primary correspondence. Fix the degree
$d_{\beta}=c_{1}(T_{X})\cdot\beta\in\mathbb{Z}$.

######  Conjecture 5.3.2.

Under the identification of variables $-q=e^{iu}$, we have the equality

$(-q)^{-d_{\beta}/2}\mathsf{Z}_{\mathsf{PT}}\left(X,D;q|\prod_{i=1}^{k}\tau_{0}(\zeta_{i})|\bm{\mu}(\delta)\right)_{\beta}=(-iu)^{d_{\beta}+\sum\ell(\mu_{j})-|\mu_{j}|}\mathsf{Z}_{\mathsf{GW}}\left(X,D;u|\prod_{i=1}^{k}\tau_{0}(\zeta_{i})|\bm{\mu}(\delta)\right)_{\beta}.$

####  5.4. Descendent conjectures

We can also formulate the descendent correspondence, in parallel with the
absolute and relative situations explained in [51, 52]. The key idea for this
correspondence is a certain universal matrix $\widetilde{\mathsf{K}}$ indexed
by partitions $\alpha$ and $\widehat{\alpha}$

$\widetilde{\mathsf{K}}_{\alpha,\widehat{\alpha}}\in{\mathbb{Q}}[i,c_{1},c_{2},c_{3}]((u)).$

We will substitute $c_{i}=c_{i}(T^{\mathrm{log}}_{X})$ the Chern classes of
the logarithmic tangent bundle of $(X,D)$ so each matrix entry is a Laurent
series with coefficients in $H^{\star}(X)$.

Before stating the correspondence, we require some further notation. As in the
definition of evaluation spaces, let $(X|D)^{\ell}$ denote the moduli space of
ordered points in expansions of $X$, associated to a choice of tropical moduli
space of ordered points in $\Sigma_{X}$.

After choosing subdivisions compatibly, if we have $k$ internal marked points,
the evaluation maps $\mathsf{ev}_{i}:\mathsf{GW}(X)\rightarrow X$ for these
points can be refined to give an evaluation map

$\mathsf{ev}:\mathsf{GW}(X)\rightarrow(X|D)^{k}.$

Let $\Delta^{\mathrm{st}}\subset(X|D)^{k}$ denote the strict transform of the
small diagonal $X\subset X^{k}$. Given $\zeta\in H^{\star}(X)$, pullback to
$\Delta^{\mathrm{st}}$ followed by pushforward defines a cohomology class
$\zeta\cdot\Delta\in H^{\star}((X|D)^{k}).$

Given a partition $\alpha$ of length $k$ we define the descendent insertion

$\tau_{\alpha}(\zeta)=\prod\psi_{i}^{\alpha_{i}-1}\cdot\mathsf{ev}^{\star}(\zeta\cdot\Delta).$

Suppose we are given a partition $\alpha$ of length $k$, and cohomology
classes $\zeta_{1},\dots,\zeta_{k}$. We will sum over all set partitions $P$
of the index set $\\{1,\dots,k\\}$. Given such a partition $P$ with an element
$S\in P$ corresponding to a subset of $\\{1,\dots,k\\}$, we set $\alpha_{S}$
to be the corresponding sub-partition of $\alpha$ and $\zeta_{S}:=\prod_{i\in
S}\zeta_{i}$.

We now define the corrected descendent insertion

$\displaystyle\overline{\tau_{\alpha_{1}-1}(\zeta_{1})\cdots\tau_{\alpha_{k}-1}(\zeta_{k})}$
$\displaystyle=\sum_{P}(-1)^{\sigma(P)}\prod_{S\in
P}\sum_{\widehat{\alpha}}\tau_{\widehat{\alpha}}(\widetilde{\mathsf{K}}_{\alpha_{S},\widehat{\alpha}}\cdot\zeta_{S}).$

Here $(-1)^{\sigma(P)}$ is a sign convention to handle odd cohomology classes
$\zeta_{i}$, which is explained in [52, Section 0.4]).

We can now state the full descendent correspondence for PT theory using these
corrected descendents.

######  Conjecture 5.4.1.

Under the identification of variables $-q=e^{iu}$, we have the equality

$\displaystyle(-q)^{-d_{\beta}/2}$
$\displaystyle\mathsf{Z}_{\mathsf{PT}}\left(X,D;q|\tau_{\alpha_{1}-1}(\zeta_{1})\cdots\tau_{\alpha_{k}-1}(\zeta_{k})|\bm{\mu}(\delta)\right)_{\beta}=$
$\displaystyle(-iu)^{d_{\beta}+\sum\ell(\mu_{j})-|\mu_{j}|}\mathsf{Z}_{\mathsf{GW}}\left(X,D;u|\overline{\tau_{\alpha_{1}-1}(\zeta_{1})\cdots\tau_{\alpha_{k}-1}(\zeta_{k})}|\bm{\mu}(\delta)\right)_{\beta}.$

##  Part II The logarithmic degeneration package

In the first part of the paper, we have dealt exclusively with the DT (or PT)
and GW theory of a pair $(X|D)$. We now explain how to use this theory to
study enumerative invariants under sufficiently nice degenerations, and in
particular, establish the compatiblity of the logarithmic curve/sheaf
conjectures with degeneration.

#### The degeneration package

We begin with a simple normal crossings degeneration over a pointed curve
$(B,0)$555Our precise assumptions on $\mathcal{Y}\to B$ are described in the
next section.

$\mathcal{Y}\to B,$

that is smooth away from the point $0\in B$; the fiber over $0$ is a normal
crossings variety denoted $Y_{0}$. We consider the GW and DT spaces to the
fibers of the family $\mathcal{Y}/B$. On the GW side, this fits into the
formalism of [26] and is studied in detail in [2]. For the DT side, the
necessary foundations are provided in the next section. We are led to moduli
spaces:

${\coprod_{\chi}\mathsf{DT}_{\beta,\chi}(\mathcal{Y}/B)}$${\coprod_{g}\mathsf{GW}_{\beta,g}(\mathcal{Y}/B)}$${B.}$

The following picture has emerged for logarithmic GW invariants in a
degeneration, from work of Abramovich, Chen, Gross, Siebert, and the second
author. Specifically, we have three aspects of the package.

  1. (i)

Deformation invariance. The virtual cycles of any two fibers of the map

$\mathsf{GW}_{\beta,g}(\mathcal{Y}/B)\to B,$

are homologically equivalent in the total space, see [26].

  2. (ii)

Decomposition. The virtual class of $\mathsf{GW}_{\beta,g}(Y_{0})$ is a sum of
virtual classes indexed by distributions of the discrete data. The discrete
data are indexed by rigid tropical stable maps. The tropical stable maps
account for the topological type of the curve as it breaks over the components
of $Y_{0}$, including the curve class distributions and the contact orders
with the strata, see [2].

  3. (iii)

Splitting formula. Given a rigid tropical map $\gamma$, the virtual class of
the associated moduli space $\mathsf{GW}_{\gamma}(Y_{0})$ can be expressed in
terms of moduli spaces of the form $\mathsf{GW}(X_{v})$ attached to the
vertices of $\gamma$, together with cohomology classes pulled back along the
evaluation maps, see [3, 61].

In this second part of the paper, we establish the same package of three
results for DT and PT theory, but also strengthen the final formula on the GW.

#### Vertex-by-vertex numerical splitting

The strengthening referred to above concerns the splitting aspect of the
package. One key feature of our splitting formula, as stated in section 8 is
that it gives both a cycle-level and numerical statement, split across
vertices. On the GW side, this is stronger than the statement established in
[61], so in Section 12 we explain how to modify the argument here to obtain
the analogous strong splitting property for GW invariants in the expanded
formalism of [61]. To emphasize the point, we state explicitly the following:

Key Consequence. The numerical logarithmic GW/DT/PT invariants of the strata
of $Y_{0}$ determine the logarithmic GW/DT/PT invariants of a general fiber
$\mathcal{Y}_{t}$, with non-vanishing cohomology666In other words, insertions
that are restricted from the total space of the degeneration. insertions.

The GW/DT/PT invariants of a stratum $W$ in $Y_{0}$ are, by definition, the
invariants of the projectivization $\mathbb{P}_{W}(N_{W/\mathcal{Y}})$ of its
normal bundle.

To be explicit, we will show deformation invariance, expressing DT invariants
of $\mathcal{Y}_{t}$ in terms of those of $Y_{0}$. Second we prove a
decomposition result, expressing the DT invariants of $Y_{0}$ as a sum of
invariants indexed by rigid Hilbert $1$-complexes $\gamma$. Finally, we prove
a splitting formula, which expresses the contribution of $\gamma$ in terms of
DT invariants associated to the vertices of $\gamma$. The first two steps,
treated in the next two sections, are essentially formal.

The splitting aspect is the heart of this part of the paper. We will proceed
in several steps: a purely combinatorial version of the splitting is
established in Section 10. After this, a gluing formula parallel to the
existing GW formulas is established in Section 11, and finally the full
formula is completed in all theories in Section 12. The full statement of the
formulas and the degeneration compatiblity of the conjectures is stated before
the proofs in Section 8 and several basic examples are recorded in Section 9.

###  6\. DT space over a base and deformation invariance

####  6.1. Moduli of vertical $1$-complexes

Let $(B,0)$ be a smooth pointed curve and let

$\mathcal{Y}\to B$

be a simple normal crossings degeneration. Our definition of a simple normal
crossings degeneration is as follows. The space $\mathcal{Y}$ will be a simple
normal crossings pair as in the first part of the paper, but it is no longer
necessarily compact. It is equipped with a morphism

${\cal Y}\to B$

such that the preimage of $0$ is contained in the boundary divisor of ${\cal
Y}$. The map is requied to be flat with reduced fibers. Note that
$\mathcal{Y}$ may contain divisors that dominate $B$, i.e. “horizontal”
divisors, this allows the possibility that the generic fiber is equipped with
a nontrivial divisor. We denote the special fiber by $Y_{0}$.

Since the family is logarithmic, there is an associated map of cone complexes:

$\pi:\Sigma_{\mathcal{Y}}\to\mathbb{R}_{\geq 0}$

We denote the fiber of $\pi$ over the point $1$ by $\Delta_{Y_{0}}$. It is a
polyhedral complex, and may include bounded cells.

The irreducible components of $Y_{0}$ will be denoted as $X_{v}$ so we have

$Y_{0}=\bigcup_{v}X_{v},$

where $v$ runs over the vertices of $\Delta_{Y_{0}}$. Typically, $\mathcal{Y}$
will be a smooth projective variety fibered over $B$ in varieties of dimension
at most $2$, but this assumption is not necessary until the virtual class is
involved.

We now construct the associated moduli stack of expansions, using the results
of [42, Section 3]. First, let us treat $\Sigma_{\mathcal{Y}}$ as a cone
complex in absolute terms, and ignore the map to $\mathbb{R}_{\geq 0}$. The
constructions in [42] give rise to a set of embedded $1$-complexes in
$\Sigma_{\mathcal{Y}}$ and this set can be given the structure of a cone
complex. Let us denote this $T^{\prime}(\Sigma_{\mathcal{Y}})$.

We say a $1$-complex in $\Sigma_{\mathcal{Y}}$ is vertical if it is contracted
by the map $\pi$ above. Let $|T(\Sigma_{\mathcal{Y}}/{\mathbb{R}}_{\geq 0})|$
denote the subset of $|T^{\prime}(\Sigma_{\mathcal{Y}})|$ parametrizing
vertical $1$-complexes.

######  Lemma 6.1.1.

The subset $|T(\Sigma_{\mathcal{Y}}/{\mathbb{R}}_{\geq 0})|$ is the set of
points in a union of cones in $T^{\prime}(\Sigma_{\mathcal{Y}})$, for any cone
complex structure on the latter.

###### Proof.

Fix a cone complex structure on $T^{\prime}(\Sigma_{\mathcal{Y}})$ and choose
a point $p$ in the relative interior of a cone $\sigma$ that parameterizes a
cone complex that fails to be vertical. If $\tau$ is a cone that contains
$\sigma$ as a face, then for any point $q$ in the relative interior of $\tau$,
the $1$-complex corresponding to $q$ is also not vertical. It follows that we
can remove the locus of non-vertical $1$-complexes by passing to a union of
cones, and so the result follows. ∎

Let us fix a cone structure $T(\Sigma_{\mathcal{Y}}/{\mathbb{R}}_{\geq 0})$ as
well as a universal family $\mathbf{G}(\Sigma_{\mathcal{Y}}/{\mathbb{R}}_{\geq
0})$ of $1$-complexes over it. The latter will be compressed to just
$\mathbf{G}$ when there is no chance of confusion.

As a consequence of the lemma, there is a canonical map

$T(\Sigma_{\mathcal{Y}}/{\mathbb{R}}_{\geq 0})\to{\mathbb{R}}_{\geq 0}$

sending a vertical $1$-complex to the point to which it is contracted.

####  6.2. Decorated tropical spaces

As in Section 2.7, we can consider vertical Chow or Hilbert $1$-complexes, or
tropical maps depending on the context. In DT/PT theory, for fixed $\beta$ and
$\chi$, we have a diagram of cone complexes

$T_{\beta,\chi}(\Sigma_{\mathcal{Y}}/{\mathbb{R}}_{\geq 0})\rightarrow
T_{\beta}(\Sigma_{\mathcal{Y}}/{\mathbb{R}}_{\geq 0})\rightarrow
T(\Sigma_{\mathcal{Y}}/{\mathbb{R}}_{\geq 0}).$

We will always choose the cone structures on these spaces compatibly, so that
each of the forgetful maps is an isomorphism upon restriction to each cone.
Note that without further restrictions these cone complexes and the maps
between them are typically not of finite type.

####  6.3. Geometric moduli spaces

From the construction in the previous section, we can convert
$T(\Sigma_{\mathcal{Y}}/{\mathbb{R}}_{\geq 0})\to{\mathbb{R}}_{\geq 0}$ into a
map of Artin fans

$\mathsf{A}(T(\Sigma_{\mathcal{Y}}/{\mathbb{R}}_{\geq
0})]\to[\mathbb{A}^{1}/\mathbb{G}_{m}].$

By also base changing along the canonical map
$B\to[\mathbb{A}^{1}/\mathbb{G}_{m}]$ associated to the Cartier divisor $0$ in
$B$, we obtain the stack of expansions of a degeneration:

$\mathsf{Exp}(\mathcal{Y}/B)\to B,$

which carries a universal family of expansions determined by
$\mathbf{G}(\Sigma_{\mathcal{Y}}/{\mathbb{R}}_{\geq 0})$. As before, we have
decorated variants $\mathsf{Exp}_{\beta}(\mathcal{Y}/B)$ and
$\mathsf{Exp}_{\beta,\chi}(\mathcal{Y}/B)$ as well, and étale maps between
them.

We can use this to build the DT space, over $B$, in this setting. Given a
scheme $S\to B$, we can consider families of expansions given by commutative
triangles:

${S}$${\mathsf{Exp}(\mathcal{Y}/B)}$${B.}$

By pulling back the universal family, we obtain a family $\mathcal{Y}_{S}\to
B$. The fiber over a point $s\in S$, that lies over $b\in B$ is an expansion
over the fiber of $\mathcal{Y}$ over $b$. As before, at each point, there is
well-defined notion of tube component – a distinguished subset of the
$\mathbb{P}^{1}$-bundle components. Note that components of the special fiber
of $\mathcal{Y}$ are never tubes.

We can similarly study subschemes

${Z_{S}}$${\mathcal{Y}_{S}}$${S,}$

in expansions over a fixed $S\to B$. At each geometric point of $S$, we can
demand the DT stability condition, as before. This gives rise to a moduli
functor

$\mathsf{DT}(\mathcal{Y}/B)\to\mathsf{Exp}(\mathcal{Y}/B)\to B.$

If we instead study PT stable pairs in expansions over a fixed $S$, we obtain
the moduli functor $\mathsf{PT}(\mathcal{Y}/B)$.

The structure results from our earlier paper [42] carry over verbatim to this
context. We summarize the outcome.

######  Theorem 6.3.1.

The space $\mathsf{DT}(\mathcal{Y}/B)$ is universally closed and separated.
For each fixed choice of curve class $\beta$ and holomorphic Euler
characteristic $\chi$, the subspace parameterizing subschemes with these data
is bounded. The map

$\mathsf{DT}(\mathcal{Y}/B)\to\mathsf{Exp}(\mathcal{Y}/B)$

is equipped with a relative perfect obstruction theory. Parallel results hold
for $\mathsf{PT}(\mathcal{Y}/B).$

###### Proof.

The functor $\mathsf{DT}(\mathcal{Y}/B)$ can be viewed as a closed subfunctor
of the space $\mathsf{DT}(\mathcal{Y})$, where the latter parameterizes
proper, at most $1$-dimensional, closed subschemes of the total space.
Precisely, we demand that at every geometric point, the subscheme
$Z_{S}\hookrightarrow\mathcal{Y}_{S}$ is contained in a fiber of the
projection to ${\mathbb{A}}^{1}$. This establishes the universal closedness,
separatedness, and boundedness after fixing numerical data. For the perfect
obstruction theory, we note that the universal expansion is always a double
point degeneration, i.e. there are no triple points. The arguments of [41,
Section 3] therefore carry over without change, as in our first paper [42,
Section 5]. ∎

As before, we have decorated variants $\mathsf{Exp}_{\beta}(\mathcal{Y}/B)$
and $\mathsf{Exp}_{\beta,\chi}(\mathcal{Y}/B)$ as well, and the forgetful map
from $\mathsf{DT}_{\beta,\chi}({\cal Y}/B)$ factors over these.

####  6.4. Deformation invariance

We state the deformation invariance property of logarithmic DT theory.

######  Theorem 6.4.1.

Let $\mathcal{Y}\to B$ be a simple normal crossings degeneration of a
threefold. The space $\mathsf{DT}_{\beta,\chi}(\mathcal{Y}/B)$ has a relative
obstruction theory over $B$. As a result, there is an associated virtual class
in Borel–Moore homology:

$[\mathsf{DT}_{\beta,\chi}(\mathcal{Y}/B)]^{\mathsf{vir}}\in\mathsf{H}_{\star}(\mathsf{DT}_{\beta,\chi}(\mathcal{Y}/B)).$

Furthermore, if $j_{b}:\\{b\\}\hookrightarrow B$ denotes the inclusion, and
$Y_{b}$ denotes the corresponding fiber of $\mathcal{Y}$, then

$j_{b}^{!}[\mathsf{DT}_{\beta,\chi}(\mathcal{Y}/B)]^{\mathsf{vir}}=[\mathsf{DT}_{\beta,\chi}(Y_{b})]^{\mathsf{vir}}.$

Parallel results hold for $[\mathsf{PT}(\mathcal{Y}/B)]^{\mathsf{vir}}.$

The same construction also gives virtual classes in Chow groups, but the
Borel–Moore statement is more useful to us here.

###### Proof.

We have already noted that there is a relative obstruction theory for the
morphism

$\mathsf{DT}_{\beta,\chi}(\mathcal{Y}/B)\to\mathsf{Exp}(\mathcal{Y}/B).$

If we pullback this obstruction theory along the inclusion $j_{b}$, we recover
the obstruction theory on the smooth fibers of $\mathcal{Y}$ from [42], and
the obstruction theory of the previous section for the fiber over $0$. We can
define Borel–Moore virtual classes by applying virtual pullback defined by
Kapranov–Vasserot [29, Section 3.2]. Alternatively, we can use the Chow
theoretic virtual pullback [37] and then use the cycle class map. The
statement of the theorem now follows from formal properties of the pullback. ∎

###  7\. Decomposition into virtual strata

####  7.1. Virtual stratification

We have constructed a moduli space equipped with structure maps:

$\mathsf{DT}_{\beta,\chi}(\mathcal{Y}/B)\to\mathsf{Exp}(\mathcal{Y}/B)\to B.$

The second map has relative dimension $0$, but the fiber over $0$ will
typically be reducible. Let us now pass to the fibers over $0$:

$\mathsf{DT}_{\beta,\chi}(Y_{0})\to\mathsf{Exp}(Y_{0})\to\operatorname{Spec}(\mathbb{C}).$

By the discussion in the previous section, the first arrow is virtually
smooth, i.e. equipped with a relative perfect obstruction theory. However, the
$0$-dimensional stack $\mathsf{Exp}(Y_{0})$ is reducible. Locally in the
smooth topology, it is isomorphic to collection of toric varieties glued along
their boundary divisors. In fact, since we are free to choose the polyhedral
structure on $\mathsf{T}(\Sigma_{\mathcal{Y}}/\mathbb{R}_{\geq 0})$ so all its
cones are smooth, locally in the smooth topology, this stack has the structure
a normal crossings variety.

Since $\mathsf{Exp}(Y_{0})$ is stratified, by pulling back along
$\mathsf{DT}_{\beta,\chi}(\mathcal{Y}/B)\to\mathsf{Exp}(\mathcal{Y}/B)$ ,we
obtain a “virtual” stratification on the moduli space of subschemes777The
virtual stratification is in fact just a stratification in the “weak” sense: a
decomposition into locally closed subschemes, such that the closure of a
stratum is contained in a union of strata. We use the term “virtual” rather
than “weak” to remind the reader that it is the pullback stratification along
the map with the perfect obstruction theory. Indeed, this makes each virtual
stratum virtually smooth..

The discussion above has a combinatorial parallel. The fiber over the point
$1$ in

$\mathsf{T}(\Sigma_{\mathcal{Y}})\to\mathbb{R}_{\geq 0}$

is a polyhedral (and not necessarily conical) complex. We denote it
$\mathsf{T}(\Delta_{Y_{0}})$.

######  Proposition 7.1.1.

The strata of $\mathsf{Exp}(Y_{0})$ are in bijection with the vertices of
$\mathsf{T}(\Delta_{Y_{0}})$.

###### Proof.

The components of the fiber over $0$ in $\mathsf{Exp}(\mathcal{Y}/B)$ are in
bijection with those rays of $\mathsf{T}(\Sigma_{\mathcal{Y}})$ that map
surjectively onto $\mathbb{R}_{\geq 0}$. These are precisely in bijection with
the vertices of $\mathsf{T}(\Delta_{Y_{0}})$. ∎

######  Definition 7.1.2.

A rigid $1$-complex are the $1$-complexes corresponding to the vertices of
$\mathsf{T}(\Delta_{Y_{0}})$. Similarly, rigid Chow $1$-complexes are those
corresponding to vertices of $\mathsf{T}_{\beta}(\Delta_{Y_{0}})$; rigid
Hilbert $1$-complexes correspond to vertices of
$\mathsf{T}_{\beta,\chi}(\Delta_{Y_{0}})$.

By the choice of our cone structures on decorated spaces, a Chow or Hilbert
$1$-complex is rigid if and only if its underlying $1$-complex is rigid.

It follows that the virtual strata of $\mathsf{DT}_{\beta,\chi}(Y_{0})$ are in
bijection with a subset of these rigid $1$-complexes. Note that this can be a
strict subset, as we have no way of guaranteeing non-emptiness of any
particular fiber of the map
$\mathsf{DT}_{\beta,\chi}(Y_{0})\to\mathsf{Exp}(Y_{0})$. Nevertheless, our
arguments will typically be insensitive to this, so allowing the possibility
of a finite set of distinct, empty strata, it is harmless to think about the
virtual strata as being in bijection with these vertices.

####  7.2. Decomposition

By the deformation invariance above, in order to understand the virtual
fundamental class of the general fiber of the DT space of $\mathcal{Y}/B$, it
suffices to understand the fiber over $0$. We know that there is a natural
morphism:

$\mathsf{DT}_{\beta,\chi}(Y_{0})\to\mathsf{Exp}_{\beta,\chi}(Y_{0}).$

As noted previously, the stack $\mathsf{Exp}_{\beta,\chi}(Y_{0})$ has several
irreducible components corresponding to rigid Hilbert $1$-complexes.

Let us now fix one such rigid Hilbert $1$-complex $\gamma$. There is an
irreducible component

$\mathsf{Exp}_{\gamma}(Y_{0})$

associated to $\gamma$. Denote the fiber product of the inclusion of this
component with $\mathsf{DT}_{\beta,\chi}(Y_{0})$ by
$\mathsf{D}T_{\gamma}(Y_{0})$. As before, the map

$\mathsf{DT}_{\gamma}(Y_{0})\to\mathsf{Exp}_{\gamma}(Y_{0})$

is equipped with a perfect obstruction theory. By pulling back the fundamental
class, we obtain a virtual class
$[\mathsf{DT}_{\gamma}(Y_{0})]^{\mathsf{vir}}$.

We can now state the decomposition theorem for the virtual class.

######  Theorem 7.2.1.

There is an equality of homology classes on $\mathsf{DT}_{\beta,\chi}(Y_{0})$
given by

$\mathsf{DT}_{\beta,\chi}(Y_{0})=\sum_{\gamma}[\mathsf{DT}_{\gamma}(Y_{0})]^{\mathsf{vir}},$

where $\gamma$ ranges over the set of rigid Hilbert $1$-complexes.

###### Proof.

The result follows from the irreducible component decomposition of
$\mathsf{Exp}_{\gamma}(Y_{0})$ and the formal properties of the virtual
pullback. Again, we can either use [29], or Manolache’s virtual pullback [37].
In the latter case, we simply note that the class descends from cycles to Chow
groups. The statement in Borel–Moore homology follows by applying the cycle
class map. ∎

######  Remark 7.2.2.

Analogous results in the GW case can be found in [2, Theorem 3.11], though we
note that much of loc. cit. concerns itself with exposing the virtual
stratification of the space of logarithmic stable maps in terms of tropical
curves. Our formalism essentially builds this into the definition, rendering
it something of a tautology. A rigid tropical stable map to $\Delta_{Y_{0}}$
is a tropical stable map whose image $1$-complex is rigid; as in the DT/PT
setting, virtual irreducible components of $\mathsf{GW}(Y_{0})$ are indexed by
rigid tropical stable maps.

Note that in GW theory, there is an additional multiplicative factor. It is
referred to in [2] as the base order and also appears in [5, 17]. If $\gamma$
is the rigid tropical stable map with induced Chow $1$-complex
$\overline{\gamma}$, then define $\ell(\gamma)$, or simple $\ell$ if $\gamma$
is clear, to be the product, taken over all edges $e$ in the Chow $1$-complex
$\overline{\gamma}$, of the least common multiplies of the dilation factors
$m_{e^{\prime}}$ of edges $e^{\prime}$ in $\gamma$ that map to $e$.

The decomposition formula in GW theory states that the virtual class
associated to the general fiber is equal to the sum of virtual classes of
spaces associated to the rigid tropical maps, weighted by $\ell(\gamma)$.

####  7.3. Uniform rigid $1$-complexes

If we fix $\beta$ but allow $\chi$ to vary, then a priori there are infinitely
many rigid $1$-complexes appearing in the decomposition formula. This would
make the degeneration formula ill-suited for studying generating functions.
Fortunately, for the appropriate choice of cone structure, we have the
following uniformity in which rigid complexes arise.

######  Proposition 7.3.1.

Fix a curve class $\beta$. There is a finite set
$\overline{\gamma_{1}},\dots,\overline{\gamma_{r}}$ of rigid $1$-complexes
such that every rigid Chow (or Hilbert) $1$-complex with curve class $\beta$
has underlying $1$-complex in this set.

In other words, all rigid Hilbert $1$-complexes arise from choosing a
$1$-complex from this finite list and distributing curve class and Euler
characteristic decorations among the vertices/edges. In particular, there are
only finitely many vertices in $\mathsf{T}_{\beta}(\Delta_{Y_{0}})$.

###### Proof.

This follows from a similar argument to Proposition 3.8.1 using pure
$1$-complexes and combinatorial flatness. Recall from Section 3.8 the notions
of pure $1$-complexes, spaces $\mathsf{T}^{\mathsf{pure}}$ of pure
$1$-complexes, and the realization of the forgetful map
$\varphi:\mathsf{T}\rightarrow\mathsf{T}^{\mathsf{pure}}$ as the relative
space of $0$-complexes, of some complete subdivision extending the universal
$1$-complex. In our setting, by taking fibers over $1$, we have polyhedral (as
opposed to conical) versions of these constructions.

For fixed $\beta$, only a bounded subcomplex of pure complexes arise, so after
choosing complex structures appropriately, we have a composition of
combinatorially flat maps

$\mathsf{T}_{\beta}(\Delta)\rightarrow\varphi^{-1}(\mathsf{T}^{\mathsf{pure}}_{\sf
bd}(\Delta))\xrightarrow{\varphi}\mathsf{T}^{\mathsf{pure}}_{\sf bd}(\Delta).$

If we fix a vertex in the last space here, the vertices in its preimage with
respect to $\varphi$ correspond to $0$-complexes obtained by distributing
points to the vertices of a complete subdivision of $\Delta$. In particular
there are only finitely many such vertices. These give us the finitely many
$1$-complexes in the statement of the proposition.

∎

######  Remark 7.3.2.

In fact, one can show that, in the statement of the proposition, we can ignore
rigid $1$-complexes associated to $\beta$ that are not pure. Indeed by the
splitting formula, if there are any free or linear $2$-valent vertices, it
must support a non-minimal Hilbert polynomial and the corresponding invariant
will vanish. In fact, one can go further and show that the only rigid
$1$-complexes that matter are those which are topologically rigid: any nearby
$1$-complex has different combinatorial type. The details for both statements
will appear elsewhere.

###  8\. Splitting formulas and compatibility of the conjectures

####  8.1. Preliminaries

Let $\overline{\gamma}$ be a fixed rigid Chow $1$-complex, indexed by a vertex
of the polyhedral complex $T_{\beta}(\Delta_{{\cal Y}})$. In turn, each vertex
$v$ of $\overline{\gamma}$ corresponds to an irreducible component $X_{v}$
which is a torus-bundle over a stratum of $Y_{0}$. The goal of the splitting
formula is to calculate the contribution of $\overline{\gamma}$ in the
decomposition formula for GW/DT/PT invariants in terms of the corresponding
logarithmic invariants associated to its vertices. More precisely, we will
split, among vertices, the contribution to DT/PT invariants of each rigid
Hilbert $1$-complex $\gamma$ lying over $\overline{\gamma}$ (and similarly
with GW invariants and rigid tropical stable maps).

We may modify ${\cal Y}$ by a subdivision and root construction in such a way
that the vertices of $\overline{\gamma}$ are a subset of the vertices of the
height $1$ slice $\Delta_{\mathcal{Y}}$ of $\Sigma_{\mathcal{Y}}$, and
similarly that the edges of $\overline{\gamma}$ are a subset of the edges of
this slice. In particular, we can assume that $\overline{\gamma}$ has no tube
vertices.

Given a vertex $v$ of $\overline{\gamma}$, let $\beta(v)\in H_{2}(X_{v})$
denote the curve class associated to $\overline{\gamma}(v)$. Let
$e_{1}(v),\ldots,e_{m}(v)$ be the edges incident to $v$, which are decorated
with integers $n_{1},\dots,n_{m}$ from the labelling of $\gamma$.

Using Proposition 3.8.1, we have a uniform combinatorial flattening of the
evaluation maps

$T_{\beta(v)}(\Sigma_{v})\rightarrow\left(\prod P(e_{i}(v)\right)^{\dagger}.$

We stress again that this map is not of finite type, and infinitely many cones
of the domain might map to a single cone in the codomain and therefore the
specific flattening construction of Proposition 3.8.1 is crucial.

####  8.2. Cycle-theoretic splitting: sheaf theory

Fix a rigid Hilbert $1$-complex $\gamma$ lying over $\overline{\gamma}$;
concretely, this means we have assigned $\chi$-labels to each of the vertices.
We use $\gamma(v)$ as the symbol used to indicate the induced discrete data
for each vertex $v$ coming from $\gamma$.

Associated to the vertex $v$, we have the product of Hilbert schemes

$\mathsf{Hilb}^{n(v)}(D_{v}):=\prod_{i=1}^{m}\mathsf{Hilb}^{n_{i}}(D_{e_{i}(v)}).$

Each factor on the right is the logarithmic Hilbert scheme of points on the
appropriate divisor, with the number of points determined by the intersection
number. From the flattened tropical evaluation map recalled above, we can take
the corresponding tropical models, so there is a combinatorially flat
evaluation map

$\mathsf{ev}_{v}:\mathsf{DT}_{\gamma(v)}(X_{v})\to(\mathsf{Hilb}^{n(v)}(D_{v}))^{\dagger},$

where $\mathsf{Hilb}^{\dagger}$ is a log modification of the product over
$e_{i}(v)$. Consolidating these over vertices, we have

$\mathsf{ev}:\prod_{v}\mathsf{DT}_{\gamma(v)}(X_{v})\to\prod_{v}(\mathsf{Hilb}^{n(v)}(D_{v}))^{\dagger}$

Note that this morphism is an external product, i.e. it is obtained by a
product of maps, one attached to each vertex $v$. This will be important in
splitting contributions across vertices.

To formulate our splitting formula, we require the following construction,
further details about which appear in Section 11.1.1.

######  Construction 8.2.1.

There is a cutting map

$\kappa:\mathsf{DT}_{\gamma}(Y_{0})\to\prod_{v\in\gamma}\mathsf{DT}_{\gamma(v)}(X_{v}).$

This map is constructed first at the combinatorial level. Given a $1$-complex
$\gamma^{\prime}$ that retracts onto $\gamma$, a $1$-complex on the factor
corresponding to a vertex $v$ of $\gamma$ is determined by a neighborhood of
$v$. This gives rise to a cutting map of cone complexes, and consequently a
map of stacks

$\mathsf{Exp}_{\gamma}(Y_{0})\to\prod_{v}\mathsf{Exp}_{\gamma(v)}(X_{v}).$

In practice, this amounts to taking an expansion of $Y_{0}$ and performing a
partial normalization prescribed by $\gamma$. Intersecting with the subscheme
gives rise to a commutative diagram involving DT spaces and stacks of
expansions

${\mathsf{DT}_{\gamma}(Y_{0})}$${\prod_{v}\mathsf{DT}_{\gamma(v)}(X_{v})}$${\mathsf{Exp}_{\gamma}(Y_{0})}$${\prod_{v}\mathsf{Exp}_{\gamma(v)}(X_{v}).}$$\scriptstyle{\kappa}$

For every edge $e$ of $\gamma$, the Hilbert scheme for the corresponding
stratum $D_{e}$ occurs twice in the product $\prod_{v}\mathsf{Hilb}(D_{v})$,
once for each adjacent vertex. As a result, we can consider the diagonal
$\Delta_{H}\subset\prod_{v}\mathsf{Hilb}_{\beta}(D_{v})$. Let
$\Delta_{H}^{\dagger}$ denote the strict transform of the diagonal
$\Delta_{H}$ in the space
$\prod_{v}\mathsf{Hilb}^{\dagger}_{\beta(v)}(D_{v})$.

We are now in a position to precisely state the strongest form of the
logarithmic DT degeneration formula.

######  Theorem 8.2.2.

There is an equality of homology classes

$\kappa_{\star}\left[\mathsf{DT}_{\gamma}(Y_{0})\right]^{\mathrm{vir}}=\left(\mathsf{ev}\right)^{\star}\Delta^{\dagger}\cap\left[\prod_{v}\mathsf{DT}_{\gamma(v)}(X_{v})\right]^{\mathsf{vir}}.$

The analogous formula holds for
$\kappa_{\star}\left[\mathsf{PT}_{\gamma}(Y_{0})\right]^{\mathrm{vir}}$

####  8.3. Cycle-theoretic splitting: curve theory

A parallel cycle theoretic splitting holds in GW theory, although some of the
combinatorial aspects are a little different.

#####  8.3.1. Combinatorial organization

We continue to fix a Chow $1$-complex $\overline{\gamma}$, but now
parameterize it using a tropical stable map $\gamma$. Precisely, $\gamma$ is a
pair $(\mathbf{G},f)$, where $\mathbf{G}$ is a decorated tropical curve in the
sense of Section 2.7 and $f$ is a map to the polyhedral complex
$\Delta_{Y_{0}}$. A map

$f:\mathbf{G}\to\Delta_{Y_{0}}$

is called rigid if, similarly to the Chow and Hilbert cases, if it corresponds
to a vertex of the moduli space of tropical stable maps to $\Delta_{Y_{0}}$
with the chosen cone complex structure. As we have noted in Section 2.7, the
tropical stable map $\gamma$ determines a Chow $1$-complex
$\overline{\gamma}$.

Let us spell out the ways in which these two could differ:

  1. (i)

For each vertex of $\overline{\gamma}$, there could be several corresponding
vertices of $\mathbf{G}$ of the tropical map that map to the given vertex. If
this happens, the degree labels on $\gamma$ are required to sum to
$\overline{\gamma}$, so this can only happen in finitely many ways.

  2. (ii)

Given vertices $u$ and $u^{\prime}$ connected by an edge $e$ then, for each
lift $v$ and $v^{\prime}$ of these vertices to $\mathbf{G}$, there could be
several edges between $v$ and $v^{\prime}$. Recall that $e$ is decorated by a
multiplicity $n_{e}$, and each edge of $\mathbf{G}$ is comes with a dilation
factor associated to the map $f$. The factor is necessarily nonzero, and the
sum of multiplicities of all edges $e^{\prime}$ over $e$ must sum to $n_{e}$.

  3. (iii)

Similarly, for each leg $e$ of $\overline{\gamma}$ there can be finitely many
legs of $\mathbf{G}$ that map to $e$. Again, the multiplicities are
necessarily positive, and must sum to $n_{e}$.

######  Remark 8.3.1 (Automorphisms).

Another combinatorial complication in GW theory is that the rigid objects
$\gamma$ may have automorphisms. This stems from point (ii) above. Precisely,
suppose we have $u$ and $u^{\prime}$ in $\overline{\gamma}$ with an edge $e$
between them, and also lifts $v$ and $v^{\prime}$ in $\mathbf{G}$ with
multiple edges between them lying over $e$. If the intersection multiplicities
along these edges are the same, then there there are automorphisms of
$\mathbf{G}$ that commute with the map to $\Delta_{Y_{0}}$.

Now for each vertex $u$ in $\overline{\gamma}$, we have a well-defined target
$X_{u}$ which is a component of $Y_{0}$. We also have a curve class
$\beta_{u}$ associated to this vertex. Moreover, for each edge or leg $e$
incident to $u$, we have an intersection multiplicity $n_{e}$ with this
divisor. By choose a lift $\gamma$ of $\overline{\gamma}$, we enhance these
data. Precisely, for each vertex, we have chosen a number of connected
components to break this curve class into, indexed by vertices $v$ in
$\mathbf{G}$ over $u$. For each vertex $v$, i.e. for each such connected
component, we have chosen a genus, an intersection multiplicity with the
boundary of $X_{u}=X_{v}$, and a partition of each intersection multiplicity
at an edge.

#####  8.3.2. The formula

With the organization clarified, we can now state the formula. For each vertex
$v$ of $\gamma$, we set

$\mathsf{Ev}_{{\bm{\mu}}(v)}:=\prod_{i=1}^{m}\mathsf{Ev}_{\mu_{e_{i}(v)}}(D_{e_{i}(v)})$

to be the product of the GW evaluation spaces from Section 3.7.

Again using the flattened tropical evaluation

$T_{\beta(v)}(\Sigma_{v})\rightarrow\left(\prod P(e_{i}(v)\right)^{\dagger},$

which is the same one we used for the DT side, we have tropical models of the
evaluation space for for each vertex $v$:

$\mathsf{Ev}_{{\bm{\mu}}(v)}^{\dagger}:=\left(\prod_{i=1}^{m}\mathsf{Ev}_{\mu_{e_{i}(v)}}(D_{e_{i}(v)})\right)^{\dagger}.$

These are the target of evaluation maps from the GW spaces, and consolidating
over all vertices, we have:

$\mathsf{ev}:\prod_{v}\mathsf{GW}_{\gamma(v)}(X_{v})\to\prod_{v}\mathsf{Ev}_{{\bm{\mu}}(v)}^{\dagger}.$

Finally, observe that associated to $\gamma$ we have a component of the space
of maps to $Y_{0}$, denoted $\mathsf{GW}_{\gamma}(Y_{0})$. Recall the
automorphisms noted in Remark 8.3.1. We can partially rigidify this stack by
taking a cover, e.g. by labelling all the edges in $\mathbf{G}$. Let
$\mathsf{GW}^{\prime}_{\gamma}(Y_{0})$ denote the corresponding stack. See [2,
Section 2.6] for a detailed treatment. Finally, as in Remark 7.2.2 let
$\ell(\gamma)$ denote the product, taken over all edges $e$ in
$\overline{\gamma}$, of the least common multiplies of the dilation factors
$m_{e^{\prime}}$ of edges $e^{\prime}$ in $\gamma$ that map to $e$.888Note
that this factor was mistakenly dropped in [61]. It appears in the cycle
theoretic splitting formula for each rigid tropical curve but cancels out in
the final numerical splitting formula due to the base change order in the
decomposition part [2].

As in the construction above for the DT moduli spaces, there is a cutting map

$\kappa:\mathsf{GW}^{\prime}_{\gamma}(Y_{0})\to\prod_{v\in\gamma}\mathsf{GW}_{\gamma(v)}(X_{v}).$

Now if we let

$\Delta_{\mathsf{Ev}_{\bm{\mu}}}^{\dagger}\subset\prod_{v}\mathsf{Ev}_{{\bm{\mu}}(v)}^{\dagger}$

denote the strict transform of the diagonal, we can state the GW splitting
formula:

######  Theorem 8.3.2.

There is an equality of homology classes

$\kappa_{\star}\left[\mathsf{GW^{\prime}}_{\gamma}(Y_{0})\right]=\prod_{e}m_{e}\cdot\frac{1}{\ell(\gamma)}\left(\mathsf{ev}\right)^{\star}\Delta^{\dagger}\cap\left[\prod_{v}\mathsf{GW}_{\gamma(v)}(X_{v})\right]^{\mathsf{vir}},$

where the indicated product is the product of all dilation factors of all
bounded edges $e$ in the rigid tropical map $\gamma$.

####  8.4. Numerical degeneration formulas

We can specialize the cycle-theoretic result to obtain numerical degeneration
formulas.

We first fix some additional notation. For each rigid Chow $1$-complex
$\gamma$, we will sum over collections of partitions $\bm{\mu}:=(\mu_{e})$ of
the edge labels $n_{e}$. For each vertex $v$, we denote by $\bm{\mu}_{v}$ the
subset of $\bm{\mu}$ given by the edges adjacent to $v$. Since the strict
transform of the diagonal $\Delta_{\mathsf{Ev}^{\bm{\mu}}}^{\dagger}$ lives in
a product, we can take its Künneth decomposition:

$[\Delta_{\mathsf{Ev}_{\bm{\mu}}}^{\dagger}]=\sum_{j}\otimes_{v}\delta^{(j)}_{v}\in\bigotimes
H^{\star}(\mathsf{Ev}_{{\bm{\mu}}(v)}^{\dagger})$

Given an index set $S$, for each $i\in S$, we fix an insertion $\zeta_{i}\in
H^{\star}(\mathcal{Y})$ and integer $a_{i}$. We also denote by $\zeta_{i}$ the
restriction of this insertion to either a fiber $Y_{t}$ or an irreducible
component $X_{v}$ of an expansion of $Y_{0}$.

######  Corollary 8.4.1.

The following numerical degeneration formulas hold for DT/PT invariants in the
setting above.

$\langle\prod_{i\in
S}\tau_{a_{i}}(\zeta_{i})\rangle^{DT,Y_{t}}_{\beta,\chi}=\sum_{\gamma}\sum_{\bm{\mu}}\sum_{S=\coprod
S_{v}}\sum_{(j)}\frac{(-1)^{\bm{\mu}}}{\mathrm{Aut}(\bm{\mu})}\cdot
m_{\bm{\mu}}\prod_{v}\langle\prod_{i\in
S_{v}}\tau_{a_{i}}(\zeta_{i})|\bm{\mu}_{v}(\delta^{(j)}_{v})\rangle^{DT,X_{v}}_{\gamma^{\prime}(v)}.$

and

$\langle\prod_{i\in
S}\tau_{a_{i}}(\zeta_{i})\rangle^{PT,Y_{t}}_{\beta,\chi}=\sum_{\gamma}\sum_{\bm{\mu}}\sum_{S=\coprod
S_{v}}\sum_{(j)}\frac{(-1)^{\bm{\mu}}}{\mathrm{Aut}(\bm{\mu})}\cdot
m_{\bm{\mu}}\prod_{v}\langle\prod_{i\in
S_{v}}\tau_{a_{i}}(\zeta_{i})|\bm{\mu}_{v}(\delta^{(j)}_{v})\rangle^{PT,X_{v}}_{\gamma^{\prime}(v)}.$

In this corollary, the first sum is over rigid Hilbert $1$-complexes $\gamma$
with discrete invariants $\beta$ and $\chi$; the second sum is over all
collections of partitions of the edge labels. The third sum is over all
possible distribution of insertions among the vertices; the last sum is over
the terms of the Kunneth decomposition for the diagonal.

###### Proof.

Since the cohomology classes $\zeta_{i}$ are restricted from $\mathcal{Y}$,
using the deformation invariance and decomposition parts of our degeneration
package, we have the equality

$\langle\prod_{i\in
S}\tau_{a_{i}}(\zeta_{i})\rangle^{DT,Y_{t}}_{\beta,\chi}=\sum_{\gamma}\mathrm{deg}\left(\prod_{i=1}^{k}\tau_{a_{i}}(\zeta_{i})[\mathsf{DT}_{\gamma}]^{\mathrm{vir}}\right)$

where $\gamma^{\prime}$ ranges over rigid Hilbert $1$-complexes with total
discrete invariants $\beta$ and $\chi$.

To split the invariant associated to $\gamma$, we pushforward along the
cutting map:

$\kappa:\mathsf{DT}_{\gamma}(Y_{0})\to\prod_{v\in\gamma}\mathsf{DT}_{\gamma(v)}(X_{v}).$

Theorem 8.2.2 tells us how the virtual class pushes forward, and Corollary
4.2.4 gives us a decomposition of the diagonal in terms of the boundary
insertions on each factor of the product. It remains to analyze the descendent
operators.

By construction, we have the partial normalization of the universal target
over $\mathsf{DT}_{\gamma^{\prime}}(Y_{0})$

$p:\coprod\mathcal{X}_{v}\rightarrow\mathcal{Y_{0}}$

where $\mathcal{X}_{v}$ denotes the pullback of the universal target from
$\mathsf{DT}_{\gamma^{\prime}(v)}(X_{v})$. . By algebraic transversality, we
have an equality of ideal sheaves

$p^{\star}(\mathcal{I}_{\gamma})=\oplus\mathcal{I}_{v}$

as well as an equality of Chern characters

$p^{\star}\mathrm{ch}_{a+2}(\mathcal{I}_{\gamma})=\sum\mathrm{ch}_{a+2}(\mathcal{I}_{v}).$

The push-pull formula gives

$\kappa_{\star}(\tau_{a}(\zeta_{i})\circ)=\sum_{v}\tau_{a}(\zeta_{i})\circ\kappa_{\star}$

As we iterate over our insertions, we obtain the statement of the corollary.

∎

For the numerical GW degeneration formula, we will state it for the
disconnected invariants.

######  Corollary 8.4.2.

The following numerical degeneration formula holds for GW invariants in the
setting above.

$\langle\prod_{i\in
S}\tau_{a_{i}}(\zeta_{i})\rangle^{\bullet,GW,Y_{t}}_{\beta,g}=\sum_{\overline{\gamma}}\sum_{g(v),\bm{\mu}}\sum_{S=\coprod
S_{v}}\sum_{(j)}\frac{(-1)^{\bm{\mu}}}{\mathrm{Aut}(\bm{\mu})}\cdot
m_{\bm{\mu}}\prod_{v}\langle\prod_{i\in
S_{v}}\tau_{a_{i}}(\zeta_{i})|\bm{\mu}_{v}(\delta^{(j)}_{v})\rangle^{\bullet,GW,X_{v}}_{\gamma(v)}.$

In this statement, the first summation is over rigid Chow $1$-complexes. The
second summation is over genus decorations for vertices and partition
decorations for edges so that the total genus agrees with $g$. For connected
invariants, one can refine this equation to a summation over rigid tropical
maps. The two formulas are related by taking account of how labelling affects
automorphism factors.

###### Proof.

The formula follows analogously, by applying the Künneth decomposed formula
for the strict diagonal. We note two features that are different from the DT
side. First, the automorphism contribution, which is slightly different from
the DT/PT side, is accounted for by Remark 8.3.1. Second, the least common
multiple multiplicity $\ell(\gamma)$ from Remark 7.2.2 does not appear in the
final numerical formula. Indeed, a factor of $\frac{1}{\ell(\gamma)}$ appears
in Theorem 8.3.2. However, for each rigid tropical map $\gamma$, the “base
order” in the decomposition formula [2] produces a multiple of $\ell(\gamma)$
as in Remark 7.2.2. These cancel in the final formula, leading to the claimed
result. ∎

######  Remark 8.4.3.

As noted in the introduction, a feature of these corollaries is that the
decomposition is “vertex by vertex”, which allows to use the Künneth
decomposition to give a significantly improved splitting as compared to the
statement in [61]. The cost, as compared to the situation of smooth pairs, is
the need for the exotic evaluation classes discussed in the introduction.
Precisely, the exotic insertions arise from the blowup step discussed in
Section 4.

####  8.5. Generating function corollaries

As a consequence of the numerical degeneration formula, we obtain degeneration
formulas for the GW/DT/PT partition functions. For these formulas to make
sense, it is important that we are summing over a fixed set of rigid Chow
$1$-complexes that only depends on $\beta$ and is independent of all other
discrete invariants (genus or Euler characteristic).

For DT and PT partition functions, the degeneration formulas are as follows.
In order to display the equations compactly, we compress
$\mathsf{Z}_{\mathsf{DT}}\left(Y_{t};q|\prod_{i\in
S}\tau_{a_{i}}(\zeta_{i})\right)_{\beta}$ to $\mathsf{Z}_{\mathsf{DT}}(q)$,
and similarly for PT and GW series, with the variable for the GW series being
$u$.

$\mathsf{Z}_{\mathsf{DT}}(q)=\sum_{\overline{\gamma}}\sum_{\bm{\mu}}\sum_{S=\coprod
S_{v}}\sum_{(j)}\frac{(-1)^{\bm{\mu}}m_{\bm{\mu}}}{\mathrm{Aut}(\bm{\mu})}\frac{1}{q^{|\bm{\mu}|}}\prod_{v}\mathsf{Z}_{\mathsf{DT}}\left(X_{v},D_{v};q|\prod_{i\in
S_{v}}\tau_{a_{i}}(\zeta_{i})|\bm{\mu}_{v}(\delta^{(j)}_{v})\right)_{\beta(v)}$

and

$\mathsf{Z}_{\mathsf{PT}}(q)=\sum_{\overline{\gamma}}\sum_{\bm{\mu}}\sum_{S=\coprod
S_{v}}\sum_{(j)}\frac{(-1)^{\bm{\mu}}m_{\bm{\mu}}}{\mathrm{Aut}(\bm{\mu})}\frac{1}{q^{|\bm{\mu}|}}\prod_{v}\mathsf{Z}_{\mathsf{PT}}\left(X_{v},D_{v};q|\prod_{i\in
S_{v}}\tau_{a_{i}}(\zeta_{i})|\bm{\mu}_{v}(\delta^{(j)}_{v})\right)_{\beta(v)}.$

For the GW partition function, we have

$\mathsf{Z}_{\mathsf{GW}}(u)=\sum_{\overline{\gamma}}\sum_{\bm{\mu}}\sum_{S=\coprod
S_{v}}\sum_{(j)}\frac{(-1)^{\bm{\mu}}m_{\bm{\mu}}}{\mathrm{Aut}(\bm{\mu})}\cdot
u^{2\ell(\bm{\mu})}\prod_{v}\mathsf{Z}_{\mathsf{GW}}\left(X_{v},D_{v};u|\prod_{i\in
S_{v}}\tau_{a_{i}}(\zeta_{i})|\bm{\mu}_{v}(\delta^{(j)}_{v})\right)_{\beta(v)}.$

In each of these expressions, the first summation is over the finite set of
rigid Chow $1$-complexes.

As a result, we have the compatibility of all of the logarithmic curve/sheaf
conjectures from Section 5. As above, we fix an snc degeneration $\mathcal{Y}$
and, for each stratum $Z_{w}$ of $Y_{0}$, let $X_{w}$ denote the
projectivization of its normal bundle in $\mathcal{Y}$, viewed, up to further
logarithmic modification, as a smooth variety with simple normal crossings
boundary. We fix insertions $\zeta_{i}\in H^{\star}(\mathcal{Y})$.

######  Corollary 8.5.1.

If each of conjectures 5.3.1, 5.3.2, and 5.4.1 hold for all components $X_{w}$
in $Y_{0}$ with insertions $\zeta_{i}$, then they also hold for $Y_{t}$ with
insertions $\zeta_{i}$.

####  8.6. A roadmap to the proof

The proof of the degeneration formula entails a number of steps. The first
task is to establish a parallel statement in a purely combinatorial context,
effectively describing the decomposition of
$\mathsf{Exp}_{\beta,\chi}(Y_{0})$, and this is done in Section 10. Next, we
prove a weak version of the formula, involving $0$-dimensional subschemes on
the universal divisor expansion over the moduli space. This is done in Section
11 and brings the DT side of the story to essentially the same level of
development as the existing logarithmic GW gluing result [61, Theorem B].
Finally, we combine these weaker statements, on both GW and DT sides, with
logarithmic intersection theory arguments to upgrade them to the claimed
“vertex-by-vertex” formula above. This latter step is done on both GW and DT
sides, and we emphasize is new even on the GW side.

###  9\. First examples

In this section, we explore the splitting formula in some basic examples, and
examine its consequences for the logarithmic curve/sheaf correspondence.

####  9.1. Degree zero invariants

The DT invariants counting zero-dimensional ideal sheaves is already very
interesting. For a pair $(X|D)$, the series for zero-dimensional subschemes
conjecture to be:

$\mathsf{Z}_{\mathsf{DT}}\left(X,D;q)\right)_{\beta=0}=M(-q)^{\int_{X}c_{3}(T^{\mathsf{log}}_{X}\otimes
K^{\mathsf{log}}_{X})},$

where

$M(q)=\prod_{n\geq 1}\frac{1}{(1-q^{n})^{n}}$

is the MacMahon function. In particular, if $X$ is a toric threefold and $D$
is its toric boundary the conjecture collapses to the statement

$\mathsf{Z}_{\mathsf{DT}}\left(X,D;q)\right)_{0}=1.$

While the general conjecture is presently open, this special case is a
pleasant consequence of the degeneration formalism. Consider the flat
degeneration of $X$ to the normal cone of a torus invariant divisor:

$\mathcal{Y}\to{\mathbb{A}}^{1}.$

The special fiber $Y_{0}$ consists of two components $X_{1}$ and $X_{2}$. Both
of them are toric, and in fact, the precise toric structure will not matter in
what follows.

Since we are dealing with $0$-dimensional subschemes, there is no step in the
formula for flattening of the evaluation map. There are also no tube
subschemes, so the formula simplifies as follows. Fix a value $\chi$ for the
holomorphic Euler characteristic. We have an induced degeneration of the
moduli space

$\mathsf{DT}_{0,n}(\mathcal{Y})\to{\mathbb{A}}^{1}.$

The special fiber decomposes into a union:

$\mathsf{DT}_{0,n}(\mathcal{Y}_{0})=\coprod_{n_{1}+n_{2}=n}\mathsf{DT}_{0,n_{1}}(X_{1})\times\mathsf{DT}_{0,n_{2}}(X_{2}).$

The decomposition is compatible with virtual classes. We can take degree, and
then pass to generating functions to get:

$\mathsf{Z}_{\mathsf{DT}}\left(X,D;q)\right)_{0}=\mathsf{Z}_{\mathsf{DT}}\left(X_{1},D_{1};q)\right)_{0}\cdot\mathsf{Z}_{\mathsf{DT}}\left(X_{2},D_{2};q)\right)_{0}$

Now, we recognize that for the appropriate choice of stack of expansion, there
is an identification of spaces

$\mathsf{DT}_{0,n}(X)\cong\mathsf{DT}_{0,n}(X_{1})\cong\mathsf{DT}_{0,n}(X_{2}),$

compatible with virtual structures. Therefore, writing $F(q)$ for the
generating series $\mathsf{Z}_{\mathsf{DT}}(X,D;q)$, we observe that

$F(q)=F(q)^{2}.$

The terms are concentrated in non-negative degrees and the constant term is
equal to $1$, so the series is invertible and we conclude that $F(q)=1$.

####  9.2. Conics in $\mathbb{P}^{2}\times\mathbb{P}^{1}$

In this example, we explain what the degeneration formula looks like in a
simple example – counting conics through $5$ points in $\mathbb{P}^{2}$. In
the following example, we consider a more sophisticated version of this
geometry.

#####  9.2.1. The invariants

We now consider the threefold $X$ to be $\mathbb{P}^{2}\times\mathbb{P}^{1}$.
We give it the toric logarithmic structure and think of the point on the
second factor as being the height. Let $h$ denote the curve class of the form
$\ell\times\\{\sf pt\\}$, where $\ell$ is a line in $\mathbb{P}^{2}$. Set
$\beta=2h$. Note in particular that this means the intersection numbers with
the divisors $\mathbb{P}^{2}\times\\{0\\}$ and
$\mathbb{P}^{2}\times\\{\infty\\}$ are $0$.

Let us describe a few of the subschemes that can appear in this space. The
minimum possible value of the Euler characteristic is $\chi=1$, and this is
the case when we have a pure curve contained in a $\mathbb{P}^{2}$ slice. On
the other hand, the union of two lines at different heights appears in the
moduli space where $\beta=2h$ but $\chi=2$. Higher Euler characteristic
subschemes can also appear by taking a conic at fixed higher, together with an
embedded point.

The virtual dimension of the moduli space is equal to $6$ for all values on
$\chi$. We can view these roughly as $5$ parameters to choose a conic in
$\mathbb{P}^{2}$ and another parameter for the height direction. We impose
conditions to extract an invariant. The simplest invariants are obtained by
forcing point incidences in the $\mathbb{P}^{2}$-direction and then imposing a
height. We can do this precisely by taking $4$ primary insertions of the

$\\{\mathsf{pt}\\}\times\mathbb{P}^{1}\subset\mathbb{P}^{2}\times\mathbb{P}^{1}.$

and one point insertion.

######  Remark 9.2.1 (Convention on conditions on $S\times\mathbb{P}^{1}$).

In the discussion, as well as the examples of the for $S\times\mathbb{P}^{1}$
that follow, we take the convention that the condition of having “$k$ point
conditions” will mean imposing $(k-1)$ conditions of the form
$\mathsf{pt}\times\mathbb{P}^{1}$ and one point condition.

#####  9.2.2. The target degeneration

In the presence of point conditions, we set the degeneration up as follows.
Choose five $\mathbb{C}[\\![t]\\!]$-points of $\mathbb{P}^{2}$, given in
homogeneous coordinates by:

$p_{i}(t)=(1\colon z^{i}_{1}(t)\colon z^{i}_{2}(t)),\ \ \textsf{for}\ \
z^{i}_{j}(t)\in\mathbb{C}[\\![t]\\!].$

We choose the elements $z^{i}_{j}(t)$ so that their $t$-valuations are
generic, e.g. they give $10$ distinct positive integers. We denote the
valuation vectors of these points by $p_{i}^{\mathsf{trop}}$. We now choose
any toric degeneration $\mathcal{P}^{2}\to{\mathbb{A}}^{1}$ of
$\mathbb{P}^{2}$ over ${\mathbb{A}}^{1}_{t}$, such that the closures of these
points lie in strata of codimension $0$ in the special fiber.

#####  9.2.3. The degeneration of the moduli space

We now consider the moduli space $\mathsf{PT}_{2h,\chi}({\cal
P}^{2}\times\mathbb{P}^{1})$ over ${\mathbb{A}}^{1}_{t}$. The virtual
irreducible components of this moduli space in the special fiber are in
bijection with rigid Hilbert $1$-complexes. Let us describe some of these.

The simplest type of rigid $1$-complex is actually a virtually birational copy
of the original moduli space
$\mathsf{PT}_{2h,\chi}(\mathbb{P}^{2}\times\mathbb{P}^{1})$. The rigid
$1$-complex is depicted in the figure below

$p^{\mathsf{trop}}_{2}$$p^{\mathsf{trop}}_{1}$$p^{\mathsf{trop}}_{3}$$p^{\mathsf{trop}}_{4}$$p^{\mathsf{trop}}_{5}$$2$$2$$2$
Figure 1. A rigid Chow $1$-complex corresponding to a component in the moduli
space $\mathsf{PT}_{2h,\chi}({\cal P}^{2}_{0}\times\mathbb{P}^{1})$. The
component is virtually birational to
$\mathsf{PT}_{2h,\chi}(\mathbb{P}^{2}\times\mathbb{P}^{1})$. The vertices
corresponding to $p_{i}^{\mathsf{trop}}$ are components of the expansion that
contain the special fibers of $p_{i}(t)$.

This rigid $1$-complex does not interact with the new components that contain
the points $p_{i}(0)$. We will see shortly that the contributions from such
vertices vanish.

In order to force some interaction with the new points, consider the same
$1$-complex as the one above, but translated to the point
$p_{1}^{\mathsf{trop}}$. If we had not subdivided to include these points,
this $1$-complex would not be rigid – we see that small translations of the
$1$-complex would have the same combinatorial type. Indeed, the combinatorial
type includes this incidence information, and as a result, the combinatorial
type changes under deformation.

More interestingly, we consider the much more degenerate rigid $1$-complex
pictured in Figure 2. In this case, the $5$ of the edges/legs of $1$-complex
pass through points $p_{i}^{\mathsf{trop}}$.

$p_{2}$$p_{1}$$p_{3}$$p_{4}$$p_{5}$ Figure 2. A rigid Chow $1$-complex
corresponding for conics in the degenerate target ${\cal
P}^{2}_{0}\times\mathbb{P}^{1}$.

Again note that it is only because we have subdivided to include
$p_{i}^{\mathsf{trop}}$ that these are rigid. For future reference, we refer
to this particular combinatorial type where $\gamma$ meets all these points as
the tropically distinguished one. Note that in this geometry, for general
valuations as above, there will be a unique rigid $1$-complex that passes
through all the points $p_{i}^{\mathsf{trop}}$, also satisfies the balancing
condition (all non-balanced ones can be ignored). This follows from standard
considerations in tropical geometry, see for instance [22, 43].

#####  9.2.4. Vanishing arguments

Now let $\gamma$ be any rigid Hilbert $1$-complex. We know from the main
splitting formula that the virtual class decomposes as:

$\left[\mathsf{PT}_{2h,\chi}({\cal P}_{0}^{2}\times\mathbb{P}^{1})\right]^{\sf
vir}=\sum_{\gamma}[\mathsf{PT}_{\gamma}]^{\sf vir}.$

We now impose the point conditions. One can easily see that if a point
constraint $p_{i}(0)$ is contained a component $X_{v}$ of the special fiber
that is not also a vertex of the rigid tropical curve $\gamma$, then the
subschemes in the space $\mathsf{PT}_{\gamma}$ are all disjoint from this
point. Therefore, admitting the tropical argument above, after imposing the
point conditions, the only relevant summand in the equation is above occurs
when $\gamma$ is the tropically distinguished one.

#####  9.2.5. The splitting

We now compute the invariant by taking apart the special fiber. In this case,
no blowups are necessary to flatten the evaluation map.

Running the degeneration formula requires splitting the diagonal. This amounts
to assigning an a point condition to two out of the three bounded edges in the
graph above. Altogether, we find that the DT invariant for conics is the
fourth power of the PT invariant for lines in
$\mathbb{P}^{2}\times\mathbb{P}^{1}$ through two points, so we find

$\mathsf{Z}_{\mathsf{PT}}(X,D;q)_{2h}=(\mathsf{Z}_{\mathsf{PT}}(X,D;q)_{h})^{4}.$

The invariant $\mathsf{Z}_{\mathsf{PT}}(X,D;q)_{h}$ can in turn be calculated
from known invariants. The underlying cycle of a stable pair with curve class
$h$ is necessarily a smooth $\mathbb{P}^{1}$. The moduli space is therefore
isomorphic to the PT moduli space of the local geometry
$\mathcal{O}_{\mathbb{P}^{1}}\oplus\mathcal{O}_{\mathbb{P}^{1}}(1)$ over
$\mathbb{P}^{1}$, relative to three fibers. The formulas (via the normalized
DT series) can be found in [49]. A similar argument applies for the usual DT
series, to give the same relation.

######  Remark 9.2.2 (A better way of imposing point conditions).

In the setting above, when we impose point condition, we have taken a “by the
book” approach. A more direct approach would be to work with a cut down moduli
space $\mathsf{PT}^{p}{\beta,\chi}(\mathcal{Y})$ which imposes the incidence
conditions to the points $p_{i}(t)$ on the whole family. One can then just use
an adapted degeneration formula in this setting that directly relates the cut
down moduli stacks. If set up in this way, all $1$-complexes are required to
pass through the points $p_{i}^{\mathsf{trop}}$. The arguments of the paper
carry over this cut down setting with cosmetic changes.

####  9.3. Flattening and exotic invariants

In the discussion above, a very simple form of the gluing formula holds. Let
us explain why, and in the process where one should “look” for correction
terms. In the process, we will see the need for exotic insertions.

Recall the setup of the degeneration package. We have a degeneration ${\cal
Y}\to B$ with special fiber $Y_{0}$. There is an associated degeneration of
the DT moduli space

$\mathsf{DT}_{\beta,\chi}({\cal Y})\to B,$

and the special fiber $\mathsf{DT}_{\beta,\chi}(Y_{0})$ consists of a number
of virtual irreducible components indexed by rigid Hilbert $1$-complexes
$\gamma$. We denote them $\mathsf{DT}_{\gamma}(Y_{0})$. We then have a map

$\prod_{v}\mathsf{DT}_{\gamma(v)}(X_{v})\to\prod_{e}\mathsf{Hilb}^{\gamma(e)}(D_{e})^{2}=\prod_{v}\mathsf{Hilb}^{\gamma(e)}(D_{v}).$

The equality on the right is simply a regrouping according to flags incident
to vertices. The cycle-level degeneration formula then states that if we
choose tropical models so that the induced map on cone spaces is
combinatorially flat, then the pullback of the strict transform of the
diagonal locus on the right is equal to the virtual class
$[\mathsf{DT}_{\beta,\chi}(Y_{0})]^{\sf vir}$, up to explicit combinatorial
multiplicities. To summarize, we note that:

  1. (i)

The tropical model of the product of Hilbert schemes over edges on the right
may not be a product of tropical models, however

  2. (ii)

the tropical model can be chosen such that the product of Hilbert schemes over
vertices on the right is still a product.

We now give two combinatorial examples of evaluation maps that exhibit (i) the
non-flatness of evaluations, and (ii) the need to flatten “jointly”, i.e. that
the edge-by-edge product structure cannot be maintained. The latter phenomenon
is responsible for the exotic invariants.

#####  9.3.1. Single divisor phenomena

We exposit and example due to Kennedy-Hunt, in his study of PT spaces on toric
surfaces relative to the full toric boundary [32]. We can again work in
dimension $2$ to see this phenomenon, and in fact, we can assume that there
are no embedded points.

Let $S$ be the toric surface whose rays are given by the positive real spans:

$\rho_{1}={\mathbb{R}}_{\geq 0}(1,0),\ \ \rho_{2}={\mathbb{R}}_{\geq
0}(-1,-1),\ \ \rho_{3}={\mathbb{R}}_{\geq 0}(1,-1).$

We can determine a curve class $\beta$ by imposing intersection numbers with
the three associated divisors

$\beta\cdot D_{1}=4,\ \ \beta\cdot D_{2}=\beta\cdot D_{3}=2.$

We let $\chi$ be the minimum possible value of the holomorphic Euler
characteristic, and then study $\mathsf{PT}_{\beta,\chi}(S)$. There is an
evaluation map

$-\cap D_{1}\colon\mathsf{PT}_{\beta,\chi}(S)\to\mathsf{Hilb}^{4}(D_{1}).$

Kennedy-Hunt observes that this map is not combinatorially flat. Indeed, the
cone complex associated to the source contains a cone dual to the tropical
curve given below:

$\cap\ D_{1}$ Figure 3. A combinatorial type exhibiting the non-flatness of
the evaluation map from the PT moduli space.

This cone maps, in codomain, to the induced expansion given by placing a
vertical line far to the right of the picture and intersecting with $3$ rays
parallel to the $x$-axis. This induces an expansion on $\mathbb{R}$. The
target cone is $2$-dimensional. However, if for example, we fix the middle of
the $3$ rays, the position of one of the rays determines the position of the
other, because the base vertices of these rays form a cycle.

#####  9.3.2. Coupling phenomena

While the above non-flatness illustrates the need to blow up, it does not show
why exotic insertions are necessary. For that we consider the following
example. We let $X=\mathbb{P}^{2}\times\mathbb{P}^{1}$ with its toric
logarithmic structure. Building on the conics case we looked at before, we
consider cubics in the $\mathbb{P}^{2}$-direction, i.e. $\beta=h\otimes{\sf
pt}$.

Let $H_{0}$, $H_{1}$, and $H_{2}$ be the pullbacks of the toric lines in
$\mathbb{P}^{2}$ and $D_{0}$ and $D_{\infty}$ the pullbacks of the toric
points on $\mathbb{P}^{1}$. We impose maximal tangency conditions, i.e. the
partition $(3)$ along the $H_{0}$ and $H_{1}$, simple tangency, i.e. the
partition $(2+1)$ along $H_{2}$ and of course have tangency $0$ along the
$D_{j}$.

The typical member of this moduli space is a plane cubic, at fixed height. The
Euler characteristic of such a subscheme is $0$. Higher Euler characteristic
subschemes can be obtained, for example by adding embedded points. The virtual
dimension of the moduli space is $9$. The tangency imposes $5$ conditions. We
then impose two bulk incidences of the form ${\sf pt}\otimes\mathbb{P}^{1}$
and one point insertion to fix the “height”.

With these conditions imposed, we can choose a degeneration of the
$\mathbb{P}^{2}$ as we did for conics, and specialize the points into
different components. Doing this, the following rigid Chow $1$-complex
appears:

$(2+1)$Newton subdivisionDual polyhedral
decomposition$(3)$$(3)$$H_{2}$$H_{1}$$H_{0}$ Figure 4. The picture on the left
is the dual Newton subdivision and the picture on the right is the dual Chow
$1$-complex. The picture has been decorated also with the choice of partition
along the legs.

In the splitting formula, we start with evaluation maps

$\prod_{v}\mathsf{DT}_{\beta,\chi}(X_{v})\to\prod_{e}{\sf Hilb}(D_{e})^{2},$

with the products taken over the three vertices in the domain, and three edges
in the codomain. The individual tropical evaluation maps

$\mathsf{DT}_{\beta,\chi}(X_{v})\to{\sf Hilb}(D_{e})$

for $v$ incident to $e$ are easily seen to be flat. Indeed in this case the
cone complex of the codomain has dimension $1$ so the flatness is automatic.
However, the consolidated map

$\mathsf{DT}_{\beta,\chi}(X_{v})\to{\sf Hilb}(D_{e_{1}})\times{\sf
Hilb}(D_{e_{2}})$

is not flat. This forces us, at least in our proof, to consider blowups of
this product that are not products of blowups. This is the source of the
exotic evaluations.

######  Remark 9.3.1 (Relevance of cycles in the $1$-complex).

If the rigid $1$-complex is a tree and the logarithmic rank of $X$ is $2$,
even if the consolidated evaluation maps are not flat, one has a fairly simple
degeneration formula, where blowups of the evaluation space can essentially be
avoided. In GW theory this is recorded in [61, Section 6] and a similar
formula holds in DT theory. More generally, if the rigid $1$-complex is a tree
in any logarithmic rank, then one can preserve the product structure on the
evaluation space, although blowups of the factors must be allows. Both these
facts are based on the fact that for a tree, one can choose an orientation on
the $1$-complex in such a way that all unbounded edges are directed towards
their incident vertex, and the orientation can be extended such that at every
vertex there is exactly one outgoing edge. By flowing along such an
orientation, gluing can be performed “one edge at a time”. Once cycles are
present in the $1$-complex this is no longer possible.

####  9.4. Trivalent vertices and constraints via degeneration

We now introduce a basic class of DT/PT invariants of toric threefolds
relative to the full toric boundary, which call the trivalent vertices. Based
on known results in the Gromov–Witten theory of exploded manifolds, it seems
reasonable to expect that the trivalent vertices determine all primary DT/PT
invariants in this geometry, see for instance [58]. In particular, it may be
possible to reduce the full GW/DT/PT equivalencet in this geometry to the
trivalent vertices.

#####  9.4.1. Setup

In what follows, we specify a projective toric surface $S$ be giving the rays
of its fan $\Sigma_{S}$, and specify a curve class $\beta$ by giving its
intersection multiplicity with each invariant divisor, i.e. an integer for
each ray.

We study DT/PT invariants on $X=S\times\mathbb{P}^{1}$, with the boundary
coming from the toric boundary. We will mainly be interested in classes of the
form $\beta\otimes{\sf pt}$.

To fix numerical data for DT/PT invariants we also fix a cohomology weighted
partition for each divisor $D_{\rho}$ in the fan, as well as “bulk” primary
insertions from $S\times\mathbb{P}^{1}$. We are interested in point insertions
in the sense of Remark 9.2.1, namely finitely many conditions of the form
$\mathsf{pt}\times\mathbb{P}^{1}$ and one point condition in the threefold
$S\times\mathbb{P}^{1}$.

######  Remark 9.4.1 (Connected vs. disconneced).

Note that if the point insertions above are imposed in logarithmic
Gromov–Witten theory, then the connected and disconnected theories coincide by
elementary arguments. This means that our conjectures, which concern the
disconnected theory, relate the invariants considered in [11], which a priori
the connected theory, with the logarithmic DT invariants defined in [42].

#####  9.4.2. The trivalent vertices

An interesting class of DT (or PT) invariants of this kind arise when $S$ is a
weighted projective plane. Given positive integers $a$ and $b$, we can take
$S_{a.b}$ to be the toric surface whose fan has rays:

$\rho_{1}={\mathbb{R}}_{\geq 0}(a,0),\ \ \rho_{2}={\mathbb{R}}_{\geq 0}(b,0),\
\ \rho_{3}={\mathbb{R}}_{\geq 0}(-a,-b).$

Note that different $a$ and $b$ can determine the same surface, e.g. $(1,1)$
and $(k,k)$ both give $S_{(a,n)}$ equal to $\mathbb{P}^{2}$. The intersection
multiplicities with the corresponding divisors $D_{1},D_{2},D_{3}$ are now
taken to be $a$, $b$, and $\mathsf{gcd}(a,b)$. This determines $\beta$.

We will define a DT invariant $N_{(a,b)}(q)$ in class $\beta$ with maximal
tangency order along each $D_{i}$, two point insertions (with the notational
convention above), and summing over all possible Euler characteristics.

#####  9.4.3. A proposed relation among trivalent vertices

We have explained above why the trivalent vertices are interesting to
consider. We give an illustration of how the degeneration formula can be used
to constrain and calculate the invariants. The basic principle is that because
different degenerations of incidence conditions must give the same answers,
one hope to extract relations by comparing them.

We propose and examine the following equality:

$N_{(2,2)}(q)\stackrel{{\scriptstyle?}}{{=}}N_{(4,1)}(q).$

The $q=1$ specializations can be checked to be equal, e.g. this is one
consequence of Mikhalkin’s correspondence theorem [43].

A notable feature of the proposal is that the moduli spaces of stable pairs in
the respective curve classes are very different. There is no obvious geometric
reason why these invariants should be the same.

#####  9.4.4. Constraining the proposal via degeneration

We proceed by computing another – non-trivalent invariant – in two different
ways using the splitting formula, so the equality forces the relation above.
This method of using consistency of the degeneration formula appears
prominently in [11, 58].

Consider the toric surface $S_{(2,3)}$. In the notation above, the
intersection multiplicities are

$\beta\cdot D_{1}=2,\ \ \beta_{2}\cdot D_{2}=3,\ \ \beta\cdot D_{3}=1.$

We impose $3$ point conditions, and in tangency conditions as follows: take
maximal tangency, i.e. the partition (2) with $D_{1}$, simple tangency, i.e.
the partition (2+1) with $D_{2}$, and necessarily tangency $1$ with $D_{3}$.

As in the example from Section 9.2, for different configurations of the three
points, we get different rigid Chow $1$-complexes. The two possibilities are
shown below:

$(2,0)$$(2,0)$$(0,2)$$(0,1)$$(0,1)$$(-2,-3)$$(-2,-3)$$(0,2)$

The invariants are defined independently of any degeneration, so they are
obviously equal. In fact, this already provides a constraint between
$N_{(2,2)}(q)$ and $N_{(4,1)}(q)$, though at this stage it is not clear how
concrete this is. We argue that it is, in fact, extremely concrete.

For each of the two rigid $1$-complexes in the figure above, we can try to
apply the splitting formula of Corollary 8.4.1 by cutting along the unique
bounded edge. In particular, one has to sum over partitions of the
multiplicity at the bounded edge, coming from the decomposition of the strict
diagonal – let us call this the gluing partition.

In the first case on the left, this weight is $2$ so we need to consider the
case where this partition is $(2)$ and where it is $(1+1)$. Note that in the
case where the gluing partition encodes maximal tangency, the gluing formula
holds without corrections. It turns out that when the gluing partition is
$(1+1)$ the contribution to the degeneration formula vanishes. This is
essentially due to the presence of the $\mathbb{P}^{1}$-direction. A detailed
argument will appear in a future paper. For now we explore the consequences,
so the reader can take the rest of the discussion as conditional if they wish.

With this at hand, the gluing formula can be applied without any correction
terms, because only maximal tangency along the divisor contributes. Since the
two degenerations are calculating the same thing. Using the
$GL_{2}({\mathbb{Z}})$ invariance of the data (i.e. applying toric
automorphisms) we have:

$N_{(2,2)}(q)\cdot N_{(2,1)}(q)=N_{(2,1)}(q)\cdot N_{(4,1)}(q).$

The terms $N_{(2,1)}(q)$ are nonzero; these can be seen by explicit
calculation in the minimal Euler characteristic case. We have the claimed
equality.

As noted, the quantity $N_{(4,1)}$ can be calculated – it is a PT invariant of
a local curve, relative to three fibers. The explicit formula for the
normalized DT series can again be found in [49].

######  Remark 9.4.2.

More generally, one can ask whether the quantity $N_{a,b}(q)$ depends only on
$ab$. We expect that this is true, and it gives a highly nontrivial relation
among PT invariants, and for example, would completely determine the trivalent
vertices. In GW theory this was done by Parker (and a closely related problem
was considered by Bousseau). A different approach to the trivalent vertices in
Gromov–Witten theory, which can also handle the more generally higher valency
vertices with descendants, has been carried out by Kennedy-Hunt, Shafi, and
Urundolil Kumaran using double ramification cycle techniques [33]. While their
ability to handle descendants is appealing, it is less clear how to transfer
this to DT/PT theory.

###  10\. Combinatorial splitting formula

The rest of this paper is devoted to proving the splitting formulas, following
the roadmap of Section 8.6.

In this section, we give a purely combinatorial version of the splitting
formula. Under the equivalence between Artin fans and cone stacks, it
describes the components of the special fiber of the stack
$\mathsf{Exp}(\mathcal{Y}/B)$ in terms of stacks of the form
$\mathsf{Exp}(X_{v})$. The argument we will present for the combinatorial
splitting formula is insensitive to the labelling of vertices and edges with
discrete data of curve classes or Euler characteristics. We will explain the
argument for $1$-complexes, noting that the decorated cases are essentially
identical.

####  10.1. Setup and notation

We have a map of cone complexes

$\pi:\Sigma_{\mathcal{Y}}\to{\mathbb{R}}_{\geq 0}.$

Let $r_{1},\ldots,r_{k}$ be the rays of $\Sigma_{\mathcal{Y}}$ that map
surjectively onto ${\mathbb{R}}_{\geq 0}$. These correspond to the irreducible
components of the normal crossings variety $Y_{0}$.

We have fixed a cone structure on the space of vertical $1$-complexes and a
universal family $\mathbf{G}$ over it. Given a point $p$ in $T(\Sigma_{Y})$ we
will write $\mathbf{G}_{p}$ for the associated $1$-complex.

In our discussion of the degeneration package, we have fixed a ray $\rho$ of
$T(\Sigma_{\mathcal{Y}})$ that maps surjectively onto the base
${\mathbb{R}}_{\geq 0}$. Furthermore, we have already subdivided
$\Sigma_{mathcalY}$ so that the associated $1$-complex has no tube vertices
and that every vertex occurring in this $1$-complex is associated to one of
the rays $r_{i}$.

We record a few observations:

  1. (i)

If $p$ is a point on $\rho$ lying over $t$ in ${\mathbb{R}}_{\geq 0}$, the
$1$-complex at $p$ can be identified with a union of vertices and edges in the
$1$-skeleton of the fiber $\pi^{-1}(t)$ of $\Sigma_{\mathcal{Y}}$ over $t$.

  2. (ii)

For any $t\in{\mathbb{R}}_{>0}$, the partially ordered sets of faces of
$\pi^{-1}(t)\subset\Sigma_{\mathcal{Y}}$ is independent of $t$.

  3. (iii)

For any point $p$ in $\rho$ lying over $t$ in ${\mathbb{R}}_{>0}$, the subset
of faces $\pi^{-1}(t)$ whose union is $\mathbf{G}_{p}$ is independent of $t$,
under the identification above.

Intuitively, over different points of ${\mathbb{R}}_{>0}$ the fibers of
$\Sigma_{\mathcal{Y}}$ differ by uniformly dilating the positions of the
vertices. Similarly, over a ray of $T(\Sigma_{\mathcal{Y}})$, the
corresponding $1$-complexes differ in exactly this manner as well.

$r_{1}$$r_{2}$ Figure 5. A schematic of the slices of of
$\Sigma_{\mathcal{Y}}$ over different points of the base ${\mathbb{R}}_{\geq
0}$.

####  10.2. Star of a cone

Let $\Sigma$ be a cone complex and $\sigma$ a cone in it. We recall from
Section 1.6 that the star of $\sigma$, denoted $\Sigma(\sigma)$, is a cone
complex obtained by taking the relative interiors of all cones adjacent to
$\sigma$ and quotienting by $\sigma$.

The motivation for considering the star fan construction is the following. If
$V$ is a simple normal crossings pair with cone complex $\Sigma$ and $W\subset
V$ is a closed stratum associated to a cone $\sigma$, then $W$ carries a
generically trivial logarithmic structure associated to the divisor on $W$
obtained by intersection with the divisorial boundary of $V$. The star fan
$\Sigma(\sigma)$ is precisely the tropicalization of $W$ with respect to this
log structure.

We are mainly interested in two instances of this star construction. First,
given $\Sigma_{\mathcal{Y}}$ and its $r_{1},\ldots,r_{k}$ lying over
${\mathbb{R}}_{\geq 0}$, the stars $\Sigma_{\mathcal{Y}}(r_{i})$ are the cone
complexes of the irreducible components of $Y_{0}$ with the induced
generically trivial logarithmic structure.

The second involves the DT moduli space itself. The standard degeneration
formula involves (i) choosing a stratum and (ii) describing it terms of
smaller moduli spaces. In what follows, we are describing the logarithmic
structure on a component of the would-be special fiber of the degeneration of
DT moduli spaces. However, we would like to work with the generically trivial
logarithmic structure, and therefore again use the star fan of a ray.

More precisely, given our fixed ray $\rho$ of $T(\Sigma_{\mathcal{Y}})$ that
maps surjectively onto the base ${\mathbb{R}}_{\geq 0}$, we consider the star
fan $T(\Sigma_{\mathcal{Y}})(\rho)$. This is a cone space which will keep
track of the tropical data associated to a virtual irreducible component of
the DT moduli space.

####  10.3. Cutting map

Our goal is to describe the star fan $T(\Sigma_{\mathcal{Y}})(\rho)$ in terms
of the moduli space of $1$-complexes on the various star fans
$\Sigma_{\mathcal{Y}}(r_{i})$. From now on, we use
$\Sigma_{1},\ldots\Sigma_{k}$ to denote these fans, i.e. we have set

$\Sigma_{i}=\Sigma_{\mathcal{Y}}(r_{i}).$

We will construct cutting maps

$\kappa_{i}:|T(\Sigma_{\mathcal{Y}})(\rho)|\to|T^{\circ}(\Sigma_{i})|\subset|T(\Sigma_{i})|$

from the star of $\rho$ to a certain subspace $T^{\circ}$ of the moduli space
of $1$-complexes in the star fans $\Sigma_{i}$, which we will specify in the
course of the construction.

In order to do this, we first discuss some preliminaries. Fix a cone $\tau$ of
the space $T(\Sigma_{\mathcal{Y}})$. We may restrict the universal family to
$\tau$ to obtain a family of $1$-complexes, compatible with the map to
${\mathbb{R}}_{\geq 0}$:

${\mathbf{G}_{\tau}}$${\Sigma_{\mathcal{Y}}}$${\tau}$${{\mathbb{R}}_{\geq
0}.}$

Fix a point $p$ in the interior of $\tau$. The fiber of $p$ in
$|\mathbf{G}_{\tau}|$ can be identified with a $1$-complex.

######  Lemma 10.3.1.

The combinatorial type of the polyhedral complex over $p$ is independent of
the choice of $p$, provided $p$ lies in the interior of $\tau$.

###### Proof.

The combinatorial structure can be described independently of $p$ as follows.
Since $\mathbf{G}_{\tau}$ is combinatorially flat over $\tau$, its cones
either map isomorphically onto the base or drop dimension by $1$. The vertices
in the preimage are in bijection with the former, while the edges and legs are
in bijection with the latter. The partially ordered set structure of the cones
of $\mathbf{G}_{\tau}$ coincides with the graph structure of the $1$-complex
over $p$, so we conclude. ∎

We denote by $G_{\tau}$ the combinatorial $1$-complex defined by the above
lemma.

Fix a ray $r_{i}$ of $\Sigma_{\mathcal{Y}}$ and the corresponding star
$\Sigma_{i}$. Given a point in $T(\Sigma_{\mathcal{Y}})$ that is contained in
the overstar of $\rho$, we will take the corresponding $1$-complex in
$\Sigma_{\mathcal{Y}}$ and construct a $1$-complex in $\Sigma_{i}$. Each
vertex of the combinatorial $1$-complex $G_{\rho}$ is assigned a cone in
$\Sigma_{\mathcal{Y}}$. By Section 10.1, this cone is in fact a ray of
$\Sigma_{\mathcal{Y}}$ that maps surjectively onto $\mathbb{R}_{\geq 0}$. Note
that there is at most one vertex $u$ of $G_{\rho}$ that is assigned to $r_{i}$
since the $1$-complex is embedded. If no such vertex exists, the cutting map
sends everything to the empty $1$-complex, so from now on we assume $u$
exists.

Given a cone $\tau$ of $T(\Sigma_{\mathcal{Y}})$ containing $\rho$ as
face,with universal family $\mathbf{G}_{\tau}\rightarrow\tau$ as above. The
inclusion $\rho\subset\tau$ determines an inclusion of complexes
$\mathbf{G}_{\rho}\subset\mathbf{G}_{\tau}$; moreover the choice of vertex $u$
determines a ray $u_{\rho}$ in $\mathbf{G}_{\rho}$ and thus a ray in
$\mathbf{G}_{\tau}$ as well. We then have a diagram of pairs of cone complexes
and rays

${\mathbf{(}G_{\tau},u_{\rho})}$${(\Sigma_{\mathcal{Y}},r_{i})}$${(\tau,\rho).}$

By taking the star fan of each pair in this diagram, we obtain a family of
complexes in $\Sigma_{i}$, and so obtain a map

$\tau(\rho)\to|T(\Sigma_{i})|.$

The construction is easily seen to be compatible with specialization of
combinatorial $1$-complexes, and therefore glues across cones. We therefore
obtain the cutting morphism

$\kappa_{i}:|T(\Sigma_{\mathcal{Y}})(\rho)|\to|T(\Sigma_{i})|.$

Given a point $p\in\tau\subset T(\Sigma_{\mathcal{Y}})(\rho)$, it follows from
this construction that the only unbounded edges of $\kappa_{i}(p)$ are
parallel to the rays of $\Sigma_{i}$, since $\mathbf{G}_{\tau}$ contains
$\mathbf{G}_{\rho}$ as a subcomplex. Let $T^{\circ}(\Sigma_{i})$ denote the
subcomplex of $T(\Sigma_{i})$ consisting of $1$-complexes with this property;
it is an open and closed subcomplex, and it contains the image of
$\kappa_{i}$.

By combining these, we have a total cutting map

$\kappa:|T(\Sigma_{\mathcal{Y}})(\rho)|\to\prod_{i}|T^{\circ}(\Sigma_{i})|.$

Here we are only writing these as maps of underlying spaces, but of course,
with appropriate subdivision, they can be lifted to maps of cone complexes.
Indeed, the arguments above apply without change to any fixed cone structure,
due to the functoriality of the star constructions.

####  10.4. Description via metric geometry

We give an equivalent description of the cutting map which can be useful when
working with explicit examples.

We begin with two constructions. First, we construct a retraction relating the
combinatorial $1$-complexes $G_{\rho}$ and $G_{\tau}$:

$\psi:G_{\tau}\to G_{\rho},$

defined as follows. Given a vertex $v$ of $G_{\tau}$, as in the proof of Lemma
10.3.1, it defines a cone $\sigma_{v}$ in $\mathbf{G}_{\tau}$ which maps
isomorphically onto $\tau$. There is a unique ray $u_{\rho}$ in
$\mathbf{G}_{\rho}$ that lies over $\rho\subset\tau$, which corresponds to a
unique vertex $u\in G_{\rho}$, which is $\psi(v)$; the map on edges and rays
can be constructed similarly (and is determined by the map on vertices).

Second, we define slice neighborhoods associated to the map
$\pi:\Sigma_{\mathcal{Y}}\rightarrow{\mathbb{R}}_{>0}$. Given any point $t$ in
${\mathbb{R}}_{>0}$, the fiber $\pi^{-1}(t)$ is a polyhedral complex
$\mathscr{P}_{t}$; its vertices are in natural bijection with those rays
$r_{1},\ldots,r_{k}$ of $\Sigma_{\mathcal{Y}}$ that are not contracted by
$\pi$. Fix a point $t$ and let $v_{i}$ be a vertex of $\mathscr{P}_{t}$. Let
$\mathscr{P}_{t}^{\circ}(v_{i})$ be the connected component containing
$v_{i}$, of the complement of faces in $\mathscr{P}_{t}$ that do not contain
$v_{i}$. This set is contained in the overstar of $r_{i}$ in
$\Sigma_{\mathcal{Y}}$, and therefore we may examine its image in
$\Sigma_{i}$. The projection from the overstar to the star maps this set
$\mathscr{P}_{t}^{\circ}(v_{i})$ isomorphically onto its image in
$\Sigma_{i}$. We refer to this as the slice neighborhood of $0$ in
$\Sigma_{i}$ at the point $t$. It follows easily from the definitions that
upon increasing $t$, the slice neighborhoods get larger, and eventually
exhaust $\Sigma_{i}$.

$v_{1}$$v_{2}$$\Sigma_{\mathcal{Y}}(r_{1})$ Figure 6. A typical slice
neighborhood of the star of $r_{1}$ is depicted on the right. The neighborhood
extends only as far as the dotted line in the $(1,1)$ direction, while the
star $\Sigma(r_{1})$ itself is a fan, and is unbounded in that direction.

Given a point $\overline{p}\in|T(\Sigma_{\mathcal{Y}})(\rho)|$, we can
describe $\kappa_{i}(\overline{p})\in T(\Sigma_{i})$ as follows. Let $u$
denote the vertex in $G_{\rho}$ associated to the ray $r_{i}$. Choose a lift
$p\in|T(\Sigma_{\mathcal{Y}})|$ lying over $t\in{\mathbb{R}}_{>0}$. Consider
the $1$-complex $\mathbf{G}_{p}$; the vertices $v$ of $\mathbf{G}_{p}$ such
that $\psi(v)=u$ define a subcomplex. It follows from the definition that
these vertices $\psi^{-1}(u)$ lie inside the slice
$\mathscr{P}_{t}^{\circ}(v_{i})\subset\mathscr{P}_{t}$, and therefore give a
finite set $S$ of points inside $\Sigma_{i}$. Any edges in $G_{\tau}$
connecting vertices in $\psi^{-1}(u)$ define corresponding edges connecting
points of $S$; similarly, any edges or rays with one boundary vertex inside
$\psi^{-1}(u)$ define unbounded rays for the corresponding point of $S$. The
result is a $1$-complex in $\Sigma_{i}$, which is precisely
$\kappa_{i}(\overline{p})$.

####  10.5. Evaluations

Our next task is to characterize the image of the total cutting map. In
parallel with geometric degeneration formulas, we will identify it with the
subset of the product space where a natural matching condition holds, given by
a diagonal condition under an evaluation map.

If $F$ is any cone complex, recall that $P(F)$ is the subcomplex of $T(F)$
parameterizing $0$-complexes, i.e. $1$-complexes without edges or rays.

If we fix a ray $\delta$ in the fan of $\Sigma_{i}$, let $\Sigma_{i}(\delta)$
denote the star fan of the ray $\delta$. Recall from Section $3$ the
evaluation map

$\mathsf{ev}_{\delta}:|T^{\circ}(\Sigma_{i})|\to|P(\Sigma_{i}(\delta)).|$

Informally, it is described as follows. Given a $1$-complex, outside of a
compact neighborhood of the origin of $\Sigma_{i}$, it consists of a
disconnected union of rays, each parallel to one of the rays of $\Sigma_{i}$.
In particular, there is a finite set of rays parallel to $\delta$, and by
taking the quotient in the direction of $\delta$, we obtain a collection of
points on $\Sigma_{i}(\delta)$.

We return to our fixed ray $\rho$ in $T(\Sigma_{\mathcal{Y}})$ with $G_{\rho}$
be the combinatorial $1$-complex over this ray. Let $e$ be an edge in
$G_{\rho}$ between vertices $v_{1}$ and $v_{2}$. The flags $(v_{i},e)$
determine rays $\delta_{1}$ and $\delta_{2}$ in star fans $\Sigma_{1}$ and
$\Sigma_{2}$ respectively. Finally, by our assumption the edge $e$ picks out
an edge in each fiber of $\Sigma_{\mathcal{Y}}$ over ${\mathbb{R}}_{>0}$. Let
$\delta_{e}$ be the $2$-dimensional cone in $\Sigma_{\mathcal{Y}}$
corresponding to $e$.

The following lemma is immediate from the definitions.

######  Lemma 10.5.1.

The following stars are canonically identified:

$\Sigma_{\mathcal{Y}}(\delta_{e})=\Sigma_{1}(\delta_{1})=\Sigma_{2}(\delta_{2}).$

Given a ray $\rho$ in $T(\Sigma)$, we have exhibited cutting maps

$\kappa:|T(\Sigma_{\mathcal{Y}})(\rho)|\to\prod_{i}|T^{\circ}(\Sigma_{i})|$

where the latter product is the space of $1$-complexes in
$\Sigma_{1},\ldots,\Sigma_{k}$. Let $G_{\rho}$ be the combinatorial
$1$-complex along $\rho$. Fix an edge $e$ with incident vertices $v_{1}$ and
$v_{2}$. Given this, and the canonical identification of Lemma 10.5.1 above,
we can construct two evaluation maps to $\Sigma_{\mathcal{Y}}(\delta_{e})$.

$\mathsf{ev}_{\delta_{i}}\circ\kappa_{i}:|T(\Sigma_{\mathcal{Y}})(\rho)|\to|P(\Sigma_{\mathcal{Y}}(\delta_{e}))|.$

Packaging them, we have a map

$\mathsf{ev}_{e}:|T(\Sigma_{\mathcal{Y}})(\rho)|\to|P(\Sigma_{\mathcal{Y}}(\delta_{e}))|\times|P(\Sigma_{\mathcal{Y}}(\delta_{e}))|.$

######  Lemma 10.5.2.

The image of the map $\mathsf{ev}_{e}$ is contained in the diagonal.

###### Proof.

Given a cone $\tau$ containing $\rho$ as a ray, the edge $e$ defines a two-
dimensional cone $\delta_{e}\subset\mathbf{G}_{\rho}\subset\mathbf{G}_{\tau}$,
and as in the definition of the cutting map, its star defines a map
$\kappa_{e}:T(\Sigma_{\mathcal{Y}})(\rho)\to
P(\Sigma_{\mathcal{Y}}(\delta_{e}))$.

On the other hand, for each $i=1,2$, the vertex $v_{i}$ defines a ray
$\rho_{i}\subset\delta_{e}\subset\mathbf{G}_{\tau}$, and we have a diagram of
triples

${\mathbf{(}G_{\tau},\delta_{e},\rho_{i})}$${(\Sigma,\delta_{e},\rho_{i})}$${(\tau,\rho,\rho)}$

By applying the compatibility of the star construction with respect to
triples, we have that
$\mathsf{ev}_{\delta_{1}}\circ\kappa_{1}=\kappa_{e}=\mathsf{ev}_{\delta_{2}}\circ\kappa_{2}$.
∎

####  10.6. The degeneration formula

Our final task is to characterize the image of the total cutting map $\kappa$
via a matching condition given by the diagonal in the tropical evaluation
space. Consider the cutting morphism

$\kappa:|T(\Sigma_{\mathcal{Y}})(\rho)|\to\prod_{i}|T^{\circ}(\Sigma_{i})|.$

There is an evaluation morphism

$\mathsf{ev}:\prod_{i}|T^{\circ}(\Sigma_{i})|\to\prod_{e}|P(\Sigma_{\mathcal{Y}}(\delta_{e}))|\times|P(\Sigma_{\mathcal{Y}}(\delta_{e}))|.$

Let $\Delta$ denote the product over all edges $e$ of the diagonal loci in
$|P(\Sigma_{\mathcal{Y}}(\delta_{e}))|\times|P(\Sigma_{\mathcal{Y}}(\delta_{e}))|$.

######  Theorem 10.6.1.

The cutting map induces an equality of sets
$\kappa:|T(\Sigma_{\mathcal{Y}})(\rho)|\xrightarrow{\sim}\mathsf{ev}^{-1}(\Delta)$.

###### Proof.

To prove injectivity and surjectivity, it is convenient to use the description
of the cutting map via slice neighborhoods, as in Section 10.4.

I. Injectivity. Given $\overline{p},\overline{p^{\prime}}\in
T(\Sigma_{\mathcal{Y}})(\rho)$, we can choose lifts $p,p^{\prime}\in
T(\Sigma_{\mathcal{Y}})$ lying over the same point $t\in{\mathbb{R}}_{>0}$.
For each ray $r_{i}$ of $\Sigma_{\mathcal{Y}}$ with corresponding vertex
$v_{i}$ of $\mathscr{P}_{t}$, the data characterizing
$\kappa_{i}(\overline{p})$ is equivalent to specifying the open subset of
$\mathbf{G}_{p}\cap\mathscr{P}_{t}^{\circ}(u_{i})$ determined by the vertices
of $G_{p}$ which retract to $v_{i}$. As $i$ varies, these open subsets cover
$\mathbf{G}_{p}$. In particular, if
$\kappa_{i}(\overline{p})=\kappa_{i}(\overline{p^{\prime}})$ for all $i$, the
corresponding open subset of the slice neighborhoods agree and, by taking the
union over $i$, we must have $\mathbf{G}_{p}=\mathbf{G}_{p^{\prime}}$ so that
$p=p^{\prime}$.

To prove surjectivity onto the pullback of the diagonal, we will explain a
reconstruction procedure: given points of each $T^{\circ}(\Sigma_{i})$ that
evaluate into the diagonal, we will construct, for any sufficiently large
$t\in{\mathbb{R}}_{>0}$, an embedded $1$-complex in $\Sigma_{\mathcal{Y}}$
that lies in the fiber of over $t$

II. Exporting $1$-complexes to the slice neighborhood. Fix a point in
$T(\Sigma_{i})$ and let $G$ be the embedded $1$-complex. On the complement of
a compact neighborhood of the origin, the $1$-complex $\mathbf{G}$ consists of
a disjoint union of unbounded rays, each parallel to one of the rays of
$\Sigma_{i}$. For future use, we refer to the subcomplex that excludes these
unbounded rays as a skeleton. In particular, there exists some sufficiently
large value of $t$ such that this slice neighborhood contains the skeleton and
intersects all the unbounded rays. We now lift the $1$-complex parameterized
by a point $T(\Sigma_{i})$ to a non-compact, $1$-dimensional polyhedral subset
$Q_{i}$ of $\mathscr{P}_{t}$, by identifying the slice neighborhood with a
subset of $\mathscr{P}_{t}$ as explained above.

III. Preparing the lifts. Now fix a point $p$ in the product
$\prod_{i}T(\Sigma_{i})$, which determines $1$-complexes in each $\Sigma_{i}$.
We have described a way to produce, for any sufficiently large $t$, a lift of
each of these $1$-complexes to polyhedral subsets of $\mathscr{P}_{t}$. If the
$t$ coordinate is sufficiently large, we can also guarantee the following
separation criterion: there is a compact neighborhood $K_{i}$ of each vertex
$v_{i}$, containing the skeleton of $Q_{i}$, such that $K_{i}\cap K_{j}$ is
empty for distinct $i$ and $j$. In preparation for gluing the lifts $Q_{i}$,
we adjust them as follows. For each $Q_{i}$, we truncate the non-compact edges
that lie outside the skeleton such that these edges remain disjoint from
$K_{j}$ for $i$ and $j$ distinct.

IV. Gluing the lifts. We now specialize to the case where $p$ is a point in
$\mathsf{ev}^{-1}(\Delta)$. By applying the procedure above, we obtain lifts
$Q_{i}$ of $1$-complexes, to non-compact polyhedral subsets of
$\mathscr{P}_{t}$, with $t$ sufficiently large as by the preceding step. For
each edge $e$ incident to a vertex $v_{i}$, the polyhedral subset $Q_{i}$ has
(a potentially empty) set of non-compact edges parallel to $e$. The lengths of
these edges may be increased or decreased, maintaining the property that the
edges belonging to $Q_{i}$ are disjoint from the neighborhood $K_{j}$ for $i$
and $j$ distinct. We now use the diagonal condition to glue the lifts together
to a $1$-complex in $\Sigma_{\mathcal{Y}}$, living over the sufficiently large
value of $t$. Consider two vertices $v_{1}$ and $v_{2}$ of $\mathscr{P}_{t}$,
sharing an edge $e$ and the lifts $Q_{1}$ and $Q_{2}$ described above. The
complement of the skeleton in each includes a set non-compact edges parallel
to $e$. In both cases, there is a natural projection from the cone containing
these non-compact edges to $\Sigma_{\mathcal{Y}}(\delta_{e})$. By
construction, the image is a $0$-complex, i.e. a point of
$P(\Sigma_{\mathcal{Y}}(\delta_{e})))$. The diagonal condition ensures that
this $0$-complex is the same for both $Q_{1}$ and $Q_{2}$. By linear geometry,
it follows that these non-compact edges in the direction of $e$ are open
subsets of the preimage of a set of points in
$\Sigma_{\mathcal{Y}}(\delta_{e})$, where the preimage is taken under the
natural projection from the overstar of $e$ to the quotient. it follows that
if the edge lengths of these non-compact edges are adjusted appropriately, the
lifts $Q_{1}$ and $Q_{2}$ may be glued. Ranging over all vertices, we now
obtain a $1$-complex in $\mathscr{P}_{t}$ whose image under the cutting map is
the chosen point $p$.

It follows from the steps in the construction that the natural

$\kappa:|T(\Sigma_{\mathcal{Y}})(\rho)|\to\prod_{i}|T(\Sigma_{i})|$

is an isomorphism onto $\mathsf{ev}^{-1}(\Delta)$ as claimed. ∎

$v_{1}$$v_{2}$ Figure 7. The dashed structure represents the polyhedral
complex $\mathscr{P}_{t}$, while the solid structure represents the lifts of
the $1$-complex from slice neighborhoods

###  11\. A first DT gluing formula

The decomposition feature of the degeneration package above isolates virtual
components $\mathsf{DT}_{\gamma}(Y_{0})$ of the special fiber
$\mathsf{DT}(Y_{0})$. In this section, we explain how to construct this space
out of the spaces $\mathsf{DT}(X_{v})$ associated to the vertices of the rigid
tropical curve $\gamma$.

The basic idea is to express a subscheme in an expansion of the broken target
$Y_{0}$ by gluing together subschemes in expansions of the components $X_{v}$.
There are two aspects of this problem. First, we construct a moduli space of
subschemes in expansions of the disjoint union of all the $X_{v}$, but with
the additional property that these expansions glue together to form an
expansion of $Y_{0}$. Next, we impose the condition that the subschemes
themselves also glue together. We deal with these in turn.

#### Conventions on terminology

For readability, in this section we will drop the decoration of moduli spaces
by discrete data whenever they are clear from context. For example, we will
use $\mathsf{DT}(X_{v})$ to mean the appropriate component of the DT spaces
associated to $X_{v}$ determined by the rigid tropical curve $\gamma$.

We also recall a piece of terminology from the first part of the paper, as it
has been a while since we’ve used it. Our moduli spaces $\mathsf{DT}(Y_{0})$,
$\mathsf{DT}(X_{v})$, and $\mathsf{Hilb}^{\mathbf{n}}(D)$ have natural strict
maps to Artin fans – respectively $\mathsf{Exp}(Y_{0})$,
$\mathsf{Exp}(X_{v})$, and $\mathsf{Exp}_{0}(D)$. Given these maps, a
distinguished class of logarithmic modifications arise as pullbacks of
modifications of this Artin fan. We called such modifications tropical models.

####  11.1. Gluing the target families

The notation is maintained from the previous section: we have a degeneration
$\mathcal{Y}\to B$, with special fiber $Y_{0}$, and components $X_{v}$. We
examine the product $\prod_{v}\mathsf{DT}(X_{v})$. Over this moduli space, we
have a disjoint union of subschemes in target expansions:

${\coprod\mathcal{Z}_{v}}$${\coprod\mathcal{X}_{v}}$${\coprod
X_{v}}$${\prod_{v}\mathsf{DT}(X_{v}).}$

Consider the locus

$\prod_{v}\mathsf{DT}^{\circ}(X_{v})\subset\prod_{v}\mathsf{DT}(X_{v})$

parameterizing subschemes where the universal target, in each factor, is
unexpanded. On this locus, the subschemes $X_{v}$ glue together to a universal
target

$Y_{0}\times\prod_{v}\mathsf{DT}^{\circ}(X_{v})\to\prod_{v}\mathsf{DT}^{\circ}(X_{v}).$

However, there may exist points of $\prod_{v}\mathsf{DT}(X_{v})$ where there
is no way to glue the expanded target families $\mathcal{X}_{v}$ to form a
family of expansions of $Y_{0}$. Indeed, as a simple example, consider a point
of the product where the universal expansion is nontrivial in one factor but
nontrivial at another factor. If the corresponding components in $Y_{0}$ meet,
the expansions should be glued but this is impossible.

Therefore it is not clear how to extend the gluing

$Y_{0}=\coprod_{v}X_{v}/\sim$

to the expansions over the product.

The solution is to look tropically. We may pass to the locus in the tropical
space of $1$-complexes where the targets can be glued – indeed, this was
already considered in the preceding Section 10, as the locus of $1$-complexes
in the fans associated to vertices of the rigid curve $\gamma$ that glue along
the maps induced by the edge directions of the initial $1$-complex $\gamma$.

#####  11.1.1. Combinatorial splitting

Let us now recall the combinatorial splitting formula of Section 10. First, we
fixed a ray $\rho$ in the space $T(\Sigma_{\mathcal{Y}})$ corresponding to a
rigid $1$-complex $\gamma$, with the assumption that the general $1$-complex
on this ray was supported on the fan of $\Sigma_{\mathcal{Y}}$.

We then constructed a total cutting map:

$\kappa:|T(\Sigma_{\mathcal{Y}})(\rho)|\to\prod_{i}|T^{\circ}(\Sigma_{i})|,$

from the star of that ray to the product of spaces of $1$-complexes associated
to stars of the rays of $\Sigma_{\mathcal{Y}}$. We also constructed an
evaluation:

$\mathsf{ev}:\prod_{i}|T^{\circ}(\Sigma_{i})|\to\prod_{e}|P(\Sigma_{\mathcal{Y}}(\delta_{e}))|\times|P(\Sigma_{\mathcal{Y}}(\delta_{e}))|.$

Finally, we identified a locus

$\Delta\subset\prod_{e}|P(\Sigma_{\mathcal{Y}}(\delta_{e}))|\times|P(\Sigma_{\mathcal{Y}}(\delta_{e}))|$

parameterizing the diagonal among pairs of $0$-complexes, one pair
corresponding to each edge $e$.

The combinatorial splitting formula asserts:

$|T(\Sigma_{\mathcal{Y}})(\rho)|\to\mathsf{ev}^{-1}(\Delta)$

is an isomorphism.

In order to use this geometrically, we fix cone structures on the spaces
involved. In particular, we fix a cone structure on
$\prod_{i}|T^{\circ}(\Sigma_{i})|$ such that $\mathsf{ev}^{-1}(\Delta)$ is a
union of cones; we will denote this cone structure by
$\bigtimes_{i}T^{\circ}(\Sigma_{i})$. We may also assume that we have chosen a
cone structure on $|T(\Sigma_{\mathcal{Y}})(\rho)|$ in such a way that

$T(\Sigma_{\mathcal{Y}})(\rho)\to\mathsf{ev}^{-1}(\Delta)\subset\bigtimes_{i}T^{\circ}(\Sigma_{i}),$

becomes an isomorphism of cone complexes onto the preimage of the diagonal.
Note that this may involve changing integral structures.

The cone structure $\bigtimes_{i}T^{\circ}(\Sigma_{i})$ on the product space
need not – and typically is not – a product of cone structures on the factors.
Nevertheless, it gives rise to a modification

$\bigtimes_{v}\mathsf{DT}(X_{v})\to\prod_{v}\mathsf{DT}(X_{v}).$

The subcomplex structure on $\mathsf{ev}^{-1}(\Delta)$ alluded to above is an
incomplete subdivision, and so it gives us an open immersion:

${\bigtimes}_{v}^{\circ}\mathsf{DT}(X_{v})\hookrightarrow\bigtimes_{v}\mathsf{DT}(X_{v}).$

#####  11.1.2. Geometric relevance

We briefly explain the geometric role of this open subspace
${\bigtimes}_{v}^{\circ}\mathsf{DT}(X_{v})$ in the next proposition and
following discussion. This discussion is not logically necessary and the
reader can skip to Section 11.2 without loss of continuity.

Recall that we have fixed a rigid $1$-complex $\gamma$.

######  Proposition 11.1.1.

Let $e$ be an edge between vertices $u$ and $v$ in $\gamma$. On the locus
${\bigtimes}_{v}^{\circ}\mathsf{DT}(X_{v})$, the universal expansion of
$D_{e}$ induced by the universal expansion $\mathcal{X}_{u}$ and by
$\mathcal{X}_{v}$ are isomorphic. More precisely, the resulting families
$\mathcal{D}_{e,u}$ and $\mathcal{D}_{e,v}$ are isomorphic as logarithmic
modifications of ${\bigtimes}_{v}^{\circ}\mathsf{DT}(X_{v})\times D_{e}$.

Let us give another way of expressing the claim, and in particular the
isomorphism “as logarithmic modifications”. The universal expansion of $D_{e}$
is a naturally a subfunctor of the functor of points on logarithmic schemes of
${\bigtimes}_{v}^{\circ}\mathsf{DT}(X_{v})\times D_{e}$. The proposition then
asserts that the two expansions $\mathcal{X}_{u}$ and $\mathcal{X}_{v}$ induce
the same subfunctor.

###### Proof.

The induced expansion of $D_{e}$ associated to $\mathcal{X}_{v}$ can be
calculated as follows. We take the universal $1$-complex, which is a family of
$1$-complexes in $\Sigma_{v}$ over the subcomplex (whose support is)
$\mathsf{ev}^{-1}(\Delta)$, and take asymptotics in the direction determined
by the flag $(v,e)$. Since it is calculated at this tropical level, the result
follows formally from the results of Section 10 and in particular, the
definition of $\mathsf{ev}^{-1}(\Delta)$: this locus is, by definition, the
locus where the $0$-complexes induced at $u$ and $v$ coincide. ∎

As a consequence of the proposition, for each edge $e$ between vertices $u$
and $v$ we have a diagram over ${\bigtimes}_{v}^{\circ}\mathsf{DT}(X_{v})$

${\mathcal{X}_{v}}$${\mathcal{D}_{e}}$${\mathcal{X}_{u}.}$

whose pushout glues the expansion of $X_{v}$ and $X_{u}$ along the expansion
of $D_{e}$. More generally, by ranging over all edges in $\gamma$, we can take
the colimit of all the spaces
$\\{\mathcal{X}_{v}\\}_{v}\cup\\{\mathcal{D}_{e}\\}_{e}$ under the natural
inclusions of the spaces $\mathcal{D}_{e}$ into the spaces $\mathcal{X}_{v}$
whenever $e$ is incident to $v$. The result is a scheme

${\cal X}^{\Join}\to{\bigtimes}_{v}^{\circ}\mathsf{DT}(X_{v}).$

Thus, over the space ${\bigtimes}_{v}^{\circ}\mathsf{DT}(X_{v})$ we can extend
the gluings of $X_{v}$ to $Y_{0}$ to gluings of expansions ${\cal X}_{v}$ to
form expansions of $Y_{0}$.

####  11.2. Evaluation sections and the family Hilbert scheme

If we fix an edge $e$ in a rigid $1$-complex $\gamma$, we have constructed a
flat family ${\cal D}_{e}$ over ${\bigtimes}^{\circ}_{v}\mathsf{DT}(X_{v})$,
which captures the induced expansion of $D_{e}$ from both vertices adjacent to
$e$.

######  Construction 11.2.1 (Boundary evaluation maps).

Let $\mathcal{D}_{e}$ be the double divisor family above. The natural morphism

$\mathcal{D}_{e}\to{\bigtimes}^{\circ}_{v}\mathsf{DT}(X_{v})$

is smooth. We may now pass to the (ordinary) Hilbert scheme of
$(D_{e}\cdot\beta)$-many points of the family $\mathcal{D}_{e}$ denoted

$\mathsf{H}(\mathcal{D}_{e}):=\mathsf{Hilb}\left(\mathcal{D}_{e}/{\bigtimes}^{\circ}_{v}\mathsf{DT}(X_{v})\right)\to{\bigtimes}^{\circ}_{v}\mathsf{DT}(X_{v}).$

We let $\mathsf{H}^{\\{2\\}}(\mathcal{D}_{e})$ be the fiber square of
$\mathsf{H}(\mathcal{D}_{e})$, subject to the following stability condition: a
point of this fiber square determines two $0$-dimensional subschemes on the
same expansion of $D_{e}$, and we demand that for every component of the
expansion, at least one of these subschemes has nontrivial support. In keeping
with the notation in the section, we drop the number of points, which is
understood to be $D_{e}\cdot\beta$.

Let $v_{1}$ and $v_{2}$ be the vertices adjacent to $e$ and let ${\cal
Z}_{v_{i}}$ be the pullbacks of the corresponding universal subschemes from
$\mathsf{DT}(X_{v})$ to ${\bigtimes}^{\circ}_{v}\mathsf{DT}(X_{v})$.
Intersecting $\mathcal{D}_{e}$ with either $\mathcal{Z}_{v_{1}}$ and
$\mathcal{Z}_{v_{2}}$ gives rise to points in $\mathsf{H}(\mathcal{D}_{e})$.
As a result, we have evaluation maps:

$\mathsf{ev}_{v_{i},e}\colon{\bigtimes}^{\circ}_{v}\mathsf{DT}(X_{v})\to\mathsf{H}(\mathcal{D}_{e}),\
\ \textrm{for $i=1,2$.}$

We can consolidate these to get a map:

$\mathsf{ev_{e}}:=\mathsf{ev}_{v_{1},e}\times\mathsf{ev}_{v_{2},e}\colon{\bigtimes}^{\circ}_{v}\mathsf{DT}(X_{v})\to\mathsf{H}^{\\{2\\}}(\mathcal{D}_{e}).$

We refer to it as the boundary evaluation map at $e$.

######  Lemma 11.2.2.

Let $\mathsf{H}^{\Delta}(\mathcal{D}_{e})$ be the relative diagonal locus in
$\mathsf{H}^{\\{2\\}}(\mathcal{D}_{e})$ over
${\bigtimes}^{\circ}_{v}\mathsf{DT}(X_{v})$. The inclusion is a regular
embedding.

###### Proof.

The Hilbert scheme of points on a surface is smooth, and as a consequence
$\mathsf{H}(\mathcal{D}_{e})$ is a smooth fibration; the diagonal is therefore
a regular embedding. ∎

Let $\mathsf{DT}_{\gamma}(Y_{0})$ denote the tropical model associated to the
chosen cone complex structure $T(\Sigma_{\mathcal{Y}})(\rho)$. By
construction, we have a map

$\mathsf{DT}_{\gamma}(Y_{0})\to{\bigtimes}_{v}\mathsf{DT}(X_{v})$

associated to the map of cone complexes
$T(\Sigma_{\mathcal{Y}})(\rho)\rightarrow\bigtimes_{i}T^{\circ}(\Sigma_{i})$.
Examining the morphism, target space contains
${\bigtimes}_{v}^{\circ}\mathsf{DT}(X_{v})$ as an open subspace. Indeed, the
map
$T(\Sigma_{\mathcal{Y}})(\rho)\rightarrow\bigtimes_{i}T^{\circ}(\Sigma_{i})$
factors through the subspace $\mathsf{ev}^{-1}(\Delta)$. We can therefore
write a more refined map

$\mathsf{DT}_{\gamma}(Y_{0})\to{\bigtimes}^{\circ}_{v}\mathsf{DT}(X_{v})$

######  Proposition 11.2.3.

There is an equality of Chow homology classes

$q_{\star}\left[\mathsf{DT}_{\gamma}(Y_{0})\right]^{\mathsf{vir}}=\bigcap_{e}\mathsf{H}^{\Delta}(\mathcal{D}_{e})\cap\left[{\bigtimes}^{\circ}_{v}\mathsf{DT}(X_{v})\right]^{\mathsf{vir}},$

where the intersection product on the right is defined by the regular
embedding in Lemma 11.2.2.

###### Proof.

We split the statement into two claims. First, an equality of spaces, and then
a compatibility with virtual classes. The scheme theoretic statement is as
follows.

Schematic Claim. Our goal is to compare the spaces

$\mathsf{DT}_{\gamma}(Y_{0})\ \ \text{and}\ \ \
\mathsf{H}^{\Delta}(\mathcal{D}_{e})\cap{\bigtimes}^{\circ}_{v}\mathsf{DT}(X_{v}).$

Earlier in this section, we have chosen a cone complex structure on the
product space $\prod_{i}|T^{\circ}(\Sigma_{i})|$ and on
$\mathsf{ev}^{-1}(\Delta)$. We consequently get Artin fans, which we denote:

${\bigtimes}_{v}\mathsf{Exp}(X_{v})\ \ \text{and}\ \ \mathsf{Exp}_{\Delta}.$

The space on the left is a blowup of the product of Exp-spaces associated to
vertices of $\gamma$, while the space on the right is an open substack of this
modification.

We now make three observations. First, there are structure maps

$\mathsf{DT}_{\gamma}(Y_{0})\to\mathsf{Exp}_{\gamma}(Y_{0}),\ \
{\bigtimes}_{v}\mathsf{DT}(X_{v})\to{\bigtimes}_{v}\mathsf{Exp}(X_{v}),\ \
\text{and}\ \
{\bigtimes}^{\circ}_{v}\mathsf{DT}(X_{v})\to\mathsf{Exp}_{\Delta}.$

Second, by the combinatorial degeneration formula, the cutting map at the
tropical level gives rise to an arrow

$\mathsf{Exp}_{\gamma}(Y_{0})\to{\bigtimes}_{v}\mathsf{Exp}(X_{v}),$

and by choosing cone structures compatibly, we can assume this is an
isomorphism onto $\mathsf{Exp}_{\Delta}$. And third, we have a natural cutting
map

${\mathsf{DT}_{\gamma}(Y_{0})}$${{\bigtimes}_{v}\mathsf{DT}(X_{v})}$${\mathsf{Exp}_{\gamma}(Y_{0})}$${\bigtimes_{v}\mathsf{Exp}(X_{v})}$

lifting the combinatorial cutting map on Artin fans, in Section 11.1.1. The
map on Artin fans is given by partial normalization of the expansion; the top
vertical arrow is then defined by pulling back the subscheme to the
components. As we have already noted before the proposition, the map
factorizes through

$\bigcap_{e}\mathsf{H}^{\Delta}(\mathcal{D}_{e})\cap{\bigtimes}^{\circ}_{v}\mathsf{DT}(X_{v})\subset{\bigtimes}_{v}\mathsf{DT}(X_{v}).$

Together, we therefore have a diagram

${\mathsf{DT}_{\gamma}(Y_{0})}$${\bigcap_{e}\mathsf{H}^{\Delta}(\mathcal{D}_{e})\cap{\bigtimes}^{\circ}_{v}\mathsf{DT}(X_{v})}$${\mathsf{Exp}_{\gamma}(Y_{0})}$${\mathsf{Exp}_{\Delta}.}$

We claim that the top horizontal is an isomorphism. Since the two bases are
naturally identified, we can verify that the functors of points relative to
the base coincide. We can see this from definitions. Indeed, if we fix an
$S$-valued point of $\mathsf{Exp}_{\gamma}(Y_{0})$ we have fixed a family of
expansions over $S$ of each component $X_{v}$, that glue. A point of the space
$\mathsf{DT}_{\gamma}(Y_{0})$ is precisely the data of subschemes in each
irreducible component that are algebraically transverse to the strata and
agree at the pairwise intersections. We conclude the required isomorphism.

The virtual structure statement now remains. The schematic claim gives an
isomorphism:

$q:\mathsf{DT}_{\gamma}(Y_{0})\to\bigcap_{e}\mathsf{H}^{\Delta}(\mathcal{D}_{e})\cap{\bigtimes}_{v}^{\circ}\mathsf{DT}(X_{v}),$

The right hand side carries a natural virtual class, obtained by pulling back
the virtual class of $\bigtimes_{v}\mathsf{DT}(X_{v})$ along a regular
embedding. We therefore make the following:

Virtual Claim. Pushforward along $q$ gives an identification of Chow homology
classes:

$q_{\star}\left[\mathsf{DT}_{\gamma}(Y_{0})\right]^{\mathsf{vir}}=\bigcap_{e}\mathsf{H}^{\Delta}(\mathcal{D}_{e})\cap\left[{\bigtimes_{v}}^{\circ}\mathsf{DT}(X_{v})\right]^{\mathsf{vir}}.$

The virtual structure claim follows from the fact that diagonal condition
coming from the gluing condition is compatible with the obstruction theory on
$\mathsf{DT}_{\gamma}(Y_{0})$. This latter result follows from [41] and [36,
Section 6], and since our gluing divisor is smooth, the same argument applies
here, so we will sketch the key points.

We denote by $\mathcal{I}_{0}$ denote the universal ideal sheaf on
$\mathsf{DT}_{\gamma}(Y_{0})$; we denote by $\mathcal{I}_{v}$ and
$\mathcal{I}_{e}$ the restriction of this sheaf to $\mathcal{X}_{v}$ and
$\mathcal{D}_{e}$ respectively. From algebraic transversality, these are
related by a short exact sequence

$0\rightarrow\mathcal{I}_{0}\rightarrow\oplus_{v}\mathcal{I}_{v}\rightarrow\oplus_{e}\mathcal{I}_{e}\rightarrow
0.$

This induces an exact triangle in the derived category:

$R\pi_{\star}\mathcal{RH}om(\mathcal{I}_{0},\mathcal{I}_{0})_{0}\rightarrow\oplus_{v}R\pi_{\star}\mathcal{RH}om(\mathcal{I}_{v},\mathcal{I}_{v})_{0}\rightarrow\oplus_{e}R\pi_{\star}\mathcal{RH}om(\mathcal{I}_{e},\mathcal{I}_{e})_{0}\rightarrow\cdots.$

Each summand in right term in this triangle is then identified, after a shift,
with the normal bundle of the relative diagonal
$\mathsf{H}^{\Delta}(\mathcal{D}_{e})\hookrightarrow\mathsf{H}^{\\{2\\}}(\mathcal{D}_{e})$.
This implies that, after taking duals, we have a compatible triangle of
obstruction theories999Abramovich has informed us that such a triangle should
be referred to as a justice of obstruction theories [3, Section 6].  and the
equality of virtual classes follows. Again, we refer the reader to [36] for
more details.

∎

###  12\. The main splitting formula

We are now in a position to relate the output of the weak splitting formula to
the statement of Theorem 8.2.2. We do this on the DT side, and then run a
parallel argument in GW theory based on the output of [61].

####  12.1. Gluing in DT theory

In the context of Proposition 11.2.3 in the previous section, we had a space
$\bigtimes_{v}\mathsf{DT}(X_{v})$, and a distinguished open subspace
${\bigtimes}^{\circ}_{v}\mathsf{DT}(X_{v})$ that supported the evaluation map
to the family Hilbert schemes $\mathsf{H}^{\\{2\\}}({\cal D}_{e})$. The
diagonal loci then allowed us to reconstruct the spaces
$\mathsf{DT}_{\gamma}(Y_{0})$.

We now manipulate this proposition using logarithmic intersection arguments,
into the claimed form in our main splitting theorem. We keep the notation from
that section, and in particular the space $\bigtimes_{v}\mathsf{DT}(X_{v})$.

Recall that we have set

$\mathsf{Hilb}_{\beta}(D_{v}):=\prod_{i=1}^{m}\mathsf{Hilb}_{\beta\cdot
D_{e_{i}(v)}}(D_{e_{i}(v)}).$

We have also chosen logarithmic modifications such that

$\mathsf{ev}:\prod_{v}\mathsf{DT}_{\gamma(v)}(X_{v})\to\prod_{v}\mathsf{Hilb}^{\dagger}_{\beta}(D_{v}),$

is combinatorially flat with reduced fibers. Let
$\bigtimes_{v}\mathsf{Hilb}(D_{v})$ be any modification of
$\prod_{v}\mathsf{Hilb}(D_{v})$ such that $\bigtimes_{v}\mathsf{DT}(X_{v})$
maps to this modification via a combinatorially flat evaluation map. Note that
this may require replacing $\bigtimes_{v}\mathsf{DT}(X_{v})$ with a further
subdivision.

######  Lemma 12.1.1.

The morphism

$\bigtimes^{\circ}_{v}\mathsf{DT}(X_{v})\to\bigtimes_{v}\mathsf{Hilb}(D_{v})$

factors as

$\bigtimes^{\circ}_{v}\mathsf{DT}(X_{v})\to{\prod_{e}}_{\bigtimes^{\circ}_{v}\mathsf{DT}(X_{v})}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\mathsf{H}^{\\{2\\}}(\mathcal{D}_{e})\xrightarrow{f}\bigtimes_{v}\mathsf{Hilb}(D_{v}).$

The second map $f$ is combinatorially flat. Note that the middle term is a
fiber product of the spaces indexed by the edges, taken over
$\bigtimes_{v}\mathsf{DT}(X_{v})$.

###### Proof.

The factorization follows easily from definitions. For the flatness, consider
the projection

$\mathsf{H}^{\\{2\\}}(\mathcal{D}_{e})\to\bigtimes_{v}\mathsf{DT}(X_{v}).$

from the fiberwise Hilbert square. We observe that the fiberwise Hilbert
square parameterized subschemes supported on the smooth locus of the
expansion, and therefore the relative logarithmic structure on this projection
map is trivial, i.e. the projection is strict. It follows that the associated
map of tropical spaces is a face morphism: every cone maps isomorphically onto
a cone of the image. Now, in order to compute the tropical map induced by $f$,
we can take a cone of $\mathsf{H}^{\\{2\\}}(\mathcal{D}_{e})$, project it
isomorphically onto a cone $\bigtimes_{v}\mathsf{DT}(X_{v})$ and apply the
evaluation map. Since the evaluation has specifically been chosen to be
combinatorially flat, the claimed combinatorial flatness follows. ∎

We now deduce the first simplification of the degeneration formula. Recall
from the context of Proposition 11.2.3, we have a space
$\bigtimes^{\circ}_{v}\mathsf{DT}(X_{v})$ defined as an open in
$\bigtimes_{v}\mathsf{DT}(X_{v})$ by the cone subspace
$\mathsf{ev}^{-1}(\Delta)$.

######  Lemma 12.1.2.

Let $\Delta^{\ddagger}$ be strict transform in
$\bigtimes_{v}\mathsf{Hilb}(D_{v})$ of the diagonal locus in
$\prod_{v}\mathsf{Hilb}(D_{v})$. The preimage of $\Delta^{\ddagger}$ along

${\bigtimes}_{v}\mathsf{DT}(X_{v})\to{\bigtimes}_{v}\mathsf{Hilb}(D_{v})$

is contained in the subspace $\bigtimes^{\circ}_{v}\mathsf{DT}(X_{v})$.

###### Proof.

The locus $\Delta^{\ddagger}$ is equal to the fine and saturated logarithmic
pullback of the diagonal in $\prod_{v}\mathsf{Hilb}(D_{v})$ under the
subdivision
$\bigtimes_{v}\mathsf{Hilb}(D_{v})\to\prod_{v}\mathsf{Hilb}(D_{v})$. We also
know that the map
${\bigtimes}_{v}\mathsf{DT}_{v}(X_{v})\to{\bigtimes}_{v}\mathsf{Hilb}(D_{v})$
is combinatorially flat with reduced fibers. Putting these together, the usual
scheme theoretic preimage in question is equal to the fine and saturated
pullback associated to the diagram

${{\bigtimes}_{v}\mathsf{DT}_{v}(X_{v})}$${\prod_{e}\mathsf{Hilb}(D_{e})}$${\prod_{e}\mathsf{Hilb}(D_{e})\times\mathsf{Hilb}(D_{e}).}$

By general arguments concerning logarithmic fiber products, this pullback is
contained in the (possibly incomplete) subdivision of
$\bigtimes_{v}\mathsf{DT}(X_{v})$ induced by fiber product of the
corresponding diagram of cone spaces. The latter fiber product is exactly the
union of cones $\mathsf{ev}^{-1}(\Delta)$ discussed in the previous section.
This defines the open subspace $\bigtimes^{\circ}_{v}\mathsf{DT}(X_{v})$ so
the claim follows. ∎

Recall the morphism

$q:\mathsf{DT}_{\gamma}(Y_{0})\to{\bigtimes}^{\circ}_{v}\mathsf{DT}_{v}(X_{v})$

used in Proposition 11.2.3. By composing with the open immersion we can
consider the map

$p:\mathsf{DT}_{\gamma}(Y_{0})\to{\bigtimes}_{v}\mathsf{DT}_{v}(X_{v}).$

We can now state the first stage of our simplified gluing formula.

######  Corollary 12.1.3.

Let $\Delta^{\ddagger}$ be strict transform in
$\bigtimes_{v}\mathsf{Hilb}(D_{v})$ of the diagonal locus in
$\prod_{v}\mathsf{Hilb}(D_{v})$. The following equality of virtual classes
holds:

$p_{\star}\left[\mathsf{DT}_{\gamma}(Y_{0})\right]^{\mathsf{vir}}=\Delta^{\ddagger}\cap\left[\bigtimes_{v}\mathsf{DT}(X_{v})\right]^{\mathsf{vir}},$

where we have suppressed the pullback of the cohomology class
$\Delta^{\ddagger}$ from the notation.

###### Proof.

The statement of Proposition 11.2.3 is the statement:

$q_{\star}\left[\mathsf{DT}_{\gamma}(Y_{0})\right]^{\mathsf{vir}}=\bigcap_{e}\mathsf{H}^{\Delta}(\mathcal{D}_{e})\cap\left[{\bigtimes}^{\circ}_{v}\mathsf{DT}(X_{v})\right]^{\mathsf{vir}},$

which is a refined intersection statement101010Refined here is used in the
sense of Fulton [21], i.e. the intersection product is supported on the scheme
theoretic intersection rather than the non-compact ambient space. supported on
the space
$\bigcap_{e}\mathsf{H}^{\Delta}(\mathcal{D}_{e})\cap{\bigtimes}^{\circ}_{v}\mathsf{DT}(X_{v})$.
This locus is proper, and by Lemma 12.1.2, the intersection class
$\bigcap_{e}\mathsf{H}^{\Delta}(\mathcal{D}_{e})$ is equal to the pullback of
$\Delta^{\ddagger}$. The corollary follows by proper pushforward and the
projection formula. ∎

The key point in the corollary above is that we have collapsed the “moving”
family of Hilbert schemes $\mathsf{H}^{\\{2\\}}(\mathcal{D}_{e})$, to a blowup
of an absolute product $\bigtimes_{v}\mathsf{Hilb}(D_{v})$.

In the next step, we use the fact that the virtual classes of the DT space are
birationally invariant, to show that we can replace
$\bigtimes_{v}\mathsf{DT}(X_{v})$ with a simpler birational model. Let
$\prod^{\diamond}_{v}\mathsf{DT}(X_{v})$ be any virtual birational model
mapping to the space $\bigtimes_{v}\mathsf{Hilb}(D_{v})$ constructed
above.111111By universal mapping properties, any such model is a subdivision
of the fine and saturated fiber product of $\prod_{v}\mathsf{DT}_{v}(X_{v})$
with $\bigtimes_{v}\mathsf{Hilb}(D_{v})$ over
$\prod_{v}\mathsf{Hilb}(D_{v})$.The cutting map again gives rise to

$p:\mathsf{DT}^{\dagger}_{\gamma}(Y_{0})\to{\prod_{v}}^{\diamond}\mathsf{DT}_{v}(X_{v}).$

######  Proposition 12.1.4.

The following formula holds:

$p_{\star}\left[\mathsf{DT}^{\dagger}_{\gamma}(Y_{0})\right]^{\mathsf{vir}}=\Delta^{\ddagger}\cap\left[{\prod_{v}}^{\diamond}\mathsf{DT}_{v}(X_{v})\right]^{\mathsf{vir}}.$

where we have suppressed the pullback of the cohomology class
$\Delta^{\ddagger}$ from the notation.

###### Proof.

We have a commutative diagram

${\bigtimes_{v}\mathsf{DT}_{v}(X_{v})}$${\bigtimes_{v}\mathsf{Hilb}(D_{v})}$${\prod_{v}^{\diamond}\mathsf{DT}_{v}(X_{v}).}$

The virtual classes are compatible under pushforward so this follows from the
projection formula. ∎

We prove the main gluing result. With the work up to this point, we have found
a particular birational model of the evaluation map such that the pullback of
the strict diagonal calculates the class of $\mathsf{DT}_{\gamma}$.

Recall we have produced a flattening of the evaluation morphism:

$\mathsf{ev}:\prod_{v}\mathsf{DT}_{\gamma(v)}(X_{v})\to\prod_{v}\mathsf{Hilb}^{\dagger}_{\beta}(D_{v}).$

Moreover, by further subdivision if required, we can assume that there is a
subdivision

$\bigtimes_{v}\mathsf{Hilb}(D_{v})\to\prod_{v}\mathsf{Hilb}^{\dagger}_{\beta}(D_{v}).$

We recall also that this map can be forced to have smooth source and target,
and therefore is a local complete intersection morphism.

######  Lemma 12.1.5.

Consider the following diagram, defined by fine and saturated logarithmic base
change:

${\prod_{v}^{\sim}\mathsf{DT}(X_{v})}$${\prod_{v}\mathsf{DT}^{\dagger}_{\gamma(v)}(X_{v})}$${\bigtimes_{v}\mathsf{Hilb}(D_{v})}$${\prod_{v}\mathsf{Hilb}^{\dagger}_{\beta}(D_{v}).}$$\scriptstyle{\mathsf{ev}^{\sim}}$$\scriptstyle{\mathsf{ev}^{\dagger}}$

The underlying map of stacks is also Cartesian.

###### Proof.

Immediate from combinatorial flatness of the evaluation map. ∎

We complete the proof of Theorem 8.2.2 using an intersection theory argument.

#### Proof of Theorem 8.2.2

We inspect the fiber diagram:

${\prod_{v}^{\sim}\mathsf{DT}(X_{v})}$${\prod_{v}\mathsf{DT}^{\dagger}_{\gamma(v)}(X_{v})}$${\bigtimes_{v}\mathsf{Hilb}(D_{v})}$${\prod_{v}\mathsf{Hilb}^{\dagger}_{\beta}(D_{v}).}$$\scriptstyle{\mathsf{ev}^{\sim}}$$\scriptstyle{\mathsf{ev}^{\dagger}}$$\scriptstyle{\alpha}$

The map $\alpha$ is a local complete intersection morphism, so Gysin pullback
is defined.

Key Claim. There is an equality of cycle classes:

$\alpha^{!}\left[\prod_{v}\mathsf{DT}^{\dagger}_{\gamma(v)}(X_{v})\right]^{\mathsf{vir}}=\left[{\prod_{v}}^{\sim}\mathsf{DT}(X_{v})\right]^{\mathsf{vir}}.$

In order to see this, observe that since $\alpha$ is a subdivision, we can
replace the bottom row of the square with the respective Artin fans:

${\prod_{v}^{\sim}\mathsf{DT}(X_{v})}$${\prod_{v}\mathsf{DT}^{\dagger}_{\gamma(v)}(X_{v})}$${\bigtimes_{v}\mathsf{Exp}_{0}(D_{v})}$${\prod_{v}\mathsf{Exp}^{\dagger}_{0}(D_{v}).}$$\scriptstyle{\mathsf{ev}^{\sim}}$$\scriptstyle{\mathsf{ev}^{\dagger}}$$\scriptstyle{a}$

The cone complexes of these spaces on the bottom row are the
$\mathsf{P}(\Sigma_{D_{e}})$ spaces discussed in Section 10 parameterizing
$0$-complexes; in fact, we can restrict to the locus with at most $m_{e}$
vertices, where $m_{e}$ the length of the putative subscheme along the edge,
determined by the class $\beta_{v}$ in the Chow $1$-complex.

We now observe that by construction, we have a factorization:

$\prod_{v}\mathsf{DT}^{\dagger}_{\gamma(v)}(X_{v})\to\prod_{v}\mathsf{Exp}^{\dagger}(X_{v})\to\prod_{v}\mathsf{Exp}^{\dagger}_{0}(D_{v}).$

By our choices of subdivision, the second arrow is flat. As a result, if we
consider the fiber square

${\prod_{v}^{\sim}\mathsf{Exp}(X_{v})}$${\prod_{v}^{\dagger}\mathsf{Exp}(X_{v})}$${\bigtimes_{v}\mathsf{Exp}_{0}(D_{v})}$${\prod_{v}\mathsf{Exp}^{\dagger}_{0}(D_{v}),}$$\scriptstyle{g}$$\scriptstyle{a}$

the vertical arrows are flat, the two horizontal arrows have the same
cotangent complex. Furthermore, the subdivision:

${\prod_{v}}^{\sim}\mathsf{DT}(X_{v})\to\prod_{v}\mathsf{DT}^{\dagger}_{\gamma(v)}(X_{v}).$

is the tropical model associated to the top arrow $g$.

Therefore, in order to prove the claim, it now suffices to show:

$g^{!}\left[\prod_{v}\mathsf{DT}^{\dagger}_{\gamma(v)}(X_{v})\right]^{\mathsf{vir}}=\left[{\prod_{v}}^{\sim}\mathsf{DT}(X_{v})\right]^{\mathsf{vir}}.$

This follows from the fact that we have a fiber square:

${\prod_{v}^{\sim}\mathsf{DT}(X_{v})}$${\prod_{v}\mathsf{DT}^{\dagger}_{\gamma(v)}(X_{v})}$${\prod_{v}^{\sim}\mathsf{Exp}(X_{v})}$${\prod_{v}^{\dagger}\mathsf{Exp}(X_{v}).}$$\scriptstyle{g}$

By [42, Section 5] the vertical arrows are equipped with the same obstruction
theory. Since $g$ is also a local complete intersection, commutativity of
virtual and Gysin pullback gives the claim.

We now return to the main statement. The arguments above produce the following
fiber diagram:

${\prod_{v}^{\sim}\mathsf{DT}(X_{v})}$${\prod_{v}\mathsf{DT}^{\dagger}_{\gamma(v)}(X_{v})}$${\bigtimes_{v}\mathsf{Hilb}(D_{v})}$${\prod_{v}\mathsf{Hilb}^{\dagger}_{\beta}(D_{v}).}$$\scriptstyle{\mathsf{ev}^{\sim}}$$\scriptstyle{\mathsf{ev}^{\dagger}}$$\scriptstyle{\alpha}$

Let $\Delta^{\ddagger}$ and $\Delta^{\dagger}$ be the strict transform of the
diagonal loci in the respective entries in the bottom row. Since
$\alpha_{\star}[\Delta^{\ddagger}]$ is the class $[\Delta^{\dagger}]$ and the
virtual class is pulled back under $\alpha$, the theorem follows compatibility
between pushforward and Gysin pullback.

∎

####  12.2. Review of the expanded GW degeneration formula

We provide a brief overview of the GW degeneration formula from [61]. The
basic setup is similar to the DT setting. We start with a degeneration
$\mathcal{Y}\to B$. There is an associated moduli space of logarithmic stable
maps:

$\mathsf{GW}_{\beta,g}(\mathcal{Y}/B)\to B.$

As explained in [61, Section 6.1], the Abramovich–Chen–Gross–Siebert
decomposition formula identifies rigid tropical maps from tropical curves,
labelled by genus and degree, to $\Delta_{\mathcal{Y}}$. These are in
bijection with the virtual irreducible components of the special fiber of
$\mathsf{GW}_{\beta,g}(\mathcal{Y}/B)\to B$.

The rigid tropical curve is a map

$\gamma\to\Delta_{\mathcal{Y}},$

and we may and do assume that the vertices and edges of $\gamma$ map onto
vertices and edges of $\Delta_{\mathcal{Y}}$. We will abuse notation slightly
by confusing the graph $\gamma$ and the parameterized map.

Each vertex $v$ determines a natural moduli space of logarithmic maps, with
the local data specified at $v$. We denote it $\mathsf{GW}(X_{v})$. The basic
version of the gluing construction is based on the natural map:

$\mathsf{ev}:\prod_{v}\mathsf{GW}(X_{v})\to\prod_{e}D_{e}\times D_{e}.$

The map is given by evaluation at the marked point corresponding to $e$ –
there is one copy for each side of the edge. So far, essentially any
birational model of the GW space may be used.

The main construction in [61] produces a modification

$\bigtimes_{v}\mathsf{GW}(X_{v})\to\prod_{v}\mathsf{GW}(X_{v}).$

The subdivision carries universal expansions $\mathcal{X}_{v}$ for each vertex
$v$. Moreover, if $e$ is an edge between $v$ and $v^{\prime}$, the induced
expansion of the divisor family $D_{e}$ is the same on $\mathcal{X}_{v}$ and
$\mathcal{X}_{v^{\prime}}$.

We denote the universal divisor expansion
$\mathcal{D}_{e}\to\bigtimes_{v}\mathsf{GW}(X_{v})$ and by
$\mathcal{D}_{e}^{\\{2\\}}$ its fiber square. We let
$\mathcal{D}_{e}^{\Delta}\subset\mathcal{D}_{e}^{\\{2\\}}$ the fiberwise
diagonal in the fibered square.

We can certainly find an appropriate model
$\mathsf{GW}^{\dagger}_{\gamma}(Y_{0})$ that maps to
$\bigtimes_{v}\mathsf{GW}(X_{v})$. The main theorem of [61, Section 6.4] is
stated below. Let $\gamma$ be a fixed tropical stable map and
$\overline{\gamma}$ its induced Chow $1$-complex. As in Remark 7.2.2 for each
rigid tropical stable map $\gamma$ with Chow $1$-complex $\overline{\gamma}$,
let $\ell(\gamma)$ denote the product, taken over all edges $e$ in
$\overline{\gamma}$, of the least common multiplies of the dilation factors
$m_{e^{\prime}}$ of edges $e^{\prime}$ in $\gamma$ that map to $e$.

######  Theorem 12.2.1.

Given the morphism:

$q:\mathsf{GW}_{\gamma}(Y_{0})\to\bigtimes_{v}\mathsf{GW}(X_{v}),$

there is an equality of Chow homology classes

$q_{\star}\left[\mathsf{GW}_{\gamma}(Y_{0})\right]^{\mathsf{vir}}=\prod_{e}m_{\cdot}\frac{1}{\ell(\gamma)}\cdot\bigcap_{e}\mathcal{D}_{e}^{\Delta}\cap\left[\bigtimes_{v}\mathsf{GW}(X_{v})\right]^{\mathsf{vir}},$

where $m_{e}$ is the tropical expansion factor along the edge $e$ in the
tropical map $\gamma$. Equivalently, $m_{e}$ is the contact order at the node
corresponding to $e$.

####  12.3. A simplified gluing formula in GW theory

We can now make a parallel set of manipulations of the degeneration formula in
GW theory as we performed in the first part of this section. Let us begin with
a statement of the theorem.

Let $\mathcal{Y}\to B$ continue to be a degeneration and let $\gamma$ be a
rigid tropical curve. We assume, as above, that the image of every vertex
(resp. edge) of $\gamma$ is a vertex (resp. vertex or edge) of
$\Delta_{\mathcal{Y}}$. Associated to each edge $e$ of $\gamma$ is a therefore
a divisor $D_{e}$ in $Y_{0}$. We set:

$\mathsf{Ev}_{\gamma}(v)=\prod_{e\colon v\prec e}D_{e},$

where the product runs over all edges and half edges adjacent to $v$. Each
vertex of $\gamma$ determines a genus, degree, and contact order for a mapping
space $\mathsf{GW}_{\gamma(v)}(X_{v})$. In the previous section, we used an
evaluation map from the moduli space $\bigtimes_{v}\mathsf{GW}(X_{v})$ the
fiber product over all edges of $\mathcal{D}_{e}^{\\{2\\}}$. There is a
natural factorization of this evaluation morphism:

$\bigtimes_{v}\mathsf{GW}(X_{v})\to{\prod_{e}}_{\bigtimes_{v}\mathsf{GW}(X_{v})}\mathcal{D}_{e}^{\\{2\\}}\to\prod_{v}\mathsf{Ev}_{\gamma}(v).$

The second factor is a fibered product ranging over the edges $e$ of $\gamma$.

By semistable reduction, we can find a subdivision of the product of
$\mathsf{Ev}_{\gamma}(v)$ over all vertices $v$, which we denote

$\bigtimes_{v}\mathsf{Ev}_{\gamma}(v)\to\prod_{v}\mathsf{Ev}_{\gamma}(v)$

such that the induced map

$\bigtimes_{v}\mathsf{GW}(X_{v})\to\bigtimes_{v}\mathsf{Ev}^{\dagger}_{\gamma}(v)$