# The local/logarithmic Correspondence and the Degeneration Formula for
Quasimaps

Alberto Cobos Rabano  School of Mathematics and Statistics, Hicks Building,
Hounsfield Rd, Sheffield S3 7RH,UK<EMAIL_ADDRESS>,  Cristina
Manolache  School of Mathematics and Statistics, Hicks Building, Hounsfield
Rd, Sheffield S3 7RH,UK<EMAIL_ADDRESS>and  Qaasim Shafi  School
of Mathematics
Watson Building
University of Birmingham
Edgbaston B15 2TT
UK<EMAIL_ADDRESS>
###### Abstract.

We study the relationship between the enumerative geometry of rational curves
in local geometries and various versions of maximal contact logarithmic curve
counts. Our approach is via quasimap theory, and we show versions of the
[vGGR19] local/logarithmic correspondence for quasimaps, and in particular for
normal crossings settings, where the Gromov-Witten theoretic formulation of
the correspondence fails. The results suggest a link between different
formulations of relative Gromov-Witten theory for simple normal crossings
divisors via the mirror map. The main results follow from a rank reduction
strategy, together with a new degeneration formula for quasimaps.

###### Contents

  1. 1 Introduction
    1. 1.1 Results
    2. 1.2 Motivation
    3. 1.3 The role of quasimaps
    4. 1.4 Quasimaps and the degeneration formula
    5. 1.5 Future directions
    6. 1.6 Acknowledgements
  2. 2 Absolute quasimaps
    1. 2.1 Quasimaps to a double point degeneration
  3. 3 Logarithmic quasimaps
    1. 3.1 The Ciocan-Fontanine–Kim obstruction theory
  4. 4 The degeneration formula
    1. 4.1 Decomposition
    2. 4.2 Gluing
  5. 5 Local/logarithmic correspondence
    1. 5.1 Smooth divisor
    2. 5.2 Simple normal crossings divisors

##  1\. Introduction

Let $X$ be a smooth projective variety and $D=D_{1}+\dots+D_{r}\subset X$ be a
simple normal crossings divisor whose irreducible components are nef. The
_local/logarithmic correspondence_ connects two enumerative theories
associated to $(X,D)$: rational curves in $X$ with maximal tangency to $D$,
and rational curves in the total space of $\mathcal{O}_{X}(-D)$. This
correspondence was conjectured by Takahashi [Tak01] and proved by Gathmann
[Gat03], for the Gromov–Witten theory of $X=\mathbb{P}^{2}$ and $D$ a smooth
cubic, and then generalised in [vGGR19] to arbitrary $X$ and $D$ any _smooth_
, nef divisor.

###  1.1. Results

We prove a quasimap local/logarithmic correspondence, for simple normal
crossings divisors, under a mild positivity assumption. Let $X=W/\\!\\!/G$ be
a GIT quotient and let $D=D_{1}+\dots+D_{r}\subset X$ be a simple normal
crossings divisor satisfying Assumptions 2.1 and 3.1.

This includes toric varieties and partial flag varieties, with any simple
normal crossings divisor, as well as complete intersections in these spaces
with divisors pulled back from the ambient space. There are two different
directions in which to generalise the local/logarithmic correspondence for
smooth divisors. When tangency is imposed with respect to all divisor
components simultaneously we prove a local/logarithmic correspondence for
arbitrary $r$.

######  Theorem A (Theorem 5.20).

Suppose $D_{1}\cap\dots\cap D_{r}\neq\emptyset$ and the components of $D$ are
very ample. Then we have the following equality of virtual classes

$[\mathcal{Q}_{0,(\underline{d},0)}^{\log}(X|D,\beta)]^{\text{\rm
vir}}=\prod_{i=1}^{r}(-1)^{d_{i}+1}\cdot\operatorname{ev}_{1}^{*}D_{i}\cap[\mathcal{Q}_{0,2}(\oplus_{i=1}^{r}\mathcal{O}_{X}(-D_{i}),\beta)]^{\text{\rm
vir}},$

where on the left hand side maximal tangency is imposed to $D_{1}\cap\dots\cap
D_{r}$ at a single point.

For $r=1$, in Gromov–Witten theory, this is the strong form of the smooth
local/logarithmic correspondence (see [TY23b, Conjecture 18]). The original
form is obtained by using the divisor equation and is the main theorem of
[vGGR19]. The authors of [vGGR19] conjectured a _different_ generalisation for
$D$ an s.n.c. divisor, where maximal tangency is imposed to the components of
the divisor at distinct points. Figure 4 exhibits the difference between the
generalisations. Although there is some numerical evidence for either
generalisation [BBvG24, BBvG22], the authors of [NR22] showed that even if
$D=D_{1}+D_{2}$, both generalisations can fail in Gromov–Witten theory. In
addition to Theorem A, we show that the strong form version of the conjecture
of [vGGR19] is true for quasimaps for $r=2$.

######  Theorem B (Theorem 5.19).

Suppose that $D=D_{1}+D_{2}$ has two components, which are very ample. Then we
have the following equality of virtual classes

$[\mathcal{Q}_{0,(d_{1},d_{2})}^{\log}(X|D_{1}+D_{2},\beta)]^{\text{\rm
vir}}=(-1)^{d_{1}+d_{2}}\cdot\prod_{i=1}^{2}\operatorname{ev}_{i}^{*}D_{i}\cap[\mathcal{Q}_{0,2}(\oplus_{i=1}^{2}\mathcal{O}_{X}(-D_{i}),\beta)]^{\text{\rm
vir}}.$

In fact, [TY23b, Conjecture 18] gives a much more general local/logarithmic
conjecture for simple normal crossings divisors. The situations in Theorems A
and B are the two extremes.

In order to prove A (as well as B), we reduce to the case where $r=1$ in
Theorems 5.21 and 5.26 by showing that, for products of projective spaces
relative toric boundary divisors, the simple normal crossings logarithmic
quasimap spaces can be obtained as fibre products of logarithmic quasimap
spaces for smooth divisors. Then, following the strategy of [vGGR19], we prove
A in the case where $r=1$ using a degeneration formula for quasimaps which we
develop.

Let $W\rightarrow\mathbb{A}^{1}$ be a toric double point degeneration. An
example is the deformation to the normal cone of a smooth projective toric
variety along a smooth toric divisor. Let $X_{0}$ denote the central fibre,
which is a union of two toric varieties $X_{1}$ and $X_{2}$ glued along a
common toric divisor $D$.

######  Theorem C (Degeneration formula for quasimaps, Theorem 4.18).

$[\mathcal{Q}_{g,n}^{\log}(X_{0},\beta)]^{\text{\rm
vir}}=\sum_{\widetilde{\Gamma}}\frac{m_{\widetilde{\Gamma}}}{|\mathrm{Aut}(\widetilde{\Gamma})|}j_{\widetilde{\Gamma},*}c_{\widetilde{\Gamma}}^{*}\,\Delta^{!}\left(\prod_{V\in
V(\widetilde{\Gamma})}[\mathcal{Q}^{\log}_{g_{V},\alpha_{V}}(X_{r(V)}|D,\beta_{V})]^{\text{\rm
vir}}\right)$

Both the proof and application of the degeneration formula for quasimaps
entail additional subtleties, when compared with stable maps. The main one is
based on the fact that quasimap spaces do not embed along closed embeddings,
see [BN21, Remark 2.3.3]. As a consequence, if $V$ is a toric subvariety of a
toric variety $X$, then quasimaps to $X$ which factor through $V$ contain more
information than quasimaps to $V$ with its toric presentation, as we
illustrate in Example 2.8. We show in Example 2.7 that this phenomenon is
relevant when studying toric double point degenerations, and it forces us to
consider non-toric presentations for the irreducible components of the special
fibre. As a result, the output of the degeneration formula, Theorem 4.18, may
contain curve classes which live on the degeneration family, rather than on
the irreducible components of the special fibre. We call such curve classes
_ghost_ classes (see Definition 2.9). For the application to the
local/logarithmic correspondence, these contributions are ruled out in
Proposition 5.17. To prove Proposition 5.17 we show that these contributions
have a trivial excess line bundle.

###  1.2. Motivation

Relative or logarithmic Gromov–Witten theory [Che14b, GS13, AC14, Li01] has
significantly shaped the landscape of enumerative geometry. The resulting
invariants (and their generalisations) have proved important in mirror
symmetry constructions [GS22], degeneration formulas [Ran22, Gro23, KLR21,
Li02, ACGS21], tropical correspondence theorems such as [MR20, Gro18, Gra22,
NS06] as well as understanding the tautological ring of the moduli space of
curves [GV05, RK23, MR23, HMP+22]. Consequently, computing logarithmic
Gromov–Witten invariants is an important problem in enumerative algebraic
geometry.

An interesting feature of the local/logarithmic correspondence is that the two
sides can be understood by qualitatively different techniques, especially when
the logarithmic divisor is genuinely simple normal crossings. Thus,
conceptually moving to one side and then using its features can be profitable.
Although there have been many numerical generalisations and variations of this
correspondence [BFGW21, vGNS24, BBvG21, BBvG24, BBvG22], the original cycle
level correspondence holds in the case where the divisor is smooth. On the
other hand, the major difficulty in computing logarithmic Gromov–Witten
invariants, at least in genus zero, lies in the case where the divisor is
genuinely simple normal crossings.

###  1.3. The role of quasimaps

There have been conjectures to generalise the local/logarithmic correspondence
to the case where the divisor is simple normal crossings, but as we have
remarked these conjectures are false in general, in Gromov–Witten theory
[NR22]. Moreover, the authors noticed [NR22, Remark 5.4] that one point of
view for the underlying reason for the failure of these conjectures was the
presence of excess components in the moduli space containing _rational tails_.
Quasimap theory [MOP11, CFK10, CFKM14] provides an alternative framework for
counting curves which excludes rational tails, and so it is conceivable that
these are no longer counterexamples in the quasimap setting.

The origins of quasimap theory are intertwined with its connections to mirror
symmetry [CFK14] and the main applications of the theory have been wall-
crossing formulas which relate Gromov–Witten invariants to quasimap invariants
[CFK14, CFK17, Zho22]. These wall-crossing formulas reduce computations in
Gromov–Witten theory to quasimap theory, which are typically easier. Indeed,
almost all results that allow us to compute genus-zero Gromov–Witten
invariants [Giv98] rely at heart on the wall-crossing formula. Recently, the
third author has developed a theory of logarithmic quasimaps [Sha24] with the
expectation of proving _logarithmic_ wall-crossing formulas, first proposed in
[BN21]. A consequence of A and B, which confirm the predictions of [NR22], is
that the techniques of ordinary Gromov–Witten/quasimap theory can be imported
over to compute logarithmic quasimap invariants in the case where the divisor
is genuinely simple normal crossings. Combining this with a conjectural wall-
crossing should be a powerful method for computing logarithmic Gromov–Witten
invariants.

###  1.4. Quasimaps and the degeneration formula

The degeneration formula for double point degenerations [Li02, Che14a, KLR21,
AF16] has been a well used tool in Gromov–Witten theory since its first
introduction to the subject [MP06, OP09, PP17]. For all the same reasons the
degeneration formula for quasimaps, C, will provide similar computational
capabilities. In this paper we use the degeneration formula to prove the
local/logarithmic correspondence for smooth pairs. There is a related
direction to this, which is the prospect of investigating holomorphic anomaly
equations in the relative setting, for which quasimaps are better suited.

In the absolute setting, for Calabi-Yau threefolds, the wall-crossing formula
coincides with the mirror map and hence quasimap invariants are exactly equal
to the B-model invariants of the mirror. This link has been exploited to give
a direct geometric proof, for local $\mathbb{P}^{2}$, of the holomorphic
anomaly equation [LP18], which gives profound structure to the Gromov–Witten
theory of a Calabi–Yau, coming from the B-model. Using the double point
degeneration formula, the authors of [BFGW21] prove a holomorphic anomaly
equation for the logarithmic Gromov–Witten theory of $\mathbb{P}^{2}$ relative
an elliptic curve. This suggests that more general holomorphic anomaly
equations in relative geometries could be pursued using the degeneration
formula for quasimaps, after generalising this beyond the toric setting.

###  1.5. Future directions

####  1.5.1. Local/logarithmic correspondence in higher genus

There are two ways in which one can generalise the original smooth
local/logarithmic correspondence. Either to increase the rank of the divisor,
as we have addressed in the present paper, or to generalise from rational
curves to higher genus curves, where on the logarithmic side one adds a
$\lambda_{g}$ class for the dimensions to match. In Gromov–Witten theory both
generalisations fail on the nose, for the higher genus situation there are
correction terms described in [BFGW21]. One can speculate about whether,
analogous to the situation for higher rank divisors, there are instances where
the correction terms vanish for the quasimap local/logarithmic correspondence
in higher genus.

####  1.5.2. Holomorphic anomaly equations

In the Gromov–Witten situation, the relationship between local and logarithmic
invariants in higher genus is used to prove a holomorphic anomaly equation for
$\mathbb{P}^{2}$ relative to an elliptic curve. Since quasimaps are better
suited to understanding holomorphic anomaly equations, a natural is to pursue
this for other geometries.

In order to accomplish this, the degeneration formula would need to be
generalised beyond the toric setting. This should be possible however. The
main difficulty in quasimap theory is the restriction of the theory to spaces
admitting a certain GIT presentation. Even the deformation to the normal cone
of a toric variety along a smooth, non-toric divisor is no longer toric. On
the other hand, one can rewrite the total space, as well as more general blow-
ups, as a GIT quotient, see, for example, [CLS22, Proposition 6.2].

####  1.5.3. Orbifold/logarithmic correspondence

The local/logarithmic correspondence concerns invariants with maximal contact
to the divisor which makes it fertile testing ground, but many of the ideas
pursued here should extend to a more general context. An alternative approach
to tangency involves replacing components of divisor with roots [Cad07], and
using the framework of orbifold Gromov–Witten theory to produce invariants
[TY23a]. There has been effort to compare the logarithmic theories with the
resulting orbifold theory [ACW17, BNR23, TY20], since the latter is
significantly easier to compute in practice. However, it computes a different
set of invariants in general, which do not immediately interact with
degeneration formulas and lacks some of the conceptual advantages of the
logarithmic theory. Furthermore, no known structure organises the difference
between the theories.

Our work here sheds some light on this difference. One can view the local
theory defined by an s.n.c. divisor as a sector of this orbifold by the
results of [BNTY23], and so our results suggest that the difference between
the computable orbifold theory and the conceptually better logarithmic theory,
might be controlled in part by mirror maps. There is a related proposal in
[You22].

Taking these issues together, it seems reasonable that orbifold and
logarithmic Gromov–Witten theory for pairs should be compared via the quasimap
theory, where at least in some sectors such as the one we study here, the
invariants coincide. The general picture appears to be more complicated, but
the significantly simplified combinatorics of quasimaps provides a clear path
to computations.

###  1.6. Acknowledgements

The suggestion of a quasimap local/logarithmic correspondence arose from
discussions with Michel van Garrel, and we would like to wholeheartedly thank
him here. We are very grateful to Yannik Schüler for his explanations on a
localisation computation in [BP05], which helped us prove Proposition 5.9. We
thank Leo Herr and Navid Nabijou for useful discussions and Dhruv Ranganathan
for discussions and very helpful comments on the draft. We thank Navid
Nabijou, Helge Ruddat and Fenglong You for their comments on a previous
version.

C.M. was partially supported by a Royal Society Dorothy Hodgkin Fellowship.
Q.S. is supported by UKRI Future Leaders Fellowship through grant number
MR/T01783X/1.

##  2\. Absolute quasimaps

Here we recall the definition of absolute GIT quasimaps from [CFKM14].

######  Assumption 2.1.

Let $Z=\operatorname{Spec}A$ be an affine algebraic variety with an action by
a reductive algebraic group $G$. Let $\theta$ be a character inducing a
linearisation for the action. We insist that

  * •

$Z^{s}=Z^{ss}$

  * •

$Z^{s}$ is nonsingular

  * •

$G$ acts freely on $Z^{s}$

  * •

$Z$ has only l.c.i singularities

Then quasimaps to $Z/\\!\\!/G$ are defined as follows.

######  Definition 2.2.

Fix non-negative integers $g,n$ and
$\beta\in\operatorname{Hom}(\operatorname{Pic}^{G}Z,\mathbb{Z})$. An
$n$-marked stable quasimap of genus $g$ and degree $\beta$ to $Z/\\!\\!/G$ is

$\left((C,p_{1},\dots,p_{n}),u\right)$

where

  1. (1)

$(C,p_{1},\dots,p_{n})$ is an $n$-marked prestable curve of genus $g$.

  2. (2)

$u:C\rightarrow[Z/G]$ of class $\beta$ (i.e. $\beta(L)=\deg_{C}u^{*}L$).

which satisfy

  1. (1)

(Non-degeneracy) there is a finite (possibly empty) set $B\subset C$, distinct
from the nodes and markings, such that $\forall c\in C\setminus B$ we have
$u(c)\in Z^{s}$.

  2. (2)

(Stability) $\omega_{C}(p_{1}+\cdots+p_{n})\otimes L_{\theta}^{\epsilon}$ is
ample $\forall\epsilon\in\mathbb{Q}_{>0}$, where $L_{\theta}$ comes from the
linearisation.

######  Theorem 2.3 ([CFKM14]).

The moduli stack of stable quasimaps $\mathcal{Q}_{g,n}(X,\beta)$ is a
Deligne–Mumford stack. If $X$ is proper then so is
$\mathcal{Q}_{g,n}(X,\beta)$. Moreover, it admits a perfect obstruction over
$\mathfrak{M}_{g,n}$ leading to a virtual fundamental class
$[\mathcal{Q}_{g,n}(X,\beta)]^{\text{\rm vir}}$.

In an earlier iteration of the theory in [CFK10], toric quasimaps are defined
for toric varieties using the description of the functor of points of a smooth
toric variety in terms of line bundle-section pairs according to rays of the
fan, proved in [Cox95]. In the general context of Definition 2.2, toric
quasimaps are recovered by taking the standard GIT presentation coming from
toric geometry.

We will prove a degeneration formula in a toric context, which means the
double point degeneration will be a _toric morphism_. However, we will still
require the more general definition of quasimaps, Definition 2.2, even in this
context, because the process of gluing a quasimap is more subtle than in the
case of maps (see the discussion and examples in Section 2.1). Specifically in
order to glue together quasimaps to the components of the central fibre we
require the GIT presentation induced by the embedding in the total space,
rather than the ordinary toric presentation.

######  Definition 2.4.

A _toric double point degeneration_ $W\rightarrow\mathbb{A}^{1}$ is a double
point degeneration with $W$ a toric variety and $W\rightarrow\mathbb{A}^{1}$ a
toric morphism. By double point degeneration we mean that
$W\rightarrow\mathbb{A}^{1}$ is a flat, proper morphism, with $W$ smooth. The
general fibre is smooth and the fibre over $0\in\mathbb{A}^{1}$ is a reduced
union of two irreducible components glued along a common smooth divisor.

######  Notation 2.5.

We denote the general fibre of a toric double point degeneration
$W\to\mathbb{A}^{1}$ by $X$. The central fibre $X_{0}$ is a union of the two
components $X_{1}$ and $X_{2}$ glued along a common divisor smooth $D$.
Necessarily the fan of $W$ has two distinguished rays which map surjectively
to the ray in the fan of $\mathbb{A}^{1}$. These rays correspond to the
divisors $X_{1}$ and $X_{2}$ of $W$, and will typically be denoted by
$\rho_{1}$ and $\rho_{2}$. All the other rays are contained in a codimension 1
sublattice, as in Figure 1. We will denote the toric GIT presentation of $W$
by $\mathbb{A}^{M}/\\!\\!/\mathbb{G}_{m}^{s}$ and the ambient stack
$[\mathbb{A}^{M}/\mathbb{G}_{m}^{s}]$ by $\mathcal{W}$.

Figure 1. A double point degeneration has two special rays
$\rho_{1},\rho_{2}$, which project to the generator of the fan of
$\mathbb{A}^{1}$ inside a line $L$. All other rays are contained in the
hyperplane $L^{\perp}$.

######  Remark 2.6.

Just as in Gromov–Witten theory one can consider quasimaps over a base. Since
this is not addressed in the literature we just make the observation that for
a double point degeneration $W\rightarrow\mathbb{A}^{1}$ we have that
$\mathcal{Q}_{g,n}(W/\mathbb{A}^{1},\beta)=\mathcal{Q}_{g,n}(W,\beta)$,
because any quasimap to $W$ necessarily lives in a fibre over
$\mathbb{A}^{1}$.

###  2.1. Quasimaps to a double point degeneration

The degeneration formula involves the study of quasimaps to a double point
degeneration $W\to\mathbb{A}^{1}$, which we assume to be toric. Let
$\mathbb{A}^{M}/\\!\\!/\mathbb{G}_{m}^{s}$ be the toric GIT presentation of
$W$. We define quasimaps to $X_{0}$ via the following Cartesian diagram.

${\mathcal{Q}_{g,n}(X_{0},\beta)}$${\mathcal{Q}_{g,n}(W,\beta)}$${0}$${\mathbb{A}^{1}}$

The degeneration formula expresses will express the virtual class of
$[\mathcal{Q}_{g,n}(X_{0},\beta)]^{\text{\rm vir}}$, defined via Gysin
pullback in the above diagram, with the virtual classes of logarithmic
quasimap spaces of $X_{1}$ and $X_{2}$ relative to $D$. For stable maps, one
has that a map to $X_{0}$ is the same as a collection of maps to $X_{i}$
gluing at the nodes. The analogue statement for quasimaps is true only if we
take the GIT presentation
$X_{i}\simeq\mathbb{A}^{M-1}/\\!\\!/\mathbb{G}_{m}^{s}$ induced by $W$, which
in general does not agree with the standard toric presentation. Example 2.7
provides a concrete example of this phenomenon.

Figure 2. The fan of $W=\operatorname{Bl}_{\infty\times
0}(\mathbb{P}^{1}\times\mathbb{A}^{1})$.

######  Example 2.7.

Consider the deformation to the normal cone of $\mathbb{P}^{1}$ relative
$\infty$, $W=\operatorname{Bl}_{\infty\times
0}(\mathbb{P}^{1}\times\mathbb{A}^{1})$. The fan of $W$, pictured in Figure 2,
induces the following GIT presentation

$W=\mathbb{A}^{4}/\\!\\!/\mathbb{G}_{m}^{2},\,\,\mathrm{where}\,\,(\lambda,\mu)\cdot(x_{0},x_{1},v_{1},v_{2})=(\lambda\mu
x_{0},\lambda x_{1},\mu v_{1},\mu^{-1}v_{2}).$

Setting $v_{1}=0$ in the presentation of $W$ induces the following
presentation of $X_{1}=\mathbb{P}^{1}$:

$X_{1}=\mathbb{A}^{3}/\\!\\!/\mathbb{G}_{m}^{2}.$

On the other hand, $\mathbb{P}^{1}$ has the following toric presentation

$\mathbb{P}^{1}=\mathbb{A}^{2}/\\!\\!/\mathbb{G}_{m}\,\,\mathrm{where}\,\,\alpha\cdot(z_{0},z_{1})=(\alpha
z_{0},\alpha z_{1}).$

There is a morphism
$f:\left[\mathbb{A}^{3}/\mathbb{G}_{m}^{2}\right]\to\left[\mathbb{A}^{2}/\mathbb{G}_{m}\right]$
given by

$\displaystyle(x_{0},x_{1},v_{2})$ $\displaystyle\mapsto(x_{0}\cdot
v_{2},x_{1})$ $\displaystyle(\lambda,\mu)$ $\displaystyle\mapsto\lambda$

and a morphism
$g:\left[\mathbb{A}^{2}/\mathbb{G}_{m}\right]\to\left[\mathbb{A}^{3}/\mathbb{G}_{m}^{2}\right]$
given by

$\displaystyle(z_{0},z_{1})$ $\displaystyle\mapsto(1,z_{1},0,z_{0})$
$\displaystyle\alpha$ $\displaystyle\mapsto(\alpha,\alpha^{-1}).$

Note that $f\circ g=\mathrm{id}$ but $g\circ f\neq\mathrm{id}$ (one must
restrict to the stable locus for $f$ and $g$ to be inverse). As a consequence,
every quasimap to $\mathbb{P}^{1}$ with the toric presentation appears as the
restriction of a quasimap to $\mathbb{P}^{1}$ with the presentation induced by
$W$, but the converse is not true.

Loosely speaking, Example 2.7 shows that a quasimap to $X_{1}$ with the
presentation induced by $W$ requires strictly more information than with the
standard presentation. For instance, the degree of a quasimap to $X_{i}$ in
the first case lies in $H_{2}(W,\mathbb{Z})$, which in general may differ from
$H_{2}(X_{i},\mathbb{Z})$.

######  Example 2.8.

For any $k\geq 0$, consider the quasimap

$\displaystyle q_{k}:\mathbb{P}^{1}$
$\displaystyle\dashrightarrow\mathbb{P}^{1}\times\mathbb{P}^{1}$
$\displaystyle[s\colon t]$ $\displaystyle\mapsto([0\colon s^{k}],[s\colon t])$

Note that $x=[0\colon 1]$ is a basepoint of $q_{k}$ for $k\geq 1$. We can
think of $q_{k}$ as giving a quasimap to $\mathbb{P}^{1}$ with the
presentation

$\mathbb{P}^{1}=\mathbb{A}^{1}\times\mathbb{A}^{2}/\\!\\!/(\mathbb{G}_{m}\times\mathbb{G}_{m})\,\,\mathrm{where}\,\,(\lambda,\mu)\cdot(x_{1},y_{0},y_{1})=(\lambda
x_{1},\mu y_{0},\mu y_{1})$

On the other hand, since $q_{k}$ factors through $[0\colon
1]\times\mathbb{P}^{1}$, there is an obvious induced quasimap to
$\mathbb{P}^{1}$ with the standard presentation given by

$\displaystyle\mathbb{P}^{1}$ $\displaystyle\dashrightarrow\mathbb{P}^{1}$
$\displaystyle[s\colon t]$ $\displaystyle\mapsto[s\colon t].$

But this is independent of $k$ and so it is impossible to recover back the
original quasimap $q_{k}$.

######  Definition 2.9.

Let $W$ be a toric variety and $\iota\colon V\hookrightarrow W$ the closure of
a torus orbit. A ghost class of $V\hookrightarrow W$ is a curve class in
$H_{2}(W,\mathbb{Z})\setminus\iota_{\ast}(H_{2}(V,\mathbb{Z}))$.

For example, the quasimap $q_{k}$ from Example 2.8 has degree $(k,1)$, which
is a ghost class unless $k=0$. The following example describes ghost classes
when the ambient $W$ is the deformation to the normal cone of a hyperplane in
projective space, which we will encounter when dealing with the degeneration
formula in Section 5.

######  Example 2.10.

Let $W(\mathbb{P}^{N},H)=\operatorname{Bl}_{H\times
0}(\mathbb{P}^{N}\times\mathbb{A}^{1})$ be the deformation to the normal cone
of a hyperplane $H$ in $\mathbb{P}^{N}$, which is a toric double point
degeneration with toric GIT presentation

$\mathbb{A}^{N+3}/\\!\\!/\mathbb{G}_{m}^{2},\,\,\mathrm{where}\,\,(\lambda,\mu)\cdot(x_{0},x_{1},\ldots,x_{N},v_{1},v_{2})=(\lambda\mu
x_{0},\lambda x_{1},\ldots,\lambda x_{N},\mu v_{1},\mu^{-1}v_{2}).$

The fan of $W(\mathbb{P}^{N},H)$ is pictured in Figure 2 for $N=1$ and in
Figure 3 for $N=2$. We compute the ghost classes of the inclusion of
$X_{1}\simeq\mathbb{P}^{N}$ in $W(\mathbb{P}^{N},H)$.

Figure 3. The fan of $W(\mathbb{P}^{2},H)=\operatorname{Bl}_{H\times
0}\mathbb{P}^{2}\times\mathbb{A}^{1}$.

Let $H_{0},\ldots,H_{N}$ be the toric divisors on $\mathbb{P}^{N}$ with
$H=H_{0}$, and let $\tilde{H}_{i}$ be the strict transform of
$H\times\mathbb{A}^{1}$. The toric divisors of $W(\mathbb{P}^{N},H)$ are:

  * •

$\widetilde{H}_{1},\ldots,\widetilde{H}_{N}$;

  * •

$\widetilde{H}$, the infinity section;

  * •

$E$, the exceptional divisor;

  * •

$\mathbb{P}^{N}$ embedded in the special fiber.

They satisfy the following relations in
$\operatorname{Pic}(W(\mathbb{P}^{N},H))$:

  * •

$[\tilde{H}_{1}]=\ldots=[\tilde{H}_{N}]=[\tilde{H}]+[E]$ and

  * •

$[E]+[\mathbb{P}^{N}]=0$.

In order to describe ghost classes of $\mathbb{P}^{N}$, we need to describe
the morphism

(1) $H_{2}(\mathbb{P}^{N},\mathbb{Z})\to
H_{2}(W(\mathbb{P}^{N},H),\mathbb{Z})$

induced by the inclusion $\mathbb{P}^{N}\hookrightarrow W(\mathbb{P}^{N},H)$
in the special fibre. On the one hand,
$H_{2}(\mathbb{P}^{N},\mathbb{Z})\simeq\mathbb{Z}\ell$, with $\ell$ the class
of a line. On the other hand,
$H_{2}(W(\mathbb{P}^{N},H),\mathbb{Z})\simeq\mathbb{Z}l\oplus\mathbb{Z}f$ with
$f$ the class of a fiber $X_{2}\to H$ and $l$ the class of a line in the
general fibre of $W(\mathbb{P}^{N},H)\to\mathbb{A}^{1}$. We fix this basis of
$H_{2}(W(\mathbb{P}^{N},H),\mathbb{Z})$ from now on. Note that a class
$(d,e)\in H_{2}(W(\mathbb{P}^{N},H),\mathbb{Z})$ is effective if and only if
$d\geq 0$ and $d+e\geq 0$.

The transpose
$\varphi\colon\operatorname{Pic}(W(\mathbb{P}^{N},H))\to\operatorname{Pic}(\mathbb{P}^{N})$
of the morphism in (1) is determined by $\varphi([\widetilde{H}_{1}])=[H_{0}]$
and $\varphi([\mathbb{P}^{N}])=-[H_{0}]$. Combining this with the fact that
the basis $\\{[\widetilde{H}_{1}],[\mathbb{P}^{N}]\\}$ of
$\operatorname{Pic}(W(\mathbb{P}^{N},H))$ and $\\{l,f\\}$ of
$H_{2}(W(\mathbb{P}^{N},H))$ are dual, we conclude that

$\displaystyle H_{2}(\mathbb{P}^{N},\mathbb{Z})$ $\displaystyle\to
H_{2}(W(\mathbb{P}^{N},H),\mathbb{Z})$ $\displaystyle\ell$
$\displaystyle\mapsto(1,-1).$

This means that an effective curve class $(d,e)\in
H_{2}(W(\mathbb{P}^{N},H),\mathbb{Z})$ is ghost for $\mathbb{P}\hookrightarrow
W(\mathbb{P}^{N},H)$ if and only if $d\neq-e$.

##  3\. Logarithmic quasimaps

In this section we will introduce logarithmic quasimaps to a toric double
point degeneration analogous to the logarithmic quasimaps of [Sha24]. We show
that this moduli space admits a virtual fundamental class using the
construction of [BF97]. First, we recall the main theorem of [Sha24] in the
specific case where the divisor is smooth.

######  Assumption 3.1.

Let $X=Z/\\!\\!/G$ be a GIT quotient and $D\subset X$ a simple normal
crossings divisor. In addition to the requirements of the previous section we
impose that $Z/\\!\\!/G$ is a subvariety of a vector space quotient
$V/\\!\\!/G$ from which $D$ is pulled back.

Suppose that $D$ is a smooth divisor and let
$\mathscr{A}=[\mathbb{A}^{1}/\mathbb{G}_{m}]$.

######  Theorem 3.2 ([Sha24, Theorems 0.1 $\&$ 0.2]).

Let, $g,n\in\mathbb{N}$, let $\beta$ a quasimap degree and let
$\alpha\in\mathbb{N}^{n}$ such that $\sum_{i=1}^{n}\alpha_{i}=D\cdot\beta$.
The moduli space $\mathcal{Q}^{\log}_{g,\alpha}(X|D,\beta)$ parametrising
logarithmic quasimaps to $(X,D)$ is a proper Deligne–Mumford stack. If $u$ is
the universal map, and $\pi$ is the projection from the universal curve, then
the complex $(\mathbf{R}\pi_{*}u^{*}\mathbb{T}^{\log}_{\mathcal{X}})^{\vee}$
is a perfect obstruction theory for the morphism
$\mathcal{Q}^{\log}_{g,\alpha}(X|D,\beta)\to\mathfrak{M}^{\log}_{g,\alpha}(\mathscr{A})$.

Now we will move to the case of a double point degneration, in the toric
setting.

Let $W\rightarrow\mathbb{A}^{1}$ be a toric double point degeneration.
Equipping $W$ (resp. $\mathbb{A}^{1}$) with the divisorial logarithmic
structure with respect to the central fibre (resp. the origin) makes
$W\rightarrow\mathbb{A}^{1}$ into a logarithmically smooth morphism. The Artin
fans [AW18] of the respective targets are
$\mathscr{A}^{2}=[\mathbb{A}^{2}/\mathbb{G}_{m}^{2}]$ and
$\mathscr{A}=[\mathbb{A}^{1}/\mathbb{G}_{m}]$. Consequently, we have a diagram

(2) ${W}$${\mathscr{A}^{2}}$${\mathbb{A}^{1}}$${\mathscr{A}}$

This diagram is not Cartesian, but $W$ maps to the fibre product
$\mathscr{A}^{\prime}=\mathscr{A}^{2}\times_{\mathscr{A}}\mathbb{A}^{1}$,
which lives over $\mathbb{A}^{1}$.

As in Notation 2.5, $W\subset\mathcal{W}$, the ambient quotient stack. It
follows that $W\rightarrow\mathscr{A}^{2}$ factors through $\mathcal{W}$.

######  Definition 3.3.

Let $C\rightarrow\mathcal{W}$ be a quasimap to $W$ (over $\mathbb{A}^{1}$).
Define the _induced map_ $C\rightarrow\mathscr{A}^{\prime}$ via the
composition

$C\rightarrow\mathcal{W}\rightarrow\mathscr{A}^{\prime}.$

The morphism $\mathscr{A}^{\prime}\rightarrow\mathbb{A}^{1}$ is a logarithmic
morphism. Write
$\mathfrak{M}^{\log}_{g,n}(\mathscr{A}^{\prime}/\mathbb{A}^{1})$ for the
moduli space of minimal prestable logarithmic maps to
$\mathscr{A}^{\prime}/\mathbb{A}^{1}$.

######  Definition 3.4.

Let $g,n$ be non-negative integers, let $\beta$ be an effective curve class on
$W$. Define $\mathcal{Q}^{\log}_{g,n}(W/\mathbb{A}^{1},\beta)$ as the fibre
product of stacks

${\mathcal{Q}^{\log}_{g,n}(W/\mathbb{A}^{1},\beta)}$${{\mathfrak{M}_{g,n}^{\log}(\mathscr{A}^{\prime}/\mathbb{A}^{1})}}$${\mathcal{Q}_{g,n}(W/\mathbb{A}^{1},\beta)}$${\mathfrak{M}_{g,n}(\mathscr{A}^{\prime}/\mathbb{A}^{1})}$$\scriptstyle{\epsilon_{2}}$$\scriptstyle{\epsilon_{1}}$

Here $\mathfrak{M}_{g,n}(\mathscr{A}^{\prime}/\mathbb{A}^{1})$ is the stack of
maps from $n$-marked, genus $g$ prestable curves to $\mathscr{A}^{\prime}$
over $\mathbb{A}^{1}$. The morphism $\epsilon_{1}$ is given by associating to
a stable quasimap the induced map to $\mathscr{A}^{\prime}$, and
$\epsilon_{2}$ is given by forgetting the logarithmic structure.

######  Lemma 3.5.

The moduli space $\mathcal{Q}^{\log}_{g,n}(W/\mathbb{A}^{1},\beta)$ is a
Deligne–Mumford stack and
$\mathcal{Q}^{\log}_{g,n}(W/\mathbb{A}^{1},\beta)\rightarrow\mathcal{Q}_{g,n}(W/\mathbb{A}^{1},\beta)$
is proper.

###### Proof.

Follows from the fact that $\epsilon_{2}$ is representable and proper (see
[Sha24, 2.6-2.8]). ∎

Next we show that $\mathcal{Q}^{\log}_{g,n}(W/\mathbb{A}^{1})$ admits a
perfect obstruction theory over
$\mathfrak{M}^{\log}_{g,n}(\mathscr{A}^{\prime}/\mathbb{A}^{1})$ following a
similar argument to [Sha24, Section 4]. Note that we have a diagram

${\mathcal{C}_{\mathcal{Q}}}$${\mathcal{C}_{\mathfrak{M}}}$${{\mathcal{W}}}$${\mathscr{A}^{\prime}}$${\mathcal{Q}^{\log}_{g,n}(W/\mathbb{A}^{1},\beta)}$${\mathfrak{M}^{\log}_{g,n}(\mathscr{A}^{\prime}/\mathbb{A}^{1})}$${\mathbb{A}^{1}}$$\scriptstyle{\pi}$$\scriptstyle{u}$

######  Lemma 3.6.

There is a morphism in the derived category of
$\mathcal{Q}^{\log}_{g,n}(W/\mathbb{A}^{1},\beta)$

$\phi:\mathbf{R}\pi_{*}(\mathbf{L}u^{*}\mathbb{L}^{\log}_{\mathcal{W}/\mathbb{A}^{1}}\otimes\omega_{\pi})\rightarrow\mathbb{L}_{\mathcal{Q}^{\log}_{g,n}(W/\mathbb{A}^{1},\beta)/\mathfrak{M}_{g,n}^{\log}(\mathscr{A}^{\prime}/\mathbb{A}^{1})}.$

###### Proof.

We have that
$\mathbb{L}^{\log}_{\mathcal{W}/\mathbb{A}^{1}}\cong\mathbb{L}_{\mathcal{W}/\mathscr{A}^{\prime}}$
by [Ols03, Corollary 5.25] and so there is a morphism

$\mathbf{L}u^{*}\mathbb{L}^{\log}_{\mathcal{W}/\mathbb{A}^{1}}\rightarrow\mathbb{L}_{\mathcal{C}_{\mathcal{Q}/\mathcal{C}_{\mathfrak{M}}}}\cong\mathbf{L}\pi^{*}\mathbb{L}_{\mathcal{Q}/\mathfrak{M}}.$

By tensoring with the dualising sheaf and applying $\mathbf{R}\pi_{*}$ we get

$\phi:\mathbf{R}\pi_{*}(\mathbf{L}u^{*}\mathbb{L}^{\log}_{\mathcal{W}/\mathbb{A}^{1}}\otimes\omega_{\pi})\rightarrow\mathbb{L}_{\mathcal{Q}/\mathfrak{M}}.\qed$

######  Proposition 3.7.

The morphism $\phi$ defines an obstruction theory and
$\mathbf{R}\pi_{*}(\mathbf{L}u^{*}\mathbb{L}^{\log}_{\mathcal{W}/\mathbb{A}^{1}}\otimes\omega_{\pi})$
is of perfect amplitude contained in $[-1,0]$.

###### Proof.

The proof follows the argument of [Sha24, Proposition 4.5]. The only change to
the argument is that we are working over the base $\mathbb{A}^{1}$ and so
[Ols03, Theorem 5.9] instead gets applied to the following diagram

${\mathcal{C}_{T}}$${\mathcal{C}_{\bar{T}}}$${\mathcal{W}\times
T}$${\mathcal{W}\times\bar{T}}$${\mathbb{A}^{1}\times
T}$${\mathbb{A}^{1}\times\bar{T}}$

Let $\mathbb{T}^{\log}_{\mathcal{W}/\mathbb{A}^{1}}$ denote the derived dual
of $\mathbb{L}_{\mathcal{W}/\mathbb{A}^{1}}^{\log}$.

To prove that $\phi$ defines a perfect obstruction theory we use an argument
similar to [Sha24, Proposition 4.7]. More precisely, by Corollary 3.9 and
generic non-degeneracy on the space of quasimaps, we have that the complex
$\mathbf{L}u^{*}\mathbb{T}^{\log}_{\mathcal{W}/\mathbb{A}^{1}}$ has cohomology
supported in $0$. This shows that the complex
$\mathbf{R}\pi_{*}\mathbf{L}u^{*}\mathbb{T}^{\log}_{\mathcal{W}/\mathbb{A}^{1}}\cong(\mathbf{R}\pi_{*}(\mathbf{L}u^{*}\mathbb{L}^{\log}_{\mathcal{W}/\mathbb{A}^{1}}\otimes\omega_{\pi}))^{\vee}$
is of perfect amplitude in $[0,1]$. ∎

Corollary 3.9 will follow from Proposition 3.8 below.

######  Proposition 3.8.

Let $W\to\mathbb{A}^{1}$ be a toric double point degeneration and let
$\mathcal{W}=[\mathbb{A}^{M}/\mathbb{G}_{m}^{s}]$ be the ambient stack. Let
$\\{D_{\rho_{i}}\\}_{i=1}^{M}$ denote the toric divisors on $W$ with
$X_{1}=D_{\rho_{1}}$ and $X_{2}=D_{\rho_{2}}$. Then the logarithmic tangent
complex of $\mathcal{W}$ relative to $\mathbb{A}^{1}$ is quasi-isomorphic to
the following complex supported in $[-1,1]$

(3) $\mathbb{T}_{\mathcal{W}/\mathbb{A}^{1}}^{\log}=[\leavevmode\hbox
to210.1pt{\vbox to19.29pt{\pgfpicture\makeatletter\hbox{\hskip
105.05034pt\lower-9.64304pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-105.05034pt}{-9.64304pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
11.07222pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.76668pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\mathcal{O}_{\mathcal{W}}^{\oplus s}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
11.07222pt\hfil&\hfil\hskip 82.9059pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-54.60039pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\oplus_{\rho\neq\rho_{1},\rho_{2}\in\Sigma_{W}(1)}\mathcal{O}_{\mathcal{W}}(D_{\rho})\oplus_{i=1}^{2}\mathcal{O}_{\mathcal{W}}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
58.90593pt\hfil&\hfil\hskip 35.07219pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.76668pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\mathcal{O}_{\mathcal{W}}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
11.07222pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} {}{ {}{}{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{-82.7059pt}{-7.14304pt}\pgfsys@lineto{-61.10582pt}{-7.14304pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-61.10582pt}{-7.14304pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-73.9962pt}{-3.42917pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{f_{1}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{ {}{}{}}{}{ {}{}{}}
{{{{{}}{ {}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{59.10593pt}{-7.14304pt}\pgfsys@lineto{80.70601pt}{-7.14304pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{80.70601pt}{-7.14304pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{67.81563pt}{-3.42917pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{f_{2}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}],$

where $f_{2}$ maps the generator of each $\mathcal{O}_{\mathcal{W}}(D_{\rho})$
to 0 and is the identity on each of the trivial factors.

###### Proof.

Recall that we have a diagram

(4)
${\mathcal{W}}$${\mathscr{A}^{\prime}}$${\mathscr{A}^{2}}$${\mathbb{A}^{1}}$${\mathscr{A}}$$\scriptstyle{\epsilon}$$\scriptstyle{\alpha}$

The logarithmic tangent complex of $\mathcal{W}$ is equivalently, the relative
tangent complex $\mathbb{T}_{\mathcal{W}/\mathscr{A}^{\prime}}$. The top row
of diagram (4) induces an exact triangle

$\mathbb{T}_{\mathcal{W}/\mathscr{A}^{\prime}}\to\mathbb{T}_{\mathcal{W}/\mathscr{A}^{2}}\to\mathbf{L}\epsilon^{*}\mathbb{T}_{\mathscr{A}^{\prime}/\mathscr{A}^{2}}\to.$

Since $\mathscr{A}^{2}\to\mathscr{A}$ is flat,
$\mathbb{T}_{\mathscr{A}^{\prime}/\mathscr{A}^{2}}=\alpha^{\ast}\mathbb{T}_{\mathbb{A}^{1}/\mathscr{A}}$
is quasi-isomorphic to $[\mathcal{O}]$.

On the other hand, we can compute $\mathbb{T}_{\mathcal{W}/\mathscr{A}^{2}}$
from the exact triangle

$\mathbb{T}_{\mathcal{W}/\mathscr{A}^{2}}\to\mathbb{T}_{\mathcal{W}}\to\mathbb{T}_{\mathscr{A}^{2}}.$

The complex $\mathbb{T}_{\mathcal{W}}$ is quasi-isomorphic to

(5) $[\mathcal{O}^{\oplus
s}\to\oplus_{\rho\in\Sigma_{W}(1)}\mathcal{O}(D_{\rho})]$

induced by the weight matrix of the action of $\mathbb{G}_{m}^{s}$ on
$\mathbb{A}^{M}$, which can be made explicit as follows. Using that $W$ is a
toric double point degeneration with special fibre
$D_{\rho_{1}}+D_{\rho_{2}}$, we can find a basis of $\operatorname{Pic}(W)$ of
the form $\\{D_{\rho_{1}},D_{\rho_{i_{2}}},\ldots,D_{\rho_{i_{s}}}\\}$ with
$i_{j}\notin\\{1,2\\}$. This choice of basis for $\operatorname{Pic}(W)$
induces a choice of generators $\\{\lambda_{1},\ldots,\lambda_{s}\\}$ of
$\operatorname{Hom}(\operatorname{Pic}(W),\mathbb{Z})\otimes_{\mathbb{Z}}\mathbb{C}^{*}\simeq\mathbb{G}_{m}^{s}$.

With this choice of generators, the action has expression

$\lambda_{i}\cdot(x_{1},\ldots,x_{M})=(\lambda_{i}^{a_{i,1}}x_{1},\ldots,\lambda_{i}^{a_{i,M}}x_{M})$

such that the weight matrix $(a_{i,j})_{1\leq i\leq s,1\leq j\leq M}$ for the
action of $\mathbb{G}_{m}^{s}$ in this basis satisfies that $a_{1,1}=1$,
$a_{1,2}=-1$ and $a_{i,1}=a_{i,2}=0$ for each $2\leq i\leq s$. Then the
morphism in eq. 5 is given by

$1_{i}\mapsto(y_{i,j})_{j=1}^{M}\text{ with
}y_{i,j}=\partial_{\lambda_{i}}(\lambda_{i}^{a_{i,j}}x_{j})\mid_{\lambda_{i}=1}=\begin{cases}0&\text{if
}a_{i,j}=0\\\ a_{i,j}x_{j}&\text{if }a_{i,j}\neq 0\end{cases}$

Using this presentation, we get that
$\mathbb{T}_{\mathcal{W}/\mathscr{A}^{2}}$ is quasi-isomorphic to the complex

$[\mathcal{O}^{\oplus
s}\to\oplus_{\rho\neq\rho_{1},\rho_{2}}\mathcal{O}(D_{\rho})\oplus\mathcal{O}^{\oplus
2}]$

with morphism

(6) $1_{i}\mapsto\begin{cases}(-y_{1,j})_{j=1}^{M}\oplus(-1,1)&\text{if
}i=1\\\ (-y_{i,j})_{j=1}^{M}\oplus(0,0)&\text{if }i\neq 1\end{cases}$

The result is obtained by computing the cone of
$\mathbb{T}_{\mathcal{W}/\mathscr{A}^{2}}\to\mathbb{T}_{\mathscr{A}^{\prime}/\mathscr{A}^{2}}$.
∎

######  Corollary 3.9.

The complex $\mathbf{L}u^{*}\mathbb{T}^{\log}_{\mathcal{W}/\mathbb{A}^{1}}$ is
a perfect complex in $[-1,0]$ and it has cohomology supported in degree $0$.

######  Remark 3.10.

The moduli space $\mathcal{Q}^{\log}_{g,n}(W/\mathbb{A}^{1},\beta)$ lives over
$\mathbb{A}^{1}$. We denote the central fibre by
$\mathcal{Q}^{\log}_{g,n}(X_{0},\beta)$, where $X_{0}$ is the central fibre of
$W$. Let $i_{0}:0\hookrightarrow\mathbb{A}^{1}$ be the natural embedding.
There is an induced pullback perfect obstruction theory on
$\mathcal{Q}^{\log}_{g,n}(X_{0},\beta)$ where the associated virtual class
coincides with
$i_{0}^{!}[\mathcal{Q}^{\log}_{g,n}(W/\mathbb{A}^{1},\beta)]^{\text{\rm
vir}}$, by functoriality of virtual pullbacks [Man12, Theorem 4.8].

###  3.1. The Ciocan-Fontanine–Kim obstruction theory

In the following we define an analogue of the obstruction theory given in
[CFK10, Section 5.3] for the situation of Theorem 3.2 as well as of
Proposition 3.7.

######  Definition 3.11.

Let $g,n,d$ be non-negative integers and let $\alpha\in\mathbb{N}^{n}$. Let
$\operatorname{\mathfrak{Pic}}_{g,\alpha,d}^{\log}(\mathscr{A})$ be defined to
be the fibre product

${\operatorname{\mathfrak{Pic}}_{g,\alpha,d}^{\log}(\mathscr{A})}$${{\operatorname{\mathfrak{Pic}}_{g,n,d}}}$${{\mathfrak{M}_{g,\alpha}^{\log}(\mathscr{A})}}$${{\mathfrak{M}_{g,n}}}$

where $\operatorname{\mathfrak{Pic}}_{g,n,d}$ is the Picard stack over
$\mathfrak{M}_{g,n}$ of degree $d$ line bundles.

######  Definition 3.12.

Let $X$ be a smooth projective toric variety with GIT presentation
$\mathbb{A}^{M}/\\!\\!/\mathbb{G}_{m}^{s}$ induced by the fan $\Sigma$. Let
$D_{i}$ denote the divisor induced by the character of $\mathbb{G}_{m}^{s}$
given by the $i^{\mathrm{th}}$ projection and let $d_{i}=\beta\cdot D_{i}$.

We define

(7)
$\operatorname{\mathfrak{Pic}}^{\log,s}_{g,\alpha,\underline{d}}(\mathscr{A}):=\operatorname{\mathfrak{Pic}}^{\log}_{g,n,d_{1}}(\mathscr{A})\times_{\mathfrak{M}^{\log}_{g,n}(\mathscr{A})}\ldots\times_{\mathfrak{M}^{\log}_{g,n}(\mathscr{A})}\operatorname{\mathfrak{Pic}}^{\log}_{g,n,d_{s}}(\mathscr{A}).$

Let
$q:\operatorname{\mathfrak{Pic}}^{\log,s}_{g,\alpha,\underline{d}}(\mathscr{A})\to\mathfrak{M}^{\log}_{g,\alpha}(\mathscr{A})$
denote the natural projection.

######  Proposition 3.13.

Let $\rho_{0}\in\Sigma(1)$, and let $D_{\rho_{0}}$ denote the associated toric
divisor. For $\rho\in\Sigma(1)$ let $\mathcal{O}_{\mathcal{X}}(D_{\rho})$
denote the associated line bundle on
$\mathcal{X}=[\mathbb{A}^{M}/\mathbb{G}_{m}^{s}]$. Then

$\mathbb{E}_{X|D_{\rho_{0}}}:=\oplus_{\rho\neq\rho_{0}\in\Sigma(1)}\mathbf{R}\pi_{*}u^{*}\mathcal{O}_{\mathcal{X}}(D_{\rho})\oplus\mathbf{R}\pi_{*}\mathcal{O}_{\mathcal{C}}$

is a dual obstruction theory for
$\nu:\mathcal{Q}^{\log}_{g,\alpha}(X|D_{\rho_{0}},\beta)\to\operatorname{\mathfrak{Pic}}^{\log,s}_{g,\alpha,\underline{d}}(\mathscr{A})$.
Moreover, this obstruction theory induces the same class as the one in Theorem
3.2.

###### Proof.

The logarithmic Euler sequence [Sha24, Equation 13] induces a distinguished
triangle

$\mathcal{O}_{\mathcal{X}}^{\oplus
s}\to\oplus_{\rho\neq\rho_{0}}\mathcal{O}_{\mathcal{X}}(D_{\rho})\oplus\mathcal{O}_{\mathcal{X}}\to\mathbb{T}_{\mathcal{X}}^{\log}\to\mathcal{O}_{\mathcal{X}}^{\oplus
s}[1].$

Taking the pullback via the universal map and then pushing down to the moduli
space gives

$\mathbf{R}\pi_{*}u^{*}\mathcal{O}_{\mathcal{X}}^{\oplus
s}\to\oplus_{\rho\neq\rho_{0}}\mathbf{R}\pi_{*}u^{*}\mathcal{O}_{\mathcal{X}}(D_{\rho})\oplus\mathbf{R}\pi_{*}\mathcal{O}_{\mathcal{C}}\to\mathbf{R}\pi_{*}\mathbf{L}u^{*}\mathbb{T}_{\mathcal{X}}^{\log}.$

We see that we have a commutative diagram

$\textstyle{\mathbb{T}_{\mathcal{Q}^{\log}_{g,\alpha}(X|D_{\rho_{0}},\beta)/\mathfrak{M}^{\log}_{g,\alpha}(\mathscr{A})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\nu^{*}\mathbb{T}_{\operatorname{\mathfrak{Pic}}^{\log,s}_{g,\alpha,\underline{d}}(\mathscr{A})/\mathfrak{M}^{\log}_{g,n}(\mathscr{A})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{R}\pi_{*}\mathbf{L}u^{*}\mathbb{T}_{\mathcal{X}}^{\log}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{R}\pi_{*}\mathcal{O}_{\mathcal{C}}^{\oplus
s}[1]}$

This induces a morphism of distinguished triangles

$\textstyle{\mathbb{T}_{\mathcal{Q}^{\log}_{g,\alpha}(X|D_{\rho_{0}},\beta)/\mathfrak{M}^{\log}_{g,\alpha}(\mathscr{A})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\nu^{*}\mathbb{T}_{\operatorname{\mathfrak{Pic}}^{\log,s}_{g,\alpha,\underline{d}}(\mathscr{A})/\mathfrak{M}^{\log,s}_{g,\alpha}(\mathscr{A})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{T}_{\mathcal{Q}^{\log}_{g,\alpha}(X|D_{\rho_{0}},\beta)/\operatorname{\mathfrak{Pic}}^{\log,s}_{g,\alpha,\underline{d}}(\mathscr{A})}[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi}$$\textstyle{\mathbf{R}\pi_{*}\mathbb{T}_{\mathcal{X}}^{\log}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{R}\pi_{*}\mathcal{O}_{\mathcal{C}}^{\oplus
s}[1]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbf{R}\pi_{*}\oplus_{\rho\neq\rho_{0}}u^{*}\mathcal{O}_{\mathcal{X}}(D_{\rho})\oplus\mathbf{R}\pi_{*}\mathcal{O}_{\mathcal{C}}[1]}$

Applying the four lemma to the long exact sequence in cohomology proves that
$\psi$ is a (dual) obstruction theory.

By functoriality of virtual pull-backs applied to the composition

${\mathcal{Q}^{\log}_{g,\alpha}(X|D_{\rho_{0}},\beta)}$${\operatorname{\mathfrak{Pic}}^{\log,s}_{g,\alpha,\underline{d}}(\mathscr{A})}$${\mathfrak{M}^{\log}_{g,\alpha}(\mathscr{A})}$$\scriptstyle{\nu}$

the result follows. ∎

Consider the stack
$\operatorname{\mathfrak{Pic}}^{\log,s}_{g,n,\underline{d}}(\mathscr{A}^{\prime})$
obtained by replacing $\mathscr{A}$ by $\mathscr{A}^{\prime}$ in Definition
3.11 and taking $X$ to be a double point degeneration in Definition 3.12. The
following result has a proof analogous to that of Proposition 3.13.

######  Proposition 3.14.

Let $W\to\mathbb{A}^{1}$ be a toric double point degeneration. Let
$\rho_{1},\rho_{2}\in\Sigma_{W}(1)$ correspond to the pieces of the central
fibre as in 2.5. The complex

$\mathbf{R}\pi_{\ast}\mathbf{L}u^{\ast}[\leavevmode\hbox to163.96pt{\vbox
to18.46pt{\pgfpicture\makeatletter\hbox{\hskip 81.97813pt\lower-9.22859pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{}{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{\offinterlineskip{}{}{{{}}{{}}}{{{}}}{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-81.97813pt}{-9.22859pt}\pgfsys@invoke{
}\hbox{\vbox{\halign{\pgf@matrix@init@row\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding&&\pgf@matrix@step@column{\pgf@matrix@startcell#\pgf@matrix@endcell}&#\pgf@matrix@padding\cr\hfil\hskip
58.90593pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-54.60039pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\oplus_{\rho\neq\rho_{1},\rho_{2}\in\Sigma_{W}(1)}\mathcal{O}_{\mathcal{W}}(D_{\rho})\oplus_{i=1}^{2}\mathcal{O}_{\mathcal{W}}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}}}&\hskip
58.90593pt\hfil&\hfil\hskip 35.07219pt\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.76668pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{${\mathcal{O}_{\mathcal{W}}}$} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}&\hskip
11.07222pt\hfil\cr}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}}{{{{}}}{{}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} {}{ {}{}{}}{}{ {}{}{}} {{{{{}}{
{}{}}{}{}{{}{}}}}}{}{{{{{}}{
{}{}}{}{}{{}{}}}}}{{}}{}{}{}{}{}{{{}{}}}{}{{}}{}{}{}{{{}{}}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.39998pt}\pgfsys@invoke{
}{}{}{}{}{{}}{}{}{{}}\pgfsys@moveto{36.03372pt}{-6.72859pt}\pgfsys@lineto{57.6338pt}{-6.72859pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{57.6338pt}{-6.72859pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} {
}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{44.74342pt}{-3.01472pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$\scriptstyle{f_{2}}$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}]$

is a dual obstruction theory for
$\mathcal{Q}^{\log}_{g,n}(W/\mathbb{A}^{1},\beta)\to\operatorname{\mathfrak{Pic}}^{\log,s}_{g,n,\underline{d}}(\mathscr{A}^{\prime})$.
Moreover, this obstruction theory induces the same class on
$\mathcal{Q}^{\log}_{g,n}(W/\mathbb{A}^{1},\beta)$ as the one constructed in
Proposition 3.7.

##  4\. The degeneration formula

In this section we prove the degeneration formula for quasimaps to a toric
double point degeneration. We do this in two steps, decomposition and gluing.
The steps follow [ACGS20] and then the arguments of [KLR21].

First we need to verify that the general fibre of
$\mathcal{Q}^{\log}_{g,n}(W/\mathbb{A}^{1},\beta)$ recovers quasimaps to the
general fibre of $W\rightarrow\mathbb{A}^{1}$.

######  Lemma 4.1.

Let $W\rightarrow\mathbb{A}^{1}$ be a toric double point degeneration, let
$t\in\mathbb{A}^{1}$ with $t\neq 0$ and let $\iota_{t}\colon
t\to\mathbb{A}^{1}$ be the corresponding embedding. Then the following diagram
is Cartesian

${{\mathcal{Q}_{g,n}(X,\beta)}}$${{\mathcal{Q}_{g,n}(W/\mathbb{A}^{1},\beta)}}$${t}$${\mathbb{A}^{1}}$$\scriptstyle{\iota_{t}}$

###### Proof.

A quasimap to $W$ consists of line-bundle section pairs $(L_{\rho},s_{\rho})$
for $\rho\in\Sigma_{W}(1)$. Let $\rho_{1},\rho_{2}$ correspond to the pieces
of the central fibre. An element of
$\mathcal{Q}_{g,n}(W/\mathbb{A}^{1},\beta)\times_{\mathbb{A}^{1}}t$ is an
element of $\mathcal{Q}_{g,n}(W/\mathbb{A}^{1},\beta)$ such that
$s_{\rho_{1}}\otimes s_{\rho_{2}}$ is constant and equal to $t$, under the
trivialisation of $L_{\rho_{1}}\otimes L_{\rho_{2}}$. This is enough to
determine the pairs $(L_{\rho_{1}},s_{\rho_{1}})$ and
$(L_{\rho_{2}},s_{\rho_{2}})$ up to a $\mathbb{G}_{m}$-action. The remaining
data is precisely the data of a quasimap to $X$. ∎

######  Lemma 4.2.

Let $W\rightarrow\mathbb{A}^{1}$ be a toric double point degeneration, let
$t\in\mathbb{A}^{1}$ with $t\neq 0$ and let $\iota_{t}\colon
t\to\mathbb{A}^{1}$ be the corresponding embedding. Then the following diagram
is Cartesian

${{\mathcal{Q}_{g,n}(X,\beta)}}$${{\mathcal{Q}_{g,n}^{\log}(W/\mathbb{A}^{1},\beta)}}$${t}$${\mathbb{A}^{1}}$$\scriptstyle{i_{t}}$

Moreover, we have that
$i_{t}^{!}[\mathcal{Q}^{\log}_{g,n}(W/\mathbb{A}^{1},\beta)]^{\text{\rm
vir}}=[\mathcal{Q}_{g,n}(X,\beta)]^{\text{\rm vir}}$

###### Proof.

There is a Cartesian diagram

${\mathbb{G}_{m}}$${\mathscr{A}^{\prime}}$${\mathbb{G}_{m}}$${\mathbb{A}^{1}}$

Since the logarithmic structure on $\mathbb{G}_{m}$ is trivial this induces a
Cartesian diagram

${{\mathfrak{M}_{g,n}\times\mathbb{G}_{m}}}$${{\mathfrak{M}_{g,n}^{\log}(\mathscr{A}^{\prime}/\mathbb{A}^{1})}}$${\mathbb{G}_{m}}$${\mathbb{A}^{1}}$

Combining Lemma 4.1 with the definition of
$\mathcal{Q}^{\log}_{g,n}(W/\mathbb{A}^{1},\beta)$ implies the first claim.
The second claim follows from the fact that the restriction of
$\mathbb{T}^{\log}_{\mathcal{W}/\mathbb{A}^{1}}$ to any non-zero fibre is
$\mathbb{T}_{\mathcal{X}}$ which follows from [Ols03, 1.1 (iv)] applied to the
diagram (in the logarithmic category)

${\mathcal{X}\times\mathbb{G}_{m}}$${{\mathcal{W}}}$${\mathbb{G}_{m}}$${\mathbb{A}^{1}.}$

∎

###  4.1. Decomposition

Lemma 4.2 tells us that the virtual class of
$\mathcal{Q}^{\log}_{g,n}(W/\mathbb{A}^{1},\beta)$ restricts to
$[\mathcal{Q}_{g,n}(X,\beta)]^{\text{\rm vir}}$ on any non-zero fibre. By
deformation invariance and Remark 3.10, this can be related to
$[\mathcal{Q}_{g,n}^{\log}(X_{0},\beta)]^{\text{\rm
vir}}={i_{0}}^{!}[\mathcal{Q}^{\log}_{g,n}(W/\mathbb{A}^{1},\beta)]^{\text{\rm
vir}}$. Now we use the decomposition formula of [ACGS20] to decompose this
class into a sum of classes coming from combinatorics of how the components of
the curve behave.

######  Definition 4.3.

Fix non-negative integers $g,n$. A _decorated bipartite graph_ , is the
following data:

  * •

a connected graph $\Gamma$ with vertex set $V(\Gamma)$ and edge set
$E(\Gamma)$;

  * •

a function $r:V(\Gamma)\rightarrow\\{1,2\\}$;

  * •

for each vertex $V$, a choice $(n_{V},g_{V})$ where $g_{V}\geq 0$ and
$n_{V}\subset\\{1,\dots,n\\}$;

  * •

for each edge $E\in E(\Gamma)$, a positive integer $w_{E}\geq 0$, called the
_weight_ ;

subject to the following conditions:

  1. (1)

for an edge connecting two vertices $V_{1}$ and $V_{2}$, $r(V_{1})\neq
r(V_{2})$;

  2. (2)

$\cup_{V}n_{V}=\\{1,\dots,n\\}$ and $\cap_{V}n_{V}=\emptyset$;

  3. (3)

$\sum_{V}g_{V}+g(\Gamma)=g$.

Abusing notation, we will refer to such a decorated bipartite graph by
$\Gamma$.

Recall that the perfect obstruction theory on
$\mathcal{Q}^{\log}_{g,n}(W/\mathbb{A}^{1},\beta)$ from Proposition 3.7 is
relative to $\mathfrak{M}_{g,n}^{\log}(\mathscr{A}^{\prime}/\mathbb{A}^{1})$.
If $\mathscr{A}_{0}=\mathscr{A}^{\prime}\times_{\mathbb{A}^{1}}0$, then there
is a Cartesian tower

(8)
${{\mathcal{Q}^{\log}_{g,n}(X_{0},\beta)}}$${{\mathcal{Q}^{\log}_{g,n}(W/\mathbb{A}^{1},\beta)}}$${{\mathfrak{M}_{g,n}^{\log}(\mathscr{A}_{0})}}$${{\mathfrak{M}_{g,n}^{\log}(\mathscr{A}^{\prime}/\mathbb{A}^{1})}}$${0}$${\mathbb{A}^{1}.}$$\scriptstyle{i_{0}}$

In Diagram (8), $\mathfrak{M}^{\log}_{g,n}(\mathscr{A}_{0})$ refers to minimal
prestable logarithmic maps to $\mathscr{A}_{0}$ _over the standard logarithmic
point_. A consequence of Diagram (8) is that the perfect obstruction theory
defining the virtual class $[\mathcal{Q}^{\log}_{g,n}(X_{0},\beta)]^{\text{\rm
vir}}=i_{0}^{!}[\mathcal{Q}^{\log}_{g,n}(W/\mathbb{A}^{1},\beta)]^{\text{\rm
vir}}$ is relative to $\mathfrak{M}^{\log}_{g,n}(\mathscr{A}_{0})$. In
[ACGS20], the authors show that the fundamental class of
$\mathfrak{M}_{g,n}^{\log}(\mathscr{A}_{0})$ decomposes into a sum of
contributions coming from decorated bipartite graphs (introduced in Definition
4.3). We denote by $\mathrm{Aut}(\Gamma)$ the subset of automorphisms of the
underlying graph $\Gamma$ that respect the extra data $r,(n_{V},g_{V})_{V}$
and $(\omega_{E})_{E}$. We let $m_{\Gamma}\in\mathbb{N}\setminus\\{0\\}$
denote the projection of the generator of the monoid
$Q_{\Gamma}^{\vee}\simeq\mathbb{N}$ to
$\Sigma(\mathrm{pt}_{\mathbb{N}})\simeq\mathbb{R}_{\geq 0}$. More explicitly,
$m_{\Gamma}$ is the least common multiple of the weights $\\{\omega_{E}\\}$.

######  Theorem 4.4.

[ACGS20, Corollary 3.8 $\&$ Lemma 3.9]

$[\mathfrak{M}_{g,n}^{\log}(\mathscr{A}_{0})]=\sum_{\Gamma}\frac{m_{\Gamma}}{|\mathrm{Aut}(\Gamma)|}i_{\Gamma,*}[\mathfrak{M}_{g,n}^{\log}(\mathscr{A}_{0},\Gamma)]$

As in the case of stable maps [ACGS20, Theorem 3.11] we will use Theorem 4.4
to decompose the virtual class
$[\mathcal{Q}^{\log}_{g,n}(X_{0},\beta)]^{\text{\rm vir}}$ into contributions
coming from quasimap moduli spaces associated to bipartite graphs with extra
information about the quasimap degree on each curve component. We first make
the formal definition.

######  Definition 4.5.

For each decorated bipartite graph $\Gamma$ we define
$\mathcal{Q}^{\log}_{g,n}(X_{0},\Gamma)$ via the Cartesian diagram

(9)
${{\mathcal{Q}^{\log}_{g,n}(X_{0},\Gamma)}}$${{\mathcal{Q}^{\log}_{g,n}(X_{0},\beta)}}$${{\mathfrak{M}_{g,n}^{\log}(\mathscr{A}_{0},\Gamma)}}$${{\mathfrak{M}_{g,n}^{\log}(\mathscr{A}_{0})}.}$$\scriptstyle{q}$$\scriptstyle{j_{\Gamma}}$$\scriptstyle{p}$$\scriptstyle{i_{\Gamma}}$

######  Lemma 4.6.

The space $\mathcal{Q}^{\log}_{g,n}(X_{0},\Gamma)$ decomposes as a disjoint
union
$\cup_{\widetilde{\Gamma}=(\Gamma,(\beta_{V})_{V})}\mathcal{Q}^{\log}_{g,n}(X_{0},\widetilde{\Gamma})$
where each $\beta_{V}$ is a quasimap degree of $W$ and a point of
$\mathcal{Q}^{\log}_{g,n}(X_{0},\widetilde{\Gamma})$ is

  * •

an $n$-pointed genus $g$ logarithmic quasimap from $C/\mathrm{pt}_{Q}$ to
$W/\mathbb{A}^{1}$ which factors through the zero fibre, i.e. an element of
$\mathcal{Q}_{g,n}^{\log}(X_{0},\beta)$;

  * •

for each vertex $V$, a stable quasimap to $X_{r(V)}$, where the GIT
presentation is induced from the embedding in $W$, from a curve $C_{V}$ with
marked points $x_{e}$ for each edge containing $V$;

such that the gluing of the curves $C_{V}$ along the points corresponding to
the edges in $E(\widetilde{\Gamma})$ give the underlying curve of $C$.
Furthermore, the underlying quasimap of the element of
$\mathcal{Q}_{g,n}^{\log}(X_{0},\beta)$ is _glued_ from the quasimaps from
each vertex.

###### Proof.

An element of $\mathcal{Q}_{g,n}^{\log}(X_{0},\Gamma)$ comes with the data of
a decomposition of the underlying curve $C$ defining the quasimap into $C_{V}$
for each $V\in V(\Gamma)$, with marked points $x_{e}$ corresponding to the
edge containing $V$ and morphisms $C_{V}\rightarrow\mathscr{A}$ such that the
gluing the $C_{V}$ according to the graph and gluing the morphisms
$C_{V}\rightarrow\mathscr{A}$ produces the underlying morphism of
$C\rightarrow\mathscr{A}_{0}$ induced by the logarithmic quasimap. On the
other hand there is a Cartesian diagram

${\mathcal{X}_{r(V)}}$${\mathcal{W}_{0}}$${\mathcal{W}}$${\mathscr{A}}$${\mathscr{A}_{0}}$${\mathscr{A}^{\prime}}$

where $\mathcal{X}_{r(V)}$ is the quotient stack inside of which $X_{r(V)}$
lives when the GIT presentation is induced by the embedding in $W$. Therefore
the morphism $C_{V}\rightarrow\mathscr{A}$ induces a morphism
$C_{V}\rightarrow\mathcal{X}_{r(V)}$, i.e. a quasimap to $X_{r(V)}$. ∎

Diagram (9) implies that the spaces
$\mathcal{Q}^{\log}_{g,n}(X_{0},\widetilde{\Gamma})$ admit a perfect
obstruction theory over $\mathfrak{M}^{\log}_{g,n}(\mathscr{A}_{0},\Gamma).$

######  Proposition 4.7.

$[\mathcal{Q}_{g,n}^{\log}(X_{0},\beta)]^{\text{\rm
vir}}=\sum_{\widetilde{\Gamma}=(\Gamma,(\beta_{V})_{V})}\frac{m_{\Gamma}}{|\mathrm{Aut}(\Gamma)|}j_{\Gamma,*}[\mathcal{Q}_{g,n}^{\log}(X_{0},\widetilde{\Gamma})]^{\text{\rm
vir}}.$

###### Proof.

We have that

$\displaystyle\,[\mathcal{Q}_{g,n}^{\log}(X_{0},\beta)]^{\text{\rm vir}}$
$\displaystyle=p^{!}[\mathfrak{M}_{g,n}^{\log}(\mathscr{A}_{0}/0)]$
$\displaystyle=\sum_{\Gamma}\frac{m_{\Gamma}}{|\mathrm{Aut}(\Gamma)|}p^{!}i_{\Gamma,*}[\mathfrak{M}_{g,n}^{\log}(\mathscr{A}_{0}/0,\Gamma)]$
$\displaystyle=\sum_{\widetilde{\Gamma}=(\Gamma,(\beta_{V})_{V})}\frac{m_{\Gamma}}{|\mathrm{Aut}(\Gamma)|}j_{\Gamma,*}[\mathcal{Q}_{g,n}^{\log}(X_{0},\widetilde{\Gamma})]^{\text{\rm
vir}}.\qed$

######  Remark 4.8.

One may wonder why in Lemma 4.6, the data of an element of
$\mathcal{Q}^{\log}_{g,n}(X_{0},\widetilde{\Gamma})$ is a quasimap to
$X_{r(V)}$ _with presentation induced by the embedding in $W$_. In fact
$X_{r(V)}$ is a toric variety and its fan induces a GIT presentation. On the
other hand a quasimap to $X_{r(V)}$ with this toric presentation contains
strictly less information and hence Lemma 4.6 would not be true with this
presentation. We explain this phenomenon in Section 2.1, see Example 2.7.

######  Remark 4.9.

Analogously to Section 3.1 we have that the obstruction theory of
$\mathcal{Q}^{\log}_{g,n}(X_{0},\widetilde{\Gamma})$ can be defined relative
to the fibre product

${\operatorname{\mathfrak{Pic}}_{g,n,\underline{d}}^{\log}(\mathscr{A}_{0},\Gamma)}$${{\operatorname{\mathfrak{Pic}}_{g,n,\underline{d}}}(\mathscr{A}_{0})}$${{\mathfrak{M}_{g,n}^{\log}(\mathscr{A}_{0},\Gamma)}}$${{\mathfrak{M}_{g,n}^{\log}(\mathscr{A}_{0})}}$

###  4.2. Gluing

Fix a stable decorated bipartite graph $\tilde{\Gamma}$. Now we relate
$[\mathcal{Q}_{g,n}^{\log}(X_{0},\widetilde{\Gamma})]^{\text{\rm vir}}$ to the
logarithmic moduli spaces corresponding to the vertices of
$\widetilde{\Gamma}$. More precisely, let $\Gamma_{V}$ denote the graph with a
single vertex, with half edges for any edge with vertex $V$ recording the
degree $\beta_{V}$, genus $g_{V}$ as well as the contact order at each half
edge $\alpha_{V}=(w_{E}:V\in E)$. For each vertex we can form the moduli space
$\mathcal{Q}_{g_{V},\alpha_{V}}^{\log}(X_{V}|D,\beta_{V})$ as in [Sha24].

######  Remark 4.10.

Strictly speaking, we need an extension of the moduli spaces defined in
[Sha24] to allow for the presentation induced here by the degenerations (as
well as non-proper targets), but this can be done and is addressed in [Sha24,
End of Section 1].

Observe that for each edge $E$ of $\tilde{\Gamma}$ there are two natural maps

$\prod_{V}\mathcal{Q}_{g_{V},\alpha_{V}}^{\log}(X_{V}|D,\beta_{V})\rightarrow
D$

because each edge is adjacent to two vertices. We build a map

$\prod_{V}\mathcal{Q}_{g_{V},\alpha_{V}}^{\log}(X_{V}|D,\beta_{V})\rightarrow\prod_{E}D^{2}.$

Define $\bigtimes_{V}\mathcal{Q}_{V}$ via the Cartesian diagram

(10)
${\bigtimes_{V}\mathcal{Q}_{V}}$${{\prod_{V\in\Gamma}\mathcal{Q}^{\log}_{g_{V},\alpha_{V}}(X_{V}|D,\beta_{V})}}$${\prod_{E}D}$${\prod_{E}D^{2}}$$\scriptstyle{\Delta}$

######  Proposition 4.11.

There is a morphism

$c_{\widetilde{\Gamma}}:\mathcal{Q}^{\log}_{g,n}(X_{0},\tilde{\Gamma})\rightarrow\bigtimes_{V}\mathcal{Q}_{V}$

which is étale of degree $\frac{\prod_{E}w_{E}}{\mathrm{lcm}(w_{E})_{E}}$.

###### Proof.

The existence of the morphism follows from the description of
$\mathcal{Q}^{\log}(X_{0},\tilde{\Gamma})$ of Lemma 4.6. Since nodes are not
basepoints, the rest of the result follows from the analogous statement for
stable maps which is proved in [KLR21, Lemma 9.2]. ∎

######  Theorem 4.12 (Gluing).

$[\mathcal{Q}^{\log}_{g,n}(X_{0},\tilde{\Gamma})]^{\text{\rm
vir}}=c_{\widetilde{\Gamma}}^{*}\Delta^{!}\prod_{V}[\mathcal{Q}^{\log}_{g_{V},\alpha_{V}}(X_{r(V)}|D,\beta_{V})]^{\text{\rm
vir}}$

As in [KLR21, Bou19], Theorem 4.12 will be proved by showing that both classes
come from compatible perfect obstruction theories with respect to different
bases.

In the following we fix a decorated bipartite graph $\widetilde{\Gamma}$ and
use the following simplified notation. Let $\mathcal{Q}_{V}$ denote
$\mathcal{Q}^{\log}_{g_{V},\alpha_{V}}(X_{r(V)}|D,\beta_{V})$. Similarly, we
denote

(11)
$\displaystyle\mathcal{Q}_{\widetilde{\Gamma}}=\mathcal{Q}^{\log}_{g,n}(X_{0},\tilde{\Gamma}),\qquad$
$\displaystyle\prod_{V}\mathcal{Q}_{V}=\prod_{V}\mathcal{Q}^{\log}_{g_{V},\alpha_{V}}(X_{r(V)}|D,\beta_{V}),$
(12)
$\displaystyle\mathfrak{M}_{\Gamma}=\mathfrak{M}_{g,n}^{\log}(\mathscr{A}_{0},\Gamma),\qquad$
$\displaystyle\prod_{V}\mathfrak{M}_{V}=\prod_{V}\mathfrak{M}^{\log}_{g_{V},\alpha_{V}}(\mathscr{A}).$

Let $\pi_{V}:\mathcal{C}_{V}\to\mathcal{Q}_{V}$ be the universal curve over
$\mathcal{Q}_{V}$, and let $\mathcal{C}_{V,\widetilde{\Gamma}}$ denote the
pieces of the partial normalisation of $\mathcal{C}$ along the nodes
corresponding to the edges of $\widetilde{\Gamma}$. With this, we have a
commutative diagram

(13)
${\mathcal{W}}$${\mathcal{X}_{r(V)}}$${\mathcal{C}}$${{\mathcal{C}_{V,\widetilde{\Gamma}}}}$${\mathcal{C}_{V}}$${\mathcal{Q}_{\widetilde{\Gamma}}}$${\mathcal{Q}_{V}}$$\scriptstyle{i_{r(V)}}$$\scriptstyle{\pi}$$\scriptstyle{u}$$\scriptstyle{j_{V}}$$\scriptstyle{\ell_{V}}$$\scriptstyle{\pi_{V}}$$\scriptstyle{u_{V}}$$\scriptstyle{g_{V}}$

as well as

${\mathcal{Q}_{\widetilde{\Gamma}}}$${{D\subset\mathcal{D}}}$${\mathcal{C}}$${\mathcal{W}}$$\scriptstyle{i_{E}}$$\scriptstyle{\operatorname{ev}_{E}}$$\scriptstyle{i}$$\scriptstyle{u}$

where $\operatorname{ev}_{E}$ and $i_{E}$ utilise the fact that the universal
curve has a distinguished node for the edge $E$.

Recall that by Theorem 3.2, for each vertex $V\in V(\widetilde{\Gamma})$, the
complex
$(\mathbf{R}\pi_{V,*}u_{V}^{*}\mathbb{T}^{\log}_{\mathcal{X}_{r(V)}})^{\vee}$
is a perfect obstruction theory for the morphism

$\mathcal{Q}^{\log}_{g_{V},\alpha_{V}}(X_{r(V)}|D,\beta_{V})\to\mathfrak{M}_{g_{V},\alpha_{V}}^{\log}(\mathscr{A}).$

Let $\mathcal{E}$ denote the complex
$u^{*}\mathbb{T}^{\log}_{\mathcal{W}/\mathbb{A}^{1}}$ on the universal curve
$\mathcal{C}_{\widetilde{\Gamma}}$ over
$\mathcal{Q}^{\log}_{g,n}(X_{0},\tilde{\Gamma})$. Let

$\mathbb{E}=(\mathbf{R}\pi_{*}\mathcal{E})^{\vee}.$

By Proposition 3.7, $\mathbb{E}$ defines a perfect obstruction theory of
$\mathcal{Q}^{\log}_{g,n}(X_{0},\tilde{\Gamma})$ relative to
$\mathfrak{M}_{g,n}^{\log}(\mathscr{A}_{0},\Gamma)$.

######  Lemma 4.13.

There is a distinguished triangle

(14) $\mathcal{E}\rightarrow\oplus_{V\in
V(\widetilde{\Gamma})}j_{V,*}\ell_{V}^{*}u_{V}^{*}\mathbb{T}^{\log}_{\mathcal{X}_{r(V)}}\rightarrow\oplus_{E\in
E(\widetilde{\Gamma})}i_{E,*}\operatorname{ev}_{E}^{*}\mathbb{T}^{\log}_{\mathcal{W}/\mathbb{A}^{1}}|_{D}\rightarrow\mathcal{E}[1]$

where
$i_{E}:\mathcal{Q}_{\widetilde{\Gamma}}\rightarrow\mathcal{C}_{\widetilde{\Gamma}}$
is the inclusion of the node corresponding to $E$ and and
$\operatorname{ev}_{E}:\mathcal{Q}_{\widetilde{\Gamma}}\rightarrow D$ is the
evaluation at the node.

###### Proof.

There is a partial normalisation exact sequence on the universal curve
$\mathcal{C}_{\widetilde{\Gamma}}$ given by

$0\rightarrow\mathcal{O}_{\mathcal{C}_{\widetilde{\Gamma}}}\rightarrow\oplus_{V\in
V(\widetilde{\Gamma})}j_{V,*}\mathcal{O}_{\mathcal{C}_{V,\widetilde{\Gamma}}}\rightarrow\oplus_{E\in
E(\widetilde{\Gamma})}i_{E,*}\mathcal{O}_{\mathcal{Q}_{\widetilde{\Gamma}}}\rightarrow
0$

We claim the result follows from tensoring the associated distinguished
triangle with $\mathcal{E}$. We have that
$i_{E,*}\mathcal{O}_{\mathcal{Q}_{\widetilde{\Gamma}}}\otimes
u^{*}\mathbb{T}^{\log}_{\mathcal{W}/\mathbb{A}^{1}}\cong
i_{E,*}(i_{E}^{*}u^{*}\mathbb{T}^{\log}_{\mathcal{W}/\mathbb{A}^{1}})\cong
i_{E,*}\operatorname{ev}_{E}^{*}\mathbb{T}^{\log}_{\mathcal{W}/\mathbb{A}^{1}}|_{D}$.
For the middle term we can notice from our explicit description above that
$\mathbb{T}^{\log}_{\mathcal{W}/\mathbb{A}^{1}}|_{\mathcal{X}_{r(V)}}\cong\mathbb{T}^{\log}_{\mathcal{X}_{r(V)}}$
using the explicit description for the latter given in [Sha22, 3.5.20]. We
have

$j_{V,*}\mathcal{O}_{\mathcal{C}_{V,\widetilde{\Gamma}}}\otimes
u^{*}\mathbb{T}^{\log}_{\mathcal{W}/\mathbb{A}^{1}}\cong
j_{V,*}j_{V}^{*}u^{*}\mathbb{T}^{\log}_{\mathcal{W}/\mathbb{A}^{1}}\cong
j_{V}^{*}\ell_{V}^{*}u_{V}^{*}\mathbb{T}^{\log}_{\mathcal{X}_{r(V)}}$

by the projection formula and commutativity of Diagram (13). This proves the
claim. ∎

######  Lemma 4.14.

We have a morphism

(15) $\oplus_{V\in
V(\widetilde{\Gamma})}j_{V,*}\ell_{V}^{*}u_{V}^{*}\mathbb{T}_{\mathcal{X}_{r(V)}}^{\log}\rightarrow\oplus_{E\in
E(\widetilde{\Gamma})}i_{E,*}\operatorname{ev}_{E}^{*}T_{D}.$

###### Proof.

Let $\mathcal{D}_{\rho}$ be the toric divisors of $\mathcal{X}_{r(V)}$ and
without loss of generality we assume that
$\mathcal{D}=\mathcal{D}_{\rho_{1}}$. The logarithmic tangent complex of
$\mathcal{X}_{r(V)}$ is

$\mathcal{O}^{\oplus
s}_{\mathcal{X}_{r(V)}}\to\oplus_{\rho\neq\rho_{1}}\mathcal{O}_{\mathcal{X}_{r(V)}}(\mathcal{D}_{\rho})\oplus\mathcal{O}_{\mathcal{X}_{r(V)}}$

and the tangent complex of $\mathcal{D}$ is

$\mathcal{O}_{\mathcal{D}}^{\oplus
s}\to\oplus_{\rho\neq\rho_{1}}\mathcal{O}_{\mathcal{D}}(\mathcal{D}_{\rho})$

To construct the morphism (15) it suffices to construct the morphism for
$\mathbb{T}^{\log}_{\mathcal{X}_{r(V)}}$ and $T_{D}$ replaced with either the
first or second terms in the above complexes. Consider the diagram

${\mathcal{C}}$${{\mathcal{C}_{V,\widetilde{\Gamma}}}}$${\mathcal{C}_{V}}$${\mathcal{Q}_{\widetilde{\Gamma}}}$${D}$${\mathcal{X}_{r(V)}}$$\scriptstyle{j_{V}}$$\scriptstyle{\ell_{V}}$$\scriptstyle{u^{\prime}_{V}}$$\scriptstyle{u_{V}}$$\scriptstyle{\operatorname{ev}_{E}}$$\scriptstyle{i_{E}}$$\scriptstyle{j^{E}_{V}}$

For the second term in the complex (the first is similar) we see that

$j_{V,*}{u^{\prime}}^{*}_{V}\left(\oplus_{\rho\neq\rho_{1}}\mathcal{O}_{\mathcal{X}_{r(V)}}(\mathcal{D}_{\rho})\oplus\mathcal{O}_{\mathcal{X}_{r(V)}}\right)\rightarrow
j_{V,*}{u^{\prime}}^{*}_{V}\left(\oplus_{\rho\neq\rho_{1}}\mathcal{O}_{\mathcal{D}}(\mathcal{D}_{\rho})\right)$

by the restriction and the projection. And then we have a morphism

$j_{V,*}{u^{\prime}}^{*}_{V}\left(\oplus_{\rho\neq\rho_{1}}\mathcal{O}_{\mathcal{D}}(\mathcal{D}_{\rho})\right)\rightarrow
j_{V,*}j^{E}_{V,*}j_{V}^{E,*}{u^{\prime}}^{*}_{V}\left(\oplus_{\rho\neq\rho_{1}}\mathcal{O}_{\mathcal{D}}(\mathcal{D}_{\rho})\right)=i_{E,*}\operatorname{ev}_{E}^{*}\left(\oplus_{\rho\neq\rho_{1}}\mathcal{O}_{D}(D_{\rho})\right)$

By adjunction and commutativity of the diagram. This procedure builds a
morphism like in (15), but only for one summand of the source and target for a
vertex $V$ contained in an edge. We form the morphism (15) by taking the
difference of sections for every pair of vertices contained in an edge. ∎

######  Notation 4.15.

Let $\mathcal{F}$ be defined by the following distinguished triangle

$\mathcal{F}\rightarrow\oplus_{V\in
V(\widetilde{\Gamma})}j_{V,*}\ell_{V}^{*}u_{V}^{*}\mathbb{T}_{\mathcal{X}_{r(V)}}^{\log}\rightarrow\oplus_{E(\widetilde{\Gamma})}i_{E,*}\operatorname{ev}_{E}^{*}T_{D}\rightarrow\mathcal{F}[1]$

on the universal curve over $\mathcal{Q}_{\widetilde{\Gamma}}$. Let
$\mathbb{F}:=(\mathbf{R}\pi_{*}\mathcal{F})^{\vee}$.

######  Lemma 4.16.

In notation as above, we have that $\mathbb{F}$ is a perfect obstruction
theory for the morphism

$\mathcal{Q}^{\log}_{g,n}(X_{0},\widetilde{\Gamma})\to\prod_{V}\mathfrak{M}^{\log}_{g_{V},\alpha_{V}}(\mathscr{A}).$

The induced class is
$c_{\widetilde{\Gamma}}^{*}\Delta^{!}\prod_{V}[\mathcal{Q}^{\log}_{g_{V},\alpha_{V}}(X_{V}|D,\beta_{V})]^{\text{\rm
vir}}$.

###### Proof.

Consider the diagram

${\mathcal{Q}_{\widetilde{\Gamma}}}$${\bigtimes_{V}\mathcal{Q}_{V}}$${{\prod_{V}\mathcal{Q}_{V}}}$${{\prod_{V}\mathfrak{M}_{V}}}$${\prod_{E\in
E(\widetilde{\Gamma})}D_{E}}$${\prod_{E\in
E(\widetilde{\Gamma})}D_{E}^{2}}$$\scriptstyle{c_{\widetilde{\Gamma}}}$$\scriptstyle{g}$$\scriptstyle{\Delta_{\mathcal{Q}}}$$\scriptstyle{\operatorname{ev}_{E}}$$\scriptstyle{\Delta}$

The map
$g:\mathcal{Q}_{\widetilde{\Gamma}}\rightarrow\prod_{V}\mathcal{Q}_{V}$ is the
composition of the étale map $c_{\widetilde{\Gamma}}$ and
$\Delta_{\mathcal{Q}}$, so it admits a perfect obstruction theory induced by
the regular embedding $\Delta$. Let

$\mathbb{B}^{\vee}:=\oplus_{E\in
E(\widetilde{\Gamma})}\operatorname{ev}_{E}^{*}[0\rightarrow T_{D}]$

in degrees $[0,1]$. Then $\mathbb{B}$ is a perfect obstruction theory for $g$.
We know that $\prod_{V}\mathcal{Q}_{V}\rightarrow\prod_{V}\mathfrak{M}_{V}$
admits a dual perfect obstruction theory

$\mathbb{A}^{\vee}:=\oplus_{V}\mathbf{R}\pi_{V,*}u_{V}^{*}\mathbb{T}^{\log}_{\mathcal{X}_{r(V)}}.$

In the following we use $\mathbb{A}$ and $\mathbb{B}$ to construct a perfect
obstruction theory for
$\mathcal{Q}_{\widetilde{\Gamma}}\rightarrow\prod_{V}\mathfrak{M}_{V}$. We
first prove that $g^{*}\mathbb{A}^{\vee}\cong\oplus_{V\in
V(\widetilde{\Gamma})}\mathbf{R}\pi_{*}j_{V,*}\ell_{V}^{*}u_{V}^{*}\mathbb{T}^{\log}_{\mathcal{X}_{r(V)}}$.
Consider the following commutative diagram with the square being Cartesian

${\mathcal{C}}$${{\mathcal{C}_{V,\widetilde{\Gamma}}}}$${\mathcal{C}_{V}}$${\mathcal{Q}_{\widetilde{\Gamma}}}$${\mathcal{Q}_{V}.}$$\scriptstyle{\pi}$$\scriptstyle{j_{V}}$$\scriptstyle{\pi_{V,\widetilde{\Gamma}}}$$\scriptstyle{\ell_{V}}$$\scriptstyle{\pi_{V}}$$\scriptstyle{g_{V}}$

The statement follows from $\pi_{V,\widetilde{\Gamma},*}=\pi_{*}j_{V,*}$ and
cohomology and base change for
$u_{V}^{*}\mathbb{T}^{\log}_{\mathcal{X}_{r(V)}}$ in the above square.

Note that
$\pi_{*}i_{E,*}\operatorname{ev}_{E}^{*}T_{D}=\operatorname{ev}^{*}_{E}T_{D}$.
With these we can apply Equation 15 to get a morphism
$g^{*}\mathbb{A}^{\vee}[-1]\to\mathbb{B}^{\vee}$. By the definition of
$\mathbb{F}$ in 4.15, we have a distinguished triangle

$\mathbb{F}\rightarrow\mathbb{B}\rightarrow
g^{*}\mathbb{A}[1]\rightarrow\mathbb{F}[1]$

Moreover, we obtain a morphism of distinguished triangles

${\mathbb{B}}$${g^{*}\mathbb{A}[1]}$${\mathbb{F}[1]}$${\mathbb{L}_{{\mathcal{Q}_{\tilde{\Gamma}}}/\prod\mathcal{Q}_{V}}}$${g^{*}\mathbb{L}_{\prod\mathcal{Q}_{V}/\prod\mathfrak{M}_{\Gamma}}[1]}$${{\mathbb{L}_{\mathcal{Q}_{\widetilde{\Gamma}}/\prod\mathfrak{M}_{V}}[1]}}$

where the commutativity of left square above is commutativity of Diagram 9.10
in [KLR21]. A repeated application of the Four Lemma implies that $\mathbb{F}$
is an obstruction theory for
$\mathcal{Q}_{\widetilde{\Gamma}}\rightarrow\prod_{V}\mathfrak{M}_{V}$.
Moreover, the diagram above shows that we have a compatible triple of
obstruction theories. By functoriality of pull-backs we have that the class
induced by $\mathbb{F}$ is equal to the class

$c_{\widetilde{\Gamma}}^{*}\Delta^{!}\prod_{V}[\mathcal{Q}^{\log}_{g_{V},\alpha_{V}}(X_{V}|D,\beta_{V})]^{\text{\rm
vir}}.\qed$

We have morphisms

${\mathcal{Q}_{\tilde{\Gamma}}}$${\mathfrak{M}_{\Gamma}}$${\prod_{V}\mathfrak{M}_{V}.}$$\scriptstyle{\delta}$

######  Proposition 4.17.

Let $\mathbb{E}^{\vee}$ be the complex
$\mathbf{R}\pi_{*}u^{*}\mathbb{T}^{\log}_{\mathcal{W}/\mathbb{A}^{1}}$. The
dotted arrow in the following diagram can be filled in to form a commutative
diagram in the derived category of $\mathcal{Q}_{\tilde{\Gamma}}$.

${\mathbb{F}}$${\mathbb{E}}$${\mathbb{L}_{\delta}[1]}$${\mathbb{L}_{{\mathcal{Q}_{\tilde{\Gamma}}}/\prod\mathfrak{M}_{V}}}$${\mathbb{L}_{{\mathcal{Q}_{\tilde{\Gamma}}}/\mathfrak{M}_{\Gamma}}}$${{\mathbb{L}_{\delta}[1]}}$

###### Proof.

If we take the pushforward of Equation (14) under $\pi$ we obtain the
distinguished triangle

(16)
$\mathbf{R}\pi_{*}u^{*}\mathbb{T}^{\log}_{\mathcal{W}/\mathbb{A}^{1}}\rightarrow\oplus_{V\in
V(\widetilde{\Gamma})}\mathbf{R}\pi_{*}j_{V,*}\ell_{V}^{*}u_{V}^{*}\mathbb{T}^{\log}_{\mathcal{X}_{V}}\rightarrow\oplus_{E\in
E(\widetilde{\Gamma})}\operatorname{ev}_{E}^{*}\mathbb{T}^{\log}_{\mathcal{W}/\mathbb{A}^{1}}|_{D}\rightarrow$

To construct the dotted arrow we use the definition of $\mathbb{F}$, the
distinguished triangle (16) and Equation 15. More precisely, we have solid
arrows which induce the dashed arrow making the following diagram commute

${\mathbb{E}^{\vee}}$${\oplus_{V\in
V(\widetilde{\Gamma})}\mathbf{R}\pi_{*}j_{V,*}\ell_{V}^{*}u_{V}^{*}\mathbb{T}^{\log}_{\mathcal{X}_{V}}}$${\oplus_{E\in
E(\widetilde{\Gamma})}\operatorname{ev}_{E}^{*}T^{\log}_{\mathcal{W}/\mathbb{A}^{1}}|_{D}}$${\mathbb{F}^{\vee}}$${\oplus_{V\in
V(\widetilde{\Gamma})}\mathbf{R}\pi_{*}j_{V,*}\ell_{V}^{*}u_{V}^{*}\mathbb{T}^{\log}_{\mathcal{X}_{V}}}$${{\oplus_{E\in
E(\widetilde{\Gamma})}\operatorname{ev}_{E}^{*}T_{D}}}$

We can complete the diagram

${\oplus_{E\in
E(\widetilde{\Gamma})}\operatorname{ev}_{E}^{*}\mathcal{O}_{D}}$${\mathbb{E}^{\vee}}$${\oplus_{V\in
V(\widetilde{\Gamma})}\mathbf{R}\pi_{*}j_{V,*}\ell_{V}^{*}u_{V}^{*}\mathbb{T}^{\log}_{\mathcal{X}_{V}}}$${\oplus_{E\in
E(\widetilde{\Gamma})}\operatorname{ev}_{E}^{*}T^{\log}_{\mathcal{W}/\mathbb{A}^{1}}|_{D}}$${\mathbb{F}^{\vee}}$${\oplus_{V\in
V(\widetilde{\Gamma})}\mathbf{R}\pi_{*}j_{V,*}\ell_{V}^{*}u_{V}^{*}\mathbb{T}^{\log}_{\mathcal{X}_{V}}}$${{\oplus_{E\in
E(\widetilde{\Gamma})}\operatorname{ev}_{E}^{*}T_{D}}}$

The octahedron axiom gives a distinguished triangle

$\mathbb{F}\to\mathbb{E}\to\oplus_{E\in
E(\widetilde{\Gamma})}\operatorname{ev}_{E}^{*}\mathcal{O}_{D}[1]\to\mathbb{F}[1].$

By [KLR21], Lemma 9.2, we have $\oplus_{E\in
E(\widetilde{\Gamma})}\operatorname{ev}_{E}^{*}\mathcal{O}_{D}\simeq\mathbb{L}_{\delta}$.
The only thing left to prove is that the map $\mathbb{F}\to\mathbb{E}$
obtained above is the same obtained by completing the morphism
$\mathbb{E}\to\mathbb{L}_{\delta}$ to a distinguished triangle. This is the
commutativity of diagram 9.12 in [KLR21]. ∎

###### Proof of Theorem 4.12.

We have a compatible triple and by functoriality of pull-backs we get the
statement. ∎

The following is the main result of this section. Let
$W\rightarrow\mathbb{A}^{1}$ be a toric double point degeneration such that
the associated quasimap moduli space is proper. Let $X_{0}=X_{1}\cup_{D}X_{2}$
denote the central fibre.

######  Theorem 4.18 (Degeneration formula).

$[\mathcal{Q}_{g,n}^{\log}(X_{0},\beta)]^{\text{\rm
vir}}=\sum_{\widetilde{\Gamma}}\frac{m_{\widetilde{\Gamma}}}{|\mathrm{Aut}(\widetilde{\Gamma})|}j_{\widetilde{\Gamma},*}c_{\widetilde{\Gamma}}^{*}\,\Delta^{!}\left(\prod_{V\in
V(\widetilde{\Gamma})}[\mathcal{Q}^{\log}_{g_{V},\alpha_{V}}(X_{r(V)}|D,\beta_{V})]^{\text{\rm
vir}}\right).$

###### Proof.

The statement follows from Proposition 4.7 together with Theorem 4.12. ∎

######  Remark 4.19.

In Section 5 we will use the degeneration formula with a certain point
condition imposed on the moduli space directly. The proof of this result is
analogous to this section. For details, see for example [Bou19, Section 7.3].

##  5\. Local/logarithmic correspondence

In this section we prove the local/logarithmic correspondence for quasimaps.
Mirroring the Gromov–Witten version [vGGR19], we will use the degeneration
formula proved in the previous section to prove this correspondence for
projective space and a coordinate hyperplane. We will then use this to pull
back the local/logarithmic correspondence to any very ample set up. We will
then show that stronger results hold than in the stable maps setting, see
Theorem 5.19 and Theorem 5.20. The proof of these theorems will utilise the
same strategy to prove these correspondences for any very ample set up.

###  5.1. Smooth divisor

######  Theorem 5.1.

Let $X$ be a GIT quotient $W/\\!\\!/G$ and let $D\subset X$ be a _smooth, very
ample_ divisor satisfying Assumptions 2.1 and 3.1. Let $\beta$ be an effective
curve class on $X$ and let $d=D\cdot\beta\geq 0$, then we have the following
equality of virtual classes

$[\mathcal{Q}_{0,(d,0)}^{\log}(X|D,\beta)]^{\text{\rm
vir}}=(-1)^{d+1}\cdot\operatorname{ev}_{2}^{*}D\cap[\mathcal{Q}_{0,2}(\mathcal{O}_{X}(-D),\beta)]^{\text{\rm
vir}}.$

As the degeneration formula for quasimaps in Theorem 4.18 only applies in a
toric setting we will first use the degeneration formula to prove

######  Theorem 5.2.

Let $X=\mathbb{P}^{N}$ for $N>1$ and $D=H$ a coordinate hyperplane, and let
$d\in\mathbb{N}$. Then

$[\mathcal{Q}_{0,(d,0)}^{\log}(\mathbb{P}^{N}|H,d)]^{\text{\rm
vir}}=(-1)^{d+1}\cdot\operatorname{ev}_{2}^{*}H\cap[\mathcal{Q}_{0,2}(\mathcal{O}_{\mathbb{P}^{N}}(-H),d)]^{\text{\rm
vir}}.$

Assuming Theorem 5.2, the proof of Theorem 5.1 is straightforward.

###### Proof of Theorem 5.1.

We have a commutative diagram

(17)
${{\mathcal{Q}^{\log}_{0,(d,0)}(X|D,\beta)}}$${{\mathcal{Q}_{0,(d,0)}^{\log}(\mathbb{P}^{N}|H,d)}}$${{\mathcal{Q}_{0,2}(X,\beta)}}$${{\mathcal{Q}_{0,2}(\mathbb{P}^{N},d)}}$$\scriptstyle{p}$$\scriptstyle{i}$$\scriptstyle{p^{\prime}}$$\scriptstyle{i^{\prime}}$

The existence of the horizontal arrows follows similarly from [CFK14, Section
3.1]. We can use the fact [Sha22, Proposition 3.5.10] that the virtual class
on $\mathcal{Q}^{\log}_{0,(d,0)}(X|D,\beta)$ can be defined relative to
$\mathcal{Q}_{0,(d,0)}^{\log}(\mathbb{P}^{N}|H,d)$ and is in fact pulled back
from the obstruction theory for $i^{\prime}$, which defines the virtual class
on $\mathcal{Q}_{0,2}(X,\beta)$. Let $\mathscr{L}$ (resp. $\mathscr{L}_{D}$)
denote the universal line bundle on the universal curve
$\mathcal{Q}_{0,2}(\mathbb{P}^{N},d)$ (resp. $\mathcal{Q}_{0,2}(X,\beta)$)
corresponding to $\mathcal{O}_{\mathbb{P}^{N}}(H)$ (resp.
$\mathcal{O}_{X}(D)$). We then have that

$\displaystyle p_{*}[\mathcal{Q}^{\log}_{0,(d,0)}(X|D,\beta)]^{\text{\rm
vir}}$
$\displaystyle=p_{*}(i^{\prime})^{!}[\mathcal{Q}_{0,(d,0)}^{\log}(\mathbb{P}^{N}|H,d)]$
$\displaystyle={i^{\prime}}^{!}{p^{\prime}}_{*}[\mathcal{Q}_{0,(d,0)}^{\log}(\mathbb{P}^{N}|H,d)]$
$\displaystyle={i^{\prime}}^{!}(-1)^{d+1}\cdot\operatorname{ev}^{*}H\cap[\mathcal{Q}_{0,2}(\mathcal{O}_{\mathbb{P}^{N}}(-H),d)]^{\text{\rm
vir}}$
$\displaystyle={i^{\prime}}^{!}(-1)^{d+1}\cdot\operatorname{ev}^{*}H\cap
e(\textbf{R}^{1}\pi_{*}\mathscr{L}^{\vee})\cap[\mathcal{Q}_{0,2}(\mathbb{P}^{N},d)]$
$\displaystyle=(-1)^{d+1}\cdot\operatorname{ev}^{*}D\cap
e(\textbf{R}^{1}\pi_{*}\mathscr{L}_{D}^{\vee})\cap[\mathcal{Q}_{0,2}(X,\beta)]^{\text{\rm
vir}}$
$\displaystyle=(-1)^{d+1}\cdot\operatorname{ev}^{*}D\cap[\mathcal{Q}_{0,2}(\mathcal{O}_{X}(-D),\beta)]^{\text{\rm
vir}}.\qed$

######  Remark 5.3.

Although we only prove Theorem 5.2 for projective spaces of dimension greater
than one, Theorem 5.1 implies the corresponding result for $\mathbb{P}^{1}$.
The reason for the restriction is that the for $\mathbb{P}^{1}$ the induced
presentation of the central fibre components are slightly more complicated so
we exclude this analysis. On the other hand, the fact that the
local/logarithmic correspondence holds for quasimaps for $\mathbb{P}^{1}$ may
be surprising. For example, a consequence is that the Gromov–Witten/quasimap
wall-crossing for $\mathcal{O}_{\mathbb{P}^{1}}(-1)$ is trivial, after capping
with a hyperplane class. This follows from the fact that the logarithmic
spaces are birational. This is not at all an obvious conclusion from [CFK17,
Theorem 4.2.1]. On the other hand there is precendent for this, in [MOP11,
Section 9] the authors show that quasimap and Gromov–Witten invariants of
$\mathcal{O}_{\mathbb{P}^{1}}{(-1)}\oplus\mathcal{O}_{\mathbb{P}^{1}}(-1)$
coincide in all genera.

In the remaining of Section 5.1, we prove Theorem 5.2. The proof, which is
deferred until the end of Section 5.1, is a combination of the equalities in
Lemma 5.7 and Propositions 5.8 and 5.9. Furthermore, Proposition 5.8 is the
result of applying the degeneration formula (Theorem 4.18) and ruling out all
the contributions except for the graph

$\textstyle{\overset{V_{1}}{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{E}$$\textstyle{\overset{V_{2}}{\bullet}}$

with $V_{1}$ an $L_{1}$-vertex, $V_{2}$ an $L_{2}$-vertex, $\omega_{E}=d$,
$\beta_{V_{1}}=(d,-d)$, $\beta_{V_{2}}=(0,d)$ and exactly one marking on each
vertex. To do so, we follow the next steps:

  1. (1)

Lemma 5.11 shows there must be two vertices with a single bounded edge
connecting them.

  2. (2)

Lemma 5.12 shows there are no contributions from graphs with _at least one_
$L_{2}$-vertex with degree different from a multiple class of a fibre.

  3. (3)

Proposition 5.17 shows there are no contributions from graphs with one edge,
but ghost classes. This uses Corollary 5.16, which follows from Lemma 5.15.

  4. (4)

Lemma 5.18 shows there are no contributions from graphs contained in $L_{1}$.

This argument follows [vGGR19], with two alterations.

The first alteration is minor. The main theorem of [vGGR19] compares the one
marked logarithmic space with the unmarked local space and involves a
forgetful morphism. Since neither of these spaces exist in quasimap theory we
use the alteration to the degeneration argument discussed in [FW21, TY23b], in
addition to adding a redundant marking.

The second alteration requires an extra argument. On the one hand the
stability condition of quasimap theory allows us to exclude certain graphs
without effort. On the other hand we require an additional vanishing of
contributions where the degrees contains “ghost” curve classes in the sense of
Definition 2.9. These occur because, as in Examples 2.7 and 2.8, the involved
spaces of quasimaps depend on the GIT presentation of the target space.

######  Notation 5.4.

With the notations of Example 2.10, let
$W(\mathbb{P}^{N},H)=\operatorname{Bl}_{H\times
0}(\mathbb{P}^{N}\times\mathbb{A}^{1})$ be the deformation to the normal cone
of $H$ in $\mathbb{P}^{N}$ for $N>1$. The central fibre is a union of
$X_{1}\simeq X=\mathbb{P}^{N}$ and
$X_{2}\simeq\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(H)|_{H})$ glued along $H$.

Let $L=\mathcal{O}(-\widetilde{H})$ denote the total space of the line bundle
over $W(\mathbb{P}^{N},H)$, where $\widetilde{H}$ is the strict transform of
$H\times\mathbb{A}^{1}$. Over a non-zero point of $\mathbb{A}^{1}$ the fibre
is just $\mathcal{O}_{\mathbb{P}^{N}}(-H)$. The central fibre $L_{0}$ is a
union of two components $L_{1}=\mathbb{P}^{N}\times\mathbb{A}^{1}$ and
$L_{2}=\mathcal{O}(-H_{\infty})$ over
$X_{2}\simeq\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(H)|_{H})$, glued along
$L_{H}=H\times\mathbb{A}^{1}$. We follow the convention from 2.5 to denote
ambient stacks with calligraphic letters, such as $\mathcal{L}$ for the stack
corresponding to $L$ and $\mathcal{L}_{0}$ for $L_{0}$.

The degree of a quasimap to $W(\mathbb{P}^{N},H)$, with its toric
presentation, is given by
$\operatorname{Pic}^{\mathbb{G}_{m}^{2}}\mathbb{A}^{5}\simeq\mathbb{Z}^{2}$.
We choose the same basis $\\{l,f\\}$ as in Example 2.10.

######  Remark 5.5.

Although the degeneration formula is written in terms of logarithmic quasimap
spaces with presentation induced by the total space we have that

$\mathcal{Q}^{\log}_{g,\alpha}(\mathbb{P}^{N}|H,(d,-d))\simeq\mathcal{Q}^{\log}_{g,\alpha}(\mathbb{P}^{N}|H,d)$

where the former space uses the presentation
$\mathbb{P}^{N}=\mathbb{A}^{N+2}/\\!\\!/\mathbb{G}_{m}^{2}$ induced by
$W(\mathbb{P}^{N},H)$ and the latter uses the standard presentation of
$\mathbb{P}^{N}$. The same is true for $L_{1}|L_{H}$.

######  Definition 5.6.

Let $\operatorname{ev}_{2}$ denote the evaluation at the second marked point.
We define $\mathcal{Q}^{\log}_{0,2}(L,(d,0))^{\widetilde{H}}$ as the fibre
product

${{\mathcal{Q}^{\log}_{0,2}(L,(d,0))^{\widetilde{H}}}}$${{\mathcal{Q}^{\log}_{0,2}(L,(d,0))}}$${\widetilde{H}}$${W(\mathbb{P}^{N},H)}$$\scriptstyle{\operatorname{ev}_{2}}$$\scriptstyle{i_{\widetilde{H}}}$

The moduli space $\mathcal{Q}^{\log}_{0,2}(L,(d,0))^{\widetilde{H}}$ comes
with a virtual class given by Gysin pullback by $i_{\widetilde{H}}$. The
following follows from the argument of [vGGR19, Lemma 2.2].

######  Lemma 5.7.

There is a morphism
$P:\mathcal{Q}^{\log}_{0,2}(L,(d,0))^{\widetilde{H}}\rightarrow\mathcal{Q}_{0,2}(\mathbb{P}^{N},d)$.
Moreover,

$P_{*}[\mathcal{Q}^{\log}_{0,2}(L_{0},(d,0))^{\widetilde{H}}]^{\text{\rm
vir}}=\operatorname{ev}_{2}^{*}H\cap[\mathcal{Q}_{0,2}(\mathcal{O}_{\mathbb{P}^{N}}(-H),d)]^{\text{\rm
vir}}.$

Next we apply the degeneration formula. We will prove that the analogue of
[vGGR19, Theorem 2.3] also applies in this situation.

Let us first introduce the notation. Consider the following diagram

(18)
${\mathcal{Q}^{\log}_{0,(d,0)}(L_{1}|L_{H},d)\times_{L_{H}}\mathcal{Q}^{\log}_{(d,0)}(L_{2}|L_{H},(0,d))^{H_{\infty}}}$${\mathcal{Q}_{\widetilde{\Gamma}}}$${{\mathcal{Q}^{\log}_{0,2}(L,(d,0))^{\widetilde{H}}}}$${{\mathcal{Q}^{\log}_{0,(d,0)}(\mathbb{P}^{N}|H,d)}}$${{\mathcal{Q}_{0,2}(\mathbb{P}^{N},d)}.}$$\scriptstyle{\mathrm{pr}_{1}}$$\scriptstyle{j_{\widetilde{\Gamma}}}$$\scriptstyle{c_{\widetilde{\Gamma}}}$$\scriptstyle{P}$$\scriptstyle{F}$

The notation $\mathcal{Q}_{\widetilde{\Gamma}}$ was introduced in Equation 11.
Here $\widetilde{\Gamma}$ is the graph with one $L_{1}$-vertex of degree
$(d,-d)$, one $L_{2}$-vertex of degree $(0,d)$ (that is, $d$ times a fibre
class), an edge of weight $d$ connecting the two vertices and an extra marking
on each vertex. In addition there is a constraint to pass through
$\widetilde{H}$ as in Definition 5.6, corresponding to the leg on the
$L_{2}$-vertex. This graph determines the spaces
$\mathcal{Q}^{\log}_{0,(d,0)}(L_{1}|L_{H},(d,-d))$ and
$\mathcal{Q}^{\log}_{0,(d,0)}(L_{2}|L_{H},(0,d))^{H_{\infty}}$ by the
degeneration formula. In general the former is a quasimap space with
$L_{1}=\mathbb{P}^{N}\times\mathbb{A}^{1}$ given by a non-toric presentation.
However, by Remark 5.5, for the degree $(d,-d)$ this space can be identified
with the logarithmic quasimap space to $L_{1}$ with its toric presentation and
with degree $d$.

######  Proposition 5.8.

Let $\Delta:L_{H}\to L_{H}\times L_{H}$ be the diagonal embedding. We have
that

$[\mathcal{Q}^{\log}_{0,2}(L_{0},(d,0))^{\widetilde{H}}]^{\text{\rm
vir}}=d\cdot
j_{\widetilde{\Gamma},*}\left(c_{\widetilde{\Gamma}}^{*}\Delta^{!}\left([\mathcal{Q}^{\log}_{0,(d,0)}(L_{1}|L_{H},d)]^{\text{\rm
vir}}\times[\mathcal{Q}^{\log}_{0,(d,0)}(L_{2}|L_{H},(0,d))^{H_{\infty}}]^{\text{\rm
vir}}\right)\right),$

###### Proof.

Applying the degeneration formula in Theorem 4.18 to $L$, we get

$[\mathcal{Q}_{0,2}^{\log}(L_{0},(d,0))^{\widetilde{H}}]^{\text{\rm
vir}}=\sum_{\widetilde{\Gamma}}\frac{m_{\widetilde{\Gamma}}}{|\mathrm{Aut}(\widetilde{\Gamma})|}j_{\widetilde{\Gamma},*}c_{\widetilde{\Gamma}}^{*}\,\Delta^{!}\left(\prod_{V\in
V(\widetilde{\Gamma})}[\mathcal{Q}^{\log}_{0,\alpha_{V}}(L_{{r(V)}}|L_{H},\beta_{V})]^{\text{\rm
vir}}\right)$

We now need to show that all terms except one vanish.

This follows from combining Lemma 5.11, Lemma 5.12, Proposition 5.17, Lemma
5.18. Finally, we see that for $\widetilde{\Gamma}$ the graph described above,
we have $\frac{m_{\widetilde{\Gamma}}}{|\mathrm{Aut}(\widetilde{\Gamma})|}=d$
since there are no automorphisms and $m_{\widetilde{\Gamma}}$ can be given by
the lowest common multiple of the weights. ∎

######  Proposition 5.9.

With the previous notations,

$\operatorname{ev}_{1,*}\left([\mathcal{Q}^{\log}_{(d,0)}(L_{2}|L_{H},(0,d))^{H_{\infty}}]^{\text{\rm
vir}}\right)=\frac{(-1)^{d+1}}{d}[H].$

###### Proof.

As in [vGGR19, Proposition 2.4], it is enough to show via localisation that

(19) $\deg\left(\operatorname{ev}_{2}^{*}(H_{\infty})\cap
e(\mathbf{R}^{1}\pi_{\ast}u^{*}\mathcal{O}_{\mathbb{P}^{1}}(-1))\cap[\mathcal{Q}^{\log}_{(d,0)}(\mathbb{P}^{1}|0,d)]^{\text{\rm
vir}}\right)=\frac{(-1)^{d+1}}{d}.$

By Lemma 5.10, there is a unique graph $\gamma$ contributing to the
localisation computation. By the proof of [BP05, Theorem 5.1], the same holds
for stable maps. Since the fixed locus corresponding to $\gamma$ lies in the
interior of both moduli spaces, we have that

(20) $\displaystyle\deg\left(\operatorname{ev}_{2}^{*}(H_{\infty})\cap
e(\mathbf{R}^{1}\pi_{\ast}u^{*}\mathcal{O}_{\mathbb{P}^{1}}(-1))\cap[\mathcal{Q}^{\log}_{(d,0)}(\mathbb{P}^{1}|0,d)]^{\text{\rm
vir}}\right)=$
$\displaystyle=\deg\left(\operatorname{ev}_{2}^{*}(H_{\infty})\cap
e(\mathbf{R}^{1}\pi_{\ast}f^{*}\mathcal{O}_{\mathbb{P}^{1}}(-1))\cap[\overline{\mathcal{M}}^{\log}_{(d,0)}(\mathbb{P}^{1}|0,d)]^{\text{\rm
vir}}\right)$

Therefore it is enough to show via localisation that

(21) $\deg\left(\operatorname{ev}_{2}^{*}(H_{\infty})\cap
e(\mathbf{R}^{1}\pi_{\ast}f^{*}\mathcal{O}_{\mathbb{P}^{1}}(-1))\cap[\overline{\mathcal{M}}^{\log}_{(d,0)}(\mathbb{P}^{1}|0,d)]^{\text{\rm
vir}}\right)=\frac{(-1)^{d+1}}{d}.$

The fixed locus associated to $\gamma$ is isomorphic to
$[\text{pt}/\mathbb{Z}_{d}]$. The equivariant Euler class of the virtual
normal bundle can be written as a product of three terms corresponding to the
vertices, edges and flags of $\gamma$, see [GP99, GV05]. We choose a
1-dimensional subtorus with equivariant parameter $\hbar$ and a lift of the
action to $\mathcal{O}_{\mathbb{P}^{1}}(-1)$ analogous to those in the proof
of [BP05, Theorem 5.1] (with the roles of 0 and $\infty$ reversed). We compute
that

$e_{\gamma}^{F}=-\frac{1}{\hbar^{2}},\quad e_{\gamma}^{v}=\hbar^{2},\quad
e_{\gamma}^{e}=(-1)^{d}\frac{d!\,\hbar^{d}}{d^{d}}.$

Therefore the equivariant Euler class of the virtual normal bundle at the
locus corresponding to $\gamma$ is

(22) $e(N_{\gamma}^{\text{\rm
vir}})=e_{\gamma}^{F}e_{\gamma}^{v}e_{\gamma}^{e}=(-1)^{d+1}\frac{d!\,\hbar^{d}}{d^{d}}.$

The contribution of the obstruction bundle is

(23)
$e(\mathbf{R}^{1}\pi_{\ast}f^{*}\mathcal{O}_{\mathbb{P}^{1}}(-1))=\frac{(d-1)!\hbar^{d-1}}{d^{d-1}}.$

Multiplying together (23), the inverse of (22), a factor $1/d$ coming from the
automorphism group of the fixed point corresponding to $\gamma$ and a factor
$\hbar$ coming from the condition that the second mark is mapped to a
hyperplane class, we get (21). We conclude by combining (19), (20) and (21). ∎

######  Lemma 5.10.

Consider the setting of Proposition 5.9. In the localisation computation of
the invariant

$\deg\left(\operatorname{ev}_{2}^{*}(H_{\infty})\cap
e(\mathbf{R}^{1}\pi_{\ast}f^{*}\mathcal{O}_{\mathbb{P}^{1}}(-1))\cap[\mathcal{Q}^{\log}_{(d,0)}(\mathbb{P}^{1}|0,d)]^{\text{\rm
vir}}\right)$

the unique graph whose contribution is non-zero is

$\gamma=\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
18.42781pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\underset{V_{1}}{\overset{\\{1\\}}{\bullet}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
63.94154pt\raise 0.0pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{$\scriptstyle{}$}}}\kern
3.0pt}}}}}}\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
25.60611pt\raise 5.43056pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{d}$}}}\kern
3.0pt}}}}}}\ignorespaces{}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
42.42781pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\underset{V_{2}}{\overset{\\{2\\}}{\bullet}}}$}}}}}}}\ignorespaces}}}}\ignorespaces$

with $V_{1}$ mapped to 0 and $V_{2}$ mapped to $\infty$.

###### Proof.

Recall that, in the context of localisation, each vertex in a graph $\gamma$
corresponds to a connected components of the preimage of the fixed points on
the target and each edge corresponds to a multiple cover of the target
$\mathbb{P}^{1}$. We decorate edges with the labels of the marked points and
edges with the degree of the cover.

We would like to do the localisation calculation directly on
$\mathcal{Q}^{\log}_{0,(d,0)}(\mathbb{P}^{1}|0,d)$. For a technical reason
this is not possible, since the base of the perfect obstruction theory is not
smooth. On the other hand one can use the procedure of [Ran22, Section 3.4] to
produce a logarithmic modification

(24)
$\widetilde{\mathcal{Q}^{\log}}_{0,(d,0)}(\mathbb{P}^{1}|0,d)\rightarrow\mathcal{Q}^{\log}_{0,(d,0)}(\mathbb{P}^{1}|0,d)$

whose obstruction theory is defined relative to the space of logarithmic maps
to expansions of $\mathscr{A}$.

Consider a fixed locus $\mathcal{Q}_{\gamma}$ in
$\widetilde{\mathcal{Q}^{\log}}_{0,(d,0)}(\mathbb{P}^{1}|0,d)$.

Since elements in $\mathcal{Q}^{\log}_{(d,0)}(\mathbb{P}^{1}|0,d)$ have
irreducible source, the morphism (24) forces quasimaps in
$\mathcal{Q}_{\gamma}$ to have irreducible source.

The tangency condition $(d,0)$ ensures that the first marked point is mapped
to 0. By the factor $\operatorname{ev}_{2}^{*}(H_{\infty})$, the second marked
point must be mapped to $\infty$. This is enough to conclude that the graph
$\gamma$ in the statement is indeed the only contributing graph. ∎

######  Lemma 5.11.

Let $\widetilde{\Gamma}$ be a decorated bipartite graph appearing in the
degeneration formula. Then if $\widetilde{\Gamma}$ is not of the form, two
vertices with a single edge connecting them, then

$[\mathcal{Q}_{0,2}^{\log}(L_{0},\widetilde{\Gamma})]^{\text{\rm vir}}=0.$

###### Proof.

The exact same argument of [vGGR19, Lemma 3.1] tells us that if
$\widetilde{\Gamma}$ contains an $L_{1}$-vertex with at least two adjacent
bounded edges then
$[\mathcal{Q}_{0,2}^{\log}(L_{0},\widetilde{\Gamma})]^{\text{\rm vir}}=0$. On
the other hand, quasimap spaces in genus zero need to have at least $2$
markings by stability. Since we have two markings in total, one of which is
forced to correspond to an $L_{2}$ vertex, we have that any graph which
contributes must have two vertices and a single bounded edge between them with
the markings distributed to either side. ∎

######  Lemma 5.12.

If $V$ is an $L_{2}$-vertex with $p_{*}\beta_{V}\neq 0$ then the class
$[\mathcal{Q}^{\log}_{0,\alpha_{V}}(L_{2}|L_{H},\beta_{V})]^{\text{\rm vir}}$
pushes forward to $0$ in $\mathcal{Q}_{0,n_{V}+r_{V}}(H,p_{*}\beta_{V})$.

###### Proof.

By the computation of virtual dimensions in [vGGR19, Lemma 5.1] we have that
the virtual dimension of
$\mathcal{Q}^{\log}_{0,\alpha_{V}}(L_{2}|L_{H},\beta_{V})$ is strictly greater
than the virtual dimension of $\mathcal{Q}_{0,n_{V}+r_{V}}(H,p_{*}\beta_{V})$
if $p_{*}\beta_{V}\neq 0$. On the other hand
$\mathcal{Q}_{0,n_{V}+r+V}(H,p_{*}\beta_{V})$ is smooth of the expected
dimension and so the result follows. ∎

######  Notation 5.13.

Let $\widetilde{\Gamma}$ be a decorated bipartite graph with two vertices and
a single edge $E$. We use the following (slightly abusive) notation for the
universal curve and the irreducible components of the universal curve.

---
$\textstyle{\mathcal{C}_{\widetilde{\Gamma}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{f}$$\textstyle{\mathcal{L}_{0}}$$\textstyle{\mathcal{Q}_{\widetilde{\Gamma}}}$
---
$\textstyle{\mathcal{C}^{i}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{i}}$$\scriptstyle{f_{i}}$$\textstyle{\mathcal{L}_{0}}$$\textstyle{\mathcal{Q}_{\widetilde{\Gamma}}}$

We denote by $P$ the section corresponding to the node.

In the following, we want to exclude the contribution from ghost classes (see
Definition 2.9). Thanks to Lemma 5.11 and Lemma 5.12, we only need to rule out
the following type of graphs. Below we use the basis $\\{l,f\\}$ of
$H_{2}(W(\mathbb{P},H))$ introduced in Example 2.10.

######  Definition 5.14.

Consider the double point degeneration $W(\mathbb{P}^{N},H)\to\mathbb{A}^{1}$.
Let $\widetilde{\Gamma}$ be a graph

$\textstyle{\overset{V_{1}}{\bullet}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{E}$$\textstyle{\overset{V_{2}}{\bullet}}$

with exactly one marking on each vertex, $V_{1}$ a $X_{1}$-vertex, $V_{2}$ an
$X_{2}$-vertex, $\omega_{E}=e$ and with curve classes $\beta_{V_{1}}=(d,-e)$
and $\beta_{V_{2}}=(0,e)$ with $d\geq e\geq 0$. We say that
$\widetilde{\Gamma}$ is a ghost graph if $d\neq e$. Equivalently,
$\widetilde{\Gamma}$ is a ghost graph if $\beta_{V_{1}}$ is a ghost class for
the embedding $\mathbb{P}^{N}\hookrightarrow W(\mathbb{P}^{N},H)$ through the
special fibre, in the sense of Definition 2.9.

######  Lemma 5.15.

Let $\widetilde{\Gamma}$ be a ghost graph. The moduli space
$\mathcal{Q}_{0,2}^{\log}(W_{0}(\mathbb{P}^{N},H),\widetilde{\Gamma})$ is
smooth over
$\operatorname{\mathfrak{Pic}}^{\log,s}_{0,2,\underline{d}}(\mathscr{A}_{0},\Gamma)$.

###### Proof.

We show that
$\mathcal{Q}_{0,2}^{\log}(W_{0}(\mathbb{P}^{N},H),\widetilde{\Gamma})$ is
unobstructed over
$\operatorname{\mathfrak{Pic}}^{\log,s}_{0,2,\underline{d}}(\mathscr{A}_{0},\Gamma)$.
Using the description of the toric divisors of $W(\mathbb{P}^{N},H)$ in
Example 2.10, we see that

(25) $\begin{aligned} \widetilde{H}_{1}\cdot(d,-e)&=d\\\
\widetilde{H}\cdot(d,-e)&=d-e\end{aligned}\qquad\qquad\begin{aligned}
\widetilde{H}_{1}\cdot(0,e)&=0\\\ \widetilde{H}\cdot(0,e)&=e.\end{aligned}$

We consider the normalization sequence of $\mathcal{C}$, the universal curve
over $\mathcal{Q}_{0,2}^{\log}(W_{0}(\mathbb{P}^{N},H),\widetilde{\Gamma})$,

$0\to\mathcal{O}_{\mathcal{C}}\to\mathcal{O}_{\mathcal{C}_{1}}\oplus\mathcal{O}_{\mathcal{C}_{2}}\to\mathcal{O}_{P}\to
0.$

Given the obstruction theory in Proposition 3.14, we take the sheaf

(26)
$\mathcal{E}(W)\coloneqq\oplus_{i=1}^{N}\mathcal{O}_{\mathcal{C}}(\widetilde{H}_{i})\oplus\mathcal{O}_{\mathcal{C}}(\widetilde{H})\oplus\mathcal{O}_{\mathcal{C}}.$

on $\mathcal{C}$ and the sheaves

$\mathcal{E}_{X_{i}|H}\coloneqq\oplus_{\rho\neq\rho_{0}\in\Sigma_{X_{i}}(1)}\mathcal{O}_{\mathcal{C}_{i}}(D_{\rho})\oplus\mathcal{O}_{\mathcal{C}_{i}}$

on $\mathcal{C}_{i}$ for $i=1,2$, with $D_{\rho_{0}}=H$ in both cases.

Tensoring the normalization sequence of $\mathcal{O}_{\mathcal{C}}$ with
$\mathcal{E}(W)$ and pushing forward to
$\mathcal{Q}_{0,2}^{\log}(W_{0}(\mathbb{P}^{N},H),\widetilde{\Gamma})$, we get
a long exact sequence

$\displaystyle 0\to$
$\displaystyle\mathbf{R}^{0}\pi_{*}\mathcal{E}(W)\to\mathbf{R}^{0}\pi^{1}_{*}\mathcal{E}_{X_{1}|H}\oplus\mathbf{R}^{0}\pi^{2}_{*}\mathcal{E}_{X_{2}|H}\stackrel{{\scriptstyle\phi}}{{\rightarrow}}\mathbf{R}^{0}\pi_{*}\mathcal{E}|_{P}\to$
$\displaystyle\to$
$\displaystyle\mathbf{R}^{1}\pi_{*}\mathcal{E}(W)\to\mathbf{R}^{1}\pi^{1}_{*}\mathcal{E}_{X_{1}|H}\oplus\mathbf{R}^{1}\pi^{2}_{*}\mathcal{E}_{X_{2}|H}\to
0.$

Since $\phi$ is surjective and since
$\mathbf{R}^{1}\pi^{1}_{*}\mathcal{E}_{X_{1}|H}\oplus\mathbf{R}^{1}\pi^{2}_{*}\mathcal{E}_{X_{2}|H}=0$
by (25), we conclude that $\mathbf{R}^{1}\pi_{*}\mathcal{E}(W)=0$. This proves
the claim. ∎

######  Corollary 5.16.

Let $\tilde{\Gamma}$ be a ghost graph. The moduli space
$\mathcal{Q}_{0,2}^{\log}(L_{0},\tilde{\Gamma})$ is smooth over
$\operatorname{\mathfrak{Pic}}^{\log,s}_{0,2,\underline{d}}(\mathscr{A}^{\prime}_{0},\Gamma)$.

###### Proof.

Since $\widetilde{\mathcal{D}}$ is effective, we have that maps from (nodal)
curves to $\mathcal{L}$ factor through the zero section. This shows that
$\mathcal{Q}_{0,2}^{\log}(W(\mathbb{P}^{N},H),\tilde{\Gamma})\simeq\mathcal{Q}_{0,2}^{\log}(L/\mathbb{A}^{1},\tilde{\Gamma})$.
Lemma 5.15 implies the claim. ∎

######  Proposition 5.17.

Let $\tilde{\Gamma}$ be a ghost graph. Then

$[\mathcal{Q}_{0,2}^{\log}(L_{0},\tilde{\Gamma})]^{\text{\rm vir}}=0$

###### Proof.

In this proof we write $W$ for $W(\mathbb{P}^{N},H)$, as there is no risk of
confusion.

Let $\mathcal{C}$ denote the universal curve over
$\mathcal{Q}_{0,2}^{\log}(L_{0},\tilde{\Gamma})\simeq\mathcal{Q}_{0,2}^{\log}(W_{0},\tilde{\Gamma})$.
By Proposition 3.14, the obstruction theory on
$\mathcal{Q}_{0,2}^{\log}(W_{0},\tilde{\Gamma})$ is determined by the sheaf
$\mathcal{E}(W)$ on $\mathcal{C}$, defined in (26), while the one on
$\mathcal{Q}_{0,2}^{\log}(L_{0},\tilde{\Gamma})$ is controlled by

(27) $\mathcal{E}(L)=\mathcal{E}(W)\oplus\mathcal{O}_{\mathcal{C}}(Z).$

Tensoring the normalization sequence of $\mathcal{O}_{\mathcal{C}}$ with
$\mathcal{E}(L)$, we get the following long exact sequence on
$\mathcal{Q}_{0,2}^{\log}(L_{0},\tilde{\Gamma})$

(28) $\displaystyle 0\to$
$\displaystyle\mathbf{R}^{0}\pi_{*}\mathcal{E}(L)\to\mathbf{R}^{0}\pi^{1}_{*}\mathcal{E}_{L_{1}|H}\oplus\mathbf{R}^{0}\pi^{2}_{*}\mathcal{E}_{L_{2}|H}\stackrel{{\scriptstyle\phi}}{{\rightarrow}}\mathbf{R}^{0}\pi_{*}\mathcal{E}(L)|_{P}\to$
$\displaystyle\to$
$\displaystyle\mathbf{R}^{1}\pi_{*}\mathcal{E}(L)\to\mathbf{R}^{1}\pi^{1}_{*}\mathcal{E}_{L_{1}|H}\oplus\mathbf{R}^{1}\pi^{2}_{*}\mathcal{E}_{L_{2}|H}\to
0$

with

(29)
$\mathcal{E}_{L_{i}|H}=\mathcal{E}_{X_{i}|H}\oplus\mathcal{O}_{\mathcal{C}_{i}}(Z).$

By Corollary 5.16,
$\mathcal{Q}_{0,2}^{\log}(W_{0},\tilde{\Gamma})\simeq\mathcal{Q}_{0,2}^{\log}(L_{0},\tilde{\Gamma})$
is smooth over
$\operatorname{\mathfrak{Pic}}^{\log,s}_{0,2,\underline{d}}(\mathscr{A}^{\prime}_{0})$
and it has $\mathbf{R}^{1}\pi_{*}\mathcal{E}(L)$ as an obstruction bundle.
This shows that

(30) $[\mathcal{Q}_{0,2}^{\log}(L_{0},\tilde{\Gamma})]^{\text{\rm
vir}}=e(\mathbf{R}^{1}\pi_{*}\mathcal{E}(L))\cap[\mathcal{Q}_{0,2}^{\log}(W_{0},\tilde{\Gamma})]^{\text{\rm
vir}}.$

In the following we show that the Euler class of
$\mathbf{R}^{1}\pi_{*}\mathcal{E}$ is zero.

We first show that we have an exact sequence

(31)
$\mathbf{R}^{0}\pi^{1}_{*}\mathcal{E}_{L_{1}|H}\oplus\mathbf{R}^{0}\pi^{2}_{*}\mathcal{E}_{L_{2}|H}\stackrel{{\scriptstyle\phi}}{{\rightarrow}}\mathbf{R}^{0}\pi_{*}\mathcal{E}(L)|_{P}\to\mathbf{R}^{0}\pi_{*}\mathcal{O}_{\mathcal{C}}(Z)|_{P}\to
0.$

For that, we use (27) and (29) to re-write the morphism $\phi$ as follows

$(\mathbf{R}^{0}\pi^{1}_{*}\mathcal{E}_{X_{1}|H}\oplus\mathbf{R}^{0}\pi^{2}_{*}\mathcal{E}_{X_{2}|H})\oplus(\mathbf{R}^{0}\pi^{1}_{*}\mathcal{O}_{\mathcal{C}_{1}}(Z)\oplus\mathbf{R}^{0}\pi^{2}_{*}\mathcal{O}_{\mathcal{C}_{2}}(Z))\stackrel{{\scriptstyle\phi}}{{\rightarrow}}(\mathbf{R}^{0}\pi_{*}\mathcal{E}(W)|_{P})\oplus\mathbf{R}^{0}\pi_{*}\mathcal{O}_{\mathcal{C}}(Z)|_{P}.$

We have that

$\mathbf{R}^{0}\pi^{1}_{*}\mathcal{E}_{X_{1}|H}\oplus\mathbf{R}^{0}\pi^{2}_{*}\mathcal{E}_{X_{2}|H}\to\mathbf{R}^{0}\pi_{*}\mathcal{E}(W)|_{P}$

is surjective and by Example 2.10 we have that

(32) $(d,-e)\cdot[Z]=e-d<0\qquad\text{and}\qquad(0,e)\cdot[Z]=-e<0,$

which shows that
$\mathbf{R}^{0}\pi^{1}_{*}\mathcal{O}_{\mathcal{C}_{1}}(Z)\oplus\mathbf{R}^{0}\pi^{2}_{*}\mathcal{O}_{\mathcal{C}_{2}}(Z)=0$.
This shows that the cokernel of $\phi$ is
$\mathbf{R}^{0}\pi_{*}\mathcal{O}_{\mathcal{C}}(Z)|_{P}$, which proves the
exactness of (31).

Combining (31) and (28), we get a short exact sequence

(33)
$0\to\mathbf{R}^{0}\pi_{*}\mathcal{O}_{\mathcal{C}}(Z)|_{P}\to\mathbf{R}^{1}\pi_{*}\mathcal{E}(L)\to\mathbf{R}^{1}\pi^{1}_{*}\mathcal{E}_{L_{1}|H}\oplus\mathbf{R}^{1}\pi^{2}_{*}\mathcal{E}_{L_{2}|H}\to
0,$

which implies that

(34) $\displaystyle
e(\mathbf{R}^{1}\pi_{*}\mathcal{E}(L))=e(\mathbf{R}^{0}\pi_{*}\mathcal{O}_{\mathcal{C}}(Z)|_{P})\cdot
e(\mathbf{R}^{1}\pi^{1}_{*}\mathcal{E}_{L_{1}|H}\oplus\mathbf{R}^{1}\pi^{2}_{*}\mathcal{E}_{L_{2}|H}).$

Therefore, it is enough to show that the Euler class of
$\mathbf{R}^{0}\pi_{*}\mathcal{O}_{\mathcal{C}}(Z)|_{P}$ is zero.

For this, we notice that $[Z]=-p^{*}[\widetilde{H}]$ with $p\colon L\to W$ the
natural projection. It follows that
$e(\mathbf{R}^{0}\pi_{*}\mathcal{O}_{\mathcal{C}}(Z)|_{P})=-\operatorname{ev}_{P}^{*}(p^{*}\widetilde{H})$.
As $p^{*}\widetilde{H}$ has a universal section which does not vanish at $P$,
we have that $(p^{*}\widetilde{H})|_{P}$ has a nowhere vanishing section and
thus $\operatorname{ev}_{P}^{*}(p^{*}\widetilde{H})$ is a trivial line bundle.
This shows that

(35) $e(\mathbf{R}^{0}\pi_{*}\mathcal{O}_{\mathcal{C}}(Z)|_{P})=0.$

The equalities (30), (34) and (35) conclude the proof. ∎

######  Lemma 5.18.

Graphs contained in $L_{1}$ do not contribute to the degeneration formula.

###### Proof.

Intersection of these with $\operatorname{ev}^{*}D_{\infty}$ is zero. ∎

###### Proof of Theorem 5.2.

By Proposition 5.8, together with commutativity of diagram (18), we have that

$\displaystyle\quad\,P_{*}[\mathcal{Q}_{0,2}^{\log}(L_{0},(d,0))^{\widetilde{H}}]^{\text{\rm
vir}}$ $\displaystyle=d\cdot
F_{*}\mathrm{pr}_{1,*}c_{\widetilde{\Gamma},*}c_{\widetilde{\Gamma}}^{*}\Delta^{!}\left([\mathcal{Q}^{\log}_{0,(d,0)}(L_{1}|L_{H},d)]^{\text{\rm
vir}}\times[\mathcal{Q}^{\log}_{0,(d,0)}(L_{2}|L_{H},(0,d))^{H_{\infty}}]^{\text{\rm
vir}}\right)$ $\displaystyle=d\cdot
F_{*}\mathrm{pr}_{1,*}\Delta^{!}\left([\mathcal{Q}^{\log}_{0,(d,0)}(L_{1}|L_{H},d)]^{\text{\rm
vir}}\times[\mathcal{Q}^{\log}_{0,(d,0)}(L_{2}|L_{H},(0,d))^{H_{\infty}}]^{\text{\rm
vir}}\right).$

On the other hand, $[\mathcal{Q}^{\log}_{0,(d,0)}(L_{1}|L_{H},d)]^{\text{\rm
vir}}$ Gysin-restricts to
$[\mathcal{Q}^{\log}_{0,(d,0)}(\mathbb{P}^{N}|H,d)]^{\text{\rm vir}}$ when
confining the evaluation map to be in $H\times\\{0\\}$. Combining this with
proposition Proposition 5.9 tells us that

$\displaystyle\mathrm{pr}_{1,*}\Delta^{!}\left([\mathcal{Q}^{\log}_{0,(d,0)}(L_{1}|L_{H},d)]^{\text{\rm
vir}}\times[\mathcal{Q}^{\log}_{0,(d,0)}(L_{2}|L_{H},(0,d))^{H_{\infty}}]^{\text{\rm
vir}}\right)=\frac{(-1)^{d+1}}{d}[\mathcal{Q}^{\log}_{0,(d,0)}(\mathbb{P}^{N}|H,d)]^{\text{\rm
vir}}.$

Finally, combining this with Lemma 5.7 tells us that

$P_{*}[\mathcal{Q}^{\log}_{0,2}(L_{0},(d,0))^{\widetilde{H}}]^{\text{\rm
vir}}=\operatorname{ev}_{2}^{*}H\cap[\mathcal{Q}_{0,2}(\mathcal{O}_{\mathbb{P}^{N}}(-H),d)]^{\text{\rm
vir}}=(-1)^{d+1}F_{*}[\mathcal{Q}^{\log}_{0,(d,0)}(\mathbb{P}^{N}|H,d)]^{\text{\rm
vir}},$

which completes the proof. ∎

###  5.2. Simple normal crossings divisors

In [vGGR19] the authors conjecture a generalisation of the local/logarithmic
correspondence to the simple normal crossings situation. In the strong form of
the correspondence, this would be

$[\overline{\mathcal{M}}_{0,((d_{1},\dots,d_{r}))}^{\log}(X|D,\beta)]^{\text{\rm
vir}}=\prod_{i=1}^{r}(-1)^{d_{i}+1}\cdot\operatorname{ev}_{i}^{*}D_{i}\cap[\overline{\mathcal{M}}_{0,r}(\oplus_{i=1}^{r}\mathcal{O}_{X}(-D_{i}),\beta)]^{\text{\rm
vir}}$

where the left hand side indicates $r$-marked curves in $X$ with maximal
tangency to $D_{i}$ at the $i^{\mathrm{th}}$ marking. The classes are viewed
as living in $\overline{\mathcal{M}}_{0,r}(X,\beta)$ but in this section we
will omit the pushforward on the left-hand side. Although this generalisation
often holds numerically, [BBvG22, BBvG24], at the level of virtual classes it
is not true in complete generality, due to counterexamples in [NR22]. However,
in [NR22, Remark 5.4] the authors note that the difference appears when there
are components in the moduli space containing rational tails and so may not be
present in the quasimap setting.

It will turn out that an easy corollary of the local/logarithmic
correspondence for smooth divisors and properties of logarithmic quasimaps is
the following.

######  Theorem 5.19.

Let $X$ be a GIT quotient $W/\\!\\!/G$ and let $D\subset X$ be a _very ample_
divisor satisfying Assumptions 2.1 and 3.1. Suppose that $D=D_{1}+D_{2}$ has
two components, which are very ample. Let $\beta$ be a curve class on $X$, let
$d_{1}=D_{1}\cdot\beta$ and let $d_{2}=D_{2}\cdot\beta$. Then we have the
following equality of virtual classes

$[\mathcal{Q}_{0,((d_{1},d_{2}))}^{\log}(X|D_{1}+D_{2},\beta)]^{\text{\rm
vir}}=(-1)^{d_{1}+d_{2}}\prod_{i=1}^{2}\operatorname{ev}_{i}^{*}D_{i}\cap[\mathcal{Q}_{0,2}(\oplus_{i=1}^{2}\mathcal{O}_{X}(-D_{i}),\beta)]^{\text{\rm
vir}}.$

In fact there is a much more general conjecture one could make for the
local/logarithmic correspondence for simple normal crossings divisors [TY23b].
The conjecture of [vGGR19] is one extreme. The other extreme, we call the
_corner_ theory. On the logarithmic side the geometry is of curves with a
single relative marking with maximal tangency to _all_ divisor components
simultaneously. Using exactly the same techniques we prove that this
generalisation holds for quasimap theory _in any rank_.

Figure 4.  The two s.n.c. maximal tangency geometries. Left-hand side depicts
[vGGR19] version and right-hand side depicts the _corner theory_.

######  Theorem 5.20.

Let $X$ be a GIT quotient $W/\\!\\!/G$ and let $D=D_{1}+\dots+D_{r}\subset X$
be a _very ample_ divisor satisfying Assumptions 2.1 and 3.1 with
$D_{1}\cap\dots\cap D_{r}\neq\emptyset$. Then

$[\mathcal{Q}_{0,(\underline{d},0)}^{\log}(X|D,\beta)]^{\text{\rm
vir}}=\prod_{i=1}^{r}(-1)^{d_{i}+1}\cdot\operatorname{ev}_{1}^{*}D_{i}\cap[\mathcal{Q}_{0,2}(\oplus_{i=1}^{r}\mathcal{O}_{X}(-D_{i}),\beta)]^{\text{\rm
vir}}.$

We start with Theorem 5.19. We first prove the equality in the following
special case.

######  Theorem 5.21.

Let $X=\mathbb{P}^{N_{1}}\times\mathbb{P}^{N_{2}}$. Let
$H_{1}^{\prime}=H_{1}\times\mathbb{P}^{N_{2}}$ for $H_{1}$ a coordinate
hyperplane in $\mathbb{P}^{N_{1}}$ and
$H_{2}^{\prime}=\mathbb{P}^{N_{1}}\times H_{2}$ for $H_{2}$ a coordinate
hyperplane in $\mathbb{P}^{N_{2}}$. Then the following diagram is Cartesian in
the category of ordinary stacks.

${{\mathcal{Q}_{0,((d_{1},d_{2}))}^{\log}(\mathbb{P}^{N_{1}}\times\mathbb{P}^{N_{2}}|H_{1}^{\prime}+H_{2}^{\prime},(d_{1},d_{2}))}}$${{\mathcal{Q}_{0,(d_{1},0)}^{\log}(\mathbb{P}^{N_{1}}\times\mathbb{P}^{N_{2}}|H_{1}^{\prime},(d_{1},d_{2}))}}$${{\mathcal{Q}_{0,(0,d_{2})}^{\log}(\mathbb{P}^{N_{1}}\times\mathbb{P}^{N_{2}}|H_{2}^{\prime},(d_{1},d_{2}))}}$${{\mathcal{Q}_{0,2}(\mathbb{P}^{N_{1}}\times\mathbb{P}^{N_{2}},(d_{1},d_{2}))}}$

###### Proof.

By stability, the boundary divisors of
$\mathcal{Q}_{0,(d_{1},0)}^{\log}(\mathbb{P}^{N_{1}}\times\mathbb{P}^{N_{2}}|H_{1}^{\prime},(d_{1},d_{2}))$
consist of quasimaps from curves with at most two components [Sha24, Example
3.1]. As a consequence the right-hand vertical morphism (or indeed the bottom
horizontal morphism) is such that the morphism on tropicalisations is
combinatorially flat. Therefore the fine and saturated fibre product coincides
with ordinary fibre product [MR24, Proposition 1.7.3], but the fine and
saturated fibre product is just
$\mathcal{Q}_{0,((d_{1},d_{2}))}^{\log}(\mathbb{P}^{N_{1}}\times\mathbb{P}^{N_{2}}|H_{1}^{\prime}+H_{2}^{\prime},(d_{1},d_{2}))$,
[Sha24, Lemma 2.9]. ∎

######  Proposition 5.22.

With the notation of Theorem 5.21, we have the following equality of (virtual)
classes

$[\mathcal{Q}_{0,((d_{1},d_{2}))}^{\log}(\mathbb{P}^{N_{1}}\times\mathbb{P}^{N_{2}}|H_{1}^{\prime}+H_{2}^{\prime},(d_{1},d_{2}))]^{\text{\rm
vir}}=(-1)^{d_{1}+d_{2}}\prod_{i=1}^{2}\operatorname{ev}_{i}^{*}H_{i}^{\prime}\,\cap\,[\mathcal{Q}_{0,2}(\oplus_{i=1}^{2}\mathcal{O}_{\mathbb{P}^{N_{1}}\times\mathbb{P}^{N_{2}}}(-H_{i}^{\prime}),(d_{1},d_{2}))]^{\text{\rm
vir}}$

###### Proof.

The local virtual class is

$[\mathcal{Q}_{0,2}(\oplus_{i=1}^{2}\mathcal{O}_{\mathbb{P}^{N_{1}}\times\mathbb{P}^{N_{2}}}(-H_{i}^{\prime}),(d_{1},d_{2}))]^{\text{\rm
vir}}=e(\oplus_{i=1}^{2}\mathbf{R}^{1}\pi_{*}(\mathscr{L}_{i}^{\vee}))\cap[\mathcal{Q}_{0,2}(\mathbb{P}^{N_{1}}\times\mathbb{P}^{N_{2}},(d_{1},d_{2})]$

where $\mathscr{L}_{i}$ denotes the line bundle over the universal curve of
$\mathcal{Q}_{0,2}(\mathbb{P}^{N_{1}}\times\mathbb{P}^{N_{2}},(d_{1},d_{2}))$
corresponding to
$\mathcal{O}_{\mathbb{P}^{N_{1}}\times\mathbb{P}^{N_{2}}}(H_{i}^{\prime})$.
But because of the Whitney sum formula, and the fact that pushforward commutes
with direct sum, we have that

$\displaystyle
e(\oplus_{i=1}^{2}\mathbf{R}^{1}\pi_{*}(\mathscr{L}_{i}^{\vee}))\cap[\mathcal{Q}_{0,2}(\mathbb{P}^{N_{1}}\times\mathbb{P}^{N_{2}},(d_{1},d_{2}))]$
$\displaystyle=$
$\displaystyle\prod_{i=1}^{2}e(\mathbf{R}^{1}\pi_{*}\mathscr{L}_{i}^{\vee})\cap[\mathcal{Q}_{0,2}(\mathbb{P}^{N_{1}}\times\mathbb{P}^{N_{2}},(d_{1},d_{2}))]$
$\displaystyle=$
$\displaystyle\prod_{i=1}^{2}[\mathcal{Q}_{0,2}(\mathcal{O}_{\mathbb{P}^{N_{1}}\times\mathbb{P}^{N_{2}}}(-H_{i}^{\prime}),(d_{1},d_{2}))]^{\text{\rm
vir}}.$

By Theorem 5.1

$\displaystyle[\mathcal{Q}_{0,(d_{1},0)}^{\log}(\mathbb{P}^{N_{1}}\times\mathbb{P}^{N_{2}}|H^{\prime}_{1},(d_{1},d_{2}))]\cdot[\mathcal{Q}_{0,(0,d_{2})}^{\log}(\mathbb{P}^{N_{1}}\times\mathbb{P}^{N_{2}}|H_{2}^{\prime},(d_{1},d_{2}))]$
$\displaystyle=(-1)^{(d_{1}+d_{2})}\prod_{i=1}^{2}\operatorname{ev}_{i}^{*}H_{i}\cdot[\mathcal{Q}_{0,2}(\oplus_{i=1}^{2}\mathcal{O}_{\mathbb{P}^{N_{1}}\times\mathbb{P}^{N_{2}}}(-H_{i}),(d_{1},d_{2}))]^{\text{\rm
vir}}$

Therefore it suffices to show that

$\displaystyle[\mathcal{Q}_{0,((d_{1},d_{2}))}^{\log}(\mathbb{P}^{N_{1}}\times\mathbb{P}^{N_{2}}|H_{1}^{\prime}+H_{2}^{\prime},(d_{1},d_{2}))]^{\text{\rm
vir}}$
$\displaystyle=[\mathcal{Q}_{0,(d_{1},0)}^{\log}(\mathbb{P}^{N_{1}}\times\mathbb{P}^{N_{2}}|H_{1}^{\prime},(d_{1},d_{2}))]\cdot[\mathcal{Q}_{0,(0,d_{2})}^{\log}(\mathbb{P}^{N_{1}}\times\mathbb{P}^{N_{2}}|H_{2}^{\prime},(d_{1},d_{2}))].$

This follows from Theorem 5.21. ∎

######  Remark 5.23.

The natural generalisation of this result for rank greater than $2$ will fail.
The introduction of additional markings will allow for boundary divisors in
the logarithmic quasimap space where the source curve contains more than two
components and so the fine and saturated product and the ordinary fibre
product may differ [Sha22, Remark 3.7.6].

######  Lemma 5.24.

Let $X=Z/\\!\\!/G$ be a GIT quotient and $D$ be a simple normal crossings
divisor on $X$ satisfying Assumptions 2.1 and 3.1. In addition suppose that
$D=D_{1}+D_{2}$ has two components, which are very ample. Let
$j:X\rightarrow\mathbb{P}=\mathbb{P}^{N_{1}}\times\mathbb{P}^{N_{2}}$ be the
induced morphism with $j^{-1}(H_{1}\times\mathbb{P}^{N_{2}})=D_{1}$ and
$j^{-1}(\mathbb{P}^{N_{1}}\times H_{2})=D_{2}$ and assume $H_{1},H_{2}$ are
coordinate hyperplanes. Then there is an induced morphism

$i:\mathcal{Q}^{\log}_{0,\alpha}(X|D_{1}+D_{2},\beta)\rightarrow\mathcal{Q}_{0,\alpha}^{\log}(\mathbb{P}|H_{1}^{\prime}+H_{2}^{\prime},(d_{1},d_{2}))$

where $H_{1}^{\prime}=H_{1}\times\mathbb{P}^{N_{2}}$,
$H_{2}^{\prime}=\mathbb{P}^{N_{1}}\times H_{2}$ and
$i_{*}\beta=(d_{1},d_{2})$. Moreover, this morphism admits a perfect
obstruction theory and the virtual pullback of the class
$[\mathcal{Q}_{0,\alpha}^{\log}(\mathbb{P}|H_{1}^{\prime}+H_{2}^{\prime},(d_{1},d_{2}))]$
coincides with the virtual class
$[\mathcal{Q}^{\log}_{0,\alpha}(X|D_{1}+D_{2},\beta)]^{\text{\rm vir}}$ as in
[Sha24, Theorem 14].

###### Proof.

This proof is similar to [Sha24, 5.10].The morphism $j$ morphism induces a
morphism of quotient stacks, which we also denote $j$,

$j:[Z/G]\rightarrow[\mathbb{A}^{N_{1}+N_{2}+2}/\mathbb{G}^{2}_{m}]$

such that $j^{-1}(\bar{H_{1}^{\prime}})=\bar{D_{1}}$ and
$j^{-1}(\bar{H_{2}^{\prime}})$. The divisors
$\bar{D_{1}}+\bar{D_{2}},\bar{H_{1}^{\prime}}+\bar{H_{2}^{\prime}}$ define
logarithmic structures via morphisms to $[\mathbb{A}^{2}/\mathbb{G}^{2}_{m}]$
such that there is a commutative diagram

${{[Z/G]}}$${{[\mathbb{A}^{N_{1}+N_{2}+2}/\mathbb{G}^{2}_{m}]}}$${[\mathbb{A}^{1}/\mathbb{G}_{m}].}$$\scriptstyle{j}$$\scriptstyle{\bar{D_{1}}+\bar{D_{2}}}$$\scriptstyle{\bar{H_{1}^{\prime}}+\bar{H_{2}^{\prime}}}$

Taking the induced distinguished triangle (on tangent complexes) gives

$\mathbb{T}_{j}\rightarrow\mathbb{T}^{\log}_{[Z/G]}\rightarrow\mathbf{L}j^{*}\mathbb{T}^{\log}_{[\mathbb{A}^{N_{1}+N_{2}+2}/\mathbb{G}^{2}_{m}]}\rightarrow\mathbb{T}_{j}[1].$

Next, consider the diagram

(36)
${\mathcal{C}_{X}}$${\mathcal{C}_{\mathbb{P}^{N}}}$${{[Z/G]}}$${{[\mathbb{A}^{N_{1}+N_{2}+2}/\mathbb{G}^{2}_{m}]}}$${\mathcal{Q}^{\log}_{0,\alpha}(X|D,\beta)}$${{\mathcal{Q}^{\log}_{0,\alpha}(\mathbb{P}|H_{1}^{\prime}+H_{2}^{\prime},(d_{1},d_{2}))}.}$$\scriptstyle{\pi}$$\scriptstyle{u}$$\scriptstyle{\pi_{\mathbb{P}}}$$\scriptstyle{u_{\mathbb{P}}}$$\scriptstyle{j}$$\scriptstyle{i}$

If we apply $\mathbf{R}\pi_{*}\circ\mathbf{L}u^{*}$ and dualise, we get a
distinguished triangle in $\mathcal{Q}^{\log}_{0,\alpha}(X|D,\beta)$

$\mathbf{L}i^{*}(\mathbf{R}(\pi_{\mathbb{P}})_{*}\mathbf{L}u_{\mathbb{P}}^{*}\mathbb{T}^{\log}_{[\mathbb{A}^{N_{1}+N_{2}+2}/\mathbb{G}^{2}_{m}]})^{\vee}\rightarrow(\mathbf{R}\pi_{*}\mathbf{L}u^{*}\mathbb{T}^{\log}_{[Z/G]})^{\vee}\rightarrow(\mathbf{R}\pi_{*}\mathbf{L}u^{*}\mathbb{T}_{j})^{\vee}\rightarrow\mathbf{L}i^{*}(\mathbf{R}(\pi_{\mathbb{P}})_{*}Lu_{\mathbb{P}}^{*}\mathbb{T}^{\log}_{[\mathbb{A}^{N_{1}+N_{2}+2}/\mathbb{G}_{m}^{2}]})^{\vee}[1].$

Note that the first two complexes in the triangle define the obstruction
theories for $\mathcal{Q}^{\log}_{0,\alpha}(X|D,\beta)$ and
$\mathcal{Q}^{\log}_{0,\alpha}(\mathbb{P}|H_{1}^{\prime}+H_{2}^{\prime},(d_{1},d_{2}))$.

We also have the commutative diagram

${{\mathcal{Q}^{\log}_{0,\alpha}(X|D,\beta)}}$${{\mathcal{Q}^{\log}_{0,\alpha}(\mathbb{P}|H_{1}^{\prime}+H_{2}^{\prime},(d_{1},d_{2}))}}$${\mathfrak{M}^{\log}_{0,\alpha}([\mathbb{A}^{2}/\mathbb{G}_{m}^{2}])}$$\scriptstyle{i}$

inducing a distinguished triangle

$\mathbf{L}i^{*}\mathbb{L}_{\mathcal{Q}^{\log}(\mathbb{P})/\mathfrak{M}^{\log}}\rightarrow\mathbb{L}_{\mathcal{Q}^{\log}(X)/\mathfrak{M}^{\log}}\rightarrow\mathbb{L}_{i}\rightarrow\mathbf{L}i^{*}\mathbb{L}_{\mathcal{Q}^{\log}(\mathbb{P})/\mathfrak{M}^{\log}}[1]$

where
$\mathcal{Q}^{\log}(X)=\mathcal{Q}^{\log}_{0,\alpha}(X|D,\beta),\,\mathcal{Q}^{\log}(\mathbb{P})=\mathcal{Q}^{\log}_{0,\alpha}(\mathbb{P}|H_{1}^{\prime}+H_{2}^{\prime},(d_{1},d_{2}))$
and
$\mathfrak{M}^{\log}=\mathfrak{M}^{\log}_{0,\alpha}([\mathbb{A}^{2}/\mathbb{G}_{m}^{2}])$.
Putting these together we get

${{\mathbf{L}i^{*}(\mathbf{R}(\pi_{\mathbb{P}})_{*}\mathbf{L}u_{\mathbb{P}}^{*}\mathbb{T}^{\log}_{[\mathbb{A}^{N_{1}+N_{2}+2}/\mathbb{G}^{2}_{m}]})^{\vee}}}$${{(\mathbf{R}\pi_{*}\mathbf{L}u^{*}\mathbb{T}^{\log}_{[Z/G]})^{\vee}}}$${(\mathbf{R}\pi_{*}\mathbf{L}u^{*}\mathbb{T}_{j})^{\vee}}$${\mathbf{L}i^{*}\mathbb{L}_{\mathcal{Q}^{\log}(\mathbb{P})/\mathfrak{M}^{\log}}}$${\mathbb{L}_{\mathcal{Q}^{\log}(X)/\mathfrak{M}^{\log}}}$${\mathbb{L}_{i}}$.$\scriptstyle{[1]}$$\scriptstyle{[1]}$

The first two (and last) vertical arrows come from the perfect obstruction
theory from [Sha24]. By [Man12, Construction 3.13] and [Man12, Remark 3.15] it
follows that $(\mathbf{R}\pi_{*}\mathbf{L}u^{*}\mathbb{T}_{i})^{\vee}$ defines
a perfect obstruction theory. By [Man12, Theorem 4.8] we have that
$i^{!}[\mathcal{Q}^{\log}_{0,\alpha}(\mathbb{P}|H_{1}^{\prime}+H_{2}^{\prime},(d_{1},d_{2}))]=[\mathcal{Q}^{\log}_{0,\alpha}(X|D,\beta)]^{\text{\rm
vir}}$. ∎

######  Remark 5.25.

One can show that this virtual pullback coincides with the pullback coming
form a perfect obstruction theory of the morphism
$\mathcal{Q}_{0,2}(X,\beta)\rightarrow\mathcal{Q}_{0,2}(\mathbb{P},(d_{1},d_{2}))$.

###### Proof of Theorem 5.19.

(37)
${{\mathcal{Q}_{0,((d_{1},d_{2}))}^{\log}(X|D_{1}+D_{2},\beta)}}$${{\mathcal{Q}_{0,((d_{1},d_{2}))}^{\log}(\mathbb{P}|H_{1}^{\prime}+H_{2}^{\prime},(d_{1},d_{2}))}}$${{\mathcal{Q}_{0,2}(X,\beta)}}$${{\mathcal{Q}_{0,2}(\mathbb{P},(d_{1},d_{2}))}}$$\scriptstyle{p}$$\scriptstyle{i}$$\scriptstyle{p^{\prime}}$$\scriptstyle{i^{\prime}}$

Let $\mathscr{L}_{i}$ (resp. $\mathscr{L}_{D_{i}}$) denote the universal line
bundle on the universal curve $\mathcal{Q}_{0,2}(\mathbb{P},\underline{d})$
(resp. $\mathcal{Q}_{0,2}(X,\beta)$) corresponding to
$\mathcal{O}_{\mathbb{P}}(H_{i}^{\prime})$ (resp. $\mathcal{O}_{X}(D_{i})$).

$\displaystyle