# Orbits of linear series on the projective line

Anand Deopurkar  Anand Deopurkar, Mathematical Sciences Institute, Australian
National University<EMAIL_ADDRESS>and  Anand Patel  Anand Patel,
Department of Mathematics, Oklahoma State University<EMAIL_ADDRESS>
###### Abstract.

We compute the equivariant fundamental class of the orbit closure of a linear
series on the projective line. We also describe the boundary of the orbit
closure and how the orbits specialise in one parameter families.

##  1\. Introduction

Let $M$ be a variety with the action of an algebraic group $G$. Given a point
$m\in M$, let $\operatorname{Orb}(m)\subset M$ be the closure of the $G$-orbit
of $m$. The fundamental class of $\operatorname{Orb}(m)$ in the equivariant
Chow ring of $M$ encodes fascinating geometric information. The class is
particularly meaningful when the $G$-orbits have geometric significance. The
main result of this paper (Theorem 1.1) is a complete and explicit expression
for these classes for
$M=\operatorname{Hom}(\mathbf{k}^{r+1},\operatorname{Sym}^{d}\mathbf{k}^{2})$
with the natural action of
$G=\operatorname{GL}(r+1)\times\operatorname{GL}(2)$. Here $\mathbf{k}$ is an
algebraically closed field in which $d!\neq 0$.

The orbits of the $\operatorname{GL}(r+1)\times\operatorname{GL}(2)$ action on
$\operatorname{Hom}(\mathbf{k}^{r+1},\operatorname{Sym}^{d}\mathbf{k}^{2})$
represent isomorphism classes of linear series of rank $r$ and degree $d$ on
the projective line. Their fundamental class turns out to depend only on the
ramification profile. For example, when $r=1$ and the linear series is base-
point free, the class is a function of the multiplicities in the ramification
divisor. Likewise, when $r=2$ and the linear series is base-point free, the
class depends on the multiplicities of cusps and hyperflexes.

The group $\operatorname{GL}(r+1)$ acts freely on the open subset of
$\operatorname{Hom}(\mathbf{k}^{r+1},\operatorname{Sym}^{d}\mathbf{k}^{2})$
consisting of injective maps. The quotient is the Grassmannian ${\bf
Gr}(r+1,\operatorname{Sym}^{d}\mathbf{k}^{2})$. So we have a surjective
homomorphism

(1)
$A^{*}_{\operatorname{GL}(r+1)\times\operatorname{GL}(2)}(\operatorname{Hom}(\mathbf{k}^{r+1},\operatorname{Sym}^{d}\mathbf{k}^{2}))\to
A^{*}_{\operatorname{GL}(2)}({\bf
Gr}(r+1,\operatorname{Sym}^{d}\mathbf{k}^{2})).$

Taking images under this map gives classes of $\operatorname{GL}(2)$-orbits in
the Grassmannian. Forgetting the group action gives a further homomorphism

(2) $A^{*}_{\operatorname{GL}(2)}({\bf
Gr}(r+1,\operatorname{Sym}^{d}\mathbf{k}^{2}))\to A^{*}({\bf
Gr}(r+1,\operatorname{Sym}^{d}\mathbf{k}^{2})).$

Taking images under this map gives non-equivariant classes of the
$\operatorname{GL}(2)$-orbits. All of these, including the non-equivariant
ones, were unknown. Table 1 shows the weighted (multiplied by the order of the
stabiliser) orbit classes for all base-point free pencils of quartics.

Ramification profile |  Weighted orbit class in the Schubert basis
---|---
$2,2,2,2,2,2$ |  $24\cdot\Omega_{3}+48\cdot\Omega_{2,1}$
$3,2,2,2,2$ |  $16\cdot\Omega_{3}+40\cdot\Omega_{2,1}$
$4,2,2,2$ |  $12\cdot\Omega_{3}+24\cdot\Omega_{2,1}$
$3,3,2,2$ |  $8\cdot\Omega_{3}+32\cdot\Omega_{2,1}$
$3,3,3$ |  $24\cdot\Omega_{2,1}$
$4,3,2$ |  $4\cdot\Omega_{3}+16\cdot\Omega_{2,1}$
Table 1. The weighted orbit classes of base-point free pencils of quartics in
${\bf Gr}(2,\operatorname{Sym}^{4}\mathbf{k}^{2})$ as a function of the
ramification profile.

Many mathematicians have studied equivariant and non-equivariant orbit
classes. We were inspired mainly by the work of Aluffi and Faber [3, 2, 1] on
the (non-equivariant) classes of orbits of points on ${\bf P}^{1}$ and curves
in ${\bf P}^{2}$. There has been exciting recent progress on extending their
work to the equivariant setting; see [13] for points on ${\bf P}^{1}$ and
quartic curves in ${\bf P}^{2}$, [12] for hyperplane arrangements in ${\bf
P}^{n}$, and [6] for generic cubics in ${\bf P}^{3}$. There is also
significant work in a more representation theoretic context. See, for example,
[11, 5] for classes of orbits of quiver representations and [4] for orbits of
matrices.

###  1.1. Main theorem

To state the main result, we need some notation. We have an identification

(3)
$A^{*}_{\operatorname{GL}(r+1)\times\operatorname{GL}(2)}(\operatorname{Hom}(\mathbf{k}^{r+1},\operatorname{Sym}^{d}\mathbf{k}^{2}))=\mathbf{Z}[\mu_{0},\dots,\mu_{r},\omega_{1},\omega_{2}]^{S_{r+1}\times
S_{2}},$

where the $\mu_{i}$’s and the $\omega_{j}$’s are the “Chern roots” of the
tautological representations of $\operatorname{GL}(r+1)$ and
$\operatorname{GL}(2)$, respectively. Given integers $a_{0}<\dots<a_{r}$ in
$\\{0,\dots,d\\}$, set

$\phi_{a_{0},\dots,a_{r}}(\mu;\omega)=\prod_{j=0}^{r}\prod_{i}((d-i)\omega_{1}+i\omega_{2}-\mu_{j}),$

where the index $i$ ranges over $\\{0,\dots,d\\}-\\{a_{0},\dots,a_{r}\\}$. Set

$\psi_{a_{0},\dots,a_{r}}(\mu;\omega)=\phi_{a_{0},\dots,a_{r}}(\mu;\omega_{1},\omega_{2})-\phi_{a_{0},\dots,a_{r}}(\mu;\omega_{2},\omega_{1}).$

Note that $\psi$ is symmetric in the $\mu$’s and anti-symmetric in the
$\omega$’s.

Given an injective
$s\in\operatorname{Hom}(\mathbf{k}^{r+1},\operatorname{Sym}^{d}\mathbf{k}^{2})$,
let $S\subset\operatorname{Sym}^{d}\mathbf{k}^{2}$ be its image. Associated to
$S$ and a point $p\in{\bf P}^{1}$, we have the vanishing sequence

$a_{0}(p)<\dots<a_{r}(p)$

characterised by the property that for each $i$, the space $S$ contains a
section vanishing to order exactly $a_{i}(p)$ at $p$. For all but finitely
many points $p\in{\bf P}^{1}$, the vanishing sequence is given by
$a_{i}(p)=i$. Let $B\subset{\bf P}^{1}$ be the points where it differs. The
ramification profile of $S$ is the multi-set

$\\{(a_{0}(b),\dots,a_{r}(b))\mid b\in B\\}.$

######  Theorem 1.1 (Theorem 5.5).

Let $\mathbf{k}$ be an algebraically closed field in which $d!\neq 0$. Let
$s\in\operatorname{Hom}(\mathbf{k}^{r+1},\operatorname{Sym}^{d}\mathbf{k}^{2})$
be an injective map with image $S\subset\operatorname{Sym}^{d}\mathbf{k}^{2}$.
Let $\\{(a_{i}(b))\mid b\in B\\}$ be the ramification profile of $S$. Let
$\Gamma\subset\operatorname{PGL}(2)$ be the stabiliser of
$S\subset\operatorname{Sym}^{d}\mathbf{k}^{2}$, and assume that it is finite.
Let $\operatorname{Orb}(s)$ be the closure of the
$\operatorname{GL}(r+1)\times\operatorname{GL}(2)$ orbit of $s$. Then the
class

$|\Gamma|\cdot[\operatorname{Orb}(s)]\in
A^{(r+1)(d-r)-3}_{\operatorname{GL}(r+1)\times\operatorname{GL}(2)}(\operatorname{Hom}(\mathbf{k}^{r+1},\operatorname{Sym}^{d}\mathbf{k}^{2}))$

is given by

$|\Gamma|\cdot[\operatorname{Orb}(s)]=\frac{1}{(\omega_{1}-\omega_{2})^{3}}\sum_{b\in
B}\psi_{a_{0}(b),\dots,a_{r}(b)}(\mu;\omega)+\frac{2-|B|}{(\omega_{1}-\omega_{2})^{3}}\cdot\psi_{0,\dots,r}(\mu;\omega).$

Despite the apparent denominators, the expression must be a polynomial
(although this is not obvious).

To get the $\operatorname{GL}(2)$-equivariant class of
$\operatorname{Orb}(S)\subset{\bf
Gr}(r+1,\operatorname{Sym}^{d}\mathbf{k}^{2})$, we apply the homomorphism (1),
which sends the $\mu$’s to the Chern roots of the universal sub-bundle. To get
the non-equivariant class of $\operatorname{Orb}(S)\subset{\bf
Gr}(r+1,\operatorname{Sym}^{d}\mathbf{k}^{2})$, we further substitute
$\omega_{1}=\omega_{2}=0$.

######  Remark 1.2.

Although Theorem 1.1 gives the non-equivariant classes by the series of
substitutions mentioned above, it does not give a simple expression for it.
Indeed, the substitution $\omega_{1}=\omega_{2}=0$ can only be performed after
dividing by the denominator in Theorem 1.1, and the result of the division is
not obvious.

######  Remark 1.3.

Given $[S]\in{\bf Gr}(r+1,\operatorname{Sym}^{d}W)$, we can consider $[S]$ as
a point of the projective space ${\bf
P}\left(\wedge^{r+1}\operatorname{Sym}^{d}W\right)$ by applying the Plucker
embedding. The class of $\operatorname{Orb}([S])$ in ${\bf
P}\left(\wedge^{r+1}\operatorname{Sym}^{d}W\right)$ depends only on the
multiplicities in the ramification divisor of $S$. The class of
$\operatorname{Orb}([S])$ in ${\bf Gr}(r+1,\operatorname{Sym}^{d}W)$ is a
richer invariant, sensitive to the full ramification profile.

###  1.2. Orbit closures

The method of proof of the main theorem also yields a description of the
boundary points of the orbit closure.

######  Theorem 1.4 (Theorem 4.5).

The orbit closure $\operatorname{Orb}(S)\subset{\bf
Gr}(r+1,\operatorname{Sym}^{d}\mathbf{k}^{2})$ is a union of finitely many
orbits. In addition to the orbit of $S$, these are the orbits of subspaces of
the form

$\langle
w_{1}^{d-a_{0}}w_{2}^{a_{0}},\dots,w_{1}^{d-a_{r}}w_{2}^{a_{r}}\rangle,$

where $a_{0}<\dots<a_{r}$ is the vanishing sequence of $S$ at some point of
${\bf P}^{1}$ and $w_{1},w_{2}$ is a basis of $\mathbf{k}^{2}$.

At a generic point of ${\bf P}^{1}$, the vanishing sequence is given by
$a_{i}=i$. The orbit of the corresponding linear series $\langle
w_{1}^{d}w_{2}^{0},\dots,w_{1}^{d-r}w_{2}^{r}\rangle$ is $1$-dimensional. At a
ramification point, the vanishing sequence is different, and the orbit of the
corresponding linear series $\langle
w_{1}^{d-a_{0}}w_{2}^{a_{0}},\dots,w_{1}^{d-a_{r}}w_{2}^{a_{r}}\rangle$ is
$2$-dimensional.

###  1.3. Orbit specialisation

Consider a one-parameter family of linear series in which some ramification
points collide. Then the flat limit of the orbit closure contains the orbit
closure of the limiting linear series. In a suitable family, the limit of the
orbit closure contains the orbit closure of a unique other linear series,
which we now describe. Let $A$ be the multi-set of vanishing sequences of the
points that collide and $B$ the multi-set of vanishing sequences of the
remaining points. Let $c=(c_{0},\dots,c_{r})$ be the vanishing sequence of the
limiting linear series at the point of collision. Set
$c^{\prime}=(d-c_{r},\dots,d-c_{0})$. Denote by $\operatorname{WOrb}$ the
weighted orbit closure, namely

$\operatorname{WOrb}=|\Gamma|\cdot\operatorname{Orb}.$

Recall that we can treat the class $[\operatorname{WOrb}]$ as a function of
the ramification profile.

######  Theorem 1.5 (Proposition 6.8).

In the notation above, we have the equality of
$\operatorname{GL}(r+1)\times\operatorname{GL}(2)$-equivariant classes

$[\operatorname{WOrb}(A\cup
B)]=[\operatorname{WOrb}(A\cup\\{c^{\prime}\\})]+[\operatorname{WOrb}(B\cup\\{c\\})].$

The statement above is a special case of a large family of linear relations
among the classes $[\operatorname{WOrb}]$; see Section 6 for more.

By repeated collisions of ramification points, we can express an arbitrary
orbit class as a non-negative linear combination of orbit classes of linear
series with 3 ramification points (the expression is not unique). For example,
in the case of quartic pencils, Figure 1 shows a sequence of collisions and
the resulting complementary orbits. As a result, we get the following equality

$\begin{split}\operatorname{WOrb}((0,2)\times 6)&=\\\
\operatorname{WOrb}((0,3)&\times 2,(1,2))+2\operatorname{WOrb}((0,2)\times
2,(1,4))+\operatorname{WOrb}((0,2)\times 2,(2,3)).\end{split}$

In the non-equivariant Chow ring of the Grassmannian, the above evaluates to
the identity

$24\cdot\Omega_{3}+48\cdot\Omega_{2,1}=\left(8\Omega_{3}+20\cdot\Omega_{2,1}\right)+2\cdot\left(8\cdot\Omega_{3}+8\cdot\Omega_{2,1}\right)+12\cdot\Omega_{2,1}.$

$(0,2)$$(0,2)$$(0,2)$$(0,2)$$(0,2)$$(0,2)$$(0,3)$$+$$(0,3)$$+$$(1,2)$$+$$(0,2)$$(0,2)$$(1,4)$$(0,2)$$(0,2)$$(1,4)$$(0,2)$$(0,2)$$(2,3)$
Figure 1. An example of orbit specialisation for a generic quartic pencil.
Collisions of ramification points create a new ramification profile and a
unique complementary orbit shown underneath.

###  1.4. Ideas in the proof

To find the class of the closure of the $G$-orbit of $m\in M$, we use a
complete orbit parametrisation for $m$. A complete orbit parametrisation
consists of a proper $G$-variety $X$ with a dense orbit and an equivariant map
$\phi\colon X\to M$ whose image contains $m$. Then, up to a scalar, the class
we seek is $\phi_{*}[X]$.

We begin with ${\bf P}^{3}={\bf
P}\operatorname{Hom}(\mathbf{k}^{2},\mathbf{k}^{2})$ with the natural
$\operatorname{GL}(2)$ action on the target $\mathbf{k}^{2}$ and the
equivariant rational map

$\phi\colon{\bf P}^{3}\dashrightarrow{\bf
Gr}(r+1,\operatorname{Sym}^{d}\mathbf{k}^{2})$

defined by $m\mapsto m(S)$. The map is only rational because when $m$ is
degenerate, the image $m(S)\subset\operatorname{Sym}^{d}\mathbf{k}^{2}$ need
not be $(r+1)$-dimensional. We prove (Theorem 4.1) that $\phi$ extends to a
regular map on a simple blow up:

(4) $\widetilde{{\bf P}}^{3}\to{\bf
Gr}(r+1,\operatorname{Sym}^{d}\mathbf{k}^{2}).$

This is the same blow-up that Aluffi and Faber use in [1] for $r=0$. We
compute the push-forward using equivariant localisation as in [13, Appendix
B]. We upgrade from the orbit class in ${\bf
Gr}(r+1,\operatorname{Sym}^{d}\mathbf{k}^{2})$ to the orbit class in
$\operatorname{Hom}(\mathbf{k}^{r+1},\operatorname{Sym}^{d}\mathbf{k}^{2})$
using general principles, outlined in Section 2.

###  1.5. Organisation

In Section 2, we discuss equivariant orbit classes in general and classes on
Grassmannians in particular. We also discuss general results about one
parameter specialisations of orbits. In Section 3, we recall vanishing
sequences, the Wronskian, and the ramification divisor of a linear series. In
Section 4, we begin the proof of the main theorem by constructing a complete
parametrisation for a linear series. In Section 5, we finish the proof of the
main theorem with a localisation computation. In Section 6, we discuss one-
parameter specialisations of orbits of linear series, and the resulting
additive decomposition of the orbit class.

###  1.6. Conventions

Throughout, $\mathbf{k}$ denotes an algebraically closed field of
characteristic 0 or of characteristic $>d$. All schemes are of finite type
over $\mathbf{k}$. A _variety_ is a reduced scheme, separated over
$\mathbf{k}$. Given a vector space $V$, the notation $\underline{V}$ denotes
the trivial vector bundle with fiber $V$ on the scheme that is clear from the
context. Given a vector bundle $V$, the _projectivisation_ ${\bf P}V$ is the
space of one-dimensional sub-bundles. Unless mentioned otherwise, _point_
means a closed point. Given a variety $X$ with a $G$-action, $A^{i}_{G}(X)$
denotes the $G$-equivariant Chow group of $X$ of cycles of co-dimension $i$,
as defined in [7].

###  1.7. Acknowledgements

We started working on the project when Anand P. visited Anand D. at the
Institute for Computational and Experimental Research in Mathematics (ICERM)
at Brown University during the semester program _Braids_ in 2022. We thank
ICERM and the organisers of _Braids_. Anand D. thanks the Australian Research
Council for the grant DE180101360 that supported a part of this project.

##  2\. Equivariant orbit classes

Let $G$ be an algebraic group and $M$ a variety with a $G$ action. A _complete
parametrisation_ of the $G$ orbit of $m\in M$ is a $G$-variety $X$ together
with a proper $G$-equivariant map $\phi\colon X\to M$ such that

  1. (1)

there exists an $x\in X$ such that $\phi(x)=m$,

  2. (2)

the $G$-orbit of $x$ is dense in $X$, and

  3. (3)

the stabiliser of $x\in X$ is trivial.

Let $\Gamma\subset G$ be the stabiliser of $m\in M$ and let $\phi\colon X\to
M$ be a complete parametrisation of the $G$ orbit of $m$. Then we have

(5) $\phi_{*}[X]=\begin{cases}|\Gamma|\operatorname{Orb}(m)&\text{ if $\Gamma$
is finite}\\\ 0&\text{otherwise}.\end{cases}$

We now specialise to two particular settings.

In the first setting, we take $M=W$, where $W$ is a non-zero
$G$-representation and the closely related setting of $M={\bf P}W$. Assume
that the image of $G\to\operatorname{GL}(W)$ contains the scalar matrices.
Suppose the map $G\to\operatorname{PGL}(W)$ factors through
$G\to\overline{G}$. (In applications, $G$ will be a general linear group and
$\overline{G}$ will be the corresponding projective general linear group.)

Let

$0\to O(-1)\to\underline{W}\to Q\to 0$

be the universal sequence on ${\bf P}W$.

######  Proposition 2.1.

Let $w\in W$ be non-zero. Let $\Gamma\subset\overline{G}$ be the stabiliser of
$[w]\in{\bf P}W$ and assume that it is finite. Let $\phi\colon X\to{\bf P}W$
be a complete parametrisation of the $\overline{G}$ orbit of $[w]\in{\bf P}W$.
Then, in $A^{*}_{G}(W)=A^{*}_{G}(\bullet)$, we have

$|\Gamma|[\operatorname{Orb}(w)]=\int_{X}c_{\rm top}(\phi^{*}Q).$

###### Proof.

Let $\pi\colon W^{*}\to{\bf P}W$ be the projection. Since the image of $G$
contains the scalars, we have

$\operatorname{Orb}(w)=\pi^{*}\operatorname{Orb}([w]),$

and hence

(6)
$|\Gamma|\cdot\operatorname{Orb}(w)=|\Gamma|\cdot\pi^{*}\operatorname{Orb}([w]).$

Since $\phi$ is a complete parametrisation of the $\overline{G}$ orbit of
$[w]$, we have

$|\Gamma|\cdot\operatorname{Orb}([w])=\phi_{*}[X].$

We now compute the right hand side.

On $X\times{\bf P}W$, let $\pi_{i}$ for $i=1,2$ be the two projections.
Consider the composite

$\pi_{2}^{*}O(-1)\to\underline{W}\to\pi_{1}^{*}\phi^{*}Q.$

Its vanishing locus is precisely the graph of $\phi\colon X\to{\bf P}W$.
Therefore, we have

(7) $\phi_{*}[X]={\pi_{2}}_{*}c_{\rm
top}\operatorname{Hom}(\pi_{2}^{*}O(-1),\pi_{1}^{*}\phi^{*}Q).$

In (6), we substitute $\phi_{*}[X]$ from (7) and use the push-pull formula on
the fiber square

${X\times W^{*}}$${W^{*}}$${X\times{\bf P}W}$${{\bf P}W.}$$\scriptstyle{\pi}$

Let $\widetilde{\pi}_{i}$ for $i=1,2$ be the two projections on $X\times
W^{*}$. Noting that the pull-back of $O(-1)$ to $W^{*}$ is trivial, in
$A^{*}_{G}(W^{*})$ we have

(8)
$\begin{split}|\Gamma|[\operatorname{Orb}(w)]&={\widetilde{\pi_{2}}}_{*}c_{\rm
top}\operatorname{Hom}(O,\widetilde{\pi}_{1}^{*}\phi^{*}Q)\\\ &=\int_{X}c_{\rm
top}(\phi^{*}Q).\end{split}$

Since $A^{*}_{G}(W^{*})=A^{*}_{G}(W)$ in co-dimension less than $\dim W$,
equation (8) also holds in $A^{*}_{G}(W)$. ∎

In the next setting, we assume that $G=\operatorname{GL}(V)\times H$ and
$M=\operatorname{Hom}(V,W)$, where $W$ is an $H$-representation. Let $\dim
V=r+1$. A closely related setting is that of $M={\bf Gr}(r+1,W)$ with the
induced action of $H$. Suppose the map $H\to\operatorname{PGL}(W)$ factors
through $H\to\overline{H}$. (In applications, $H$ will be a general linear
group and $\overline{H}$ will be the corresponding projective general linear
group.)

Let

$0\to E\to\underline{W}\to Q\to 0$

be the universal sequence on the Grassmannian.

######  Proposition 2.2.

Let $s\in\operatorname{Hom}(V,W)$ be an injective homomorphism with image $S$.
Let $\Gamma\subset\overline{H}$ be the stabiliser of $[S]\in{\bf Gr}(r+1,W)$
and assume that it is finite. Let $\phi\colon X\to{\bf Gr}(r+1,W)$ be a
complete parametrisation of the $\overline{H}$-orbit of $[S]$. Then, in
$A^{*}_{G}(\operatorname{Hom}(V,W))=A^{*}_{G}(\bullet)$, we have

$|\Gamma|[\operatorname{Orb}(s)]=\int_{X}c_{\rm
top}\operatorname{Hom}(\underline{V},\phi^{*}Q).$

###### Proof.

Consider the $H$-equivariant bundle $\phi^{*}E$ on $X$. Set

$F=\operatorname{Hom}(\underline{V},\phi^{*}E).$

Then $F$ admits an action of $G=\operatorname{GL}V\times H$. Set $Y={\bf P}F$;
let $\pi\colon Y\to X$ be the projection; and let $\psi=\pi\circ\phi$. The
composite

(9)
$O_{Y}(-1)\xrightarrow{i}\operatorname{Hom}(\underline{V},\psi^{*}E)\xrightarrow{j}\operatorname{Hom}(\underline{V},\underline{W})$

yields a $G$-equivariant map

$Y\to{\bf P}\operatorname{Hom}(V,W).$

This map is a complete parametrisation of the
$\operatorname{PGL}V\times\overline{H}$-orbit of $[s]$. It is easy to check
that the stabiliser of $[s]$ in $\operatorname{PGL}V\times\overline{H}$ maps
isomorphically to $\Gamma\subset\overline{H}$ under the second projection. By
Proposition 2.1, in $A^{*}_{G}(\operatorname{Hom}(V,W))$ we have

$|\Gamma|[\operatorname{Orb}(s)]=\int_{Y}c_{\rm
top}(\operatorname{coker}(j\circ i)).$

From the sequence in (9), we have an exact sequence

$0\to\operatorname{coker}i\to\operatorname{coker}(j\circ
i)\to\operatorname{coker}j\to 0.$

and hence the equality of top Chern classes

$c_{\rm top}(\operatorname{coker}j\circ i)=c_{\rm
top}(\operatorname{coker}i)\cdot c_{\rm top}(\operatorname{coker}j).$

Observe that

$\operatorname{coker}j=\pi^{*}\operatorname{Hom}(\underline{V},\phi^{*}Q).$

Since $Y\to X$ is a projective space bundle and $i$ is its universal sub-
bundle, we have

$\int_{Y/X}c_{\rm top}(\operatorname{coker}i)=[X].$

Now, the push-pull formula gives

$\int_{Y/X}c_{\rm top}(\operatorname{coker}i)\cdot c_{\rm
top}(\operatorname{coker}j)=c_{\rm
top}(\operatorname{Hom}(\underline{V},\phi^{*}Q)).$

Integrating the two sides along $X$ gives the result. ∎

We now study specialisations of orbits in families. The results are
essentially from [13, § 2.5], but we state and prove them more generally.

Fix an action of an algebraic group $G$ on a variety $M$. For simplicity,
assume that $G$ is irreducible. Given $m\in M$, define the $G$-equivariant
cycle

$\operatorname{WOrb}(m)=\begin{cases}|\operatorname{Stab}(m)|\cdot\operatorname{Orb}(m)&\text{
if $\operatorname{Stab}(m)$ is finite,}\\\ 0&\text{otherwise}.\end{cases}$

Let $\Delta$ be the spectrum of a DVR with general point $\eta$ and special
point $0$. Fix a section $m_{\eta}{\colon}\eta\to M_{\eta}$. Let $\Sigma$ be
the cycle on $M$ obtained as the limit of the cycle
$\operatorname{WOrb}(m_{\eta})$ on $M_{\eta}$. Our goal is to understand
$\Sigma$. We begin with an easy observation.

######  Proposition 2.3.

Suppose $m_{\eta}$ has a limit $\overline{m}\in M$. Then the cycle
$\Sigma-\operatorname{WOrb}(\overline{m})$ is effective.

###### Proof.

Let $X\supset G$ be an equivariant compactification. Consider the map
$\alpha\colon G_{\eta}\to M_{\eta}$ defined by $g\mapsto gm_{\eta}$, and let
$Y\subset X\times M\times\Delta$ be the closure of the graph of $\alpha$. Then
$Y$ is irreducible of dimension $\dim G+1$. Let $\pi\colon Y\to M$ be the
projection. Observe that we have

${\pi}_{*}Y_{\eta}=\operatorname{WOrb}(m_{\eta})\text{ and
}{\pi}_{*}Y_{0}=\Sigma.$

By construction, $Y$ is invariant under the $G$-action on $X\times
M\times\Delta$. Since $y=(1,\overline{m},0)\in Y_{0}$, we have a copy of
$G\subset Y_{0}$ given by $G\cdot y$. Let $Z\subset Y_{0}$ be its closure.
Note that ${\pi}_{*}Z=\operatorname{WOrb}(\overline{m})$. Furthermore, since
$\dim Z=\dim Y_{0}$, we see that $Z\subset Y_{0}$ is an irreducible component.
Then $\Sigma-\operatorname{WOrb}(\overline{m})$ is the push-forward of the
effective cycle $[Y_{0}]-[Z]$. ∎

We have the following strengthening of Proposition 2.3 (see [13, Proposition
2.13]).

######  Proposition 2.4.

Consider sections $g_{1},\dots,g_{n}{\colon}\eta\to G_{\eta}$ such that for
all $i\neq j$, the section $g_{i}g_{j}^{-1}$ does not extend to a section
$\Delta\to G_{\Delta}$. Suppose $\overline{m}_{1},\dots,\overline{m}_{n}\in M$
are limits of $g_{1}m_{\eta},\dots,g_{n}m_{\eta}$. Then the cycle
$\Sigma-\operatorname{WOrb}(\overline{m}_{1})-\dots-\operatorname{WOrb}(\overline{m}_{n})$
is effective.

###### Proof.

Let $X\supset G$ be an equivariant compactification. Consider the map
$\alpha\colon G_{\eta}\to G_{\eta}^{n}\times M_{\eta}$ given by

$\alpha\colon g\mapsto(gg_{1}^{-1},\dots,gg_{n}^{-1},gm_{\eta})$

and let $Y\subset X^{n}\times M\times\Delta$ be the closure of its image.
Denoting by $\pi$ the projection onto the $M$ factor, we have

${\pi}_{*}Y_{\eta}=\operatorname{WOrb}(m_{\eta})\text{ and
}{\pi}_{*}Y_{0}=\Sigma.$

Let $\overline{g}_{i,j}\in X$ be the limit of $g_{i}g_{j}^{-1}$. Then
$\overline{g}_{i,j}\not\in G$ for $i\neq j$. Then the limit of $\alpha(g_{i})$
is the point

$y_{i}=(\overline{g}_{i,1},\dots,\overline{g}_{i,i-1},1,\overline{g}_{i,i+1},\overline{g}_{i,n},\overline{m}_{i})\in
Y_{0}.$

Each $y_{i}$ gives a copy of $G\subset Y_{0}$, say $G_{i}$, defined by
$G_{i}=G\cdot y_{i}$. Since $\overline{g}_{i,j}\not\in G$ for $i\neq j$, the
$j$-th components of the points of $G_{i}$ do not lie in $G$ for $i\neq j$ and
lie in $G$ for $i=j$. In particular, for $i\neq j$, the $G_{i}$ and $G_{j}$
are disjoint. Let $Z_{i}$ be the closure of $G_{i}$. Then the $Z_{i}$ are
distinct irreducible components of $Y_{0}$. The cycle
$\Sigma-\operatorname{WOrb}(\overline{m}_{1})-\dots-\operatorname{WOrb}(\overline{m}_{n})$
is the $\pi$-pushforward of the effective cycle $Y_{0}-Z_{1}-\dots-Z_{n}$. ∎

##  3\. Vanishing sequences and the Wronskian

Let $C$ be a smooth curve, $L$ a line bundle of degree $d$ on $C$, and
$S\subset H^{0}(C,L)$ a subspace of dimension $r+1$. The pair $(S,L)$ is
called a _linear series of rank $r$ and degree $d$_ or a $g^{r}_{d}$ on $C$.
Given a point $p\in C$, there exists a unique sequence of integers
$a_{0}(p)<\dots<a_{r}(p)$ in $\\{0,\dots,d\\}$ such that $S$ contains a
section $s_{i}$ that vanishes to order exactly $a_{i}(p)$ at $p$. The sequence
$a_{i}(p)$ is called the _vanishing sequence_ of $S$ at $p$. The sections
$s_{0},\dots,s_{r}$ form a basis of $S$.

Denote by $j_{r+1}L$ the bundle of jets of order $r+1$ of $L$; that is

$j_{r+1}L={\pi_{2}}_{*}\left(\pi_{1}^{*}L\otimes O_{(r+1)\Delta}\right).$

Here $\Delta\to C\times C$ is the diagonal and the $\pi_{i}$ are the two
projections. The linear series $S\subset H^{0}(C,L)$ induces an evaluation map

$\underline{S}\to L,$

and also a map

(10) $\underline{S}\to j_{r+1}L.$

The _Wronskian_ of $S$ is the determinant of (10).

######  Proposition 3.1.

The Wronskian is non-zero and vanishes to order $\sum a_{i}(p)-i$ at $p$.

###### Proof.

The proof is a part of [9, Proposition 1.1]. We give it for completeness.

Let $s_{i}\in S$ vanish to order $a_{i}=a_{i}(p)$ at $p$. Denote by
$\widehat{M}$ the completion of an $O_{C}$-module $M$ at $p$. Choose an
isomorphism of $\mathbf{k}\llbracket t\rrbracket$-modules
$\widehat{L}\cong\mathbf{k}\llbracket t\rrbracket$. Then we have an
isomorphism

$\widehat{j_{r+1}L}\cong\mathbf{k}\llbracket t,u\rrbracket/(u-t)^{r+1}.$

If the section $s\in\widehat{L}\cong\mathbf{k}\llbracket t\rrbracket$ is given
by the power series $s(t)$, then the induced section $j_{r+1}(s)$ of
$\widehat{j_{r+1}L}$ is given by

$j_{r+1}(s)=s(t)+s^{\prime}(t)(u-t)+\dots+s^{(r)}(t)\frac{1}{r!}(u-t)^{r}.$

Let $s_{i}$ correspond to the power series
$s_{i}(t)=t^{a_{i}}z_{i}\in\mathbf{k}\llbracket t\rrbracket$, where
$z_{i}\in\mathbf{k}\llbracket t\rrbracket$ is a unit. With respect to the
bases $s_{0},\dots,s_{r+1}$ of $S$ and $1,(u-t),\dots,(u-t)^{r}/r!$ of
$\widehat{j_{r+1}L}$, the matrix of the map (10) is given by

$\begin{pmatrix}t^{a_{0}}z_{0}&t^{a_{1}}z_{1}&\dots&t^{a_{r}}z_{r}\\\
(t^{a_{0}}z_{0})^{\prime}&(t^{a_{1}}z_{1})^{\prime}&\dots&(t^{a_{r}}z_{r})^{\prime}\\\
\vdots&\vdots&\ddots&\vdots\\\
(t^{a_{0}}z_{0})^{(r)}&(t^{a_{1}}z_{1})^{(r)}&\dots&(t^{a_{r}}z_{r})^{(r)}.\end{pmatrix}$

The lowest degree term in its determinant is $t^{\sum a_{i}-i}$ and its
coefficient is the determinant of the $(r+1)\times(r+1)$ matrix $C$ with
$C_{ij}=a_{i}(a_{i}-1)\cdots(a_{i}-j+1)$. It remains to show that the
coefficient is non-zero. By a sequence of invertible row transformations, the
matrix $C$ reduces to the Vandermonde matrix $V$ with $V_{ij}=a_{i}^{j}$.
Since the $a_{i}$’s are distinct integers in $\\{0,\dots,d\\}$, their images
are distinct in $\mathbf{k}$, which has characteristic 0 or characteristic
$>d$. Hence the determinant of $V$, and hence of $C$, is non-zero. ∎

Let $\rho(S)\subset C$ be the vanishing locus of the Wronskian of $S$. We call
$\rho(S)$ the _ramification divisor_ of $S$ and its points the _ramification
points_.

######  Corollary 3.2.

For all $p\in C-\rho(S)$, we have $a_{i}(p)=i$. In particular, for all but
finitely many $p$, we have $a_{i}(p)=i$.

###### Proof.

Follows immediately from Proposition 3.1. ∎

Note that $\rho(S)$ is the zero-locus of a section of $\det j_{r+1}L$. Recall
that we have a sequence of surjective maps

$j_{r+1}L\to j_{r}L\to\dots\to j_{2}L\to j_{1}L=L\to 0$

with

$\ker(j_{i+1}L\to j_{i}L)=L\otimes\Omega_{C}^{i}.$

As a result, we get

(11) $\det j_{r+1}L=L^{r+1}\otimes\Omega_{C}^{r(r+1)/2}.$

Consider the Grassmannian ${\bf Gr}={\bf Gr}(r+1,H^{0}(C,L))$ with the
universal sub-bundle

$S\subset\underline{H^{0}(C,L)}.$

On ${\bf Gr}\times C$, we have the universal evaluation map

$\pi_{1}^{*}S\to\pi_{2}^{*}L,$

which induces a map

$\pi_{1}^{*}S\to\pi_{2}^{*}j_{r+1}L.$

Taking the determinant of the last map and applying ${\pi_{1}}_{*}$ gives the
following map on ${\bf Gr}$:

$\det S\to\underline{H^{0}(C,\det j_{r+1}L)}.$

By Proposition 3.1, the map above is non-zero at every point of ${\bf Gr}$,
and hence yields a morphism

(12) $\rho\colon{\bf Gr}(r+1,H^{0}(C,L))\to{\bf P}H^{0}(C,\det j_{r+1}L).$

######  Proposition 3.3.

The map $\rho$ in (12) is finite. For $C={\bf P}^{1}$, it is also surjective.

###### Proof.

Assume that $\dim H^{0}(C,L)>r+1$; otherwise, the statement is vacuous.

Since the Grassmannian has Picard rank one, any morphism from it is either
finite or constant. It is easy to see that $\rho$ is not constant; so it must
be finite.

For $C={\bf P}^{1}$ and $L=O(d)$, we have by (11)

$\det j_{r+1}L\cong O((r+1)d)\otimes O(-r(r+1))=O((r+1)(d-r)).$

Hence, both the source and target of $\rho$ have the same dimension, namely
$(r+1)(d-r)$. ∎

##  4\. Complete parametrisation of the orbit of a linear series

Fix a $2$-dimensional vector space $V$ and an $(r+1)$-dimensional subspace
$S\subset\operatorname{Sym}^{d}V$.

On ${\bf P}V\cong{\bf P}^{1}$, let $O(-1)\subset\underline{V}$ be the
universal sub-bundle and $Q$ the quotient $\underline{V}/O(-1)$. Then we have
a canonical identification $\operatorname{Sym}^{d}V=H^{0}({\bf P}V,Q^{d})$. We
regard $S$ as a linear series associated to the line bundle $Q^{d}$.

Let $W$ be another $2$-dimensional vector space. Consider the

${\bf P}^{3}={\bf P}\operatorname{Hom}(V,W).$

We have a rational map

$\phi\colon\mathbf{P}^{3}\dashrightarrow{\bf Gr}(r+1,\operatorname{Sym}^{d}W)$

defined by

(13) $m\mapsto[m(S)].$

Note that the map is equivariant with respect to the obvious
$\operatorname{GL}W$ action. The goal of this section is to understand a
$\operatorname{GL}W$-equivariant resolution of $\phi$.

Let $B\subset\mathbf{P}V$ be the ramification divisor of $S$. For every $b\in
B$, let $L_{b}\subset\mathbf{P}^{3}$ be the linear subspace

(14) $L_{b}=\\{[m]\mid m(b)=0\\}.$

Then $L_{b}$ is a copy of ${\bf P}^{1}$ linearly embedded in ${\bf P}^{3}$,
and for $b_{1}\neq b_{2}$, the lines $L_{b_{1}}$ and $L_{b_{2}}$ are disjoint.
Let

$\beta\colon\widetilde{{\bf P}}^{3}\to{\bf P}^{3}$

be the blow-up at $\bigsqcup_{b\in B}L_{b}$. Note that the lines $L_{b}$ are
$\operatorname{GL}W$-invariant and hence the blow-up $\widetilde{{\bf P}}^{3}$
inherits an action by $\operatorname{GL}W$. See Figure 2 for a sketch of ${\bf
P}^{3}$ and the lines $L_{b}$.

Figure 2.  The map $\phi\colon{\bf P}^{3}\dashrightarrow{\bf
Gr}(r+1,\operatorname{Sym}^{d}W)$ becomes regular after blowing up the lines
$L_{b}$ (red) on the discriminant quadric $\Delta$. Along the lines of the
opposite ruling (blue, dashed), the map is constant.

The key observation is the following consequence of a result of Aluffi and
Faber [1].

######  Theorem 4.1.

The rational map

$\phi\colon{\bf P}^{3}\dashrightarrow{\bf Gr}(r+1,\operatorname{Sym}^{d}W)$

extends to a regular map

$\widetilde{\phi}\colon\widetilde{{\bf P}}^{3}\to{\bf
Gr}(r+1,\operatorname{Sym}^{d}W).$

For the proof, we need a simple lemma.

######  Lemma 4.2.

Let $\rho\colon X\to Y$ be a finite map between varieties. Let $P$ be an
irreducible normal variety with a rational map $P\dashrightarrow X$. If the
composite $P\dashrightarrow Y$ extends to a regular map $P\to Y$, then so does
$P\dashrightarrow X$.

###### Proof.

Let $\Gamma_{X}\subset P\times X$ and $\Gamma_{Y}\subset P\times Y$ be the
closures of the graphs of $P\dashrightarrow X$ and $P\dashrightarrow Y$,
respectively. Then the projections $\Gamma_{X}\to P$ and $\Gamma_{Y}\to P$ are
birational. In fact, by the hypothesis on $P\dashrightarrow Y$, the projection
$\Gamma_{Y}\to P$ is an isomorphism. The map $\Gamma_{X}\to\Gamma_{Y}$ induced
by $\rho$ is finite and birational. Since the target $\Gamma_{Y}\cong P$ is
normal, we conclude that $\Gamma_{X}\to\Gamma_{Y}$ is an isomorphism. Hence
the projection $\Gamma_{X}\to P$ is also an isomorphism. We deduce that
$P\dashrightarrow X$ extends to a regular map. ∎

###### Proof of Theorem 4.1.

Set

${\bf Gr}={\bf Gr}(r+1,\operatorname{Sym}^{d}W)$

and

${\bf P}={\bf P}H^{0}({\bf P}^{1},\det j_{r+1}O(d))={\bf
P}\operatorname{Sym}^{(r+1)(d-r)}W.$

Recall that we have the finite map

$\rho\colon{\bf Gr}\to{\bf P}$

defined by the Wronskian. By definition, $\rho$ is
$\operatorname{GL}W$-equivariant.

Let $R$ be the Wronksian of $S$. Consider the rational map

${\bf P}^{3}\dashrightarrow{\bf P}$

defined by

$m\mapsto[m(R)].$

Then we have the commutative diagram

${\widetilde{\bf P}^{3}}$${{\bf Gr}}$${\widetilde{\bf P}^{3}}$${{\bf
P}.}$$\scriptstyle{\rho}$

By [1, Proposition 1.2], the map $\widetilde{\bf P}^{3}\dashrightarrow{\bf P}$
extends to a regular map $\widetilde{\bf P}^{3}\to{\bf P}$. By Lemma 4.2, we
conclude the same about $\widetilde{\bf P}^{3}\dashrightarrow{\bf Gr}.$ (In
[1], the base field is assumed to have characteristic 0, but Proposition 1.2
and its proof hold in any characteristic.) ∎

Let $\Delta\subset{\bf P}^{3}={\bf P}\operatorname{Hom}(V,W)$ be the locus of
degenerate maps. Then $\Delta$ is a quadric hypersurface, defined by the
vanishing of the determinant. We have an isomorphism

$\Delta\xrightarrow{\sim}{\bf P}V\times{\bf P}W$

given by

(15) $m\mapsto(\ker m,\operatorname{im}m).$

Under this isomorphism, the line $L_{b}$ defined in (14) corresponds to the
line $\\{b\\}\times{\bf P}W$.

We now do a simple calculation that allows us to understand the map

$\widetilde{\phi}\colon\widetilde{\bf P}\to{\bf
Gr}(r+1,\operatorname{Sym}^{d}W).$

Choose bases $\langle v_{1},v_{2}\rangle$ for $V$ and $\langle
w_{1},w_{2}\rangle$ for $W$. Let us write elements of
$\operatorname{Hom}(V,W)$ as $2\times 2$-matrices with respect to these bases.
Consider the one-parameter family of homomorphisms given by

(16) $m_{t}=\begin{pmatrix}1&0\\\ 0&t\end{pmatrix},\quad t\in{\bf A}^{1}.$

For $t\neq 0$, let

$S_{t}=m_{t}(S)\subset\operatorname{Sym}^{d}W,$

and

$S_{0}=\lim_{t\to 0}S_{t}\subset\operatorname{Sym}^{d}W.$

The following proposition describes $S_{0}$.

######  Proposition 4.3.

Let $S_{0}$ be defined as above. Let $b\in{\bf P}V$ be the point corresponding
to $[v_{2}]\subset V$ and let $a_{0}<\dots<a_{r}$ be the vanishing sequence of
$S$ at $b$. Then $S_{0}\subset\operatorname{Sym}^{d}W$ is given by

$S_{0}=\langle
w_{1}^{d-a_{0}}w_{2}^{a_{0}},\dots,w_{1}^{d-a_{r}}w_{2}^{a_{r}}\rangle.$

###### Proof.

Let $s_{i}\in S$ vanish to order $a_{i}$ at $b$. Then there exists a non-zero
$c_{i}\in\mathbf{k}$ and a homogeneous polynomial $F_{i}(v_{1},v_{2})$ of
degree $d-a_{i}-1$ such that

$s_{i}=c_{i}v_{1}^{d-a_{i}}v_{2}^{a_{i}}+v_{2}^{a_{i}+1}F_{i}(v_{1},v_{2}).$

We see that

$m_{t}(s_{i})=c_{i}t^{a_{i}}w_{1}^{d-a_{i}}w_{2}^{a_{i}}+t^{a_{i}+1}w_{2}^{a_{i}+1}F_{i}(w_{1},tw_{2}).$

Therefore, we get

$\lim_{t\to 0}\langle m_{t}(s_{i})\rangle=\langle
w_{1}^{d-a_{i}}w_{2}^{a_{i}}\rangle,$

and hence

$\lim_{t\to 0}m_{t}(S)=\langle
w_{1}^{d-a_{0}}w_{2}^{a_{0}},\dots,w_{1}^{d-a_{r}}w_{2}^{a_{r}}\rangle.$

∎

######  Corollary 4.4.

If $b\in{\bf P}V$ is not a ramification point of $S$, then $S_{0}$ depends
only on the subspace $\operatorname{im}m_{0}\subset W$.

###### Proof.

In this case, the vanishing sequence at $b$ is $0<\dots<r$. From Proposition
4.3, we see that $S_{0}\subset\operatorname{Sym}^{d}W$ is the subspace of
sections that vanish to order at least $d-r$ at
$\operatorname{im}m_{0}=[w_{1}]\in{\bf P}W$. ∎

Corollary 4.4 implies that the map $\widetilde{\phi}$ is constant on the
proper transforms of the lines ${\bf P}V\times\\{w\\}\subset\Delta$ (see
Figure 2).

######  Theorem 4.5.

Let $S^{\prime}\subset\operatorname{Sym}^{d}W$ be the image of
$S\subset\operatorname{Sym}^{d}V$ under an isomorphism $V\to W$. The
$\operatorname{GL}W$-orbit of $[S^{\prime}]\in{\bf
Gr}(r+1,\operatorname{Sym}^{d}W)$ contains finitely many orbits in its
closure. In addition to the orbit of $[S^{\prime}]$, these are the orbits of
$\langle
w_{1}^{d-a_{0}}w_{2}^{a_{0}},\dots,w_{1}^{d-a_{r}}w_{2}^{a_{r}}\rangle$ where
$a_{0}<\dots<a_{r}$ is the vanishing sequence of $S$ at some $p\in{\bf P}V$.

###### Proof.

The closure of the $\operatorname{GL}W$-orbit of $[S^{\prime}]$ is precisely
the image of the morphism

$\widetilde{\phi}\colon\widetilde{\bf P}\to{\bf
Gr}(r+1,\operatorname{Sym}^{d}W).$

Let $\widetilde{\Delta}\subset\widetilde{\bf P}$ be the pre-image of
$\Delta\subset{\bf P}^{3}$. Then $\widetilde{\bf P}-\widetilde{\Delta}$ is a
copy of $\operatorname{PGL}_{3}$, and its image is the orbit of
$[S^{\prime}]$.

Consider $\widetilde{m}\in\widetilde{\Delta}$ and let $m\in\Delta$ be its
image. Suppose $m$ does not lie on any line $L_{b}$ and let
$\operatorname{im}m=[w_{1}]\in{\bf P}W$. Then by Corollary 4.4, the image
$\widetilde{\phi}(\widetilde{m})\in{\bf Gr}(r+1,\operatorname{Sym}^{d}W)$
corresponds to

(17) $\langle w_{1}^{d}w_{2}^{0},\dots,w_{1}^{d-r}w_{2}^{r}\rangle,$

where $w_{2}\in W$ is any vector linearly independent from $w_{1}$.

Suppose $m$ lies on $L_{b}$. Let $w=\operatorname{im}m\in{\bf P}W$, so that
$m\in\Delta$ corresponds to $(b,w)$ in the isomorphism (15). Suppose $v_{2}\in
V$ is such that $b=[v_{2}]$ and $w_{1}\in W$ such that $w=[w_{1}]$. The point
$\widetilde{m}$ corresponds to a direction in the normal bundle $N_{L_{b}/{\bf
P}^{3}}$ at $p$. Consider the direction in $N_{L_{b}/{\bf P}^{3}}|_{p}$
corresponding to the line ${\bf P}V\times\\{w\\}\subset\Delta$ through $m$. By
Corollary 4.4, the map $\widetilde{\phi}$ is constant on this line with value
given by the subspace (17). If $\widetilde{m}$ corresponds to this direction,
then $\widetilde{\phi}(\widetilde{m})$ corresponds to the same subspace. All
the other directions in $N_{L_{b}/{\bf P}^{3}}|_{p}$ can be realised by
tangent directions of families $m_{t}$ for suitable choices of $v_{1}\in V$
and $w_{2}\in W$. If $\widetilde{m}$ corresponds to the tangent direction
given by $m_{t}$, then Proposition 4.3 implies that
$\widetilde{\phi}(\widetilde{m})$ corresponds to the subspace

(18) $\langle
w_{1}^{d-a_{0}}w_{2}^{a_{0}},\dots,w_{1}^{d-a_{r}}w_{2}^{a_{r}}\rangle.$

In conclusion, we see that the image of every point of $\widetilde{\Delta}$ is
of the form (17), which corresponds to the generic vanishing sequence, or
(18), which corresponds to the vanishing sequence of a ramification point. ∎

##  5\. Localisation

The goal of this section is to compute the equivariant orbit class of a linear
series using localisation on $\widetilde{{\bf P}}^{3}$. The calculation is
similar to the one in [13, Appendix B].

Choose a basis $\langle w_{1},w_{2}\rangle$ of $W$ and let
$T\subset\operatorname{GL}(W)$ be the diagonal torus with respect to this
basis. Choose coordinates $t_{1},t_{2}$ on $T$ so that the action on $W$ is

$(t_{1},t_{2})\colon(w_{1},w_{2})\mapsto(t_{1}w_{1},t_{2}w_{2}).$

Then the $T$-fixed locus of ${\bf P}^{3}={\bf P}\operatorname{Hom}(V,W)$
consists of two skew lines $\Lambda_{1}$ and $\Lambda_{2}$ given by

$\Lambda_{i}={\bf P}\operatorname{Hom}(V,\langle w_{i}\rangle).$

Note that the lines $\Lambda_{i}\subset{\bf P}^{3}$ lie in the discriminant
quadric $\Delta$, and are in the opposite ruling as the lines $L_{b}$ blown up
to get $\widetilde{{\bf P}}^{3}$. The $T$-fixed locus of $\widetilde{{\bf
P}}^{3}$ consists of the proper transforms $\widetilde{\Lambda}_{i}$ of
$\Lambda_{i}$ and points $p_{i,b}$ for $i=1,2$ and $b\in B$. The point
$p_{i,b}$ maps under the blow-down to the point $\Lambda_{i}\cap L_{b}$. We
now describe it in local coordinates.

Choose a basis $v_{1},v_{2}$ of $V$ such that $[v_{2}]=b\in{\bf P}V$. In the
bases $v_{1},v_{2}$ of $V$ and $w_{1},w_{2}$ of $W$, we represent
$m\in\operatorname{Hom}(V,W)$ by the matrix

$m=\begin{pmatrix}X_{11}&X_{12}\\\ X_{21}&X_{22}\end{pmatrix}.$

The matrix entries $X_{ij}$ are functionals on $\operatorname{Hom}(V,W)$. The
torus $T$-acts by

$(t_{1},t_{2})\colon X_{ij}\mapsto t_{i}^{-1}X_{ij}.$

The line $\Lambda_{1}\subset{\bf P}^{3}$ is defined by $X_{21}=X_{22}=0$ and
the line $L_{b}\subset{\bf P}^{3}$ by $X_{12}=X_{22}=0.$ Their intersection is
the point $X_{12}=X_{21}=X_{22}=0$. Choose affine coordinates
$x_{ij}=X_{ij}/X_{11}$ around this point. Then the blow-up of ${\bf P}^{3}$ in
$L_{b}$ has the coordinate charts

$(x_{21},x_{12},x_{22},u\mid x_{12}=ux_{22})\text{ and
}(x_{21},x_{12},x_{22},v\mid x_{22}=vx_{12}).$

The point $p_{1,b}$ is contained in the second chart, and is given by

(19) $p_{1,b}:x_{21}=x_{22}=v=0.$

Given $(m_{1},m_{2})\in\mathbf{Z}$, let $\chi(m_{1},m_{2})$ be the character
of $T$ given by

$\chi(m_{1},m_{2})\colon(t_{1},t_{2})\mapsto t_{1}^{m_{1}}\cdot
t_{2}^{m_{2}}.$

The next two propositions give the normal bundles of the $T$-fixed loci.

######  Proposition 5.1.

Let $N_{1,b}$ be the tangent space to $p_{1,b}\in\widetilde{{\bf P}}^{3}$. As
a $T$-representation, we have

$N_{1,b}\cong\chi(-1,1)\oplus\chi(-1,1)\oplus\chi(1,-1).$

###### Proof.

From (19), we see that the co-tangent space at $p_{1,b}$ is generated by
$x_{21},x_{22},v$, on which $T$ acts by

$(t_{1},t_{2})\colon x_{21}\mapsto t_{1}t_{2}^{-1}x_{21}\quad x_{22}\mapsto
t_{1}t_{2}^{-1}x_{22}\quad v\mapsto t_{1}^{-1}t_{2}v.$

We now take the dual. ∎

######  Proposition 5.2.

Let $\widetilde{N}_{1}$ be the normal bundle of
$\widetilde{\Lambda}_{1}\subset\widetilde{{\bf P}}^{3}$. In the
$T$-equivariant Grothendieck group of $\widetilde{\Lambda}_{1}={\bf P}^{1}$
(with the trivial $T$-action), we have

$[N_{1}]=\left([O]+[O\left(2-|B|\right)]\right)\otimes\chi(-1,1).$

###### Proof.

Let $N_{1}$ be the normal bundle of $\Lambda_{1}\subset{\bf P}^{3}$. The
defining equations show that $\Lambda_{1}$ is the zero locus of sections of
$O_{{\bf P}^{3}}(1)\otimes\chi(0,1)$ and the restriction of $O_{{\bf
P}^{3}}(1)$ to $\Lambda_{1}$ is $O(1)\otimes\chi(-1,0)$. Therefore, we get

$N_{1}\cong\left(O(1)\oplus O(1)\right)\otimes\chi(-1,1).$

Recall that $\beta\colon\widetilde{\bf P}^{3}\to{\bf P}^{3}$ is the blow-up
along $\bigsqcup_{b}L_{b}$. We have the exact sequence

$0\to\widetilde{N}_{1}\to\beta^{*}N_{1}\to Q\to 0,$

where $Q\cong O_{\operatorname{supp}B}\otimes\chi(-1,1)$. The statement
follows. ∎

Let $\widetilde{S}$ be the pull-back to $\widetilde{\bf P}^{3}$ of the
universal sub-bundle of ${\bf Gr}(r+1,\operatorname{Sym}^{d}W)$. The following
two propositions describe $\widetilde{S}$ on the $T$-fixed locus.

######  Proposition 5.3.

Let $p=p_{1,b}$ and let $a_{0}<\dots<a_{r}$ be the vanishing sequence of $S$
at $b$. As a $T$-representation, we have

$\widetilde{S}|_{p}\cong\bigoplus_{i=0}^{r}\chi(d-a_{i},a_{i}).$

###### Proof.

Consider the family $m_{t}$ as in (16), where $[v_{2}]=b$. Then, in
$\widetilde{\bf P}$ we have

$\lim_{t\to 0}m_{t}=p.$

Therefore, we have

$\widetilde{S}|_{p}=\lim_{t\to 0}m_{t}(S).$

Proposition 4.3 gives the limit and yields

$\widetilde{S}|_{p}=\langle
w_{1}^{d-a_{0}}w_{2}^{a_{0}},\dots,w_{1}^{d-a_{r}}w_{2}^{a_{r}}\rangle.$

The statement follows. ∎

######  Proposition 5.4.

As a $T$-equivariant sheaf on $\widetilde{\Lambda}_{1}$, we have

$[\widetilde{S}]|_{\widetilde{\Lambda}_{1}}=\bigoplus_{i=0}^{r}\chi(d-i,i).$

###### Proof.

By Corollary 4.4, the map $\widetilde{\phi}$ is constant on
$\widetilde{\Lambda}_{1}$, and its value corresponds to the subspace

$\langle
w_{1}^{d}w_{2}^{0},\dots,w_{1}^{d-r}w_{2}^{r}\rangle\subset\operatorname{Sym}^{d}W.$

The statement follows. ∎

We are now ready to prove the main theorem of the paper, which we restate.

######  Theorem 5.5.

Let
$s\in\operatorname{Hom}(\mathbf{k}^{r+1},\operatorname{Sym}^{d}\mathbf{k}^{2})$
be an injective map with image $S\subset\operatorname{Sym}^{d}\mathbf{k}^{2}$.
Let $\\{(a_{i}(b))\mid b\in B\\}$ be the ramification profile of $S$. Let
$\Gamma\subset\operatorname{PGL}(2)$ be the stabiliser of
$S\subset\operatorname{Sym}^{d}\mathbf{k}^{2}$, and assume that it is finite.
Let $\operatorname{Orb}(s)$ be the closure of the
$\operatorname{GL}(r+1)\times\operatorname{GL}(2)$ orbit of $s$. Then the
class

$|\Gamma|\cdot[\operatorname{Orb}(s)]\in
A^{(r+1)(d-r)-3}_{\operatorname{GL}(r+1)\times\operatorname{GL}(2)}(\operatorname{Hom}(\mathbf{k}^{r+1},\operatorname{Sym}^{d}\mathbf{k}^{2}))$

is given by

$|\Gamma|\cdot[\operatorname{Orb}(s)]=\frac{1}{(\omega_{1}-\omega_{2})^{3}}\sum_{b\in
B}\psi_{a_{0}(b),\dots,a_{r}(b)}(\mu;\omega)+\frac{2-|B|}{(\omega_{1}-\omega_{2})^{3}}\cdot\psi_{0,\dots,r}(\mu;\omega).$

###### Proof.

Set $V=W=\mathbf{k}^{2}$ and consider the rational map

${\bf P}^{3}={\bf P}\operatorname{Hom}(V,W)\dashrightarrow{\bf
Gr}(r+1,\operatorname{Sym}^{d}W)$

given by

$m\mapsto m(S).$

Let $\phi\colon\widetilde{{\bf P}}^{3}\to{\bf
Gr}(r+1,\operatorname{Sym}^{d}W)$ be its resolution provided by Theorem 4.1.
Then $\phi$ is a complete parametrisation of the $\operatorname{PGL}W$-orbit
of $[S]$. By Proposition 2.2, we have

$|\Gamma|\cdot[\operatorname{Orb}(s)]=\int_{\widetilde{{\bf
P}}^{3}}c_{(r+1)(d-r)}\operatorname{Hom}(\underline{\mathbf{k}}^{r+1},\phi^{*}Q),$

where $Q$ is the universal quotient bundle on the Grassmannian. In our case,
we have

$\phi^{*}Q=\operatorname{Sym}^{d}\underline{W}/\widetilde{S}.$

Let $T\subset\operatorname{GL}(2)$ be a maximal torus. By the equivariant
localisation formula [8, Corollary 1], we have

(20) $\int_{\widetilde{{\bf
P}}^{3}}c_{(r+1)(d-r)}\operatorname{Hom}(\underline{\mathbf{k}}^{r+1},\phi^{*}Q)=\sum_{F}\int_{F}\frac{c_{(r+1)(d-r)}\operatorname{Hom}(\underline{\mathbf{k}}^{r+1},\phi^{*}Q)}{e(N_{F})},$

where the sum is over the $T$-fixed loci $F$ and $e(N_{F})$ is the top Chern
class of the equivariant normal bundle of $F\subset\widetilde{{\bf P}}^{3}$.
In our case, the fixed loci consist of the lines $\widetilde{\Lambda}_{i}$ for
$i=1,2$ and the points $p_{i,b}$ for $i=1,2$ and $b\in B$.

Let us compute the contribution from $F=\widetilde{\Lambda}_{1}$. Proposition
5.2 give us the numerator of the integrand:

$\prod_{j=0}^{r}\prod_{i=r+1}^{d}((d-i)\omega_{1}+i\omega_{2}-\mu_{j})=\phi_{0,\dots,r}(\mu;\omega).$

Proposition 5.4 gives us the denominator of the integrand:

$(\omega_{2}-\omega_{1})(\omega_{2}-\omega_{1}+(2-|B|)h),$

where $h=c_{1}(O(1))$. The integral is the coefficient of $h$ in the power
series expansion of the quotient. We may compute it by formally
differentiating with respect to $h$ and setting $h=0$. The result is

(21)
$\int_{\widetilde{\Lambda}_{1}}\frac{c_{(r+1)(d-r)}\operatorname{Hom}(\mathbf{k}^{r+1},\phi^{*}Q)}{e(N_{\widetilde{\Lambda}_{1}})}=\frac{2-|B|}{(\omega_{1}-\omega_{2})^{3}}\cdot\phi_{0,\dots,r}(\mu;\omega).$

The contribution from $F=\widetilde{\Lambda}_{2}$ is the same as above with
$\omega_{1}$ and $\omega_{2}$ swapped.

Let us compute the contribution from $F=p_{1,b}$. Proposition 5.3 gives us the
numerator:

$\prod_{j=0}^{r}\prod_{i}((d-i)\omega_{1}+i\omega_{2}-\mu_{j})=\phi_{a_{0}(b),\dots,a_{r}(b)}(\mu;\omega).$

On the left hand side above, the index $i$ ranges in the set
$\\{0,\dots,d\\}-\\{a_{0}(b),\dots,a_{r}(b)\\}$. Proposition 5.1 gives us the
denominator:

$(\omega_{1}-\omega_{2})^{3}.$

Taking the integral is trivial in this case, so the result is

(22)
$\int_{p_{1,b}}\frac{c_{(r+1)(d-r)}\operatorname{Hom}(\mathbf{k}^{r+1},\phi^{*}Q)}{e(N_{p_{1,b}})}=\frac{1}{(\omega_{1}-\omega_{2})^{3}}\cdot\phi_{a_{0}(b),\dots,a_{r}(b)}(\mu;\omega).$

The contribution from $F=p_{2,b}$ is the same as above with $\omega_{1}$ and
$\omega_{2}$ swapped.

By combining (21), (22), and their analogues for $\widetilde{\Lambda}_{2}$ and
$p_{2,b}$ we see that the integral in (20) is as claimed. ∎

##  6\. Orbit specialisation

The goal of this section is to understand limits of the orbit closure of a
linear series under one-parameter specialisations. Recall that
$\operatorname{Orb}$ denotes the orbit closure and $\operatorname{WOrb}$ the
weighted orbit closure, namely the cycle obtained by multiplying the orbit
closure by the order of the stabiliser group in $\operatorname{PGL}(2)$.

Fix a positive integer $m$ and for each $j=1,\dots,m$, fix an increasing
sequence of non-negative integers $a_{0}(j)<\dots<a_{r}(j)$. Assume that

(23) $\sum_{j,i}(a_{i}(j)-i)=(r+1)(d-r).$

We consider one-parameter degenerations of linear series on ${\bf P}^{1}$ with
ramification profile $\\{(a_{0}(j),\dots,a_{r}(j))\mid j=1,\dots,m\\}$. We use
limit linear series, for which we refer the reader to [10]. Although [10]
assumes characteristic 0, the theory holds without modification for
characteristic $>d$; see [14].

Let $\Delta$ be the spectrum of a DVR with special point $0$ and generic point
$\eta$. Let $p_{1,\eta},\dots,p_{m,\eta}$ be distinct sections $\eta\to{\bf
P}^{1}_{\eta}$ and let $(S_{\eta},O(d))$ be a linear series that has vanishing
sequence $a_{0}(j)<\dots<a_{r}(j)$ at $p_{j,\eta}$. Let
$(\mathcal{X}\to\Delta,p_{1},\dots,p_{m})$ be a semi-stable extension of
$({\bf P}^{1}_{\eta}\to\eta,p_{1,\eta},\dots,p_{m,\eta})$ such that
$\mathcal{X}$ is non-singular. Then the central fiber
$(X,\overline{p}_{1},\dots,\overline{p}_{m})$ is a semi-stable pointed
rational curve. The linear series $S_{\eta}$ yields a refined limit linear
series $\mathcal{S}$ on $\mathcal{X}$ (use the implication $3\implies 1$ of
[10, Proposition 2.5]). Let $S$ be the reduction of $\mathcal{S}$ to the
central fiber.

We record the ramification profile of the limit linear series $S$ using a
decorated graph. We begin with the dual graph $\Gamma$ of the pointed curve
$(X,\overline{p}_{1},\dots,\overline{p}_{m})$. Recall that $\Gamma$ is
obtained by drawing a vertex $\nu$ for every irreducible component $X_{\nu}$
of $X$, an edge $\mu$ to $\nu$ for every node of $X$ on $X_{\mu}$ and
$X_{\nu}$, and a half-edge $i$ on $\nu$ if the marked point $\overline{p}_{i}$
lies on $X_{\nu}$. We think of the full-edge from $\mu$ to $\nu$ as a
combination of two half-edges, one at $\mu$ and one at $\nu$. We use the term
_dangling half-edge_ when we refer to a true half-edge, and not a part of a
full-edge. On $\Gamma$ we record the ramification information of $S$ as
follows. We label a half-edge $e$ incident to a vertex $\nu$ by the vanishing
sequence of the $X_{\nu}$-aspect of $S$ at the point of $X_{\nu}$ represented
by $e$. If $e$ is a full edge, joining $\mu$ and $\nu$ say, then it gets two
labels, one at $\mu$, say $(a_{0},\dots,a_{r})$, and one at $\nu$, say
$(b_{0},\dots,b_{r})$. Since $S$ is a refined series, the two labels are
complementary; that is, they satisfy $a_{i}+b_{r-i}=d$. Figure 3 shows an
example of a labelled dual graph $\Gamma$ arising from a limit linear series
of rank $2$ and degree $4$.

$(0,2)$$(0,2)$$(0,2)$$(0,2)$$(0,2)$$(0,2)$$(0,3)$$(1,4)$$(0,3)$$(1,4)$$(1,2)$$(2,3)$
Figure 3.  We represent the ramification information of a refined limit
$g^{r}_{d}$ by a labelled dual graph as above, where $r=1$ and $d=4$. The
labels on the half-edges are vanishing sequences, and the two labels on a
full-edge are complementary.

For a vertex $\nu$ of $\Gamma$, let $S(\nu)$ be the $X_{\nu}$-aspect of the
limit linear series $S$.

######  Theorem 6.1.

In the notation above, let $\Sigma$ be the cycle on ${\bf
Gr}(r+1,\operatorname{Sym}^{d}\mathbf{k}^{2})$ obtained as the limit of the
cycle $\operatorname{WOrb}(S_{\eta})$. Then we have the equality of
$\operatorname{GL}(2)$-invariant cycles

$\Sigma=\sum_{\nu}\operatorname{WOrb}(S(\nu)).$

###### Proof.

We first show that the cycle

$D=\Sigma-\sum_{\nu\in\Gamma}\operatorname{WOrb}(S(\nu))$

is effective using Proposition 2.4.

For every vertex $\nu$ of $\Gamma$, let $\mathcal{X}_{\nu}$ be the surface
obtained by blowing down all components of $X$ except $X_{\nu}$. Then we have
an isomorphism $\mathcal{X}_{\nu}\cong{\bf P}^{1}\times\Delta$; fix such an
isomorphism and identify the two. Choose an arbitrary vertex of $\Gamma$ and
call it $0$. For every $\nu$, let $g_{\nu}\in\operatorname{PGL}_{2}(\eta)$ be
the unique element such that

$\mathcal{X}_{0,\eta}={\bf P}^{1}\times\eta\xrightarrow{g_{\nu}}{\bf
P}^{1}\times\eta=\mathcal{X}_{\nu,\eta}$

represents the identity map

$\mathcal{X}_{0,\eta}=\mathcal{X}_{\eta}\to\mathcal{X}_{\nu,\eta}=\mathcal{X}_{\eta}.$

For $\mu\neq\nu$, the identity map on the generic fiber does not extend to an
isomorphism $\mathcal{X}_{\mu}\to\mathcal{X}_{\nu}$. Therefore,
$g_{\mu}g_{\nu}^{-1}\in\operatorname{PGL}_{2}(\eta)$ does not extend to an
element of $\operatorname{PGL}_{2}(\Delta)$. On $\mathcal{X}_{\nu}={\bf
P}^{1}\times\Delta$, the limit of $[S_{\eta}]{\colon}\eta\to{\bf
Gr}(r+1,\operatorname{Sym}^{d}\mathbf{k}^{2})$ is $[S(\nu)]\in{\bf
Gr}(r+1,\operatorname{Sym}^{d}\mathbf{k}^{2}).$ Now, Proposition 2.4 applied
to $M={\bf Gr}(r+1,\operatorname{Sym}^{d}\mathbf{k}^{2})$ and
$G=\operatorname{PGL}_{2}$ implies that

$D=\Sigma-\sum_{\nu}\operatorname{WOrb}(S(\nu))$

is effective.

Let $N=(d-r)(r+1)$. We apply the Wronskian map

$\rho{\colon}{\bf Gr}(r+1,\operatorname{Sym}^{d}\mathbf{k}^{2})\to{\bf
P}\operatorname{Sym}^{N}\mathbf{k}^{2}$

and see that the cycle

$\rho_{*}D=\rho_{*}\Sigma-\sum_{\nu}\rho_{*}\operatorname{WOrb}(S(\nu))$

is effective. Let $R_{\eta}$ be the ramification divisor of $S_{\eta}$ and
$R(\nu)$ the ramification divisor of $S(\nu)$. Then the cycle $\rho_{*}\Sigma$
on ${\bf P}\operatorname{Sym}^{N}\mathbf{k}^{2}$ is the limit of the cycle
$\operatorname{WOrb}(R_{\eta})$ and $\rho_{*}\operatorname{WOrb}(S(\nu))$ is
the cycle $\operatorname{WOrb}(R(\nu))$. Aluffi and Faber compute the degree
of the weighted orbit closure for any divisor on ${\bf P}^{1}$ (see [1,
Proposition 1.3] and note that in their paper this degree is called the “pre-
degree”). Using their formula, it is straightforward to check that
$\deg\rho_{*}D=0$ (see Remark 6.2). Since $\rho_{*}D$ is effective, we
conclude that $\rho_{*}D=0$ and since $\rho$ is finite, that $D=0$. ∎

######  Remark 6.2.

We indicate how to check that the pre-degree of the orbit of $R_{\eta}$ is the
sum of the pre-degrees of the orbits of $R(\nu)$ using [1, Proposition 1.3].
The pre-degree of the orbit closure of a divisor depends only on the
multiplicities of the points in the divisor. More precisely, by [1,
Proposition 1.3], the pre-degree for a divisor of degree $N$ with
multiplicities $m_{1},\dots,m_{s}$ is given by

(24) $\begin{split}p(m_{1},\dots,m_{s})&=N^{3}-3N\sum m_{i}^{2}+2\sum
m_{i}^{3}\\\ &=\sum_{i\neq j\neq k}m_{i}m_{j}m_{k}.\end{split}$

To understand the multiplicities in the divisors $R(\nu)$, consider the
labelled dual graph $\Gamma$ and replace the vanishing sequences by their
weights, where the weight of the sequence $(a_{0},\dots,a_{r})$ is
$\sum_{i}(a_{i}-i)$. Then the multiplicities in $R_{\eta}$ are the labels of
the dangling half-edges and the multiplicities of $R(\nu)$ are the labels of
the half-edges incident to $\nu$. The sum of the labels of all the half-edges
incident to a vertex is $N=(d-r)(r+1)$, and the sum of the two labels of a
full-edge is also $N$. Consider the operation of contracting an edge of
$\Gamma$ as shown below

$\leavevmode\hbox to248.02pt{\vbox
to31.94pt{\pgfpicture\makeatletter\hbox{\hskip
212.2458pt\lower-22.05939pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.71387pt}{0.0pt}\pgfsys@curveto{4.71387pt}{2.60342pt}{2.60342pt}{4.71387pt}{0.0pt}{4.71387pt}\pgfsys@curveto{-2.60342pt}{4.71387pt}{-4.71387pt}{2.60342pt}{-4.71387pt}{0.0pt}\pgfsys@curveto{-4.71387pt}{-2.60342pt}{-2.60342pt}{-4.71387pt}{0.0pt}{-4.71387pt}\pgfsys@curveto{2.60342pt}{-4.71387pt}{4.71387pt}{-2.60342pt}{4.71387pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}{{}}{}{{}}{}{{}}{}{{}}{}\pgfsys@moveto{-28.45276pt}{-14.22638pt}\pgfsys@moveto{0.0pt}{-14.22638pt}\pgfsys@moveto{28.45276pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-32.5058pt}{-15.10178pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize$m_{1}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.75pt}{-16.72638pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\ldots}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{24.46097pt}{-15.31139pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize$m_{s}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{ {}{}{}}
{{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{-4.39507pt}{-2.19765pt}\pgfsys@lineto{-20.86671pt}{-10.43394pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{ {}{}{}}
{{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{4.39507pt}{-2.19765pt}\pgfsys@lineto{20.92796pt}{-10.46457pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\par\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{}{{}}{}{{}}{}\pgfsys@moveto{-184.94292pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{-180.22905pt}{0.0pt}\pgfsys@curveto{-180.22905pt}{2.60342pt}{-182.3395pt}{4.71387pt}{-184.94292pt}{4.71387pt}\pgfsys@curveto{-187.54634pt}{4.71387pt}{-189.65678pt}{2.60342pt}{-189.65678pt}{0.0pt}\pgfsys@curveto{-189.65678pt}{-2.60342pt}{-187.54634pt}{-4.71387pt}{-184.94292pt}{-4.71387pt}\pgfsys@curveto{-182.3395pt}{-4.71387pt}{-180.22905pt}{-2.60342pt}{-180.22905pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-184.94292pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-184.94292pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}{{}}{}{{}}{}{{}}{}{{}}{}\pgfsys@moveto{-204.85976pt}{-14.22638pt}\pgfsys@moveto{-184.94292pt}{-14.22638pt}\pgfsys@moveto{-165.02608pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-208.9128pt}{-15.10178pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize$m_{1}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-188.69292pt}{-16.72638pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\ldots}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-168.91579pt}{-15.05276pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{\scriptsize$m_{\ell}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{ {}{}{}}
{{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{-188.94135pt}{-2.85634pt}\pgfsys@lineto{-197.27371pt}{-8.80864pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{ {}{}{}}
{{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{-180.94449pt}{-2.85634pt}\pgfsys@lineto{-172.44879pt}{-8.9253pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{}{{}}{}{{}}{}\pgfsys@moveto{-113.81102pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{-109.09715pt}{0.0pt}\pgfsys@curveto{-109.09715pt}{2.60342pt}{-111.2076pt}{4.71387pt}{-113.81102pt}{4.71387pt}\pgfsys@curveto{-116.41444pt}{4.71387pt}{-118.52489pt}{2.60342pt}{-118.52489pt}{0.0pt}\pgfsys@curveto{-118.52489pt}{-2.60342pt}{-116.41444pt}{-4.71387pt}{-113.81102pt}{-4.71387pt}\pgfsys@curveto{-111.2076pt}{-4.71387pt}{-109.09715pt}{-2.60342pt}{-109.09715pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{-113.81102pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-113.81102pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}{{}}{}{{}}{}{{}}{}{{}}{}\pgfsys@moveto{-133.72786pt}{-14.22638pt}\pgfsys@moveto{-113.81102pt}{-14.22638pt}\pgfsys@moveto{-93.89418pt}{-14.22638pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-140.12201pt}{-14.88943pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{\scriptsize$m_{\ell+1}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-117.56102pt}{-16.72638pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\ldots}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-97.88597pt}{-15.31139pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize$m_{s}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{ {}{}{}}
{{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{-117.80975pt}{-2.85597pt}\pgfsys@lineto{-125.49179pt}{-8.34254pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{ {}{}{}}
{{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{-109.81259pt}{-2.85634pt}\pgfsys@lineto{-101.41898pt}{-8.85239pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
{{}}{}{}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{}{{{}{}}}{}\pgfsys@moveto{-180.02905pt}{0.0pt}\pgfsys@lineto{-118.72488pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-169.59052pt}{3.533pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize$v$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-133.58542pt}{3.533pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize$w$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-61.90552pt}{-1.8894pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\leadsto$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>
Since

$N=v+w=m_{1}+\dots+m_{\ell}+v=m_{\ell+1}+\dots+m_{s}+w,$

we have $v=m_{\ell+1}+\dots+m_{s}$ and $w=m_{1}+\dots+m_{\ell}$. From (24) it
follows that

$p(m_{1},\dots,m_{\ell},v)+p(m_{\ell+1},\dots,m_{s},w)=p(m_{1},\dots,m_{s}).$

By contracting the edges one by one, we reduce the sum of the pre-degrees for
$R(\nu)$ to a single sum, which is the pre-degree for $R_{\eta}$.

######  Corollary 6.3.

In the setup of Theorem 6.1, in $A^{*}_{\operatorname{GL}(2)}({\bf
Gr}(r+1,\operatorname{Sym}^{d}\mathbf{k}^{2}))$ we have

$[\operatorname{WOrb}(S_{\eta})]=\sum_{\nu}[\operatorname{WOrb}(S(\nu))].$

###### Proof.

Since $[\operatorname{WOrb}(S_{\eta})]=[\Sigma]$, the statement follows from
Theorem 6.1. ∎

Recall that the class $\operatorname{WOrb}(S)$ depends only on the
ramification profile of $S$. The ramification profile of $S_{\eta}$ is
recorded by the labels on the dangling half-edges of $\Gamma$. The
ramification profile of $S(\nu)$ is recorded by the labels on the half-edges
incident to $\nu$. Thus, the equality in Corollary 6.3 can be read off purely
from the labelled graph $\Gamma$.

A natural question is: which labelled graphs $\Gamma$ arise from degenerations
of linear series? Here, the obvious necessary conditions are also sufficient.
Let $\Gamma$ be a tree with $m$ dangling half-edges. Suppose all the half-
edges of $\Gamma$, including the full-edges considered as two half-edges, are
labelled with increasing $(r+1)$-tuples of integers in $\\{0,\dots,d\\}$.
Assume that the two labels on a full-edge are complementary.

######  Proposition 6.4.

Let $\Gamma$ be a labelled graph as above. Suppose for every vertex $\nu$ of
$\Gamma$, there is a linear series $S(\nu)$ of rank $r$ and degree $d$ on
${\bf P}^{1}$ whose ramification profile agrees with the multi-set of labels
of half-edges incident to $\nu$. Then $\Gamma$ arises from a degeneration.
That is, there exists a $\Delta$, a semi-stable $m$-pointed curve
$(\mathcal{X}\to\Delta,p_{1},\dots,p_{m})$ and a refined linear series
$\mathcal{S}$ on $\mathcal{X}$ such that the labelled dual graph of its
central fiber is $\Gamma$.

###### Proof.

The proof is a simple application of the smoothing theorem for $g^{r}_{d}$’s
[10, Theorem 3.4].

Let $(X,\overline{p}_{1},\dots,\overline{p}_{m})$ be a semi-stable curve with
dual graph $\Gamma$. We assemble the linear series $S(\nu)$ together to get a
refined limit linear series $S$ on $X$. Note that the expected dimension of
the space $G^{r}_{d}$ on a rational curve with a prescribed ramification
divisor is $0$. Since the Wronskian map is finite and the ramification
divisors of all aspects of $S$ are prescribed, $S$ is an isolated point of
$G^{r}_{d}(X,\overline{p}_{1},\dots,\overline{p}_{m})$. By [10, Theorem 3.4],
there exists a $\Delta$ and a smoothing
$(\mathcal{X}\to\Delta,p_{1},\dots,p_{m},\mathcal{S})$ of
$(X,\overline{p}_{1},\dots,\overline{p}_{m},S)$. ∎

######  Remark 6.5.

The existence of a $g^{r}_{d}$ on ${\bf P}^{1}$ with prescribed ramification
profile is a Schubert theoretic condition on ${\bf
Gr}(r+1,\operatorname{Sym}^{d}\mathbf{k}^{2})$. Such a series exists if and
only if the product of certain Schubert classes on the Grassmannian is non-
zero. See the remark after [9, Theorem 2.3] for a precise statement.

Theorem 6.1 asserts an equality of $\operatorname{GL}(2)$-invariant cycles on
${\bf Gr}(r+1,\operatorname{Sym}^{d}\mathbf{k}^{2})$. It can be easily
upgraded to an equality of
$\operatorname{GL}(r+1)\times\operatorname{GL}(2)$-invariant cycles on
$\operatorname{Hom}(\mathbf{k}^{r+1},\operatorname{Sym}^{d}\mathbf{k}^{2})$.

######  Proposition 6.6.

In the notation of Theorem 6.1, let
$s_{\eta}\colon\operatorname{Hom}(\mathbf{k}^{r+1},\operatorname{Sym}^{d}\mathbf{k}^{2})_{\eta}$
be a homomorphism with image $S_{\eta}$. Let
$s_{\nu}\in\operatorname{Hom}(\mathbf{k}^{r+1},\operatorname{Sym}^{d}\mathbf{k}^{2})$
be homomorphisms with images $S(\nu)$. Let $\Sigma$ be the flat limit of
$\operatorname{WOrb}(s_{\eta})$. Then, on
$\operatorname{Hom}(\mathbf{k}^{r+1},\operatorname{Sym}^{d}\mathbf{k}^{2})$,
we have the equality of
$\operatorname{GL}(r+1)\times\operatorname{GL}(2)$-invariant cycles

$\Sigma=\sum_{\nu}\operatorname{WOrb}(s_{\nu}).$

###### Proof.

Let
$U\subset\operatorname{Hom}(\mathbf{k}^{r+1},\operatorname{Sym}^{d}\mathbf{k}^{2})$
be the open subset of injective homomorphisms. The action of
$\operatorname{GL}(r+1)$ on $U$ is free and the map

$\operatorname{im}\colon U\to{\bf
Gr}(r+1,\operatorname{Sym}^{d}\mathbf{k}^{2})$

taking a homomorphism to its image is the quotient by
$\operatorname{GL}(r+1)$. On $U$, the assertions follow from Theorem 6.1.

It suffices to prove that $\Sigma$ does contain any $(r+1)^{2}+3$-dimensional
components in the complement of $U$. We claim that $\Sigma$ is the closure of
$\Sigma\cap U$, from which the statement follows.

To see this, take $x\in\Sigma$. Then, possibly after a base change, we can
choose
$x_{\eta}\in\operatorname{Hom}(\mathbf{k}^{r+1},\operatorname{Sym}^{d}\mathbf{k}^{2})_{\eta}$
in the $\operatorname{GL}(r+1)\times\operatorname{GL}(2)$-orbit of $s_{\eta}$
such that the flat limit of $x_{\eta}$ is $x$. Let
$S\subset\operatorname{Sym}^{d}\mathbf{k}^{2}$ be the $(r+1)$-dimensional
subspace corresponding to the flat limit of $\operatorname{im}(x_{\eta})$ in
${\bf Gr}(r+1,\operatorname{Sym}^{d}\mathbf{k}^{2})$. Then it is easy to see
that $S$ contains the image of $x$. Let $\ell$ be the dimension of the image
of $x$. By making a change of coordinates on $\mathbf{k}^{r+1}$ if necessary,
assume that $x$ sends the last $r+1-\ell$ standard basis vectors to 0. Let
$s^{\prime}\in\operatorname{Hom}(\mathbf{k}^{r+1},\operatorname{Sym}^{d}\mathbf{k}^{2})$
be a map that agrees with $x$ on the first $\ell$ basis vectors and whose
image is $S$. Then $s^{\prime}\in\Sigma\cap U$. Consider the diagonal one
parameter subgroup $\lambda_{t}\to\operatorname{GL}(r+1)$ with diagonal
entries

$(\underbrace{1,\dots,1}_{\ell},\underbrace{t,\dots,t}_{r+1-\ell}).$

Then, by construction, $\lim_{t\to 0}\lambda_{t}(s^{\prime})=x$. Since
$\Sigma\cap U$ is $\operatorname{GL}(r+1)$-invariant, and
$s^{\prime}\in\Sigma\cap U$, we get that $\lambda_{t}(s^{\prime})\in\Sigma\cap
U$ for $t\neq 0$. We conclude that $x$ lies in the closure of $\Sigma\cap U$.
∎

######  Corollary 6.7.

In the setup of Proposition 6.6, in
$A^{*}_{\operatorname{GL}(r+1)\times\operatorname{GL}(2)}(\operatorname{Hom}(\mathbf{k}^{r+1},\operatorname{Sym}^{d}\mathbf{k}^{2}))$,
we have

$[\operatorname{WOrb}(s_{\eta})]=\sum_{\nu}[\operatorname{WOrb}(s_{\nu})].$

###### Proof.

Since $[\operatorname{WOrb}(s_{\eta})]=[\Sigma]$, the statement follows from
Proposition 6.6. ∎

Finally, we discuss the specialisation mentioned in the introduction (Theorem
1.5). Let $p_{1},\dots,p_{m}$ be sections $\Delta\to{\bf P}^{1}\times\Delta$
that are distinct on the generic fiber but $p_{1}=\dots=p_{\ell}$ on the
central fiber (with no other coincidences). Let $S_{\eta}$ be a linear series
on the generic fiber whose ramification points are $p_{1},\dots,p_{m}$ and
whose vanishing sequence at $p_{i}$ is $a(i)=(a_{0}(i),\dots,a_{r}(i))$. Let
$S$ be the flat limit of $S_{\eta}$. It is easy to see that the vanishing
sequence of $S$ at $p_{i}$ for $i>\ell$ is $a(i)$. Let
$c=(c_{0},\dots,c_{r+1})$ be the vanishing sequence of $S$ at
$p=p_{1}=\dots=p_{\ell}$. Let $c^{\prime}=(d-c_{r},\dots,d-c_{0})$ be the
complementary sequence.

######  Proposition 6.8.

In the setup above, there exists a linear series of rank $r$ and degree $d$ on
${\bf P}^{1}$ with ramification profile $\\{a(1),\dots,a(\ell),c^{\prime}\\}$.
Furthermore, on
$\operatorname{Hom}(\mathbf{k}^{r+1},\operatorname{Sym}^{d}\mathbf{k}^{2})$ we
have the equality of
$\operatorname{GL}(r+1)\times\operatorname{GL}(2)$-equivariant classes

$[\operatorname{WOrb}(a(1),\dots,a(m))]=[\operatorname{WOrb}(a(1),\dots,a(\ell),c^{\prime})]+[\operatorname{WOrb}(a(\ell+1),\dots,a(m),c)].$

###### Proof.

The existence of the linear series can be deduced using Schubert calculus. We
give an alternate geometric proof using the smoothing theorem.

Consider the pointed stable limit of $({\bf P}^{1}_{\eta},p_{1},\dots,p_{m})$.
The dual graph of limit, together with the labelling given by the limit linear
series, has the form

$\leavevmode\hbox to164.33pt{\vbox
to42.54pt{\pgfpicture\makeatletter\hbox{\hskip
32.52437pt\lower-31.57028pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{9.66846pt}{0.0pt}\pgfsys@curveto{9.66846pt}{5.3398pt}{5.3398pt}{9.66846pt}{0.0pt}{9.66846pt}\pgfsys@curveto{-5.3398pt}{9.66846pt}{-9.66846pt}{5.3398pt}{-9.66846pt}{0.0pt}\pgfsys@curveto{-9.66846pt}{-5.3398pt}{-5.3398pt}{-9.66846pt}{0.0pt}{-9.66846pt}\pgfsys@curveto{5.3398pt}{-9.66846pt}{9.66846pt}{-5.3398pt}{9.66846pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-3.61632pt}{-3.41666pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$T$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{{{}}}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{104.85446pt}{0.0pt}\pgfsys@curveto{104.85446pt}{2.60342pt}{102.74402pt}{4.71387pt}{100.1406pt}{4.71387pt}\pgfsys@curveto{97.53717pt}{4.71387pt}{95.42673pt}{2.60342pt}{95.42673pt}{0.0pt}\pgfsys@curveto{95.42673pt}{-2.60342pt}{97.53717pt}{-4.71387pt}{100.1406pt}{-4.71387pt}\pgfsys@curveto{102.74402pt}{-4.71387pt}{104.85446pt}{-2.60342pt}{104.85446pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{100.1406pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{100.1406pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{{{
{}}}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{}}{}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-29.19136pt}{-21.7968pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize$a(1)$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{{{}}}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.625pt}{-26.83728pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize\ldots}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{{{
{}}}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ }}{ }
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{16.5468pt}{-21.7968pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize$a(\ell)$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{ {}{}{}}
{{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{-7.42091pt}{-6.50493pt}\pgfsys@lineto{-14.84631pt}{-13.0138pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{ {}{}{}}
{{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{-9.86845pt}\pgfsys@lineto{0.0pt}{-18.40428pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{ {}{}{}}
{{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{7.3795pt}{-6.55222pt}\pgfsys@lineto{14.65704pt}{-13.0138pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{{{
{}}}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{}}{}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{61.64719pt}{-18.29337pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{\scriptsize$a(\ell+1)$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{{{}}}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{97.5156pt}{-21.88269pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize\ldots}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{{{
{}}}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ }}{ }
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{113.18396pt}{-18.29337pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize$a(m)$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{ {}{}{}}
{{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{96.0054pt}{-2.6545pt}\pgfsys@lineto{85.32564pt}{-9.51036pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{ {}{}{}}
{{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{100.1406pt}{-4.91386pt}\pgfsys@lineto{100.1406pt}{-13.44969pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{ {}{}{}}
{{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{103.9785pt}{-3.0686pt}\pgfsys@lineto{112.03514pt}{-9.51036pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{}{{{}{}}}{}\pgfsys@moveto{9.86845pt}{0.0pt}\pgfsys@lineto{95.22673pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{24.8862pt}{3.533pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{\scriptsize$c^{\prime}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{76.6407pt}{3.533pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize$c$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>
In this diagram, $T$ is a labelled tree with dangling half-edges labelled
$a(1),\dots,a(\ell)$, and one of its vertices, say $t$, is connected to the
remaining vertex, say $o$, by an edge that has the label $c^{\prime}$ near $t$
and $c$ near $o$. We apply Proposition 6.4 to $T$ together with the dangling
edges labelled $a(1),\dots,a(\ell)$ and $c^{\prime}$. A geometric general
fiber of the asserted degeneration gives a linear series of rank $r$ and
degree $d$ on ${\bf P}^{1}$ with ramification profile
$\\{a(1),\dots,a(\ell),c^{\prime}\\}$.

By applying Proposition 6.4 to the two vertex graph

$\leavevmode\hbox to155.87pt{\vbox
to37.58pt{\pgfpicture\makeatletter\hbox{\hskip
29.02094pt\lower-26.61569pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.71387pt}{0.0pt}\pgfsys@curveto{4.71387pt}{2.60342pt}{2.60342pt}{4.71387pt}{0.0pt}{4.71387pt}\pgfsys@curveto{-2.60342pt}{4.71387pt}{-4.71387pt}{2.60342pt}{-4.71387pt}{0.0pt}\pgfsys@curveto{-4.71387pt}{-2.60342pt}{-2.60342pt}{-4.71387pt}{0.0pt}{-4.71387pt}\pgfsys@curveto{2.60342pt}{-4.71387pt}{4.71387pt}{-2.60342pt}{4.71387pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{{{}}}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{99.89987pt}{0.0pt}\pgfsys@curveto{99.89987pt}{2.60342pt}{97.78943pt}{4.71387pt}{95.186pt}{4.71387pt}\pgfsys@curveto{92.58258pt}{4.71387pt}{90.47214pt}{2.60342pt}{90.47214pt}{0.0pt}\pgfsys@curveto{90.47214pt}{-2.60342pt}{92.58258pt}{-4.71387pt}{95.186pt}{-4.71387pt}\pgfsys@curveto{97.78943pt}{-4.71387pt}{99.89987pt}{-2.60342pt}{99.89987pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{95.186pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{95.186pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{{{
{}}}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{}}{}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-25.68793pt}{-18.29337pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize$a(1)$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{{{}}}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.625pt}{-21.88269pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize\ldots}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{{{
{}}}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ }}{ }
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{13.04337pt}{-18.29337pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize$a(\ell)$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{ {}{}{}}
{{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{-3.73637pt}{-3.1915pt}\pgfsys@lineto{-11.13383pt}{-9.51036pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{ {}{}{}}
{{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{0.0pt}{-4.91386pt}\pgfsys@lineto{0.0pt}{-13.44969pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{ {}{}{}}
{{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{3.71223pt}{-3.21947pt}\pgfsys@lineto{10.96613pt}{-9.51036pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{{{
{}}}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{}}{}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{56.6926pt}{-18.29337pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{\scriptsize$a(\ell+1)$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{{{{}}}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{
{}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{92.561pt}{-21.88269pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize\ldots}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{{{
{}}}}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ }}{ }
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{108.22937pt}{-18.29337pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize$a(m)$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{ {}{}{}}
{{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{91.05081pt}{-2.6545pt}\pgfsys@lineto{80.37105pt}{-9.51036pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{ {}{}{}}
{{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{95.186pt}{-4.91386pt}\pgfsys@lineto{95.186pt}{-13.44969pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{ {}{}{}}
{{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{99.02391pt}{-3.0686pt}\pgfsys@lineto{107.08055pt}{-9.51036pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{}{{{}{}}}{}\pgfsys@moveto{4.91386pt}{0.0pt}\pgfsys@lineto{90.27214pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{19.93161pt}{3.533pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{\scriptsize$c^{\prime}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{
{}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{71.68611pt}{3.533pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\scriptsize$c$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},$

we see that the graph above arises from a degeneration. We now apply Corollary
6.7 to this degeneration. ∎

## References

  * [1] P. Aluffi and C. Faber.  Linear orbits of $d$-tuples of points in $\mathbb{P}^{1}$.  J. Reine Angew. Math., 445:205–220, 1993.
  * [2] P. Aluffi and C. Faber.  Linear orbits of smooth plane curves.  J. Algebr. Geom., 2(1):155–184, 1993.
  * [3] P. Aluffi and C. Faber.  Linear orbits of arbitrary plane curves.  Michigan Math. J., 48:1–37, 2000.
  * [4] A. Berget and A. Fink.  Equivariant Chow classes of matrix orbit closures.  Transform. Groups, 22(3):631–643, 2017.
  * [5] A. S. Buch and R. Rimányi.  A formula for non-equioriented quiver orbits of type $A$.  J. Algebr. Geom., 16(3):531–546, 2007.
  * [6] A. Deopurkar, A. Patel, and D. Tseng.  A universal formula for counting cubic surfaces, 2021.  Pre-print, arXiv:2109.12672.
  * [7] D. Edidin and W. Graham.  Equivariant intersection theory (With an appendix by Angelo Vistoli: The Chow ring of ${\mathcal{M}}_{2}$).  Invent. Math., 131(3):595–644, 1998.
  * [8] D. Edidin and W. Graham.  Localization in equivariant intersection theory and the Bott residue formula.  Am. J. Math., 120(3):619–636, 1998.
  * [9] D. Eisenbud and J. Harris.  Divisors on general curves and cuspidal rational curves.  Invent. Math., 74:371–418, 1983.
  * [10] D. Eisenbud and J. Harris.  Limit linear series: Basic theory.  Invent. Math., 85:337–371, 1986.
  * [11] L. M. Fehér, R. Rimányi, and A. Weber.  Characteristic classes of orbit stratifications, the axiomatic approach.  In Schubert calculus and its applications in combinatorics and representation theory. Selected papers presented at the “International Festival in Schubert Calculus”, Guangzhou, China, November 6–10, 2017, pages 223–249. Singapore: Springer, 2020.
  * [12] M. Lee, A. Patel, H. Spink, and D. Tseng.  Orbits in $(\mathbb{P}^{r})^{n}$ and equivariant quantum cohomology.  Adv. Math., 362:79, 2020.
  * [13] M. Lee, A. Patel, and D. Tseng.  Equivariant degenerations of plane curve orbits, 2019.  Pre-print, arXiv:1903.10069.
  * [14] B. Osserman.  A limit linear series moduli scheme.  Ann. Inst. Fourier, 56(6):1165–1205, 2006.