}$, with a geometric quotient
$X^{s}/H$ which is an open subset of a so-called enveloping quotient
$X\\!\mathbin{\text{\rotatebox[origin={c}]{70.0}{\scalebox{1.2}{$\approx$}}}}\\!H$
with an $H$-invariant morphism $\phi:X^{ss}\to
X\\!\mathbin{\text{\rotatebox[origin={c}]{70.0}{\scalebox{1.2}{$\approx$}}}}\\!H$,
and if the algebra of invariants $\bigoplus_{k\geq 0}H^{0}(X,L^{\otimes
k})^{H}$ is finitely generated then
$X\\!\mathbin{\text{\rotatebox[origin={c}]{70.0}{\scalebox{1.2}{$\approx$}}}}\\!H$
is the associated projective variety [4, 16]. But in general the enveloping
quotient
$X\\!\mathbin{\text{\rotatebox[origin={c}]{70.0}{\scalebox{1.2}{$\approx$}}}}\\!H$
is not necessarily projective, the morphism $\phi$ is not necessarily
surjective (and its image may be only a constructible subset, not a
subvariety, of
$X\\!\mathbin{\text{\rotatebox[origin={c}]{70.0}{\scalebox{1.2}{$\approx$}}}}\\!H$).
In addition there are in general no obvious analogues of the Hilbert–Mumford
criteria.

However when $H=U\rtimes R$ has internally graded unipotent radical $U$ and
acts linearly on a projective variety $X$, then provided that we are willing
to modify the linearisation of the action by replacing the line bundle $L$ by
a sufficiently divisible tensor power and multiplying by a suitable character
of $H$ (which will not change the action of $H$ on $X$), many of the key
features of classical GIT still apply.

Let such an $H$ act linearly on an irreducible projective variety $X$ with
respect to a very ample line bundle $L$. Let $\chi:H\to{\mathbb{G}}_{m}$ be a
character of $H$. Its kernel contains $U$, and its restriction to $\hat{U}$
can be identified with an integer so that the integer 1 corresponds to the
character of $\hat{U}$ which fits into the exact sequence
$U\hookrightarrow\hat{U}\to\lambda({\mathbb{G}}_{m})$. Let $\omega_{\min}$ be
the minimal weight for the $\lambda({\mathbb{G}}_{m})$-action on
$V:=H^{0}(X,L)^{*}$ and let $V_{\min}$ be the weight space of weight
$\omega_{\min}$ in $V$. Suppose that
$\omega_{\min}=\omega_{0}<\omega_{1}<\cdots<\omega_{\max}$ are the weights
with which the one-parameter subgroup $\lambda:{\mathbb{G}}_{m}\leq\hat{U}\leq
H$ acts on the fibres of the tautological line bundle
$\mathscr{O}_{{\mathbb{P}}((H^{0}(X,L)^{*})}(-1)$ over points of the connected
components of the fixed point set
${\mathbb{P}}((H^{0}(X,L)^{*})^{{\mathbb{G}}_{m}}$ for the action of
${\mathbb{G}}_{m}$ on ${\mathbb{P}}((H^{0}(X,L)^{*})$; since $L$ is very ample
$X$ embeds in ${\mathbb{P}}((H^{0}(X,L)^{*})$ and the line bundle $L$ extends
to the dual $\mathscr{O}_{{\mathbb{P}}((H^{0}(X,L)^{*})}(1)$ of the
tautological line bundle on ${\mathbb{P}}((H^{0}(X,L)^{*})$. Note that we can
assume that there exist at least two distinct such weights since otherwise the
action of the unipotent radical $U$ of $H$ on $X$ is trivial, and so the
action of $H$ is via an action of the reductive group $R=H/U$.

######  Definition 5.2.

Let $c$ be a positive integer such that

$\frac{\chi}{c}=\omega_{\min}+\epsilon$

where $\epsilon>0$ is sufficiently small; we will call rational characters
$\chi/c$ with this property well adapted to the linear action of $H$, and we
will call the linearisation well adapted if
$\omega_{\min}<0\leq\omega_{\min}+\epsilon$ for sufficiently small
$\epsilon>0$. How small $\epsilon$ is required to be will depend on the
situation; more precisely, we will say that some property P holds for well
adapted linearisations if there exists $\epsilon(P)>0$ such that property P
holds for any linearisation for which
$\omega_{\min}<0\leq\omega_{\min}+\epsilon(P)$.

######  Remark 6.

In [10] it is shown that under hypotheses which will be satisfied in our
situation it suffices to take $0<\epsilon<1$.

The linearisation of the action of $H$ on $X$ with respect to the ample line
bundle $L^{\otimes c}$ can be twisted by the character $\chi$ so that the
weights $\omega_{j}$ are replaced with $\omega_{j}c-\chi$; let
$L_{\chi}^{\otimes c}$ denote this twisted linearisation. Let
$X^{s,{\mathbb{G}}_{m}}_{\min+}$ denote the stable subset of $X$ for the
linear action of ${\mathbb{G}}_{m}$ with respect to the linearisation
$L_{\chi}^{\otimes c}$; by the theory of variation of (classical) GIT [15,
41], if $L$ is very ample then $X^{s,{\mathbb{G}}_{m}}_{\min+}$ is the stable
set for the action of ${\mathbb{G}}_{m}$ with respect to any rational
character $\chi/c$ such that $\omega_{\min}<\chi/c<\omega_{\min+1}$. Let

$Z_{\min}:=X\cap{\mathbb{P}}(V_{\min})=\left\\{\begin{array}[]{c|c}\hbox{\multirowsetup$x\in
X$}&\text{$x$ is a ${\mathbb{G}}_{m}$-fixed point and}\\\
&\text{${\mathbb{G}}_{m}$ acts on $L^{*}|_{x}$ with weight
$\omega_{\min}$}\end{array}\right\\}$

and

$X_{\min}^{0}:=\\{x\in X\mid p(x)\in Z_{\min}\\}\quad\mbox{ where }\quad
p(x)=\lim_{\begin{subarray}{c}t\to 0\\\
t\in{\mathbb{G}}_{m}\end{subarray}}t\cdot x\quad\mbox{ for }x\in X.$

######  Definition 5.3.

(cf. [6]) With this notation, we define the following condition for the
$\hat{U}$-action on $X$:

($*$) $\operatorname{Stab}_{U}(z)=\\{e\\}\textrm{ for every }z\in Z_{\min}.$

Note that ($*$ ‣ 5.3) holds if and only if we have
$\operatorname{Stab}_{U}(x)=\\{e\\}$ for all $x\in X^{0}_{\min}$. This is also
referred to as the condition that ‘semistability coincides with stability’ for
the action of $\hat{U}$ (or, when $\lambda:{\mathbb{G}}_{m}\to R$ is fixed,
for the linear action of $U$); see Definition 5.7 below.

######  Definition 5.4.

When ($*$ ‣ 5.3) holds for a well adapted action of $\hat{U}$ the min-stable
locus for the $\hat{U}$-action is

$X^{s,{\hat{U}}}_{\min+}=X^{ss,{\hat{U}}}_{\min+}=\bigcap_{u\in
U}uX^{s,\lambda({\mathbb{G}}_{m})}_{\min+}=X^{0}_{\min}\setminus UZ_{\min}.$

######  Definition 5.5.

A well-adapted linear action of the linear algebraic group $H$ on an
irreducible projective variety consists of the data $(X,L,H,\hat{U},\chi)$
where

  1. (1)

$H$ is a linear algebraic group with internally graded unipotent radical $U$,

  2. (2)

$H$ acts linearly on $X$ with respect to a very ample line bundle $L$, while
$\chi:H\to{\mathbb{G}}_{m}$ is a character of $H$ and $c$ is a positive
integer such that the rational character $\chi/c$ is well adapted for the
linear action of $\hat{U}=U\rtimes{\mathbb{G}}_{m}$ on $X$.

We will often refer to this set-up simply as a well-adapted action of $H$ on
$X$.

######  Theorem 5.6 ([6]).

Let $(X,L,H,\hat{U},\chi)$ be a well-adapted linear action satisfying
condition ($*$ ‣ 5.3). Then

  1. (1)

the algebras of invariants

$\oplus_{m=0}^{\infty}H^{0}(X,L_{m\chi}^{\otimes cm})^{\hat{U}}\text{ and
}\oplus_{m=0}^{\infty}H^{0}(X,L_{m\chi}^{\otimes
cm})^{H}=(\oplus_{m=0}^{\infty}H^{0}(X,L_{m\chi}^{\otimes cm})^{\hat{U}})^{R}$

are finitely generated;

  2. (2)

the enveloping quotient
$X\\!\mathbin{\text{\rotatebox[origin={c}]{70.0}{\scalebox{1.2}{$\approx$}}}}\\!\hat{U}$
is the projective variety associated to the algebra of invariants
$\oplus_{m=0}^{\infty}H^{0}(X,L_{m\chi}^{\otimes cm})^{\hat{U}}$ and is a
geometric quotient of the open subset $X^{s,\hat{U}}_{\min+}$ of $X$ by
$\hat{U}$;

  3. (3)

the enveloping quotient
$X\\!\mathbin{\text{\rotatebox[origin={c}]{70.0}{\scalebox{1.2}{$\approx$}}}}\\!H$
is the projective variety associated to the algebra of invariants
$\oplus_{m=0}^{\infty}H^{0}(X,L_{m\chi}^{\otimes cm})^{{H}}$ and is the
classical GIT quotient of
$X\\!\mathbin{\text{\rotatebox[origin={c}]{70.0}{\scalebox{1.2}{$\approx$}}}}\\!\hat{U}$
by the induced action of $R/\lambda({\mathbb{G}}_{m})$ with respect to the
linearisation induced by a sufficiently divisible tensor power of $L$.

######  Definition 5.7.

Let $X$ be a projective variety which has a well adapted linear action of a
linear algebraic group $H=U\rtimes R$ with internally graded unipotent radical
$U$. When ($*$ ‣ 5.3) holds we denote by $X^{s,{{H}}}_{\min+}$ and
$X^{ss,{{H}}}_{\min+}$ the pre-images in
$X^{s,{{\hat{U}}}}_{\min+}=X^{ss,{{\hat{U}}}}_{\min+}$ of the stable and
semistable loci for the induced linear action of the reductive group
$H/\hat{U}=R/\lambda({\mathbb{G}}_{m})$ on
$X\\!\mathbin{\text{\rotatebox[origin={c}]{70.0}{\scalebox{1.2}{$\approx$}}}}\\!\hat{U}=X^{s,\hat{U}}_{\min+}/\hat{U}$.

By $H$-stability=$H$-semistability we mean that ($*$ ‣ 5.3) holds and
$X^{s,{{H}}}_{\min+}=X^{ss,{{H}}}_{\min+}$. The latter is equivalent to the
requirement that $\operatorname{Stab}_{H}(x)$ is finite for all $x\in
X^{ss,{{H}}}_{\min+}$; then the projective variety
$X\\!\mathbin{\text{\rotatebox[origin={c}]{70.0}{\scalebox{1.2}{$\approx$}}}}\\!H$
is a geometric quotient of the open subset
$X^{s,{{H}}}_{\min+}=X^{ss,{{H}}}_{\min+}$ of $X$ by the action of $H$.

######  Remark 7.

When the conditions of Theorem 5.6 hold, we call
$X\\!\mathbin{\text{\rotatebox[origin={c}]{70.0}{\scalebox{1.2}{$\approx$}}}}\\!H$
(respectively
$X\\!\mathbin{\text{\rotatebox[origin={c}]{70.0}{\scalebox{1.2}{$\approx$}}}}\\!\hat{U}$)
the GIT quotient and we denote it by $X/\\!/H$ (respectively $X/\\!/\hat{U}$).

It is shown in [4] that if $(X,L,H,\hat{U},\chi)$ is a well-adapted linear
action satisfying $H$-stability=$H$-semistability, then

  1. (1)

there is a sequence of blow-ups of $X$ along $H$-invariant projective
subvarieties resulting in a projective variety $\hat{X}$ with a well adapted
linear action of $H$ which satisfies the condition ($*$ ‣ 5.3), so that
Theorem 5.6 applies, giving us a projective geometric quotient

$\hat{X}/\\!/\hat{U}=\hat{X}^{s,\hat{U}}_{\min+}/\hat{U}$

and its (reductive) GIT quotient
$\hat{X}/\\!/H=(\hat{X}/\\!/\hat{U})/\\!/R=(\hat{X}/\\!/\hat{U})/\\!/R$ where
$R=H/U$;

  2. (2)

there is a sequence of further blow-ups along $H$-invariant projective
subvarieties resulting in a projective variety $\tilde{X}$ satisfying the same
conditions as $\hat{X}$ and in addition $\tilde{X}/\\!/H={\rm
Proj}(\oplus_{m=0}^{\infty}H^{0}(X,L_{m\chi}^{\otimes cm})^{H})$ is the
geometric quotient by $H$ of the $H$-invariant open subset
$\tilde{X}^{s,H}_{\min+}$.

##  6\. Moment maps and cohomology of non-reductive quotients

In this section we briefly summarise the results of [8], which generalise
results of the second author [28] and Martin [29] to the cohomology of GIT
quotients by non-reductive groups with internally graded unipotent radicals.

First let us recall the reductive picture. Let $X$ be a nonsingular complex
projective variety acted on by a complex reductive group $G$ with respect to
an ample linearisation. Then we can choose a maximal compact subgroup $K$ of
$G$ and a $K$-invariant Fubini–Study Kähler metric on $X$ with corresponding
moment map $\mu:X\to\mathfrak{k}^{*}$, where $\mathfrak{k}$ is the Lie algebra
of $K$ and $\mathfrak{k}^{*}=\,{\rm
Hom}\,_{{\mathbb{R}}}(\mathfrak{k},{\mathbb{R}})$ is its dual.
$\mathfrak{k}^{*}$ embeds naturally in the complex dual
$\mathfrak{g}^{*}=\,{\rm Hom}\,_{{\mathbb{C}}}(\mathfrak{k},{\mathbb{C}})$ of
the Lie algebra $\mathfrak{g}=\mathfrak{k}\otimes{\mathbb{C}}$ of $G$, as
$\mathfrak{k}^{*}=\\{\xi\in\mathfrak{g}^{*}:\xi(\mathfrak{k})\subseteq{\mathbb{R}}\\}$;
using this identification we can regard $\mu:X\to\mathfrak{g}^{*}$ as a
‘moment map’ for the action of $G$, although of course it is not a moment map
for $G$ in the traditional sense of symplectic geometry.

In [28] it is shown that the norm-square $f=||\mu||^{2}$ of the moment map
$\mu:X\to\mathfrak{k}^{*}$ induces an equivariantly perfect Morse
stratification of $X$ such that that the open stratum which retracts
equivariantly onto the zero level set $\mu^{-1}(0)$ of the moment map
coincides with the GIT semistable locus $X^{ss}$ for the linear action of $G$
on $X$. In particular this tells us that the restriction map

$H^{*}_{G}(X;{\mathbb{Q}})\to H^{*}_{G}(X^{ss};{\mathbb{Q}})$

is surjective; we also have an isomorphism (of vector spaces though not of
algebras) $H^{*}_{G}(X;{\mathbb{Q}})\cong H^{*}(X;{\mathbb{Q}})\otimes
H^{*}(BG;{\mathbb{Q}}).$ Moreover, $\mu^{-1}(0)$ is $K$-invariant and its
inclusion in $X^{ss}$ induces a homeomorphism

(17) $\mu^{-1}(0)/K\cong X/\\!/G.$

When $X^{s}=X^{ss}$ the $G$-equivariant rational cohomology of $X^{ss}$
coincides with the ordinary rational cohomology of its geometric quotient
$X^{ss}/G$, which is the GIT quotient $X/\\!/G$, and we get expressions for
the Betti numbers of $X/\\!/G$ in terms of the equivariant Betti numbers of
the unstable GIT strata, which can be described inductively, and of $X$ [28].
In order to describe the ring structure on the rational cohomology of
$X/\\!/G$, the surjectivity of the composition

$\kappa:H^{*}_{G}(X;{\mathbb{Q}})\to H^{*}_{G}(X^{ss};{\mathbb{Q}})\cong
H^{*}(X/\\!/G;{\mathbb{Q}})$

can be combined with Poincaré duality on $X/\\!/G$ and the nonabelian
localisation formulas for intersection pairings on $X/\\!/G$ given in [24].

Martin [29] used (17) to obtain formulas for the intersection pairings on the
quotient $X/\\!/G$ in a different way, by relating these pairings to
intersection pairings on the associated quotient $X/\\!/T_{\mathbb{C}}$, where
$T_{\mathbb{C}}\subseteq G$ is a maximal torus. He proved a formula expressing
the rational cohomology ring of $X/\\!/G$ in terms of the rational cohomology
ring of $X/\\!/T_{\mathbb{C}}$ and an integration formula relating
intersection pairings on the cohomology of $X/\\!/G$ to corresponding pairings
on $X/\\!/T_{\mathbb{C}}$. This integration formula, combined with methods
from abelian localisation, leads to residue formulas for pairings on $X/\\!/G$
which are closely related to those of [24] (see also [44]).

In [8] similar results are obtained for non-reductive actions. Let $X$ be a
nonsingular complex projective variety with a linear action of a complex
linear algebraic group $H=U\rtimes R$ with internally graded unipotent radical
$U$ with respect to an ample line bundle $L$; then the Levi subgroup $R$ is
the complexification of a maximal compact subgroup $Q$ of $H$. The unipotent
radical $U$ of $H$ is internally graded by a central 1-parameter subgroup
$\lambda:{\mathbb{C}}^{*}\to Z(R)$ of $R$. Let
$\hat{U}=U\rtimes\lambda({\mathbb{C}}^{*})\subseteq H$; then
$\lambda(S^{1})\subseteq\lambda({\mathbb{C}}^{*})\subseteq\hat{U}$ is a
maximal compact subgroup of $\hat{U}$. Assume also that semistability
coincides with stability for the $\hat{U}$-action, in the sense of Definition
5.7. Using the embedding $X\subseteq{\mathbb{P}}^{n}$ defined by a very ample
tensor power of $L$, and a corresponding Fubini–Study Kähler metric invariant
under the maximal compact subgroup $Q$ of $H$, an $H$-moment map
$\mu_{H}:X\to\mathrm{Lie}H^{*}=\,{\rm
Hom}\,_{\mathbb{C}}(\mathrm{Lie}H,{\mathbb{C}})$ is defined in [8] by
composing the $G=\mathrm{GL}(n+1)$-moment map $\mu_{G}:X\to\mathfrak{g}^{*}$
with the map of complex duals $\mathfrak{g}^{*}\to\mathrm{Lie}(H)^{*}$ coming
from the representation $H\to\mathrm{GL}(n+1)$. It is shown in [8] that if the
linearisation of the action of $H$ on $X$ is well-adapted (which can be
achieved by adding a suitable central constant to the moment map) and if
$H$-stability=$H$-semistability (see Definition 5.7), then
$H\mu_{H}^{-1}(0)=X^{s,H}=X^{ss,H}$ and the embedding of $\mu_{H}^{-1}(0)$ in
$X^{ss,H}$ induces a homeomorphism

$\mu_{H}^{-1}(0)/Q\simeq X/\\!/H=X^{s,H}/H.$

In particular when $H=\hat{U}=U\rtimes{\mathbb{C}}^{*}$, this tells that the
embedding $\mu_{\hat{U}}^{-1}(0)\hookrightarrow X^{ss,\hat{U}}$ induces a
homeomorphism $\mu_{\hat{U}}^{-1}(0)/S^{1}\simeq X/\\!/\hat{U}$. Indeed to
have an embedding of $\mu_{H}^{-1}(0)$ in $X^{ss,H}$ and an induced
homeomorphism $\mu_{H}^{-1}(0)/Q\simeq X/\\!/H$, the condition that
$H$-stability=$H$-semistability can be weakened to the requirement that
$\hat{U}$-stability=$\hat{U}$-semistability.

Similar results hold more generally when $X$ is compact Kähler but not
necessarily projective. Suppose that $Y$ is a compact Kähler manifold acted on
by a complex reductive Lie group $G$ such that $G$ is the complexification of
a maximal compact subgroup $K$, so their Lie algebras satisfy
${\mathfrak{g}}={\mathfrak{k}}\oplus i{\mathfrak{k}}$. Let $B\subseteq G$ be a
Borel subgroup such that $G=KB$ and $K\cap B=T$ is a maximal torus in $K$. We
fix $\hat{U}=U\rtimes\lambda({\mathbb{C}})^{*}\subseteq B$ where
$\lambda:{\mathbb{C}}^{*}\to T_{\mathbb{C}}$ grades the unipotent subgroup $U$
of $B$; then $K\cap\hat{U}=S^{1}$ is a maximal compact subgroup of $\hat{U}$.
The Lie algebra of $\hat{U}$ decomposes as a real vector space as

(18) $\hat{{\mathfrak{u}}}={\mathbb{R}}\oplus
i{\mathbb{R}}\oplus{\mathfrak{u}}$

where $\mathrm{Lie}(K\cap\hat{U})={\mathbb{R}}$ and ${\mathfrak{u}}$ is the
Lie algebra of the complex unipotent group $U$. The set of positive roots
$\Delta^{+}\subseteq\Delta$ contains the weights of the adjoint action of $G$
on the Lie algebra of the unipotent radical of $B$, that is, the Cartan
decomposition has the form

${\mathfrak{g}}={\mathfrak{g}}^{-}\oplus{\mathfrak{t}}_{{\mathbb{C}}}\oplus{\mathfrak{g}}^{+}\text{
where
}\mathfrak{b}={\mathfrak{t}}_{{\mathbb{C}}}\oplus{\mathfrak{g}}^{+}\text{ and
}{\mathfrak{g}}^{\pm}=\oplus_{\alpha\in\Delta^{\pm}}e(\alpha).$

Suppose that $H=U\rtimes R\subseteq G$ where $R$ is the complexification of
$Q=K\cap H$ and $\lambda({\mathbb{C}}^{*})$ is central in $R$, so that $H$ has
internally graded unipotent radical with $\hat{U}\subseteq H$. Suppose also
that $X\subseteq Y$ is a compact complex submanifold invariant under the $H$
action, and that $K$ preserves the Kähler structure on $Y$, so
$S^{1}=K\cap\hat{U}$ and $Q=K\cap H$ preserve the induced Kähler structure on
$X$. Then we can define (generalised) moment maps $\mu_{\hat{U}}$ and
$\mu_{H}$ from $X$ to the complex duals of the Lie algebras of ${\hat{U}}$ and
$H$ by composing the restriction maps from $\mathfrak{g}^{*}$ to these duals
with the $G$-moment map on $Y$ and the inclusion of $X$ in $Y$.

In the present paper we will work with actions of the diffeomorphism group
(see §4.1)

$\hat{U}=\mathrm{Diff}_{k}(1)=\left\\{\left(\begin{array}[]{ccccc}\alpha_{1}&\alpha_{2}&\alpha_{3}&\ldots&\alpha_{k}\\\
0&\alpha_{1}^{2}&2\alpha_{1}\alpha_{2}&\ldots&2\alpha_{1}\alpha_{k-1}+\ldots\\\
0&0&\alpha_{1}^{3}&\ldots&3\alpha_{1}^{2}\alpha_{k-2}+\ldots\\\
0&0&0&\ldots&\cdot\\\
\cdot&\cdot&\cdot&\ldots&\alpha_{1}^{k}\end{array}\right)\begin{array}[]{c}:\alpha_{1},\ldots,\alpha_{k}\in{\mathbb{C}},\\\
\alpha_{1}\neq 0\end{array}\right\\}$

on projective varieties. Therefore we only state the results of [8] for the
$H=\hat{U}$ case. In this situation the symplectic description of the GIT
quotient $X/\\!/\hat{U}$ as $X/\\!/\hat{U}=\mu_{\hat{U}}^{-1}(0)/S^{1}$ fits
into the diagram

(19)
$\textstyle{\mu_{K}^{-1}(0)/S^{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\textstyle{\mu_{\hat{U}}^{-1}(0)/S^{1}=X/\\!/\hat{U}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$$\textstyle{\mu_{S^{1}}^{-1}(0)/S^{1}=X/\\!/{\mathbb{C}}^{*}}$

######  Definition 6.1.

For a weight $\alpha$ of ${\mathbb{C}}^{*}\subseteq\hat{U}$, let
${\mathbb{C}}_{\alpha}$ denote the corresponding $1$-dimensional complex
representation of ${\mathbb{C}}^{*}$ and let

$L_{\alpha}:=\mu_{S^{1}}^{-1}(0)\times_{S^{1}}{\mathbb{C}}_{\alpha}\to
X/\\!/{\mathbb{C}}^{*},$

denote the associated line bundle whose Euler class is denoted by
$e(\alpha)\in H^{2}(X/\\!/{\mathbb{C}}^{*})\simeq H_{\mathbb{C}}^{2}(X)$. For
a ${\mathbb{C}}^{*}$-invariant complex subspace
$\mathfrak{a}\subseteq\mathfrak{b}$ let

$V_{\mathfrak{a}}=\mu_{S^{1}}^{-1}(0)\times_{S^{1}}\mathfrak{a}\to
X/\\!/{\mathbb{C}}^{*}$

denote the corresponding vector bundle.

Then we have

######  Proposition 6.2 ([8], Proposition 5.11).

  1. (1)

The vector bundle $V_{\mathfrak{u}}^{*}\to X/\\!/{\mathbb{C}}^{*}$ has a
$C^{\infty}$-section $s$ which is transverse to the zero section and whose
zero set is the submanifold $\mu_{\hat{U}}^{-1}(0)/\\!/S^{1}\subseteq
X/\\!/{\mathbb{C}}^{*}$. Therefore the ${\mathbb{C}}^{*}$-equivariant normal
bundle is

$\mathscr{N}(i)\simeq V_{\mathfrak{u}}^{*}.$

  2. (2)

Let ${\mathfrak{b}}=\hat{{\mathfrak{u}}}\oplus\mathfrak{v}$ be a decomposition
invariant under the adjoint ${\mathbb{C}}^{*}$ action. Then the complex vector
bundle $V^{*}_{{\mathfrak{v}}}\to X/\\!/{\mathbb{C}}^{*}$ has a transversal
section whose zero set is the submanifold $\mu_{K}^{-1}(0)/\\!/S^{1}$.
Therefore the ${\mathbb{C}}^{*}$-equivariant normal bundles are

$\mathscr{N}(j)\simeq V^{*}_{{\mathfrak{v}}}\text{ and }\mathscr{N}(i\circ
j)\simeq V^{*}_{{\mathfrak{v}}\oplus{\mathfrak{u}}}$

This leads us to the following theorems:

######  Theorem 6.3 ([8], Theorem 5.12).

Let $X$ be a smooth projective variety endowed with a well-adapted action of
$\hat{U}=U\rtimes{\mathbb{C}}^{*}$ such that
$\hat{U}$-stability=$\hat{U}$-semistability holds. Then there is a natural
ring isomorphism

$H^{*}(X/\\!/\hat{U},{\mathbb{Q}})\simeq\frac{H^{*}(X/\\!/{\mathbb{C}}^{*},{\mathbb{Q}})}{ann(\mathrm{Euler}(V_{\mathfrak{u}})}.$

Here $\mathrm{Euler}(V_{\mathfrak{u}})\in H^{*}(X/\\!/{\mathbb{C}}^{*})$ is
the Euler class of the bundle $V_{\mathfrak{u}}$ and

$\mathrm{ann}(\mathrm{Euler}(V_{\mathfrak{u}}))=\\{c\in
H^{*}(X/\\!/{\mathbb{C}}^{*},{\mathbb{Q}})|c\cup\mathrm{Euler}(V_{\mathfrak{u}})=0\\}\subseteq
H^{*}(X/\\!/{\mathbb{C}}^{*},{\mathbb{Q}}).$

is the annihilator ideal.

######  Theorem 6.4 ([8], Theorem 5.13).

Let $X$ be a smooth projective variety endowed with a well-adapted action of
$\hat{U}=U\rtimes{\mathbb{C}}^{*}$ such that
$\hat{U}$-stability=$\hat{U}$-semistability holds. Assume that the stabiliser
in $\hat{U}$ of a generic $x\in X$ is trivial. Given a cohomology class $a\in
H^{*}(X/\\!/\hat{U})$ with a lift $\tilde{a}\in
H^{*}(X/\\!/{\mathbb{C}}^{*})$, then

$\int_{X/\\!/\hat{U}}a=\int_{X/\\!/{\mathbb{C}}^{*}}\tilde{a}\cup\mathrm{Euler}(V_{\mathfrak{u}}),$

where $\mathrm{Euler}(V_{\mathfrak{u}})$ is the cohomology class defined in
Theorem 6.3. Here we say that $\tilde{a}\in H^{*}(X/\\!/{\mathbb{C}}^{*})$ is
a lift of $a\in H^{*}(X/\\!/\hat{U})$ if $a=i^{*}\tilde{a}$.

######  Remark 8.

Theorem 6.4 can be generalised to allow the triviality assumption for the
stabiliser in $\hat{U}$ of a generic $x\in X$ to be omitted; then the sizes of
the stabilisers in $\hat{U}$ and ${\mathbb{C}}^{*}$ of a generic $x\in X$ are
included in the formula for $\int_{X/\\!/\hat{U}}a$.

Finally, we have residue formulas for the intersection pairings on the
quotient $X/\\!/{\mathbb{C}}^{*}$. There are two surjective ring homomorphisms

$\kappa_{{\mathbb{C}}^{*}}:H_{S^{1}}^{*}(X;{\mathbb{Q}})\to
H^{*}(X/\\!/{\mathbb{C}}^{*};{\mathbb{Q}})\text{ and
}\kappa_{\hat{U}}:H_{\hat{U}}^{*}(X;{\mathbb{Q}})=H_{S^{1}}^{*}(X;{\mathbb{Q}})\to
H^{*}(X/\\!/\hat{U};{\mathbb{Q}})$

from the $S^{1}$-equivariant cohomology of $X$ to the ordinary cohomology of
the corresponding GIT quotients. The fixed points of the maximal compact
subgroup $S^{1}$ of $\hat{U}$ on $X\subseteq{\mathbb{P}}^{n}$ correspond to
the weights of the ${\mathbb{C}}^{*}$ action on $X$, and since this action is
well-adapted, these weights satisfy

$\omega_{\min}=\omega_{0}<0<\omega_{1}<\ldots<\omega_{n}.$

We can represent elements of
$H_{\hat{U}}^{*}(X;{\mathbb{Q}})=H_{S^{1}}^{*}(X;{\mathbb{Q}})$ as polynomial
functions on the Lie algebra of ${\mathbb{C}}^{*}$ whose coefficients are
differential forms on $X$ and which are equivariantly closed.

######  Theorem 6.5 ([8], Theorem 5.14 and Corollary 5.15).

Let $X$ be a smooth projective variety endowed with a well-adapted action of
$\hat{U}=U\rtimes{\mathbb{C}}^{*}$ such that
$\hat{U}$-stability=$\hat{U}$-semistability holds (in the sense of Definitions
5.3 and 5.5). Let $z$ be the standard coordinate on the Lie algebra of
${\mathbb{C}}^{*}$. Given any $\hat{U}$-equivariant cohomology class $\eta$ on
$X$ represented by an equivariant differential form $\eta(z)$ whose degree is
the dimension of $X/\\!/\hat{U}$, we have

$\int_{X/\\!/\hat{U}}\kappa_{\hat{U}}(\eta)=n_{{\mathbb{C}}^{*}}\operatornamewithlimits{Res}_{z=\infty}\int_{F_{\min}}\frac{i_{F_{\min}}^{*}(\eta(z)\cup\mathrm{Euler}(V_{\mathfrak{u}})(z))}{\mathrm{Euler}(\mathscr{N}_{F_{\min}})(z)}dz$

where $F_{\min}$ is the union of those connected components of the fixed point
locus $X^{{\mathbb{C}}^{*}}$ on which the $S^{1}$-moment map takes its minimum
value $\omega_{\min}$, and $n_{\hat{U}}$ is the positive integer which is the
order of the stabiliser in $\hat{U}$ of a generic $x\in X$.

##  7\. Equivariant localisation and multidegrees

This section is a brief introduction to equivariant cohomology and
localisation. For more details, we refer the reader to Berline–Getzler–Vergne
[12] and Bérczi–Szenes [11].

Let ${K}\cong U(1)^{n}$ be the maximal compact subgroup of
$T\cong({\mathbb{C}}^{*})^{n}$, and denote by $\mathfrak{t}$ the Lie algebra
of ${K}$. Identifying $T$ with the group ${\mathbb{C}}^{n}$, we obtain a
canonical basis of the weights of $T$:
$\lambda_{1},\ldots,\lambda_{n}\in\mathfrak{t}^{*}$.

For a manifold $M$ endowed with the action of ${K}$, one can define a
differential $d_{K}$ on the space $S^{\bullet}\
mathfrak{t}^{*}\otimes\Omega^{\bullet}(M)^{K}$ of polynomial functions on
$\mathfrak{t}$ with values in ${K}$-invariant differential forms by the
formula:

$[d_{K}\alpha](X)=d(\alpha(X))-\iota(X_{M})[\alpha(X)],$

where $X\in\mathfrak{t}$, and $\iota(X_{M})$ is contraction by the
corresponding vector field on $M$. A homogeneous polynomial of degree $d$ with
values in $r$-forms is placed in degree $2d+r$, and then $d_{K}$ is an
operator of degree 1. The cohomology of this complex–the so-called equivariant
de Rham complex, denoted by $H^{\bullet}_{T}(M)$, is called the
$T$-equivariant cohomology of $M$. Elements of $H_{T}^{\bullet}(M)$ are
therefore polynomial functions $\mathfrak{t}\to\Omega^{\bullet}(M)^{K}$ and
there is an integration (or push-forward map) $\int:H_{T}^{\bullet}(M)\to
H_{T}^{\bullet}(\mathrm{point})=S^{\bullet}\mathfrak{t}^{*}$ defined as

$(\int_{M}\alpha)(X)=\int_{M}\alpha^{[\mathrm{dim}(M)]}(X)\text{ for all
}X\in\mathfrak{t}$

where $\alpha^{[\mathrm{dim}(M)]}$ is the differential-form-top-degree part of
$\alpha$. The following proposition is the Atiyah-Bott-Berline-Vergne
localisation theorem in the form of [12], Theorem 7.11.

######  Theorem 7.1 ((Atiyah-Bott [2], Berline-Vergne [13])).

Suppose that $M$ is a compact complex manifold and $T$ is a complex torus
acting smoothly on $M$, and the fixed point set $M^{T}$ of the $T$-action on M
is finite. Then for any cohomology class $\alpha\in H_{T}^{\bullet}(M)$

$\int_{M}\alpha=\sum_{f\in
M^{T}}\frac{\alpha^{[0]}(f)}{\mathrm{Euler}^{T}(T_{f}M)}.$

Here $\mathrm{Euler}^{T}(T_{f}M)$ is the $T$-equivariant Euler class of the
tangent space $T_{f}M$, and $\alpha^{[0]}$ is the differential-form-degree-0
part of $\alpha$.

The right hand side in the localisation formula considered in the fraction
field of the polynomial ring of
$H_{T}^{\bullet}(\mathrm{point})=H^{\bullet}(BT)=S^{\bullet}\mathfrak{t}^{*}$
(see more on details in Atiyah–Bott [2] and [12]). Part of the statement is
that the denominators cancel when the sum is simplified.

###  7.1. Equivariant Poincaré duals and multidegrees

Restricting the equivariant de Rham complex to compactly supported (or quickly
decreasing at infinity) differential forms, one obtains the compactly
supported equivariant cohomology groups $H^{\bullet}_{{K},\mathrm{cpt}}(M)$.
Clearly $H^{\bullet}_{{K},\mathrm{cpt}}(M)$ is a module over
$H^{\bullet}_{K}(M)$. For the case when $M=W$ is an $N$-dimensional complex
vector space, and the action is linear, one has
$H^{\bullet}_{K}(W)=S^{\bullet}\mathfrak{t}^{*}$ and
$H^{\bullet}_{{K},\mathrm{cpt}}(W)$ is a free module over $H^{\bullet}_{K}(W)$
generated by a single element of degree $2N$:

(20)
$H^{\bullet}_{{K},\mathrm{cpt}}(W)=H^{\bullet}_{{K}}(W)\cdot\mathrm{Thom}_{{K}}(W)$

Fixing coordinates $y_{1},\dots,y_{N}$ on $W$, in which the $T$-action is
diagonal with weights $\eta_{1},\ldots,\eta_{N}$, one can write an explicit
representative of $\mathrm{Thom}_{{K}}(W)$ as follows:

$\mathrm{Thom}_{{K}}(W)=e^{-\sum_{i=1}^{N}|y_{i}|^{2}}\sum_{\sigma\subset\\{1,\ldots,N\\}}\prod_{i\in\sigma}\eta_{i}/2\cdot\prod_{i\notin\sigma}dy_{i}\,d\bar{y}_{i}$

We will say that an algebraic variety has dimension $d$ if its maximal-
dimensional irreducible components are of dimension $d$. A $T$-invariant
algebraic subvariety $\Sigma$ of dimension $d$ in $W$ represents
${K}$-equivariant $2d$-cycle in the sense that

  * •

a compactly-supported equivariant form $\mu$ of degree $2d$ is absolutely
integrable over the components of maximal dimension of $\Sigma$, and
$\int_{\Sigma}\mu\in S^{\bullet}\mathfrak{t}$;

  * •

if $d_{K}\mu=0$, then $\int_{\Sigma}\mu$ depends only on the class of $\mu$ in
$H^{\bullet}_{{K},\mathrm{cpt}}(W)$,

  * •

and $\int_{\Sigma}\mu=0$ if $\mu=d_{K}\nu$ for a compactly-supported
equivariant form $\nu$.

######  Definition 7.2.

Let $\Sigma$ be an $T$-invariant algebraic subvariety of dimension $d$ in the
vector space $W$. Then the equivariant Poincaré dual of $\Sigma$ is the
polynomial on $\mathfrak{t}$ defined by the integral

(21)
$\mathrm{eP}[\Sigma]=\frac{1}{(2\pi)^{d}}\int_{\Sigma}\mathrm{Thom}_{{K}}(W).$

######  Remark 9.

  1. (1)

An immediate consequence of the definition is that for an equivariantly closed
differential form $\mu$ with compact support, we have

$\int_{\Sigma}\mu=\int_{W}\mathrm{eP}[\Sigma]\cdot\mu.$

This formula serves as the motivation for the term equivariant Poincaré dual.

  2. (2)

This definition naturally extends to the case of an analytic subvariety of
${\mathbb{C}}^{n}$ defined in the neighborhood of the origin, or more
generally, to any $T$-invariant cycle in ${\mathbb{C}}^{n}$.

Another terminology for the equivariant Poincaré dual is multidegree, which is
close in spirit to the original construction of Joseph [25]. Let
$\Sigma\subset W$ be a $T$-invariant subvariety. Then we have

$\mathrm{eP}[\Sigma,W]_{T}=\mathrm{mdeg}[I(\Sigma),{\mathbb{C}}[y_{1},\ldots,y_{N}]].$

Some basic properties of the equivariant Poincaré dual are listed in [11],
these are: Positivity, Additivity, Deformation invariance, Symmetry and a
formula for complete intersections. Using these properties one can easily
describe an algorithm for computing $\mathrm{mdeg}[I,S]$ as follows (see
Miller–Sturmfels [30, §8.5], Vergne [43] and [11] for details).

An ideal $M\subset S$ generated by a set of monomials in $y_{1},\ldots,y_{N}$
is called a _monomial ideal_. Since $\mathrm{in}_{<}(I)$ is such an ideal, by
the deformation invariance it is enough to compute $\mathrm{mdeg}[M]$ for
monomial ideals $M$. If the codimension of $\Sigma(M)$ in $W$ is $s$, then the
maximal dimensional components of $\Sigma(M)$ are codimension-$s$ coordinate
subspaces of $W$. Such subspaces are indexed by subsets
$\mathbf{i}\in\\{1\ldots N\\}$ of cardinality $s$; the corresponding
associated primes are $\mathfrak{p}[\mathbf{i}]=\langle
y_{i}:i\in\mathbf{i}\rangle$. Then

$\mathrm{mult}(\mathfrak{p}[\mathbf{i}],M)=\left|\left\\{\mathbf{a}\in{\mathbb{Z}}_{+}^{[{\mathbf{i}}]};\;\mathbf{y}^{\mathbf{a}+\mathbf{b}}\notin
M\text{ for all
}\mathbf{b}\in{\mathbb{Z}}_{+}^{[\hat{\mathbf{i}}]}\right\\}\right|,$

where
${\mathbb{Z}}_{+}^{[\mathbf{i}]}=\\{\mathbf{a}\in{\mathbb{Z}}_{+}^{N};a_{i}=0\text{
for }i\notin\mathbf{i}\\}$, $\hat{\mathbf{i}}=\\{1\ldots
N\\}\setminus\mathbf{i}$, and $|\cdot|$, as usual, stands for the number of
elements of a finite set. By the normalization and additivity axiom we have

(22)
$\mathrm{mdeg}[M,S]=\sum_{|\mathbf{i}|=s}\mathrm{mult}(\mathfrak{p}[\mathbf{i}],M)\prod_{i\in\mathbf{i}}\eta_{i}.$

By definition, the weights $\eta_{1},\ldots\eta_{N}$ on $W$ are linear forms
of $\lambda_{1},\ldots\lambda_{r}$, the basis of $({\mathbb{C}}^{*})^{r}$, and
we denote the coefficient of $\lambda_{j}$ in $\eta_{i}$ by
$\mathrm{coeff}(\eta_{i},j)$, $1\leq i\leq N,1\leq j\leq r$, and introduce

$\deg(\eta_{1},\ldots,\eta_{N};m)=\\#\\{i;\;\mathrm{coeff}(\eta_{i},m)\neq
0\\}\\}.$

It is clear from the formula (22) that

$\deg_{\lambda_{m}}\mathrm{mdeg}[I,S]\leq\deg(\eta_{1},\ldots,\eta_{N};m)$

holds for any $1\leq m\leq r$. We need a slightly stronger result in the next
section which we formulate and prove here.

######  Proposition 7.3.

Let $W$ be an $N$-dimensional complex vector space with coordinates
$y_{1},\dots,y_{N}$ endowed with an diagonal action of
$({\mathbb{C}}^{*})^{r}$ acting with weights $\eta_{1}\ldots\eta_{N}$. Let
$X\subset W$ be a $({\mathbb{C}}^{*})^{r}$-invariant irreducible subvariety
not contained in the coordinate hyperplanes $\\{y_{i}=0\\}$ for $1\leq i\leq
N$. Let $I=I(X)\subset S$ be its $({\mathbb{C}}^{*})^{r}$-invariant prime
ideal. Then

$\deg_{\lambda_{m}}\mathrm{mdeg}[I,S]\leq\deg(\eta_{1},\ldots,\eta_{N};m)-1$

###### Proof.

By the positivity property of the multidegree $\mathrm{mdeg}[I,S]$ is indeed a
polynomial of the weights $\eta_{i},i=1,\ldots,N$. Let

$\mathrm{coeff}(\eta_{i},m)\neq 0\text{ for }1\leq i\leq s;\
\mathrm{coeff}(\eta_{s+1},m)=\ldots=\mathrm{coeff}(\eta_{N},m)=0.$

The idea of the proof is to choose an appropriate monomial order on the
polynomial ring $S={\mathbb{C}}[y_{1},\ldots,y_{N}]$ to ensure that $y_{1}$
does not appear in the corresponding initial ideal.

To that end recall, that a weight function is a linear map
$\rho:{\mathbb{Z}}^{N}\to{\mathbb{Z}}$. This defines a partial order
$>_{\rho}$ on the monomials of $S$, called the weight order associated to
$\rho$, by the rule $m=y^{a}>_{\rho}n=y^{b}$ iff $\rho(a)>\rho(b)$. Here
$a=(a_{1},\ldots,a_{N}),b=(b_{1},\ldots,b_{N})$ are arbitrary multiindices.
Any weight order can be extended to a compatible monomial order $>$ (see
Eisenbud [17, Ch 15.2]), which means that $m>_{\rho}n$ implies $m>n$. For our
purposes define

$\rho(y_{1})=-1,\rho(y_{2})=\ldots=\rho(y_{N})=0$

and let $>$ denote arbitrary compatible monomial order on $S$. By definition
for a monomial $m\in S$

(23) $\rho(m)<0\Longleftrightarrow y_{1}|m.$

Let $p=m_{1}+\ldots+m_{t}\in I$ be a polynomial where the $m_{i}$’s are
monomials in the variables $y_{1},\ldots,y_{N}$ of the same weighted degree.
We claim that if $in_{<}(p)$ is a basis element of the initial ideal
$in_{<}(I)$ then not all monomials $m_{i}$ of $p$ are divisible by $y_{1}$.
Indeed, if $y_{1}|m_{i}$ for $i=1,\ldots,t$ then $y_{1}|p$. But $p\neq y_{1}$
by assumption because this would mean that $X\subset\\{y_{1}=0\\}$. Therefore
$p^{\prime}=p/y_{1}\in I$ and $y_{1}\cdot in_{>}(p^{\prime})=in_{>}(p)$ holds
and therefore $in_{<}(p)$ is not among the generators of $in_{>}(I)$. So there
is a monomial of $p$ not containing $y_{1}$, and by (23) the weight of this
monomial is strictly bigger to the weight of any other containing $y_{1}$.
Consequently, $y_{1}$ does not divide any of the generators of $in_{>}(I)$,
and by (22), $\mathrm{mdeg}[I,S]$ does not depend on $\eta_{1}$. The only
possible variables containing $\lambda_{m}$ are therefore
$\eta_{2},\ldots,\eta_{s}$, giving a maximum total degre $s-1$. ∎

###  7.2. The Rossman formula

The Rossmann equivariant localisation formula is an improved version of the
Atiyah-Bott/Berline-Vergne localisation for singular varieties sitting in a
smooth ambient space. Let $Z$ be a complex manifold with a holomorphic
$T$-action, and let $M\subset Z$ be a $T$-invariant analytic subvariety with
an isolated fixed point $p\in M^{T}$. Then one can find local analytic
coordinates near $p$, in which the action is linear and diagonal. Using these
coordinates, one can identify a neighborhood of the origin in
$\mathrm{T}_{p}Z$ with a neighborhood of $p$ in $Z$. We denote by
$\hat{T}_{p}M$ the part of $\mathrm{T}_{p}Z$ which corresponds to $M$ under
this identification; informally, we will call $\hat{T}_{p}M$ the $T$-invariant
tangent cone of $M$ at $p$. This tangent cone is not quite canonical: it
depends on the choice of coordinates; the equivariant dual of
$\Sigma=\hat{T}_{p}M$ in $W=\mathrm{T}_{p}Z$, however, does not. Rossmann
named this the equivariant multiplicity of $M$ in $Z$ at $p$:

(24)
$\mathrm{emult}_{p}[M,Z]\overset{\mathrm{def}}{=}\mathrm{eP}[\hat{T}_{p}M,\mathrm{T}_{p}Z].$

######  Remark 10.

In the algebraic framework one might need to pass to the tangent scheme of $M$
at $p$ (cf. Fulton [21]). This is canonically defined, but we will not use
this notion.

The analog of the Atiyah-Bott formula for singular subvarieties of smooth
ambient manifolds is the following statement.

######  Proposition 7.4 (Rossmann’s localisation formula [38]).

Let $\mu\in H_{T}^{*}(Z)$ be an equivariant class represented by a holomorphic
equivariant map $\mathfrak{t}\to\Omega^{\bullet}(Z)$. Then

(25) $\int_{M}\mu=\sum_{p\in
M^{T}}\frac{\mathrm{emult}_{p}[M,Z]}{\mathrm{Euler}^{T}(\mathrm{T}_{p}Z)}\cdot\mu^{[0]}(p),$

where $\mu^{[0]}(p)$ is the differential-form-degree-zero component of $\mu$
evaluated at $p$.

##  8\. NRGIT quotients versus curvilinear Hilbert schemes

The moduli of $k$-jets in ${\mathbb{C}}^{n}$ is a quasi-affine non-reductive
quotient. The NRGIT compactification and
$\mathrm{CHilb}^{k+1}({\mathbb{C}}^{n})$ are two different compactifications.
In this section we explain the strategy of our argument: we find a blow-up of
the NRGIT model which admits a morphism to
$\mathrm{CHilb}^{k+1}({\mathbb{C}}^{n})$, so tautological integrals over the
curvilinear Hilbert scheme can be pulled back to equivariant integrals over
the NRGIT model. We set up the notations here, and explain the initial blow-
up: fibration over the flag manifold, and introduce the coordinates
$\beta_{ij}$.

Recall the test curve model from Theorem 4.4, which says that for any $k,n$

$\mathrm{CHilb}^{k+1}({\mathbb{C}}^{n})=\overline{\mathrm{im}(\phi)}\subset\mathrm{Grass}_{k}(J_{k}(n,1)^{*})$

with the $\mathrm{Diff}_{k}$-invariant morphism
$\phi:J_{k}^{reg}(1,n)\to\mathrm{Grass}_{k}(J_{k}(n,1)^{*})$ (which we will
refer to in short as the test curve morphism) defined as

$\phi(v_{1},\ldots,v_{k})=v_{1}\wedge(v_{2}+v_{1}^{2})\wedge(v_{3}+2v_{1}v_{2}+v_{1}^{3})\wedge\ldots\wedge(\sum_{i_{1}+\ldots+i_{r}=k}v_{i_{1}}\ldots
v_{i_{r}})$

Let $k\leq n$ and fix a basis $\\{e_{1},\ldots,e_{n}\\}$ of
${\mathbb{C}}^{n}$. Then, according to Theorem 4.4, the curvilinear Hilbert
scheme is the closure of the $\mathrm{GL}(n)$ orbit of
$p_{k,n}=\phi(e_{1},\ldots,e_{k})$:

$\overline{\,{\rm Im}\,(\phi)}=\overline{\mathrm{GL_{n}}\cdot p_{k,n}}.$

In order to avoid working with singular centers of blow-ups we slightly modify
the strategy outlined in §2. Let $P_{n,k}\subset\mathrm{GL}_{n}$ denote the
parabolic subgroup which preserves the flag

$\mathbf{f}=(\mathrm{Span}(e_{1})\subset\mathrm{Span}(e_{1},e_{2})\subset\ldots\subset\mathrm{Span}(e_{1},\ldots,e_{k})\subset{\mathbb{C}}^{n}).$

######  Definition 8.1.

Define the partial desingularization

$\widetilde{\mathrm{CHilb}}^{k+1}({\mathbb{C}}^{n})=\mathrm{GL}_{n}\times_{P_{n,k}}\overline{P_{n,k}\cdot
p_{k,n}}$

with the resolution map
$\rho:\widetilde{\mathrm{CHilb}}^{k+1}({\mathbb{C}}^{n})\to\mathrm{CHilb}^{k+1}({\mathbb{C}}^{n})$
given by $\rho(g,x)=g\cdot x$. In short, this is the blow-up of the
curvilinear Hilbert scheme at the linear part. Note that the blown-up space
fibers over the complete flag manifold on ${\mathbb{C}}^{n}$:

$\pi:\widetilde{\mathrm{CHilb}}^{k+1}({\mathbb{C}}^{n})\to\mathrm{GL}_{n}/P_{n,k}=\mathrm{Flag}_{1,2,\ldots,k}({\mathbb{C}}^{n})$

Equivalently, let $J_{k}^{\mathrm{nondeg}}({1},{n})\subset
J_{k}^{\mathrm{reg}}({1},{n})$ be the set of test curves with
$\gamma^{\prime},\ldots,\gamma^{(k)}$ linearly independent. These correspond
to the nonsingular $n\times k$ matrices in $\,{\rm
Hom}\,({\mathbb{C}}^{k},{\mathbb{C}}^{n})$, and they fibre over the set of
complete flags in ${\mathbb{C}}^{n}$:

(26) $J_{k}^{\mathrm{nondeg}}({1},{n})/\mathrm{Diff}_{k}(1)\to\,{\rm
Hom}\,({\mathbb{C}}^{k},{\mathbb{C}}^{n})/B_{k}=\mathrm{Flag}_{1,\ldots,k}({\mathbb{C}}^{n})$

where $B_{k}\subset\mathrm{GL}(k)$ is the upper Borel. The image of the fibres
under $\phi$ are isomorphic to $P_{n,k}\cdot p_{k,n}$, and therefore
$\widetilde{\mathrm{CHilb}}^{k+1}({\mathbb{C}}^{n})$ is the fibrewise
compactification of $J_{k}^{\mathrm{nondeg}}({1},{n})$ over
$\mathrm{Flag}_{1\ldots k}({\mathbb{C}}^{n})$.

The strategy outlined in §2 will be refined as follows:

(27)
---
$\textstyle{\mathrm{Jet}_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\mathrm{Jet}_{k}^{ss}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\phi}}$$\textstyle{\mathrm{GL}(n)\times_{P_{n,k}}{\mathbb{P}}({\mathbb{C}}\oplus\,{\rm
Hom}\,^{\mathbf{f}}({\mathbb{C}}^{k},{\mathbb{C}}^{n}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{\widetilde{\mathrm{CHilb}}^{k+1}({\mathbb{C}}^{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Flag}_{1\ldots
k}({\mathbb{C}}^{n})}$

Hence we need to describe the iterated blow-up process of the fiber over
${\mathbf{f}}$. Here $\,{\rm
Hom}\,^{\mathbf{f}}({\mathbb{C}}^{k},{\mathbb{C}}^{n})$ are formed by $n\times
k$ matrices whose column vectors $v_{1},\ldots,v_{k}$ sit over $\mathbf{f}$,
that is they have the form

(28) $\displaystyle v_{1}=$ $\displaystyle v_{11}e_{1}$ $\displaystyle v_{2}=$
$\displaystyle v_{22}e_{2}+v_{12}e_{1}$ $\displaystyle\cdots$ $\displaystyle
v_{k}=$ $\displaystyle v_{kk}e_{k}+v_{{k-1}k}e_{k-1}+\ldots+v_{1k}e_{1}$

The rational map

$\phi:{\mathbb{P}}[{\mathbb{C}}\oplus\,{\rm
Hom}\,^{\mathbf{f}}({\mathbb{C}}^{k},{\mathbb{C}}^{n})]\dasharrow{\mathbb{P}}(\wedge^{k}\mathrm{Sym}^{\leq
k}{\mathbb{C}}^{n})$

on the compactified fiber over $\mathbf{f}$ reads as

$[x:v_{1},\ldots
v_{k}]\mapsto[v_{1}\wedge(xv_{2}+v_{1}^{2})\wedge(x^{2}v_{3}+xv_{1}v_{2}+v_{1}^{3})\wedge\ldots\wedge(\sum_{i_{1}+\ldots+i_{r}=k}x^{k-r}v_{i_{1}}\ldots
v_{i_{r}})]$

##  9\. Equivariant localisation and the Residue Vanishing Theorem

In this subsection we develop a two step equivariant localisation method on
$\widetilde{\mathrm{CHilb}}^{k+1}({\mathbb{C}}^{n})$ which is a stronger
version of our iterated residue in [11]. We need an important restriction on
the parameters to make this method work, namely we assume that $k\leq n$ in
this section. The partial resolution
$\rho:\widetilde{\mathrm{CHilb}}^{k+1}({\mathbb{C}}^{n})\to\mathrm{CHilb}^{k+1}({\mathbb{C}}^{n})$
fibers over the flag manifold $\mathrm{Flag}_{k}({\mathbb{C}}^{n})$

(29)
$\textstyle{\widetilde{\mathrm{CHilb}}^{k+1}({\mathbb{C}}^{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\rho}$$\scriptstyle{\mu}$$\textstyle{\mathrm{CHilb}^{k+1}({\mathbb{C}}^{n})\subset\mathrm{Grass}_{k}(\mathrm{Sym}^{\leq
k}{\mathbb{C}}^{n})}$$\textstyle{\,{\rm
Hom}\,^{\mathrm{nondeg}}({\mathbb{C}}^{k},{\mathbb{C}}^{n})/B_{k}=\mathrm{Flag}_{k}({\mathbb{C}}^{n})}$

where the fibres of $\mu$ are isomorphic to $\overline{P_{k,n}\cdot
p_{k,n}}\subset\mathrm{Grass}_{k}(\mathrm{Sym}^{\leq k}{\mathbb{C}}^{n})$.

Let $e_{1},\ldots,e_{n}\in{\mathbb{C}}^{n}$ be an eigenbasis of
${\mathbb{C}}^{n}$ for the $T^{n}\subset\mathrm{GL}(n)$ action with weights
$\lambda_{1},\ldots,\lambda_{n}\in\mathfrak{t}^{*}$ and let
${\mathbf{f}}=(\langle e_{1}\rangle\subset\langle
e_{1},e_{2}\rangle\subset\ldots\subset\langle
e_{1},\ldots,e_{k}\rangle\subset{\mathbb{C}}^{n})$ denote the standard flag in
${\mathbb{C}}^{n}$ fixed by the upper Borel as before.

Since $\widetilde{\mathrm{CHilb}}^{k+1}({\mathbb{C}}^{n})$ fibers over the
flag manifold $\mathrm{Flag}_{k}({\mathbb{C}}^{n})$, the ABBV localisation
formula of Proposition 7.1 reads as

(30)
$\int_{\widetilde{\mathrm{CHilb}}^{k+1}({\mathbb{C}}^{n})}\alpha=\sum_{\sigma\in\mathscr{S}_{n}/\mathscr{S}_{n-k}}\frac{\alpha_{\sigma({\mathbf{f}})}}{\prod_{1\leq
m\leq k}\prod_{i=m+1}^{n}(\lambda_{\sigma\cdot i}-\lambda_{\sigma\cdot m})},$

where

  * •

$\sigma$ runs over the ordered $k$-element subsets of $\\{1,\ldots,n\\}$
labeling the fixed flags $\sigma({\mathbf{f}})=(\langle
e_{\sigma(1)}\rangle\subset\ldots\subset\langle
e_{\sigma(1)},\ldots,e_{\sigma(k)}\rangle\subset{\mathbb{C}}^{n})$ in
${\mathbb{C}}^{n}$.

  * •

$\prod_{1\leq m\leq
k}\prod_{i=m+1}^{n}(\lambda_{\sigma(i)}-\lambda_{\sigma(m)})$ is the
equivariant Euler class of the tangent space of
$\mathrm{Flag}_{k}({\mathbb{C}}^{n})$ at $\sigma({\mathbf{f}})$.

  * •

if $X_{\sigma({\mathbf{f}})}=\mu^{-1}(\sigma({\mathbf{f}}))$ denotes the fibre
then
$\alpha_{\sigma({\mathbf{f}})}=(\int_{X_{\sigma({\mathbf{f}})}}\alpha)^{[0]}(\sigma({\mathbf{f}}))\in
S^{\bullet}\mathfrak{t}^{*}$ is the differential-form-degree-zero part
evaluated at $\sigma({\mathbf{f}})$.

In particular, the Chern roots of the tautological bundle over
$\mathrm{Grass}_{k}(\mathrm{Sym}^{\leq k}{\mathbb{C}}^{n})$ at the fixed point
$\sigma({\mathbf{f}})$ are represented by
$\lambda_{\sigma(1)},\ldots,\lambda_{\sigma(k)}\in\mathfrak{t}^{*}$ and
therefore if $\lambda$ is a Chern polynomial of the tautological bundle then

(31)
$\alpha_{\sigma({\mathbf{f}})}=\sigma\cdot\alpha_{\mathbf{f}}=\alpha_{\mathbf{f}}(\lambda_{\sigma(1)},\ldots,\lambda_{\sigma(k)})\in
S^{\bullet}\mathfrak{t}^{*},$

is the $\sigma$-shift of the polynomial
$\alpha_{{\mathbf{f}}}=(\int_{X_{{\mathbf{f}}}}\alpha)^{[0]}({\mathbf{f}})\in
S^{\bullet}\mathfrak{t}^{*}$ corresponding to the distinguished fixed flag
${\mathbf{f}}$.

###  9.1. Transforming the localisation formula into iterated residue

We transform the right hand side of (30) into an iterated residue motivated by
Bérczi–Szenes [11]. This step turns out to be crucial in handling the
combinatorial complexity of the fixed point data in the Atiyah-Bott
localisation formula and condense the symmetry of this fixed point data in an
efficient way which enables us to prove the vanishing of the contribution of
all but one of the fixed points.

To describe this formula, we will need the notion of an iterated residue (cf.
e.g. [39]) at infinity. Let $\omega_{1},\dots,\omega_{N}$ be affine linear
forms on ${\mathbb{C}}^{k}$; denoting the coordinates by $z_{1},\ldots,z_{k}$,
this means that we can write
$\omega_{i}=a_{i}^{0}+a_{i}^{1}z_{1}+\ldots+a_{i}^{k}z_{k}$. We will use the
shorthand $h(\mathbf{z})$ for a function $h(z_{1}\ldots z_{k})$, and
$\,d\mathbf{z}$ for the holomorphic $n$-form $dz_{1}\wedge\dots\wedge dz_{k}$.
Now, let $h(\mathbf{z})$ be an entire function, and define the iterated
residue at infinity as follows:

(32)
$\operatornamewithlimits{Res}_{z_{1}=\infty}\operatornamewithlimits{Res}_{z_{2}=\infty}\dots\operatornamewithlimits{Res}_{z_{k}=\infty}\frac{h(\mathbf{z})\,\,d\mathbf{z}}{\prod_{i=1}^{N}\omega_{i}}\overset{\mathrm{def}}{=}\left(\frac{1}{2\pi
i}\right)^{k}\int_{|z_{1}|=R_{1}}\ldots\int_{|z_{k}|=R_{k}}\frac{h(\mathbf{z})\,\,d\mathbf{z}}{\prod_{i=1}^{N}\omega_{i}},$

where $1\ll R_{1}\ll\ldots\ll R_{k}$. The torus $\\{|z_{m}|=R_{m};\;m=1\ldots
k\\}$ is oriented in such a way that
$\operatornamewithlimits{Res}_{z_{1}=\infty}\ldots\operatornamewithlimits{Res}_{z_{k}=\infty}\,d\mathbf{z}/(z_{1}\cdots
z_{k})=(-1)^{k}$. We will also use the following simplified notation:
$\operatornamewithlimits{Res}_{\mathbf{z}=\infty}\overset{\mathrm{def}}{=}\operatornamewithlimits{Res}_{z_{1}=\infty}\operatornamewithlimits{Res}_{z_{2}=\infty}\dots\operatornamewithlimits{Res}_{z_{k}=\infty}.$

In practice, one way to compute the iterated residue (32) is the following
algorithm: for each $i$, use the expansion

(33)
$\frac{1}{\omega_{i}}=\sum_{j=0}^{\infty}(-1)^{j}\frac{(a^{0}_{i}+a^{1}_{i}z_{1}+\ldots+a_{i}^{q(i)-1}z_{q(i)-1})^{j}}{(a_{i}^{q(i)}z_{q(i)})^{j+1}},$

where $q(i)$ is the largest value of $m$ for which $a_{i}^{m}\neq 0$, then
multiply the product of these expressions with $(-1)^{k}h(z_{1}\ldots z_{k})$,
and then take the coefficient of $z_{1}^{-1}\ldots z_{k}^{-1}$ in the
resulting Laurent series.

######  Proposition 9.1 (Bérczi–Szenes [11], Proposition 5.4).

For any homogeneous polynomial $Q(\mathbf{z})$ on ${\mathbb{C}}^{k}$ we have

(34)
$\sum_{\sigma\in\mathscr{S}_{n}/\mathscr{S}_{n-k}}\frac{Q(\lambda_{\sigma(1)},\ldots,\lambda_{\sigma(k)})}{\prod_{1\leq
m\leq k}\prod_{i=m+1}^{n}(\lambda_{\sigma\cdot i}-\lambda_{\sigma\cdot
m})}=\operatornamewithlimits{Res}_{\mathbf{z}=\infty}\frac{\prod_{1\leq
m<l\leq
k}(z_{m}-z_{l})\,Q(\mathbf{z})\,d\mathbf{z}}{\prod_{l=1}^{k}\prod_{i=1}^{n}(\lambda_{i}-z_{l})}.$

######  Remark 11.

Changing the order of the variables in iterated residues, usually, changes the
result. In this case, however, because all the poles are normal crossing,
formula (34) remains true no matter in what order we take the iterated
residues.

This together with (30) and (31) gives

######  Proposition 9.2.

Let $k\leq n$ and $\alpha(\eta_{1},\ldots,\eta_{k})$ be a Chern polynomial in
the Cherns roots of the tautological rank $k$ bundle $\mathscr{E}$ over
$\mathrm{Grass}_{k}(\mathrm{Sym}^{\leq k}{\mathbb{C}}^{n})$. Then

$\int_{\widetilde{\mathrm{CHilb}}^{k+1}({\mathbb{C}}^{n})}\alpha(u)=\operatornamewithlimits{Res}_{\mathbf{z}=\infty}\frac{\prod_{1\leq
m<l\leq
k}(z_{m}-z_{l})\,\alpha_{{\mathbf{f}}}(z_{1},\ldots,z_{k})\,d\mathbf{z}}{\prod_{l=1}^{k}\prod_{i=1}^{n}(\lambda_{i}-z_{l})}$

To calculate
$\alpha_{\mathbf{f}}(z_{1},\ldots,z_{k})=\int_{X_{\mathbf{f}}}\alpha$ on the
fiber

$X_{\mathbf{f}}=\mu^{-1}({\mathbf{f}})\simeq\overline{P_{k,n}\cdot
p_{k,n}}\subset\mathrm{Flag}_{k}(\mathrm{Sym}^{\leq k}{\mathbb{C}}^{n})$

we proceed a second equivariant localisation on $X_{\mathbf{f}}$. Note that

$X_{\mathbf{f}}=\overline{\phi(\,{\rm
Hom}\,^{\mathbf{f}}({\mathbb{C}}^{k},{\mathbb{C}}^{n})}$

where

$\,{\rm
Hom}\,^{\mathbf{f}}({\mathbb{C}}^{k},{\mathbb{C}}^{n})=\\{\psi\in\,{\rm
Hom}\,({\mathbb{C}}^{k},{\mathbb{C}}^{n}):\psi(e_{i})\subset{\mathbb{C}}_{[i]}\text{
for }i=1,\ldots,k\\}$

and ${\mathbb{C}}_{[i]}\subset{\mathbb{C}}^{n}$ is the subspace spanned by
$e_{1},\ldots,e_{i}$. The $\mathrm{GL}(n)$ action on $\,{\rm
Hom}\,^{\mathbf{f}}({\mathbb{C}}^{k},{\mathbb{C}}^{n})$ reduces to
$\mathrm{GL}({\mathbb{C}}_{[k]})\subset\mathrm{GL}(n)$, and we perform a
second $\mathrm{GL}({\mathbb{C}}_{[k]})$-equivariant localisation on
$X_{\mathbf{f}}$ to calculate $\alpha_{\mathbf{f}}$. In the residue formula
the Chern roots of the torus $T^{k}\subset\mathrm{GL}({\mathbb{C}}_{[k]})$ of
$\mathscr{E}$ on $X_{\mathbf{f}}$ are $z_{1},\ldots,z_{k}$.

We perform this second localisation on the blown-up fibers of
$\mathrm{Jet}_{k}$. We arrive to this space after an iterated blow-up process
encoded by a rooted blow-up tree, and the localization formula will give a sum
over all fixed points, where fixed points correspond to leaves of the tree.
For each leaf $L$ we will have a cluster of weights $\\{z^{L}(\beta):\beta\in
B\\}$ associated to a fixed finite parametrising set $B$, and these will give
the tangent weights of the blown-up space at this fixed point. For a leaf
$L\in\mathscr{L}_{k}$, the image $\tilde{\phi}$ under the blown-up morphism is
a fixed point on $\mathrm{Grass}_{k}(\mathrm{Sym}^{\leq k}{\mathbb{C}}^{n})$
with tautological weights $z^{L}_{1},\ldots,z^{L}_{k}$. Both the $z^{L}_{i}$
and $z^{L}(\beta)$ weights are linear form in the $z_{i}$’s. We arrive at

######  Proposition 9.3.

Let $k\leq n$ and $\alpha(\eta_{1},\ldots,\eta_{k})$ be a Chern polynomial in
the Cherns roots of the tautological rank $k$ bundle over
$\mathrm{Grass}_{k}(\mathrm{Sym}^{\leq k}{\mathbb{C}}^{n})$. Let
$\mathscr{L}_{k}$ denote the set of leaves in the blow-up tree. Then

(35)
$\int_{\mathrm{Jet}_{k}/\\!/\mathrm{Diff}_{k}}\tilde{\phi}^{*}\alpha=\sum_{L\in\mathscr{L}_{k}}\operatornamewithlimits{Res}_{\mathbf{z}=\infty}\frac{(k-1)!z^{k-1}\prod_{m<l}(z_{m}-z_{l})\alpha(z^{L}_{1},\ldots,z^{L}_{k})}{\mathrm{Euler}^{L}(\mathrm{Jet}_{k})\prod_{l=1}^{k}\prod_{i=1}^{n}(\lambda_{i}-z_{l})}\,\,d\mathbf{z}$

where $\mathrm{Euler}^{L}(\mathrm{Jet}_{k})=\prod_{\beta\in B}z^{L}(\beta)$ is
the equivariant Euler class of the tangent space at $L$.

######  Remark 12.

The formula (35) contains a few unexplained ingredients: the set of ’leaves’ ,
and the weights $z_{i},z_{i}^{L}$. Still, below we will be able to prove the
residue vanishing theorem using only the following minimal information on
these data:

  1. (1)

$z_{i}^{L},z^{L}(\beta)\text{ are linear forms in }z_{1},\ldots,z_{k}\text{
for all }1\leq i\leq k,L\in\mathscr{L}_{k},\beta\in B$

  2. (2)

Since $\tilde{\phi}$ is equivariant, is sends the $T^{k}$-fixed point $L$ on
$\mathrm{Jet}_{k}$ to a $T^{k}$ fixed point on
${\mathbb{P}}(\wedge^{k}\mathrm{Sym}^{\leq k}{\mathbb{C}}^{n})$. Assume that
$\tilde{\phi}(L)=e_{\boldsymbol{\pi}}=e_{1}\wedge e_{\pi_{2}}\ldots\wedge
e_{\pi_{k}}\in{\mathbb{P}}(\wedge^{k}\mathrm{Sym}^{\leq k}{\mathbb{C}}^{n})$
is the torus fixed point corresponding to the sequence of partitions
${\boldsymbol{\pi}}=(\pi_{1}=(1),\pi_{2},\ldots,\pi_{k})$. Then

$z_{i}^{L}=z_{\pi_{i}}=\Sigma_{p\in\pi_{i}}z_{p}\text{ for }i=1,\ldots,k.$

  3. (3)

The cardinality of $B$ is $\dim(\mathrm{Jet}^{\mathbf{f}}_{k})=k(k+1)/2$ and
therefore $\mathrm{Euler}^{L}(\mathrm{Jet}_{k})$ is the product of $k(k+1)/2$
linear forms for all $L$.

###  9.2. The residue vanishing theorem

In this section we prove that all but one term on the right hand side of (35)
vanish. This key feature of the iterated residue has already appeared in
Bérczi-Szenes [11] but here we need to prove a slightly different version. We
devote the rest of this section to the proof of the following theorem.

######  Theorem 9.4 ((The Residue Vanishing Theorem)).

Let $k\leq n$ and $m\leq 2n-(k+1)k/2$. Let

$\mathrm{Euler}(\mathscr{E}\otimes{\mathbb{C}}^{m})=\Pi_{j=1}^{m}\Pi_{i=1}^{k}(\theta_{j}+z_{i})$

be the Euler class in Theorem 4.5. Let $\mathscr{L}_{k}$ denote the set of
leaves in the blow-up tree. Then all terms but the ones corresponding to
leaves $L\in\mathscr{L}_{k}$ which are mapped to
${\mathbf{g}}=([1],[2],\ldots,[k])\in\mathrm{Grass}_{k}(\mathrm{Sym}^{\leq
k}{\mathbb{C}}^{n})$ vanish in (35). For these leaves $z^{L}_{i}=z_{i}$ for
$i=1,\ldots k$, leaving us with

(36)
$\int_{\mathrm{Jet}_{k}/\\!/\mathrm{Diff}_{k}}\mathrm{Euler}(\mathscr{E}\otimes{\mathbb{C}}^{m})=\sum_{L\in\tilde{\phi}^{-1}({\mathbf{g}})}\operatornamewithlimits{Res}_{\mathbf{z}=\infty}\frac{(k-1)!z^{k-1}\prod_{m<l}(z_{m}-z_{l})\Pi_{j=1}^{m}\Pi_{i=1}^{k}(\theta_{j}+z_{i})}{\mathrm{Euler}^{L}(\mathrm{Jet}_{k})\prod_{l=1}^{k}\prod_{i=1}^{n}(\lambda_{i}-z_{l})}\,\,d\mathbf{z}$

where $\mathrm{Euler}^{L}(\mathrm{Jet}_{k})=\prod_{\beta\in B}z^{L}(\beta)$ is
the equivariant Euler class of the tangent space at $L$.

######  Remark 13.

Since the Thom polynomial $\mathrm{Tp}_{k}^{n,m}$ only depends on $k$ and
$m-n$, the numerative conditions on these parameters in the Residue Vanishing
Theorem do not affect our formula: for any fixed codimension $m-n$ it is
possible to choose $n$ big enough so that these conditions are met.

For the preparation of the proof, we describe following [11] $\S 6.2$ the
conditions under which iterated residues of the type appearing in the sum in
(35) vanish and we prove Theorem 9.4.

We start with the 1-dimensional case, where the residue at infinity is defined
by (32) with $d=1$. By bounding the integral representation along a contour
$|z|=R$ with $R$ large, one can easily prove the following lemma.

######  Lemma 9.5.

Let $p(z),q(z)$ be polynomials of one variable. Then

$\operatornamewithlimits{Res}_{z=\infty}\frac{p(z)\,dz}{q(z)}=0\quad\text{if
}\deg(p(z))+1<\deg(q).$

Consider now the multidimensional situation. Let $p(\mathbf{z}),q(\mathbf{z})$
be polynomials in the $k$ variables $z_{1}\ldots z_{k}$, and assume that
$q(\mathbf{z})$ is the product of linear factors $q=\prod_{i=1}^{N}L_{i}$, as
in (35). We continue to use the notation $\,d\mathbf{z}=dz_{1}\dots dz_{k}$.
We would like to formulate conditions under which the iterated residue

(37)
$\operatornamewithlimits{Res}_{z_{1}=\infty}\operatornamewithlimits{Res}_{z_{2}=\infty}\dots\operatornamewithlimits{Res}_{z_{k}=\infty}\frac{p(\mathbf{z})\,\,d\mathbf{z}}{q(\mathbf{z})}$

vanishes. Introduce the following notation:

  * •

For a set of indices $S\subset\\{1\ldots k\\}$, denote by
$\deg(p(\mathbf{z});S)$ the degree of the one-variable polynomial $p_{S}(t)$
obtained from $p$ via the substitution $z_{m}\to\begin{cases}t\text{ if }m\in
S,\\\ 1\text{ if }m\notin S.\end{cases}$.

  * •

For a nonzero linear form $L=a_{0}+a_{1}z_{1}+\ldots+a_{k}z_{k}$, denote by
$\mathrm{coeff}(L,z_{i})=a_{i}$ the coefficient $a_{i}$.

  * •

Finally, for $1\leq m\leq k$, set

$\mathrm{lead}(q(\mathbf{z});m)=\\#\\{i;\;\max\\{l;\;\mathrm{coeff}(L_{i},z_{l})\neq
0\\}=m\\}$

which is the number of those factors $L_{i}$ in which the coefficient of
$z_{m}$ does not vanish, but the coefficients of $z_{m+1},\ldots,z_{k}$ are
$0$.

We can group the $N$ linear factors of $q(\mathbf{z})$ according to the
nonvanishing coefficient with the largest index; in particular, for $1\leq
m\leq k$ we have

$\deg(q(\mathbf{z});m)\geq\mathrm{lead}(q(\mathbf{z});m))\text{ and
}\sum_{m=1}^{k}\mathrm{lead}(q(\mathbf{z});m)=N.$

Now applying Lemma 9.5 to the first residue in (37), we see that

$\operatornamewithlimits{Res}_{z_{d}=\infty}\frac{p(z_{1},\ldots,z_{d-1},z_{d})\,d\mathbf{z}}{q(z_{1},\ldots,z_{d-1},z_{d})}=0$

whenever $\deg(p(\mathbf{z});d)+1<\deg(q(\mathbf{z}),d)$; in this case, of
course, the entire iterated residue (37) vanishes.

Now we suppose the residue with respect to $z_{d}$ does not vanish, and we
look for conditions of vanishing of the next residue:

(38)
$\operatornamewithlimits{Res}_{z_{k-1}=\infty}\operatornamewithlimits{Res}_{z_{k}=\infty}\frac{p(z_{1},\ldots,z_{k-2},z_{k-1},z_{k})\,d\mathbf{z}}{q(z_{1},\ldots,z_{k-2},z_{k-1},z_{k})}.$

Now the condition $\deg(p(\mathbf{z});k-1)+1<\deg(q(\mathbf{z}),k-1)$ will
insufficient; for example,

(39)
$\operatornamewithlimits{Res}_{z_{k-1}=\infty}\operatornamewithlimits{Res}_{z_{k}=\infty}\frac{dz_{k-1}dz_{k}}{z_{k-1}(z_{k-1}+z_{k})}=\operatornamewithlimits{Res}_{z_{k-1}=\infty}\operatornamewithlimits{Res}_{z_{k}=\infty}\frac{dz_{k-1}dz_{k}}{z_{k-1}z_{k}}\left(1-\frac{z_{k-1}}{z_{k}}+\ldots\right)=1.$

After performing the expansions (33) to $1/q(\mathbf{z})$, we obtain a Laurent
series with terms $z_{1}^{-i_{1}}\ldots z_{k}^{-i_{k}}$ such that
$i_{k-1}+i_{k}\geq\mathrm{deg}(q(z);k-1,k)$, hence the condition

(40) $\deg(p(\mathbf{z});k-1,k)+2<\deg(q(\mathbf{z});k-1,k)$

will suffice for the vanishing of (38). This argument easily generalizes to
the following statement.

######  Proposition 9.6 ([11] Proposition 6.3).

Let $p(\mathbf{z})$ and $q(\mathbf{z})$ be polynomials in the variables
$z_{1}\ldots z_{k}$, and assume that $q(\mathbf{z})$ is a product of linear
factors: $q(\mathbf{z})=\prod_{i=1}^{N}L_{i}$; set $\,d\mathbf{z}=dz_{1}\dots
dz_{k}$. Then

$\operatornamewithlimits{Res}_{z_{1}=\infty}\operatornamewithlimits{Res}_{z_{2}=\infty}\dots\operatornamewithlimits{Res}_{z_{k}=\infty}\frac{p(\mathbf{z})\,d\mathbf{z}}{q(\mathbf{z})}=0$

if for some $l\leq k$, the following holds:

$\deg(p(\mathbf{z});d,d-1,\dots,l)+d-l+1<\deg(q(\mathbf{z});d,d-1,\dots,l)$

Note that the equality $\deg(q(\mathbf{z});l)=\mathrm{lead}(q(\mathbf{z});l)$
means that

(41) $\text{ for each }i=1\ldots N\text{ and
}m>l)(\mathrm{coeff}(L_{i},z_{l})\neq 0\text{ implies
}\mathrm{coeff}(L_{i},z_{m})=0.$

We are ready to proof the Residue Vanishing Theorem. Recall that our goal is
to show that all the terms of the sum in (35) vanish except for the ones
corresponding to leaves L such that
$\tilde{\phi}(L)={\boldsymbol{\pi}}_{\mathrm{dst}}=e_{1}\wedge\ldots\wedge
e_{k}$. The plan is to apply Proposition 9.6 in stages to show that the
itrated residue vanishes unless $z^{L}_{i}=[i]$ holds, starting with $i=k$ and
going backwards.

Fix a sequence
${\boldsymbol{\pi}}=(\pi_{1},\dots,\pi_{k})\in{\boldsymbol{\Pi}_{k}}$, and
consider the iterated residue on the right hand side of (35) corresponding to
a leaf $L$ with $\tilde{\phi}(L)=e_{\boldsymbol{\pi}}$ The expression under
the residue is the product of two fractions:

$\frac{p(\mathbf{z})}{q(\mathbf{z})}=\frac{p_{1}(\mathbf{z})}{q_{1}(\mathbf{z})}\cdot\frac{p_{2}(\mathbf{z})}{q_{2}(\mathbf{z})}$

where

(42)
$\frac{p_{1}(\mathbf{z})}{q_{1}(\mathbf{z})}=\frac{(k-1)!z^{k-1}\,\prod_{m<l}(z_{m}-z_{l})}{\prod_{\beta\in
B}z^{L}(\beta)}\text{ and
}\frac{p_{2}(\mathbf{z})}{q_{2}(\mathbf{z})}=\frac{\Pi_{j=1}^{m}\Pi_{i=1}^{k}(\theta_{j}+z_{\pi_{i}})}{\prod_{l=1}^{k}\prod_{i=1}^{n}(\lambda_{i}-z_{l})}.$

Note that $p(\mathbf{z})$ is a polynomial, while $q(\mathbf{z})$ is a product
of linear forms. As a first step we show that if $\pi_{k}\neq[k]$, then
already the first residue in the corresponding term on the right hand side of
(35) – the one with respect to $z_{k}$ – vanishes. Indeed, if
$\pi_{k}\neq[k]$, then $\deg(q_{2}(\mathbf{z});k)=n$, while $z_{k}$ does not
appear in $p_{2}(\mathbf{z})$. On the other hand,
$\deg(p_{1}(\mathbf{z});k)=k-1$, hence

(43) $\deg(p(\mathbf{z});k)=k-1\text{ and }\deg(q(\mathbf{z});k)\geq n$

and $k\leq n$, so $\deg(p(\mathbf{z}))\leq\deg(q(\mathbf{z}))-2$ holds and we
can apply Lemma 9.5.

We can thus assume that $\pi_{k}=[k]$, and proceed to the next step and take
the residue with respect to $z_{k-1}$. Assume that $\pi_{k-1}\neq[k-1]$. Then

(44) $\deg(q_{2}(\mathbf{z}),k-1,k)=2n,\deg(p_{2}(\mathbf{z});k-1,k)=m.$

and

(45)
$\deg(q_{1}(\mathbf{z}),k-1,k)\leq\\#B=(k+1)k/2,\deg(p_{1}(\mathbf{z});k-1,k)=2k-3.$

Hence for $m<2n+2k-5-(k+1)k/2$ we can apply Proposition 9.6 with $l=k-1$ to
deduce the vanishing of the residue with respect to $k-1$.

Continuing this argument in the same way provides an inductive proof of the
the Residue Vanishing Theorem.

###  9.3. Initial blow-up to achieve stability-semistability for the
$\mathrm{Diff}_{k}$ action

Recall the rational map

$\phi:{\mathbb{P}}={\mathbb{P}}[{\mathbb{C}}\oplus\,{\rm
Hom}\,^{\mathbf{f}}({\mathbb{C}}^{k},{\mathbb{C}}^{n})]\dasharrow{\mathbb{P}}(\wedge^{k}\mathrm{Sym}^{\leq
k}{\mathbb{C}}^{n})$

on the compactified fiber over $\mathbf{f}$ has reads as

$[x:v_{1},\ldots
v_{k}]\mapsto[v_{1}\wedge(xv_{2}+v_{1}^{2})\wedge(x^{2}v_{3}+xv_{1}v_{2}+v_{1}^{3})\wedge\ldots]$

Here $v_{1},\ldots,v_{k}\in{\mathbb{C}}^{n}$ are vectors of the form (46)
below, representing the columns of a matrix $M\in\,{\rm
Hom}\,^{\mathbf{f}}({\mathbb{C}}^{k},{\mathbb{C}}^{n})$ and $x$ is the
compactifying coordinate, while $\mathrm{Diff}_{k}$ acts via the right action

$[x:M]\cdot\hat{u}=[x:M\hat{u}]\text{ for }\hat{u}\in\mathrm{Diff}_{k}$

or equivalently via the left action

$\hat{u}\cdot[x:M]=[x:M(\hat{u})^{-1}]\text{ for
}\hat{u}\in\mathrm{Diff}_{k}.$

We want to apply the results of non-reductive GIT described in $\S$ 5, which
are stated for left actions. For this we need a one-parameter subgroup
$\lambda:{\mathbb{C}}^{*}\to\mathrm{Diff}_{k}$ whose adjoint action on the Lie
algebra of $U\subset\mathrm{Diff}_{k}$ has only strictly positive weights; we
can take

$\lambda(t)=\left(\begin{array}[]{ccccc}t^{-1}&0&0&\ldots&0\\\
0&t^{-2}&0&\ldots&0\\\ 0&0&t^{-3}&\ldots&0\\\ 0&0&0&\ldots&\cdot\\\
\cdot&\cdot&\cdot&\ldots&t^{-k}\end{array}\right),$

and then

$\lambda(t)\cdot[x:M]=[x:M\left(\begin{array}[]{ccccc}t&0&0&\ldots&0\\\
0&t^{2}&0&\ldots&0\\\ 0&0&t^{3}&\ldots&0\\\ 0&0&0&\ldots&\cdot\\\
\cdot&\cdot&\cdot&\ldots&t^{k}\end{array}\right)].$

The weights of this (left) action of the one-parameter subgroup of
$\mathrm{Diff}_{k}$ defined by $\lambda$ are $\\{0,1,2,\ldots,k\\}$. The
minimal weight space is the point

$Z_{\min}=\\{[1:0:\ldots:0]\\}$

and the $U$-stabiliser of this point is $U$. Thus the non-reductive GIT blow-
up process described at the end of $\S$3 starts with blowing up the projective
space ${\mathbb{P}}$ along $Z_{\min}$ to get

${\tilde{\mathbb{P}}}=\mathrm{Bl}_{[1:0:\ldots:0]}{\mathbb{P}}=\\{([x:v_{1},\ldots,v_{k}],[w_{1},\ldots
w_{k}]):w_{i}\otimes v_{j}=w_{j}\otimes v_{i}\mbox{ for }1\leq i<j\leq k\\}$

embedded in ${\mathbb{P}}^{{k\choose 2}}\times{\mathbb{P}}^{{k\choose
2}-1}\subset{\mathbb{P}}^{({k\choose 2}+1){k\choose 2}-1}$. Here

(46) $\displaystyle v_{1}=$ $\displaystyle\beta_{11}e_{1}$ $\displaystyle
v_{2}=$ $\displaystyle\beta_{22}e_{2}+\beta_{12}e_{1}$ $\displaystyle\cdots$
$\displaystyle v_{k}=$
$\displaystyle\beta_{kk}e_{k}+\beta_{{k-1}k}e_{k-1}+\ldots+\beta_{1k}e_{1}$

We fix the ample linearisation $L=\mathscr{O}_{{\mathbb{P}}^{{k\choose
2}}}(1)\otimes\mathscr{O}_{{\mathbb{P}}^{{k\choose 2}-1}}(1)$ on
${\mathbb{P}}^{{k\choose 2}}\times{\mathbb{P}}^{{k\choose 2}-1}$ and restrict
it to ${\tilde{\mathbb{P}}}$. The minimal weight space for the action of
$\lambda({\mathbb{C}}^{*})$ on ${\tilde{\mathbb{P}}}$ is the intersection
$\tilde{Z}_{\min}$ of the exceptional divisor $E$ and the strict transform of
${\mathbb{P}}[x:v_{1}:0:\cdots:0]\subset{\mathbb{P}}$:

$\tilde{Z}_{\min}=\\{([1:0:\ldots:0],[w_{1}:0:\ldots:0]):w_{1}\in{\mathbb{C}}^{n},w_{1}\neq
0\\}\subset E\subset{\tilde{\mathbb{P}}}.$

and its contracting set is

${\tilde{\mathbb{P}}}_{\min}^{0}=\\{([x:v_{1}:\ldots:v_{k}],[w_{1}:w_{2}:\ldots:w_{k}]):w_{1}\neq
0,\text{ and not all of }w_{2},\ldots,w_{k}\text{ is
zero}\\}\subset{\tilde{\mathbb{P}}}$

The $U$-stabiliser of any point in $\tilde{Z}_{\min}$ is trivial, and hence
stability coincides with semistability for the induced $\mathrm{Diff}_{k}$
action on ${\tilde{\mathbb{P}}}$. This remains valid after blow-ups, hence
semistability=stability holds for all spaces in this blow-up process.

Due to the form (46) of the vectors, on
${\tilde{\mathbb{P}}}^{ss}={\tilde{\mathbb{P}}}_{\min}^{0}\setminus\tilde{Z}_{\min}$
we have $w_{1}=\beta_{11}e_{1}$ and on the affine chart
$U_{0}\subset{\tilde{\mathbb{P}}}$, where $\beta_{11}\neq 0$, we set
$\beta_{11}=1$. Note that ${\tilde{\mathbb{P}}}^{ss}\subset U_{0}$. We get
relations $\beta_{11}w_{i}=v_{i}$, and hence the affine coordinates on $U_{0}$
are $\beta_{ij}$ for $1\leq i\leq j\leq k$. We will use the notation
$t=\beta_{11}$ for this distinguished coordinate. The exceptional divisor is
$\\{t=0\\}\subset U_{0}$ and the rational map

$\phi:{\tilde{\mathbb{P}}}^{ss}\dasharrow{\mathbb{P}}(\wedge^{k}\mathrm{Sym}^{\leq
k}{\mathbb{C}}^{n})$

can be written as

(47) $\phi(t:w_{1},\ldots
w_{k})=[te_{1}\wedge(tw_{2}+(tw_{1})^{2})\wedge(tw_{3}+t^{2}w_{1}w_{2}+(tv_{1})^{3})\wedge\ldots]=\\\
e_{1}\wedge(w_{2}+tw_{1}^{2})\wedge\ldots\wedge(\sum_{i_{1}+\ldots+i_{r}=k}t^{r-1}w_{i_{1}}\ldots
w_{i_{r}})$

##  10\. Examples: Thom polynomials for $k\leq 5$

In this section we work out the strategy for small examples, and we compute
Thom polynomials of $A_{3},A_{4}$ and $A_{5}$ singularities to demonstrate the
key ingredients of out new approach.

###  10.1. Thom polynomials of $A_{3}$ singularities

The affine coordinates over the flag
${\mathbf{f}}=(e_{1},e_{2},e_{3})\in\mathrm{Flag}_{3}({\mathbb{C}}^{n})$ are

$\displaystyle w_{1}=$ $\displaystyle e_{1}$ $\displaystyle w_{2}=$
$\displaystyle\beta_{22}e_{2}+\beta_{12}e_{1}$ $\displaystyle w_{3}=$
$\displaystyle\beta_{33}e_{3}+\beta_{23}e_{2}+\beta_{13}e_{1}$

We sort the map
$\phi_{3}:U_{0}=\mathrm{Spec}({\mathbb{C}}[t,\beta_{ij}])\to{\mathbb{P}}(\mathrm{Sym}^{\leq
3}{\mathbb{C}}^{n})$ by the degree of $t$:

$\displaystyle\phi_{3}(t,w_{1},w_{2},w_{3})=$ $\displaystyle
w_{1}\wedge(w_{2}+tw_{1}^{2})\wedge(w_{3}+tw_{1}w_{2}+t^{2}w_{1}^{3})=$
$\displaystyle[\beta_{22}\beta_{33}e_{1}\wedge e_{2}\wedge e_{3}]+$
$\displaystyle t$ $\displaystyle[\beta_{33}e_{1}\wedge e_{1}^{2}\wedge
e_{3}+(\beta_{23}-\beta_{12}\beta_{22})e_{1}\wedge e_{1}^{2}\wedge
e_{2}+\beta_{22}^{2}e_{1}\wedge e_{2}\wedge e_{1}e_{2}]+$ (48) $\displaystyle
t^{2}$ $\displaystyle[\beta_{22}e_{1}\wedge e_{2}\wedge
e_{1}^{3}+\beta_{22}e_{1}\wedge e_{1}^{2}\wedge e_{1}e_{2}]+$ $\displaystyle
t^{3}$ $\displaystyle[e_{1}\wedge e_{1}^{2}\wedge e_{1}^{3}]$

The ideal of indeterminacy locus of $\phi_{3}$ is generated by the
coefficients:

$I(\phi_{3})=(\beta_{22}\beta_{33},\beta_{33}t,\beta_{22}^{2}t,(\beta_{23}-\beta_{12}\beta_{22})t,\beta_{22}t^{2},t^{3})$

We blow-up along the indeterminacy locus of the monomial map

$\phi_{3}^{mon}(t,w_{1},w_{2},w_{3})=[\beta_{22}\beta_{33},\beta_{33}t,\beta_{22}^{2}t,\beta_{23}t,\beta_{12}\beta_{22}t,\beta_{22}t^{2},t^{3}]\in{\mathbb{P}}^{6}$

formed by all monomials in the $\phi_{3}$. The indeterminacy locus is
$\mathrm{Spec}(M(\phi_{3}))$ with

$M(\phi_{3})=(\beta_{22}\beta_{33},\beta_{33}t,\beta_{22}^{2}t,\beta_{23}t,\beta_{12}\beta_{22}t,\beta_{22}t^{2},t^{3}]\in{\mathbb{P}}^{6})$

is generated by the coordinates all monomials in $\phi_{3}$. Now
$\mathrm{Spec}(M(\phi_{3}))$ is the union of coordinate subspaces with
multiplicities, determined by the primary decomposition of the monomial ideal
$M(\phi_{3})$:

$M(\phi_{3})=(t,\beta_{22})\cap(t,b_{33})\cap(t^{3},\beta_{22},\beta_{23},\beta_{33})\cap(t^{2},\beta_{12},\beta_{22}^{2},\beta_{23},\beta_{33})$

We proceed by blowing up along one of the components which is i) reduced for
the blow-up to be smooth ii) $\mathrm{Diff}_{3}$-invariant and iii) of maximal
dimension. Then we choose the affine chart corresponding to the minimal weight
space. We pick $(\underline{t},\beta_{33})$, and the affine chart $t\neq 0$ as
the only choice as the weight of $t$ is $1$ and the weight of $\beta_{33}$ is
$2$. Let
$\tilde{\phi}_{3}:\mathrm{Bl}_{(t,b_{3}3),t}U_{0}\to{\mathbb{P}}(\mathrm{Sym}^{\leq
3}{\mathbb{C}}^{n})$ denote the blown-up rational map, then its monomial ideal
is

$M(\tilde{\phi}_{3})=(t\beta_{22}\tilde{\beta}_{33},t^{2}\tilde{\beta}_{33},t\beta_{22}^{2},t\beta_{23},t\beta_{12}\beta_{22},t^{2}\beta_{22},t^{3})$

which defines the same monomial map to ${\mathbb{P}}(\mathrm{Sym}^{\leq
3}{\mathbb{C}}^{n})$ as the one by dividing all monomials by $t$, and we take
the primary decomposition again:

$M(\tilde{\phi}_{3})=(\beta_{22}\tilde{\beta}_{33},t\tilde{\beta}_{33},\beta_{22}^{2},\beta_{23},\beta_{12}\beta_{22},t\beta_{22},t^{2})=(t,\beta_{22},\beta_{23})\cap(t^{2},\beta_{22},\beta_{23},\tilde{\beta}_{33})\cap(t,\beta_{12},\beta_{22}^{2},\beta_{23},\tilde{\beta}_{33})$

Then we pick the ideal $(t,\beta_{22},\beta_{23})$ and note that the weight of
$t$ and $\beta_{22}$ is $1$ and the weight of $\beta_{23}$ is $2$. So we have
two minimal weight charts, and the blow-up maps have the following monomial
ideals:

$\displaystyle M(\tilde{\phi}_{3}^{\beta_{22}})$
$\displaystyle=(\tilde{\beta}_{33},\tilde{t}\tilde{\beta}_{33},\beta_{22},\tilde{\beta}_{23},\beta_{12},\tilde{t}\beta_{22},\tilde{t}^{2}\beta_{22})=(\tilde{\beta}_{33},\beta_{22},\tilde{\beta}_{23},\beta_{12})$
$\displaystyle M(\tilde{\phi}_{3}^{t})$
$\displaystyle=(\tilde{\beta}_{22}\tilde{\beta}_{33},\tilde{\beta}_{33},t\tilde{\beta}_{22}^{2},\tilde{\beta}_{23},\beta_{12}\tilde{\beta}_{22},t\tilde{\beta}_{22},t)$

On the first affine chart our blow-up process terminates: there is only one
reduced component, and we can write the blown-up map on this chart as

$\displaystyle\tilde{\tilde{\phi}}_{3}(\tilde{t},\beta_{12},\beta_{13},\beta_{22},\tilde{\beta}_{23},\tilde{\beta}_{33})=\tilde{\beta}_{33}e_{1}\wedge
e_{2}\wedge e_{3}+\tilde{t}\tilde{\beta}_{33}e_{1}\wedge e_{1}^{2}\wedge