# Wall-crossing of universal Brill-Noether classes

Alex Abreu  Universidade Federal Fluminense, Instituto de Matemática
<EMAIL_ADDRESS>and  Nicola Pagani  Department of Mathematical Sciences,
University of Liverpool<EMAIL_ADDRESS>
###### Abstract.

We give an explicit graph formula, in terms of decorated boundary strata
classes, for the wall-crossing of universal Brill-Noether classes.

More precisely, fix $n>0$ and $d<g$ , and two stability conditions
$\phi^{-},\phi^{+}$ for degree $d$ compactified universal (over
$\overline{\mathcal{M}}_{g,n}$) Jacobians that lie on opposite sides of a
stability hyperplane. Our main result is a formula for the difference between
$\mathsf{w}_{d}(\phi^{+})$ and the pullback of $\mathsf{w}_{d}(\phi^{-})$
along the (rational) identity map
$\mathsf{Id}\colon\overline{\mathcal{J}}^{d}_{g,n}(\phi^{+})\dashrightarrow\overline{\mathcal{J}}^{d}_{g,n}(\phi^{-})$.
The calculation involves constructing a resolution of the identity map by
means of subsequent blow-ups.

###### Contents

  1. 1 Introduction
    1. 1.a Related work
    2. 1.b Acknowledgments
  2. 2 Notation and preliminaries
    1. 2.a Posets
    2. 2.b Graphs
    3. 2.c Families of curves and sheaves
    4. 2.d Moduli spaces and graphs
  3. 3 Compactified Jacobians and Universal Brill–Noether Classes
    1. 3.a The universal stability space
    2. 3.b The stability hyperplanes
    3. 3.c Compactified Jacobians, universal and semistable family
    4. 3.d Brill–Noether classes
  4. 4 Normal crossing stratification categories and blowups
    1. 4.a Categories of resolved strata for a normal crossing stratification
    2. 4.b Normal crossing stratifications
    3. 4.c Intersection theory formulas
    4. 4.d Blow-up
  5. 5 Combinatorial aspects of Wall-Crossing
    1. 5.a Extremal sets, vine functions and full forests
    2. 5.b The stratification categories
    3. 5.c The case of “good” hyperplanes.
  6. 6 Nonsingular resolution of the identity
  7. 7 Wall–Crossing Formulas
    1. 7.a The case of disjoint blowups
    2. 7.b Wall-crossing in low codimension
    3. 7.c Pullbacks via Abel–Jacobi sections

##  1\. Introduction

The Brill–Noether theory of line bundles on nonsingular algebraic curves is a
classical pillar of XIX century algebraic geometry, which has been
rediscovered and reused to prove important contemporary results. Broadly
speaking, the theory is about studying the space of line bundles of a fixed
degree having a fixed number of linearly independent global sections (see
[ACGH85] and references therein for a survey of the classical results).

For fixed integers $g,n$ (we will assume for uniformity of notation, that
$g\geq 2$ and $n\geq 1$), and $d$ there exists a universal Jacobian
$\mathcal{J}_{g,n}^{d}\to\mathcal{M}_{g,n}$, a moduli space that parameterizes
isomorphism classes of degree $d$ line bundles over smooth, $n$-pointed curves
of genus $g$. From now on we assume $d<g$ and define the universal
Brill–Noether class $\mathsf{w}_{d}$ as the fundamental class in
$\mathcal{J}^{d}_{g,n}$ of the locus $\mathsf{W}_{d}$ of line bundles that
admit a nonzero global section. This locus has fiberwise codimension $g-d$
over $\mathcal{M}_{g,n}$ and it is empty for $d<0$. In this paper we study
extensions of this class to different compactifications of the universal
Jacobian.

The moduli space $\mathcal{M}_{g,n}$ admits a natural, modular and well-
studied compactification $\overline{\mathcal{M}}_{g,n}$ obtained by adding
(Deligne–Mumford) _stable_ pointed curves. On the other hand, there are
several natural compactifications of $\mathcal{J}_{g,n}^{d}$ over
$\overline{\mathcal{M}}_{g,n}$. In the words of Oda–Seshadri [OS79], this
should not be seen as a drawback of the theory, but rather a merit.

In [KP19] Kass–Pagani constructed an affine space of stability conditions
$V_{g,n}^{d}$ with an explicit hyperplane arrangement, with the property that
every $\phi\in V_{g,n}^{d}$ produces a compactification
$\overline{\mathcal{J}}^{d}_{g,n}(\phi)$ of the universal Jacobian, with good
properties (it is a nonsingular DM stack) when $\phi$ is not on a hyperplane.
This space comes with a natural origin — a canonical stability — and so far
most of the attention has been devoted to compactified Jacobians corresponding
to this particular value (or to its perturbations when the latter belongs to
some hyperplanes), see [GZ14], [HMP+].

In this paper we study how the Brill–Noether classes, suitably extended to
classes $\mathsf{w}_{d}(\phi)$ on $\overline{\mathcal{J}}^{d}_{g,n}(\phi)$,
vary in $\phi$. What we mean by this is the following: for different stability
conditions $\phi_{1},\phi_{2}$, the identity on the common open set
$\mathcal{J}^{d}_{g,n}$ of line bundles on smooth curves defines a rational
map

$\mathsf{Id}\colon\overline{\mathcal{J}}^{d}_{g,n}(\phi_{1})\dashrightarrow\overline{\mathcal{J}}^{d}_{g,n}(\phi_{2}),$

and we can then compute the difference
$\mathsf{w}_{d}(\phi_{2})-\mathsf{Id}^{*}\mathsf{w}_{d}(\phi_{1})$. By
“compute”, we mean produce an explicit “graph formula”, as in the case of
tautological classes on the moduli space of curves
$\overline{\mathcal{M}}_{g,n}$, which can all be expressed as linear
combinations of “decorated boundary strata classes” (see [Pan18]). While an
established theory of a tautological ring for
$\overline{\mathcal{J}}^{d}_{g,n}(\phi)$ is not yet available (a large
literature is available for the case of a single curve, the case of the
universal moduli space has recently been the subject of important results in
[Yin16], [BHP+], [HMP+]), there are several natural classes on each
compactified universal Jacobian, and “decorated boundary strata classes”,
supported on the boundary of $\overline{\mathcal{J}}^{d}_{g,n}(\phi)$, may be
defined in complete analogy with the case of $\overline{\mathcal{M}}_{g,n}$.
In fact, an important underlying motivation for our work is to develop a
categorical and wall–crossing framework for a theory of tautological classes
over compactified universal Jacobians.

We now discuss what we mean by “a suitable extension” for the class
$\mathsf{w}_{d}(\phi)$. One possible approach is to take the Zariski closure,
but this is very hard to control, and it does not have good formal properties
(for example, it does not commute with base change). Another approach is to
consider sheaves in $\overline{\mathcal{J}}^{d}_{g,n}(\phi)$ that admit a
nonzero global section, but that locus is, in general, not of the expected
dimension and not equidimensional. Our extension instead is by means of the
Thom–Porteous’ formula. By virtue of its universal property, there is a
tautological (or Poincaré) sheaf $\mathcal{L}_{\text{tau}}(\phi)$ on the
universal curve
$\pi\colon\overline{\mathcal{C}}_{g,n}\to\overline{\mathcal{J}}^{d}_{g,n}(\phi)$.
We define the extension as the degeneracy class

(1.1)
$\mathsf{w}_{d}(\phi):=c_{g-d}(-R^{\bullet}\pi_{\ast}\mathcal{L}_{\text{tau}}(\phi)),$

as in [Ful98, Chapter 14]. By the Thom–Porteous formula (see _loc.cit._), the
restriction of $\mathsf{w}_{d}(\phi)$ to $\mathcal{J}^{d}_{g,n}$ equals the
(Poincaré dual of the) original Brill–Noether class $\mathsf{w}_{d}$. We
compare (1.1) with the class of the Zariski closure in Proposition 3.38. The
class (1.1) is supported on the universal Brill–Noether locus, but in general
the latter does not have the expected codimension, hence its fundamental class
does not coincide with (1.1) (more details in Proposition 4.18).

The class (1.1) is the formal analogue of the $\lambda_{g-d}$ class on
$\overline{\mathcal{M}}_{g,n}$ (the Hodge bundle
$R^{\bullet}\pi_{\ast}(\omega_{\pi})$ being replaced by
$-R^{\bullet}\pi_{\ast}\mathcal{L}$). Given the important role that the
$\lambda$-classes have played in the enumerative geometry of curves /
intersection theory for moduli of curves, it is legitimate to expect that the
same will be true of $\mathsf{w}_{d}(\phi)$.

In this paper we assume that $\phi^{+}$ and $\phi^{-}$ are on opposite sides
of a stability hyperplane (Definition 5.1), and we give an explicit graph
formula for the difference

$\mathsf{w}_{d}(\phi^{+})-\mathsf{Id}^{*}\mathsf{w}_{d}(\phi^{-}).$

In order to achieve this, we first produce a nonsingular resolution of the
identity

${\widetilde{\mathcal{J}}^{d}_{g,n}(\phi^{+},\phi^{-})}$${\overline{\mathcal{J}}^{d}_{g,n}(\phi^{+})}$${\overline{\mathcal{J}}^{d}_{g,n}(\phi^{-})}$$\scriptstyle{p}$$\scriptstyle{p_{-}}$$\scriptstyle{\mathsf{Id}}$

by an explicit sequence of blow-ups of
$\overline{\mathcal{J}}^{d}_{g,n}(\phi^{+})$. We use this resolution to give,
in Theorem 7.4, an explicit and closed graph formula for the difference
$p^{*}(\mathsf{w}_{d}(\phi^{+}))-p_{-}^{*}(\mathsf{w}_{d}(\phi^{-}))$ in the
cohomology of $\widetilde{\mathcal{J}}^{d}_{g,n}(\phi^{+},\phi^{-})$. Finally,
we calculate the push-forward of that formula via $p$ to write a formula
(again a graph formula, explicit and closed) for the difference
$\mathsf{w}_{d}(\phi^{+})-\mathsf{Id}^{*}\mathsf{w}_{d}(\phi^{-})$.

Our construction of $\widetilde{\mathcal{J}}^{d}_{g,n}(\phi^{+},\phi^{-})$ and
our formulas are complicated by the fact that, for some of the hyperplanes,
the locus where the identity is undefined fails to be irreducible. In those
cases, the space $\widetilde{\mathcal{J}}^{d}_{g,n}(\phi^{+},\phi^{-})$ is
constructed as an explicit sequence of blow ups along centers that have
transversal self-intersection, and this construction plays an important part
in our paper.

In this introduction we describe the particular case of our construction and
formula when the indeterminacy locus is irreducible (this occurs in many
cases, and in some sense in most cases as long as $n>1$). Then the
indeterminacy locus
${\mathcal{J}}_{\beta}^{\prime}\subset\overline{\mathcal{J}}^{d}_{g,n}(\phi^{+})$
generically parameterizes curves with $2$ nonsingular components of genus,
say, $g_{X}$ and $g_{Y}$, carrying markings $S$ and $S^{c}$, and joined at a
certain number of nodes, say $t$, together with line bundles of some fixed
bidegree, say, $(d-d_{Y},d_{Y})$. The locus ${\mathcal{J}}_{\beta}^{\prime}$
can be parameterized by a “resolved stratum”

$f_{\beta}\colon{\mathcal{J}}_{\beta}\rightarrow{\mathcal{J}}_{\beta}^{\prime}\hookrightarrow\overline{\mathcal{J}}^{d}_{g,n}(\phi^{+})$

(which we simply call “a stratum” in the main body of the paper), where the
$t$ nodes are parameterized: a general point of ${\mathcal{J}}_{\beta}$ is a
triple of a $(|S|+t)$-pointed curve of genus $g_{X}$, a $(|S^{c}|+t)$-pointed
curve of genus $g_{Y}$, and a line bundle of bidegree $(d-d_{Y},d_{Y})$. The
conormal bundle to $f_{\beta}$ has rank $t$ and it splits as a direct sum of
line bundles, whose first Chern classes we call $\Psi_{1},\ldots,\Psi_{t}$
(see Remark 7.30 for more details on how these relate to the “classical”
$\psi$-classes in $\overline{\mathcal{M}}_{g,n}$). The base change to
${\mathcal{J}}_{\beta}$ of the universal family
$\pi\colon\overline{\mathcal{C}}_{g,n}\to\overline{\mathcal{J}}^{d}_{g,n}(\phi)$
consists of two irreducible components, say $\pi_{X}\colon
X\to{\mathcal{J}}_{\beta}$ and $\pi_{Y}\colon Y\to{\mathcal{J}}_{\beta}$, of
genus $g_{X}$ and $g_{Y}$ respectively, each carrying a tautological sheaf
$L_{X}$ and $L_{Y}$ (obtained by pulling back $\mathcal{L}_{\text{tau}}$).

Our main result for the difference
$\mathsf{w}_{d}(\phi^{+})-\mathsf{Id}^{*}\mathsf{w}_{d}(\phi^{-})$, in this
special case, reads (Corollary 7.33 with $m=1$):

$\sum_{\begin{subarray}{c}s+j+\lambda\\\
=g-d-t\end{subarray}}\binom{g_{Y}-d_{Y}-j-1}{g-d-
j-s}\frac{f_{{\beta}*}}{t!}\Bigg{(}c_{s}(-R^{\bullet}\pi^{X}_{\ast}L_{X}(-X\cap
Y))\cdot c_{j}(-R^{\bullet}\pi^{Y}_{\ast}L_{Y})\cdot
h_{\lambda}(\Psi_{1},\ldots,\Psi_{t})\Bigg{)}$

where $h_{\lambda}$ denotes the complete homogeneous polynomial of degree
$\lambda$ in $t$ variables.

The special case $d=g-1$ of the above formula, when the Brill–Noether class is
a divisor (the theta divisor), was discovered in [KP17, Theorem 4.1]. In that
case the calculation was massively simplified by the fact that the classes
have codimension $1$, and therefore, because the total space is nonsingular,
no blowup is required.

Theorem 7.4 is the main result in this paper. It computes the pull-back to the
resolution $\widetilde{\mathcal{J}}^{d}_{g,n}(\phi^{+},\phi^{-})$ of the
difference $\mathsf{w}_{d}(\phi^{+})-\mathsf{Id}^{*}\mathsf{w}_{d}(\phi^{-})$
in terms of decorated boundary strata classes. The formula for the difference
in $\overline{\mathcal{J}}^{d}_{g,n}(\phi^{+})$ is obtained by pushing the
latter forward along a blow-down morphism, which generates more complicated
coefficients.

The starting point to construct the resolution
$\widetilde{\mathcal{J}}^{d}_{g,n}(\phi^{+},\phi^{-})$ is the observation that
the tautological sheaf for $\phi^{+}$ is not $\phi^{-}$-stable, and the locus
${\mathcal{J}}_{\beta}^{\prime}$ where it fails $\phi^{-}$ stability
generically parameterizes curves with $2$ nonsingular irreducible components
(throughout called a “vine curve”) and a fixed bidegree. When the locus
${\mathcal{J}}_{\beta}^{\prime}$ is irreducible, the resolution is constructed
by blowing up $\overline{\mathcal{J}}^{d}_{g,n}(\phi^{+})$ at
${\mathcal{J}}_{\beta}^{\prime}$. The two components $X^{\prime}\cup
Y^{\prime}$ of the pull-back of the universal curve to the exceptional divisor
$E$ are now divisors (“universal twistors”) in the blowup of the universal
curve, and after suitably tensoring by one of them, the sheaf
$\mathcal{L}_{\text{tau}}(\phi^{+})$ becomes $\phi^{-}$-stable. The latter
sheaf defines the other morphism
$\widetilde{\mathcal{J}}^{d}_{g,n}(\phi^{+},\phi^{-})\to\overline{\mathcal{J}}^{d}_{g,n}(\phi^{-})$
by the universal property. The main technical difficulty is to suitably
identify a sequence of blowups at centers that have transversal self-
intersection, which allows one to generalize the above reasoning to the case
when the base locus ${\mathcal{J}}_{\beta}^{\prime}$ is not irreducible.

Note that other resolutions of the identity map may also be constructed
following existing literature ([AP20], [Hol21], [MW20], [HMP+], [CGH+]), but
those constructions yield _singular_ spaces.

In Section 3 we introduce the objects we work with, compactified universal
Jacobians and Brill–Noether classes. In Section 4 we write axioms for
“resolved” strata of normal crossing stratifications, and then prove some
general intersection theory results that are valid in this context. The main
geometric ideas here are not entirely new, but we could not find a suitable
reference in this generality, and we believe that this axiomatic point of view
will prove helpful in the current research landscape. In Section 5 we discuss
the combinatorial aspects that arise from a wall-crossing situation where
there are stability conditions $\phi^{\pm}$ are on opposite sides of a given
stability hyperplane (Definition 5.1). Our paper is concerned with the case of
rank $1$ sheaves on nodal curves, and the combinatorics of Section 5 should be
the shadow of a theory for higher dimension and rank. The central definition
is that, for each graph $G$ and divisor $D$ on $G$ and choice of stability
conditions $\phi^{\pm}$ on opposite sides of a hyperplane, of a poset
$\operatorname{Ext}(G,D)$ of “extremal” subsets of the vertices of $G$.
Section 6 gives the construction of the resolution
$\widetilde{\mathcal{J}}^{d}_{g,n}(\phi^{+},\phi^{-})$. Finally, in Section 7
we are then ready to employ intersection theory techniques and calculate the
wall-crossing term. At the end of Section 7 we explain how the pullback of the
wall-crossing term via an Abel–Jacobi section can be explicitly calculated in
terms of decorated boundary strata classes in $\overline{\mathcal{M}}_{g,n}$
by employing the main result of [PRvZ20].

In the background of this work, we produce two results that we believe are of
independent interest. The first is Theorem 3.29, where we interpret the
universal quasistable family over $\overline{\mathcal{J}}^{d}_{g,n}(\phi)$
(also known in the literature as Caporaso’s family from [Cap94], see also
[MMUV22] and [EP16]) as a fine compactified universal Jacobian
$\overline{\mathcal{J}}^{d}_{g,n+1}(\phi^{\prime})$ with one extra point.

Secondly, as part of Proposition 3.38, we describe the collection of stability
conditions for $d~{}<~{}0$ such that $\mathsf{w}_{d}(\phi)=0$. One can choose
a suitable Abel–Jacobi section
$\sigma\colon\overline{\mathcal{M}}_{g,n}\to\overline{\mathcal{J}}^{d}_{g,n}(\phi)$
and obtain a zero class $\sigma^{*}\mathsf{w}_{d}(\phi)$. A different formula
for the latter as a linear combination of standard tautological classes was
given in [PRvZ20] by means of the GRR formula. This gives tautological
relations in $\overline{\mathcal{M}}_{g,n}$ (see Remark 3.42 for the details).
Note that these relations are in degree larger than $g$ (the degree is $g-d$
for negative $d$), the same range of Pixton’s double ramification relations
(proven by Clader-Janda in [CJ18]).

###  1.a. Related work

An important motivation that we have not mentioned in the above discussion, is
its relation with the (possibly twisted) double ramification cycle. For a
review of the latter and related literature, we address the reader to [BHP+]
and [HMP+, Section 1.1]. We refer to [PRvZ20, Section 3.3] for how the double
ramification cycle relates to the Brill–Noether classes discussed here. As
pointed out in _loc.cit._ , the theory on how these classes are extended to
the boundary and then pulled back to $\overline{\mathcal{M}}_{g,n}$ via some
Abel–Jacobi section is trivial for nodal curves of compact type (i.e. the
moduli space of multidegree zero line bundles is compact) and for curves with
$1$ node, and the complement of the locus of all such curves is generically
parameterized by vine curves.

In [BHP+] the authors discuss the theory of a “universal double ramification
cycle” as an operational class of degree $g$ in the Artin stack of families of
line bundles on families of nodal curves, which correspond to “universally
intersecting with the closure of the zero section”. Our extensions (1.1) could
also be described in that language, and in fact the construction of an
operational class would avoid a lot of technical difficulties owing to the
fact that the classes (1.1) obviously commute with base change.

In this paper we do not discuss a modular description of our resolution
$\widetilde{\mathcal{J}}^{d}_{g,n}(\phi^{+},\phi^{-})$. We expect that one
such description should be possible following the recent work [Mol22b] by
Molcho. The same author has also recently proved in [Mol22a] that the pull-
back of the Brill–Noether classes $\mathsf{w}_{d}(\phi)$ via all Abel–Jacobi
_rational_ sections is tautological in $\overline{\mathcal{M}}_{g,n}$ (this
was conjectured in [PRvZ20, Section 4.1]).

###  1.b. Acknowledgments

To be added after the refereeing process.

##  2\. Notation and preliminaries

###  2.a. Posets

In this paper we will work with many posets (typically, the one underlying
some category of stratifications, and some of its subposets). Here we
recollect the relevant notation.

######  Definition 2.1.

Let $P$ be a finite partially ordered set (or a poset).

A subset $C$ of $P$ is called a _chain_ , if the partial order on $C$ induced
by $P$ is a total order on $C$.

A poset is _ranked_ if for every element $a$, all maximal chains having $a$ as
the largest element have the same length (called the _rank_ of $a$).

The poset $P$ is called a _forest_ , if for every $a\in P$ the lower set
$\\{b\leq a\\}$ is a chain. More generally, we say that a subset $F\subseteq
P$ is a forest if $F$ together with the partial order induced by $P$ is a
forest.

If $a>b$ and there exists no $c$ such that $a>c>b$, then we say that $a$
_covers_ $b$, and write $a\gtrdot b$.

###  2.b. Graphs

By a _graph_ we mean a finite, connected, undirected multigraph, decorated
with a genus function and markings (see for example [CCUW20, Section 3.1] and
[MMUV22, Section 2.1] for a precise definition).

If $G$ is a graph, we write $V(G)$ for its set of vertices and $E(G)$ for its
set of edges, we write $g\colon V(G)\to\mathbb{N}$ for the genus function and
$\operatorname{leg}\colon\\{1,\ldots,n\\}\to V(G)$ for the markings function.

If $S\subseteq V(G)$, we write $G(S)$ for the complete subgraph of $G$ on the
vertices $S$, and say that $G(S)$ is the subgraph of $G$ induced by $S$.

Given $V_{1},V_{2}\subseteq V(G)$, we write $E(V_{1},V_{2})$ for the edges
that have one endpoint in $V_{1}$ and another in $V_{2}$ (if the edge is a
loop, we include it if and only if its adjacent vertex is in both $V_{1}$ and
$V_{2}$).

If $G$ is a graph and $E\subseteq E(V(G))$, we denote by $G^{E}$ the graph
obtained from $G$ by adding exactly $1$ vertex, denoted $v_{e}$, in the
“interior” of each edge $e\in E$. We call each such $v_{e}$ an _exceptional
vertex_ of $G^{E}$.

A graph $G$ is _stable_ if

$2g(v)-2+|E(\\{v\\},\\{v\\}^{c})|+|\operatorname{leg}^{-1}(v)|>0$

for every vertex $v\in V(G)$.

###  2.c. Families of curves and sheaves

A _nodal curve_ $C$ is a reduced and connected projective scheme of dimension
$1$ over some fixed algebraically closed field, with singularities that are at
worst ordinary double points. The (arithmetic) _genus_ of $C$ is
$p_{a}(C)=h^{1}(C,\mathcal{O}_{C})$. A _subcurve_ $X$ of $C$ is a connected
union of irreducible components of $C$. Its complement $X^{c}$ is the union of
the other components of $C$.

A _$n$ -pointed curve_ is a tuple $(C,p_{1},\ldots,p_{n})$ where $C$ is a
nodal curve, and $p_{1},\ldots,p_{n}$ are pairwise distinct nonsingular points
of $C$. Its _dual graph_ $G(C)$ has the irreducible components of $C$ as
vertices, the nodes of $C$ as edges, the geometric genus (resp. the marked
points) of each component as the genus (resp. the markings) decoration.

A morphism $f\colon C^{\prime}\to C$ of nodal curves is _a semistable
modification_ if it is obtained by contracting some subcurves, not necessarily
irreducible, $E\subset C^{\prime}$ such that $g(E)=0$ and $|E\cap E^{c}|=2$.
Every subcurve $E\subset C^{\prime}$ contracted by $f$ is called _an
exceptional curve of $f$_. A semistable modification such that every
exceptional curve is irreducible is called a _quasistable modification_.

A coherent sheaf on a nodal curve $C$ has _rank $1$_ if its localization at
each generic point of $C$ has length $1$. It is _torsion-free_ if it has no
embedded components. If the stalk of a torsion-free sheaf $F$ over $C$ fails
to be locally free at a point $P\in C$, which must necessarily be a node, we
will say that $F$ is _singular_ at $P$. If $F$ is a rank $1$ torsion-free
sheaf on $C$ we say that $F$ is _simple_ if its automorphism group is
$\mathbb{G}_{m}$ or, equivalently, if removing from $C$ the singular points of
$F$ does not disconnect $X$.

A _family of nodal curves_ over a $\mathbb{C}$-scheme $S$ is a proper and flat
morphism $\mathcal{C}\to S$ whose fibers are nodal curves. (Throughout, all
families $\mathcal{C}/S$ will admit a distinguished section in the $S$-smooth
locus of $\mathcal{C}$). A _semistable (resp. a quasistable) modification_ of
the family $\mathcal{C}/S$ is another family $\mathcal{C}^{\prime}/S$ with a
$S$-morphism $f\colon\mathcal{C}^{\prime}\to\mathcal{C}$ that is a semistable
(resp. a quasistable) modification (as defined above) on all geometric points
$s\in S$.

If $T$ is a $S$-scheme, a _family of rank $1$ torsion-free simple sheaves_
parameterized by $T$ over a family of curves $\mathcal{C}\to S$ is a coherent
sheaf $F$ of rank $1$ on $\mathcal{C}\times_{S}T$, flat over $T$, whose fibers
over the geometric points are torsion-free and simple.

If $F$ is a rank $1$ torsion-free sheaf on a nodal curve $C$, the _(total)
degree_ of $F$ is $\deg_{C}(F):=\chi(F)-1+p_{a}(C)$.If $X\subseteq C$ is a
subcurve, we denote by $F_{X}$ the maximal torsion-free quotient of
$F\otimes\mathcal{O}_{X}$. The total degree and the degree of $F_{X}$ and
$F_{X^{c}}$ are related by the formula

(2.2) $\deg_{C}(F)=\deg_{X}F+\deg_{X^{c}}F+\delta_{X\cap X^{c}}(F),$

where $\delta_{S}(F)$ is the number of points in $S$ where the stalk of $F$
fails to be locally free.

A line bundle $F^{\prime}$ on a semistable modification $f\colon C^{\prime}\to
C$ is called _positively admissible_ (see [EP16]) if $\deg_{E}(F^{\prime})$ is
either $0$ or $1$ on every exceptional subcurve of $f$. The following results
follow from [EP16, Section 5].

######  Proposition 2.3.

Let $\pi\colon\mathcal{C}\to S$ and $\pi^{\prime}\colon\mathcal{C}^{\prime}\to
S$ be families of nodal curves and $f\colon\mathcal{C}^{\prime}\to X$ be a
semistable modification. Let $F^{\prime}$ be a positively admissible sheaf on
$\mathcal{C}^{\prime}$ and set $F=f_{*}(F^{\prime})$.

  1. (1)

The sheaf $F$ is a torsion free rank-$1$ sheaf and $R^{1}f_{*}(F^{\prime})=0$,
in particular $f_{*}(F^{\prime})$ commutes with base change. Moreover, we have
that $R^{\bullet}\pi_{*}(F)=R^{\bullet}\pi^{\prime}_{*}(F^{\prime})$.

  2. (2)

The sheaf $f_{*}(F^{\prime})$ is invertible if and only if
$\deg_{E}(F^{\prime})=0$ on every exceptional subcurve of $f$. Moreover, in
this case, $F^{\prime}=f^{*}f_{*}(F^{\prime})$.

  3. (3)

If $f$ is a quasistable modification and $\deg_{E}(F^{\prime})=1$ for every
exceptional subcurve, then
$\mathcal{C}^{\prime}=\mathbb{P}_{\mathcal{C}}(F^{\vee})$ and $F^{\prime}$ is
isomorphic to the tautological line bundle
$\mathcal{O}_{\mathbb{P}_{\mathcal{C}}(F^{\vee})}(1)$.

  4. (4)

More generally, we have that $f$ factors as
$X^{\prime}\xrightarrow{g}\mathbb{P}_{\mathcal{C}}(F^{\vee})\to X$, and
$\mathcal{O}(1)\cong g_{*}(F^{\prime})$ and $F^{\prime}\cong
g^{*}(\mathcal{O}(1))$.

In particular, we have the following.

######  Corollary 2.4.

Let $\mathcal{C}\to S$ be a family of nodal curves. Taking the direct image
under the quasistable modification gives a bijection between isomorphism
classes of positively admissible line bundles on quasistable modifications of
$\mathcal{C}/S$, and isomorphism classes of families of rank $1$ torsion free
sheaves on $\mathcal{C}$.

We now define the multidegree of a sheaf on a nodal curve as the multidegree
of the unique positively admissible line bundle as in the above corollary.

A degree $d$ _pseudodivisor_ on a graph $G$ is a pair $(E,D)$ where
$E\subseteq E(G)$ and $D\in\operatorname{Div}^{d}(G^{E})$ satisfies
$D(v^{\prime})=1$ for each exceptional vertex $v^{\prime}$. When $E$ is empty,
we simply write $D$ in place of the pair $(\varnothing,D)$.

Given a degree-$d$ rank $1$ torsion free sheaf $F$ on a curve $C$, we define
the multidegree $\underline{\deg}(F)$ of $F$ as the pseudodivisor $(E,D)$ on
the dual graph $G(C)$ of $C$ as follows. The set $E$ is the set of edges of
$G(C)$ that correspond to nodes of $C$ where $F$ is not locally free. The
divisor $D$ on $G(C)^{E}$ is defined by $D(v)=\deg_{C_{v}}(F_{C_{v}})$ if
$v\in V(G(C))\subseteq V(G(X)^{E})$ and $D(v)=1$ for every exceptional vertex
$v$. By Equation (2.2), we have that $(E,D)$ is a degree-$d$ pseudodivisor.

Note also that a rank $1$ torsion free sheaf on $C$ is simple if and only if
its multidegree $(E,D)$ has the property that $E$ does not disconnect the
graph $G(C)$.

###  2.d. Moduli spaces and graphs

Here we discuss some general notation on moduli spaces of curves. We refer the
reader to [ACG11] for more details on nodal curves and their dual graphs.

A $n$-pointed curve $(C,p_{1},\ldots,p_{n})$ is _stable_ if
$|\operatorname{Aut}(C,p_{i})|<\infty$. We will sometimes abuse notation and
write $C$ for $(C,p_{i})$. For example, we will say that the genus of
$(C,p_{i})$ is $g$ to mean that the $h^{1}(C,\mathcal{O}_{C})=g$, the
arithmetic genus of the underlying curve $C$ is $g$.

We will denote by $\overline{\mathcal{M}}_{g,n}$ the moduli spaces of stable
$n$-pointed curves of genus $g$. The moduli space comes admits a
stratification by dual graphs, which we now discuss.

####  2.d.1. Stable graphs

We denote by $G_{g,n}$ the small category of stable, $n$-pointed graphs of
genus $g$ (where we have fixed a choice of $1$ object for each isomorphism
class). Morphisms $G\to G^{\prime}$ are given by an edge contraction followed
by an isomorphism. (More details in [ACG11] and [MMUV22]). There is a natural
functor $G_{g,n+1}\to G_{g,n}$ that forgets the last point and stabilizes the
graph.

####  2.d.2. Stratification of moduli of stable curves.

For $G\in G_{g,n}$ there is a gluing morphism

$f_{G}\colon\overline{\mathcal{M}}_{G}:=\prod_{v\in
V(G)}\overline{\mathcal{M}}_{g(v),n(v)}\rightarrow\Big{[}\prod_{v\in
V(G)}\overline{\mathcal{M}}_{g(v),n(v)}/\operatorname{Aut}(G)\Big{]}\to\overline{\mathcal{M}}_{G}^{\prime}\hookrightarrow\overline{\mathcal{M}}_{g,n}.$

We say that $G$, or $\overline{\mathcal{M}}_{G}$, or $f_{G}$, is a (resolved)
stratum of $\overline{\mathcal{M}}_{g,n}$. We regard
$\overline{\mathcal{M}}_{G}$ as a “resolved stratum” and its image
$\overline{\mathcal{M}}_{G}^{\prime}$ as the corresponding “embedded stratum”.

The codimension $1$ strata are the following divisors generically
parameterizing curves with $1$ node:

  1. (1)

the divisor $\Delta_{\text{irr}}$, generically parameterizing irreducible
curves

  2. (2)

for $0\leq i\leq g$ and $S\subseteq[n]$ (except $i=0$ and $|S|<2$ and $i=g$
and $|S|>n-2$), the divisor $\Delta_{i,S}=\Delta_{g-i,S^{c}}$ generically
parameterizing curves with $2$ components, of which one of genus $i$ carrying
the marked points in $S$.

On the (resolved) stratum the normal bundle to $f_{G}$ splits as a direct sum
of line bundles

$N_{f_{G}}=\bigoplus_{e\in E(G)}\mathbb{L}_{e}.$

We denote by $\Psi_{e}=-c_{1}(\mathbb{L}_{e})$. (Recall that, if $e$ is the
edge whose half edges $h(e),h^{\prime}(e)$ are based at $v,v^{\prime}\in
V(G)$, then the cotangent line bundles to $h(e)$ and $h^{\prime}(e)$ are
denoted by $\mathbb{L}_{h(e)}$ and $\mathbb{L}_{h^{\prime}(e)}$ and its first
Chern classes $\psi_{h(e)}$ and $\psi_{h^{\prime}(e)}$. We then have
$\mathbb{L}_{e}=\mathbb{L}_{h(e)}^{\vee}\boxtimes\mathbb{L}_{h^{\prime}(e)}^{\vee}$
and so $\Psi_{e}=\psi_{h(e)}+\psi_{h^{\prime}(e)}$, but this will not play a
role.)

In Section 4 we will define what the category of (resolved) strata induced by
normal crossing divisors on a DM stack, and will interpret the category
$G_{g,n}$ as the category of strata of the nonsingular DM-stack
$\overline{\mathcal{M}}_{g,n}$ induced by the normal crossing divisor
$\Delta=\Delta_{\text{irr}}\cup\bigcup_{i,S}\Delta_{i,S}$.

##  3\. Compactified Jacobians and Universal Brill–Noether Classes

In this chapter we introduce the basic objects of study in this paper,
compactified universal Jacobians, and extensions of universal Brill–Noether
classes by means of Thom–Porteous formula. We also recall the results on the
stability space of compactified universal Jacobians that we will need later.

###  3.a. The universal stability space

Here we recall the definition and first results on the stability space of a
single curve and on the universal stability space $V_{g,n}^{d}$ from [KP19].

######  Definition 3.1.

For a fixed graph $G$, we define the space of polarizations

$V^{d}_{\text{stab}}(G):=\left\\{\phi\in\mathbb{R}^{V(G)}:\ \sum_{v\in
V(G)}\phi(v)=d\right\\}\subset\mathbb{R}^{V(G)}.$

For $V\subseteq V(G)$, we write $\phi(V)$ for $\sum_{v\in V}\phi(v)$.

Every morphism $f\colon G\rightarrow G^{\prime}$ of graphs induces a morphism
$f_{*}\colon V^{d}_{\text{stab}}(G)\rightarrow
V^{d}_{\text{stab}}(G^{\prime})$ by setting

(3.2) $f_{*}\phi(v^{\prime})=\sum\limits_{f(v)=v^{\prime}}\phi(v)$

and we define the space of universal polarizations as the limit (or inverse
limit)

$V^{d}_{g,n}:=\varprojlim_{G\in G_{g,n}}V^{d}_{\text{stab}}(G),$

i.e. as the space of assignments $\left(\phi(G)\in
V^{d}_{\text{stab}}(G)\colon\ G\in G_{g,n}\right)$ that are compatible with
all graph morphisms.

We now present a simple description of the universal stability space
$V_{g,n}^{d}$ that follows from [KP19, Corollary 4.3]. The result requires
that we introduce some notation for graphs of “vine curves”.

######  Definition 3.3.

A _vine curve triple_ $(i,t,S)$ consists of two natural numbers $i,t$ and a
subset $S\subseteq[n]$, such that $0\leq i\leq g$, $1\leq t$, $i+t\leq g+1$,
and such that if $(i,t)=(0,1)$ then $|S|\geq 2$, if $(i,t)=(0,2)$ then
$|S|\geq 1$, if $(i,t)=(g,1)$ then $|S^{c}|\geq 2$ and if $(i,t)=(g-1,2)$ then
$|S^{c}|\geq 1$.

A _vine curve_ is a stable graph $G(i,t,S)$ associated to a vine curve triple,
which consists of two vertices of genus $i$ and $g-i$ respectively connected
by $t$ edges, and with marking $S$ on the first vertex and $S^{c}$ on the
second vertex. We will always assume that $S$ contains the first marked point.

The stability space $V^{d}_{\text{stab}}(G(i,t,S))$ is an affine subspace of
$\mathbb{R}^{2}$. We can parameterize it by means one variable $x_{i,t,S}$ by
taking the inverse image under the projection onto the first factor. That
means, we describe

$V^{d}_{\text{stab}}(G(i,t,S))=\\{(x_{i,t,S},d-x_{i,t,S}):x_{i,t,S}\in\mathbb{R}\\}\subset\mathbb{R}^{2}.$

We now introduce the stability space of “vine curves” using the previous
definition.

######  Definition 3.4.

We let

$T_{g,n}^{d}:=\prod_{\begin{subarray}{c}(i,t,S)\\\ \textrm{ a vine curve
triple }\end{subarray}}V^{d}_{\text{stab}}(G(i,t,S)).$

Then we define:

  1. (1)

The vector space $C^{d}_{g,n}$ as the quotient of $T^{d}_{g,n}$ obtained as
the product of all vine curve triples of the form
$V^{d}_{\text{stab}}(G(i,1,S))$.

  2. (2)

The vector space $D^{d}_{g,n}$ is the quotient of $T^{d}_{g,n}$ obtained as
the product of all $V^{d}_{\text{stab}}(G(0,2,\\{j\\})$ for $j=1\ldots,n$.

Throughout we will use the coordinates $x_{i,t,S}$ introduced in the end of
Definition 3.3 on the spaces $T_{g,n}^{d}$ and on its quotients $C_{g,n}^{d}$
and $D_{g,n}^{d}$.

There are natural restriction affine linear maps:

$\tau_{d}\colon V_{g,n}^{d}\to T_{g,n}^{d},\quad\rho_{d}\colon V_{g,n}^{d}\to
C_{g,n}^{d}\times D_{g,n}^{d}$

One of the main results of [KP19, Section 3] is that the universal stability
space embeds into the “vine curves” stability space, and that $\rho_{d}$ is an
isomorphism.

######  Proposition 3.5.

([KP19, Lemma 3.8, Corollary 3.4]) The affine linear map $\tau_{d}$ is
injective. The vector space homomorphism $\rho_{0}$ is an isomorphism. Each
morphism $\rho_{d}$ is an isomorphism of affine spaces.

###  3.b. The stability hyperplanes

We will later see in Section 3.c that for every universal stability condition
$\phi\in V_{g,n}^{d}$ there exists a compactified universal Jacobian
parameterizing $\phi$-stable (rank $1$, torsion free) sheaves on every (flat)
family of $n$-pointed stable curves of genus $g$. Here we combinatorially
introduce the degenerate locus of $V_{g,n}^{d}$, which will later be seen to
be the locus of $\phi$’s such that there exist strictly semistable sheaves on
some stable curves. We will introduce the degenerate locus as a union of
hyperplanes (which one could think of as a finite, non-centered, toric
hyperplane arrangement). This explicit description is taken from [KP19,
Section 5].

######  Definition 3.6.

We say that a polarization $\phi\in V^{d}_{\text{stab}}(G)$ is _degenerate_ if
for some subset $\varnothing\subsetneq V_{0}\subsetneq V(G)$ the quantity

(3.7) $\frac{\left|E(V_{0},V_{0}^{c})\right|}{2}+\sum\limits_{v\in
V_{0}}\phi(v)$

is an integer.

We say that a universal stability condition $\phi\in V_{g,n}^{d}$ is
_degenerate_ if for some $G\in{G}_{g,n}$, the $G$-component $\phi(G)$ is
degenerate in $V^{d}_{\text{stab}}(G)$.

The degenerate locus is a locally finite union of affine hyperplanes, and we
will soon describe these hyperplane explicitly. Let us start with a simple
example.

######  Example 3.8.

(Vine curves) If $G$ is a vine curve, after identifying
$V^{d}_{\text{stab}}(G)=\mathbb{R}$ by projecting onto the first factor (as
done in the end of Definition 3.3), we have that the degenerate locus is a
locally finite collection of points that only depends on the parity of the
number of nodes $t$. If $t$ is even, the degenerate locus corresponds to the
$\mathbb{Z}\subset\mathbb{R}$. If $t$ is odd, the degenerate locus corresponds
to the $\frac{1}{2}+\mathbb{Z}\subset\mathbb{R}$.

We now give an explicit description of the degenerate locus in $V_{g,n}^{d}$,
based on [KP19, Section 5]. By Proposition 3.5, we have that
$V_{g,n}^{d}\subset T_{g,n}^{d}$, where the latter is the stability space of
vine curves (one for each topological type), with coordinates $x_{i,t,S}$ for
each vine curve triple $(i,t,S)$ (see Definition 3.3).

For each vine curve triple $(i,t,S)$ and integer $k$, define the (translate of
the coordinate) hyperplane

(3.9) $T_{g,n}^{d}\supset
H(i,t,S;k):=\begin{cases}\\{x_{i,t,S}=k\\}&\textrm{for even }t,\\\
\\{x_{i,t,S}=\frac{1}{2}+k\\}&\textrm{for odd }t.\end{cases}$

One main result of [KP19, Section 5] is that the degenerate locus in the
universal stability space is the pull-back of translates of coordinate
hyperplanes in the stability space of vine curves. More precisely:

######  Proposition 3.10.

([KP19, Lemma 5.8]) The degenerate locus in $V_{g,n}^{d}$ is a union of
hyperplanes. Each hyperplane is the inverse image via the affine linear
embedding $\tau_{d}\colon V_{g,n}^{d}\subset T_{g,n}^{d}$ of a hyperplane of
the form $H(i,t,S;k)$.

This description hides the difficulty that the embedding $\tau_{d}$ has, in
general, a very high codimension.

A more explicit description of the degenerate locus can be obtained via the
isomorphism $V_{g,n}^{d}\cong C_{g,n}^{d}\times D_{g,n}^{d}$. When expressing
the hyperplanes of (3.9) in terms of the coordinates $x_{i,1,S}$ and the
coordinates $x_{j}:=x_{0,2,\\{j\\}}$, by [KP19, Theorem 2] we have111Note that
the formula in loc.cit is translated by the coordinates of a “degree-$d$
canonical stability condition” – a choice of an origin in $V_{g,n}^{d}$ that
we do not discuss here.

(3.11) $x_{i,t,S}=\frac{2g-2i-t}{2g-2}\cdot\sum_{j\in
S}x_{j}+\frac{2i-2+t}{2g-2}\cdot\Bigg{(}d-\sum_{j\notin S}x_{j}\Bigg{)}$

whenever $t\geq 2$. Therefore, the stability hyperplanes take the following
form

(3.12) $H(i,1,S;k)=\left\\{x_{i,1,S}=k+\frac{1}{2}\right\\}$

for all vine curve triples (Definition 3.3) of the form $(i,1,S)$ (the
boundary divisors in $\overline{\mathcal{M}}_{g,n}$ that generically
parameterize curves with $2$ components)

(3.13) $H(i,t,S;k)=\left\\{\frac{2g-2i-t}{2g-2}\cdot\sum_{j\in
S}x_{j}+\frac{2i-2+t}{2g-2}\cdot\Bigg{(}d-\sum_{j\notin
S}x_{j}\Bigg{)}=k+\frac{t}{2}\right\\}$

for all vine curve triples $(i,t,S)$ with $t\geq 2$.

Note that the degenerate locus parameterized by the hyperplanes in (3.12) and
(3.13) may come with multiplicities. In other words, there exist
$(i_{1},t_{1},S_{1};k_{1})\neq(i_{2},t_{2},S_{2};k_{2})$ such that
$H(i_{1},t_{1},S_{1};k_{1})=H(i_{2},t_{2},S_{2};k_{2})$. We will now analyse
these hyperplanes and study when they may coincide.

It is immediate to observe that a necessary condition for two hyerplanes of
this form to coincide, is that their corresponding subset of marked points
must also coincide:

######  Proposition 3.14.

If any two hyperplanes $H(i_{1},t_{1},S_{1};k_{1})$ and
$H(i_{2},t_{2},S_{2};k_{2})$ coincide, then $S_{1}=S_{2}$.

###### Proof.

Straightforward. ∎

First we deal with the hyperplanes of (3.12), occurring on compact type vine
curves (or divisorial vine curves). Those are all simple:

######  Proposition 3.15.

The hyperplanes in (3.12) are pairwise distinct and each of them is distinct
from any of the hyperplanes in (3.13).

###### Proof.

Straightforward. ∎

The next proposition is about hyperplanes of the form (3.13) with $S\neq[n]$.
As we shall discuss in Section 5.c, a stability hyperplane of this type
witnesses a change of stability on loci of vine curves that are _disjoint_.

######  Proposition 3.16.

If $S\neq[n]$ and $(i_{1},t_{1};k_{1})\neq(i_{2},t_{2};k_{2})$ are such that
$H(i_{1},t_{1},S;k_{1})$ and $H(i_{2},t_{2},S;k_{2})$ are equal, then
$2i_{1}+t_{1}=2i_{2}+t_{2}$.

###### Proof.

Straightforward. ∎

The most interesting vine curves from the point of view of the stability
decomposition are those with $S=[n]$. Over those vine curves it can occur that
two stability hyperplanes of the form (3.13) coincide. For example, if $d=0$,
by fixing $\sum_{j\in[n]}x_{j}=g-1$ one sees that all hyperplanes of the form
$H(i,t,[n];k)$ with $i+\lceil t/2\rceil+k=g$ coincide (note that this is a
finite collection, because of the constraints $i+t\leq g+1$, $i\geq 0$ and
$t\geq 2)$.

###  3.c. Compactified Jacobians, universal and semistable family

Here we define, for every nondegenerate $\phi\in V_{g,n}^{d}$, a fine
compactified universal Jacobian $\overline{\mathcal{J}}^{d}_{g,n}(\phi)$,
parameterizing $\phi$-stable sheaves. The construction is taken from [KP19,
Section 4], in the language of pseudodivisors from [AP20, Section 4]. Each
fine compactified Jacobian will come with a normal crossing stratification
category (an abstract definition of this notion will be given and discussed in
the next section).

We also describe (Theorem 3.29) a quasistable modification of the universal
curve
$\overline{\mathcal{C}}_{g,n}(\phi)\to\overline{\mathcal{J}}^{d}_{g,n}(\phi)$
as a certain $(n+1)$-universal Jacobian
$\overline{\mathcal{J}}^{d}_{g,n+1}(\alpha(\phi))$.

######  Definition 3.17.

For $\phi\in V^{d}_{\text{stab}}(G)$ we say that a pseudodivisor $(E,D)$ is
_$\phi$ -semistable_ if

(3.18) $\phi(V_{0})-\deg_{V_{0}}(D)+\frac{|E(V_{0},V_{0}^{c})|}{2}\geq 0$

for every $V_{0}\subseteq V(G^{E})$. We say that $(E,D)$ is _$\phi$ -stable_
if the inequality above is strict for every $V_{0}$ such that $V_{0}\neq
V(G^{E})$ and $V_{0}$ is not contained in the set of exceptional vertices.
Given $v_{0}\in V(G)$, we say that $(E,D)$ is _$(\phi,v_{0})$ -quasistable_ if
the inequality is strict for every $V_{0}$ such that $V_{0}\neq V(G^{E})$ and
$v_{0}\in V_{0}$.

As stipulated in Section 2.c, when $E=\varnothing$, we will simply write $D$
for $(\varnothing,D)$.

######  Remark 3.19.

By [AP20, Proposition 4.6] if a pseudodivisor $(E,D)$ on $G$ is
$(\phi,v_{0})$-quasistable for some $(\phi,v_{0})$, then $E\subseteq E(G)$
does not disconnect $G$.

######  Remark 3.20.

We have introduced the degenerate locus of $V^{d}_{\text{stab}}(G(X))$ and of
$V_{g,n}^{d}$ in Definition (3.6). We claim that, in both cases, an element
$\phi$ is nondegenerate if and only if all semistable pseudodivisors are
stable. The “only if” is immediate. The other implication is proved in [KP19,
Section 5].

We now define stability for rank $1$ torsion free sheaves on curves.

######  Definition 3.21.

([KP19, Definition 4.2]) Let $C$ be a nodal curve with dual graph $G(C)$ and
let $\phi\in V^{d}_{\text{stab}}(G(C))$. A rank $1$ torsion-free sheaf $F$ of
degree $d$ on $C$ is _$\phi$ -(semi)stable_ if its multidegree
$\underline{\deg}(F)$ is a $\phi$-(semi)stable pseudodivisor.

If $P\in C^{\text{sm}}$ is a nonsingular point of $C$ in the component
$C_{v_{0}}$, we say that $F$ is _$(\phi,P)$ -quasistable_ if
$\underline{\deg}(F)$ is $(\phi,v_{0})$-quasistable.

If $C^{\prime}\to C$ is a semistable modification of $C$ and $C^{\prime}$ is a
positively admissible line bundle on $C^{\prime}$, we say that $F^{\prime}$ is
$\phi$-(semi)stable or $(\phi,P)$-quasistable if so is $f_{*}(F^{\prime})$.

For $\phi\in V^{d}_{\text{stab}}(G(C))$ and $P\in C$, we define
$\overline{\mathcal{J}}^{d}_{\phi,P}(C)$ to be the subscheme of
$\operatorname{Simp}^{d}(C)$ parameterizing $(\phi,P)$-quasistable sheaves.

Note that if $F$ is a rank $1$ torsion free sheaf on $C$ then (1) if $F$ is
$(\phi,P)$-quasistable then it is simple, and (2) the sheaf $F$ is simple if
and only if its multidegree $(E,D)$ has the property that $E\subseteq G(C)$ is
nondisconnecting.

######  Remark 3.22.

Let $\phi\in V^{d}_{\text{stab}}(G(C))$ and $p\in C^{\text{sm}}$ be as above.
Let $\phi^{\prime}\in V^{d}_{\text{stab}}(G(C))$ be a small perturbation of
$\phi$ obtained by subtracting a small $\epsilon>0$ to $\phi$ on the vertex of
$G(C)$ containing $P$, and by subtracting a small positive amount on all other
components (so that $\sum\phi^{\prime}(v)=\sum\phi(v)=d$).

Then $(\phi,P)$-quasistability coincides with $\phi^{\prime}$-stability which
in turn coincides with $\phi^{\prime}$-semistability,

We are now ready to introduce the notion of _universal_ polarizations and
compactified Jacobians. Each universal polarization will give rise to a fine
compactified Jacobian, and to a stratification category. Recall that, for any
$1\leq i\leq n$, we denote by
$\sigma_{i}\colon\overline{\mathcal{M}}_{g,n}\to\overline{\mathcal{C}}_{g,n}$
the $i$-th smooth section.

######  Definition 3.23.

Let $\phi\in V_{g,n}^{d}$ be a universal polarization.

We define ${\mathfrak{C}}_{g,n}(\phi)$ to be the category whose objects are
$(G,(E_{G},D_{G}))$ where $G$ is an object of $G_{g,n}$ and $(E_{G},D_{G})$ is
a $\phi$-semistable pseudodivisor on $G$. A morphism
$(G,(\varnothing,D_{G}))\to(G^{\prime},(\varnothing,D_{G^{\prime}}))$ in
${\mathfrak{C}}_{g,n}(\phi)$ is a morphism
$f\in\operatorname{Mor}_{G_{g,n}}(G,G^{\prime})$ such that the induced
homomorphism
$f_{\ast}\colon\operatorname{Div}(G)\to\operatorname{Div}(G^{\prime})$ on
divisors satisfies $f_{\ast}(D)=D^{\prime}$. We refer to [AP20, Section 2.1]
for the notion of a morphism
$(G,(E_{G},D_{G}))\to(G^{\prime},(E_{G^{\prime}},D_{G^{\prime}}))$ when
$E_{G},E_{G^{\prime}}$ are nonempty.

Similarly, we define $\mathfrak{C}_{g,n}(\phi,\sigma_{i})$ to be the category
whose objects are $(G,(E_{G},D_{G}))$ and $(E_{G},D_{G})$ is
$(\phi,\sigma_{i})$-quasistable. (By abuse of notation, $\sigma_{i}$ gives the
choice of the element $\operatorname{leg}_{G}(i)\in V(G)$ for each stable
graph $G$).

We say that a family of rank $1$ torsion-free simple sheaves of degree $d$ on
a family of stable curves is _$\phi$ -(semi)stable_ or
$(\phi,\sigma_{i})$-quasistable if that property holds on all geometric
fibers. We define $\overline{\mathcal{J}}^{d}_{g,n}(\phi)$ to be the moduli
stack parameterizing $\phi$-semistable sheaves on families of stable curves.
We define $\overline{\mathcal{J}}^{d}_{g,n}(\phi,\sigma_{i})$ to be the moduli
stack parameterizing $(\phi,\sigma_{i})$-quasistable sheaves on families of
stable curves.

######  Notation 3.24.

If $\phi\in V_{g,n}^{d}$ is nondegenerate then, by Remark 3.20 we have that
all semistable sheaves are stable. It follows that for every $1\leq i\leq n$
we have the equalities
${\mathfrak{C}}_{g,n}(\phi)=\mathfrak{C}_{g,n}(\phi,\sigma_{i})$ and
$\overline{\mathcal{J}}^{d}_{g,n}(\phi)=\overline{\mathcal{J}}^{d}_{g,n}(\phi,\sigma_{i})$,
where
$\sigma_{i}\colon\overline{\mathcal{M}}_{g,n}\to\overline{\mathcal{C}}_{g,n}$
is the $i$-th section.

######  Remark 3.25.

For $\phi\in V_{g,n}^{d}$ degenerate and for all $1\leq i\leq n$ we can
describe $\mathfrak{C}_{g,n}(\phi,\sigma_{i})$ (resp.
$\overline{\mathcal{J}}^{d}_{g,n}(\phi,\sigma_{i})$) as
${\mathfrak{C}}_{g,n}(\phi)$ (resp.
$\overline{\mathcal{J}}^{d}_{g,n}(\phi^{\prime}_{i})$) for some nondegenerate
perturbation $\phi_{i}^{\prime}$ of $\phi$. (As done in Remark 3.22 for a
single curve).

In order to achieve this, we let $\phi^{\prime}_{i}$ by subtracting from
$\phi$ an arbitrarily small $\epsilon>0$ on each curve on its irreducible
component containing the section $\sigma_{i}$, and by subtracting a small
quantity on all other components (so for all curves, the sum over all
irreducible components of the values of $\phi$ and of $\phi_{i}^{\prime}$
coincide). The fact that such $\phi_{i}^{\prime}$ can be constructed in a way
that is compatible with graph morphisms follows by using Proposition 3.5.

The following guarantees the existence of universal moduli spaces.

######  Theorem 3.26.

([KP19, Corollary 4.4] and [Est01]/[Mel19]) For all $\phi\in V_{g,n}^{d}$ and
$1\leq i\leq n$ the stack $\overline{\mathcal{J}}^{d}_{g,n}(\phi,\sigma_{i})$
is a nonsingular Deligne–Mumford stack, and the forgetful morphism
$\overline{\mathcal{J}}^{d}_{g,n}(\phi,\sigma_{i})\to\overline{\mathcal{M}}_{g,n}$
is representable, proper and flat.

The moduli stacks of Theorem 3.26 are called _fine compactified universal
Jacobians_.

As observed in [KP19, Remark 4.6], the fine compactified (universal) Jacobians
produced by this construction are the same as those defined by Esteves and
Melo [Est01, Mel19].

By virtue of its universal property, the universal family
$\pi\colon\overline{\mathcal{C}}_{g,n}(\phi)\to\overline{\mathcal{J}}^{d}_{g,n}(\phi)$
carries some tautological (or Poincaré) sheaves $F_{\text{tau}}(\phi)$. These
are of fiberwise total degree $d$ and $\phi$-stable. They are not unique, but
the difference of any two of them is the pullback of a line bundle from
$\overline{\mathcal{J}}^{d}_{g,n}(\phi)$. One way to make a definite choice of
a tautological sheaf is to assume that it is trivial along a given smooth
section.

Note that, as described in [CMKV15], the total space
$\overline{\mathcal{C}}_{g,n}(\phi)$ is singular. A natural desingularization
of $\overline{\mathcal{C}}_{g,n}(\phi)$, carrying a tautological line bundle
$L_{\text{tau}}(\phi)$, was provided by Esteves–Pacini in [EP16] by using a
semistable modification of the universal family. Here we will give an
alternative description of it using a compactified universal Jacobian with one
extra point.

######  Remark 3.27.

We observe that there is a natural map $\alpha\colon V_{g,n}^{d}\to
V_{g,n+1}^{d}$, with image in the degenerate locus, defined as follows.

If $G$ is the stable graph obtained as the stabilization of the
$(n+1)$-pointed graph $G^{\prime}$ after the $n+1$ marking is removed, then
there is a natural bijection between the vertices of $G^{\prime}$ and those of
$G$, except possibly for $1$ extra genus $0$ vertex of $G^{\prime}$. Then
define $\alpha(\phi)$ as the assignment on $G^{\prime}$ that is defined by
this bijection and that is $0$ on the extra genus $0$ vertex of $G^{\prime}$
(when that exists). The extra genus $0$ vertex could be a tail (when it is
connected to the complement by $1$ edge) or a bridge (when it is connected to
the complement by $2$ edges). The fact that $\phi$ is compatible for graph
morphisms implies the same property for $\alpha(\phi)$.

######  Notation 3.28.

We will slightly abuse the notation and, for $\phi\in V_{g,n}^{d}$, we will
simply write $\phi\in V_{g,n+1}^{d}$ in place of $\alpha(\phi)\in
V_{g,n+1}^{d}$.

We now show that a quasistable modification of the universal curve can be
described as the morphism
$\pi^{\prime}\colon\overline{\mathcal{J}}^{d}_{g,n+1}(\phi,\sigma_{i})\to\overline{\mathcal{J}}^{d}_{g,n}(\phi,\sigma_{i})$
that forgets the last point and stabilizes, thus mapping each
$(C^{\prime},p_{1},\ldots,p_{n+1},F)$ to $(C,p_{1},\ldots,p_{n},f_{*}F)$,
where $f\colon C^{\prime}\to C$ is the stabilization of
$(C,p_{1},\ldots,p_{n})$. In order to do that, for each fixed $i=1,\ldots,n$,
we define a morphism
$\psi\colon\overline{\mathcal{J}}^{d}_{g,n+1}(\phi,\sigma_{i})\to\overline{\mathcal{C}}_{g,n}(\phi,\sigma_{i})$
by

$(C^{\prime},p_{1},\ldots,p_{n+1},L)\mapsto(C,p_{1},\ldots,p_{n},f_{*}L,f(p_{n+1}))$

where $f\colon C^{\prime}\to C$ is the stabilization of the curve
$(C^{\prime},p_{1},\ldots,p_{n})$. Then we show that $\psi$ is the
stabilization of $\pi^{\prime}$.

######  Theorem 3.29.

For each $1\leq i\leq n$, the forgetful morphism $\psi$ defined above is the
unique positively admissible quasistable modification of the universal curve
over $\overline{\mathcal{J}}^{d}_{g,n}$. A tautological line bundle on
$\overline{\mathcal{J}}^{d}_{g,n+1}(\phi,\sigma_{i})$ is

(3.30)
$\mathcal{L}_{\textnormal{tau}}:=\sigma_{n+1}^{*}(\mathcal{F}^{\prime}_{\textnormal{tau}})\otimes\sigma_{1}^{*}(\mathcal{F}^{\prime-1}_{\textnormal{tau}}),$

where
$\mathcal{F}^{\prime}_{\textnormal{tau}}=\mathcal{F}^{\prime}_{\textnormal{tau}}(\phi)$
is a tautological sheaf on the universal curve
$\pi\colon\overline{\mathcal{C}}_{g,n+1}(\phi,\sigma_{i})\to\overline{\mathcal{J}}^{d}_{g,n+1}(\phi,\sigma_{i})$.

###### Proof.

We apply Proposition 2.3 to show that the morphism $\psi$ is a quasistable
modification of the universal curve.

Let us begin by proving that $\psi$ is a quasistable modification of
$\mathcal{C}_{g,n}(\phi,\sigma_{i})$. Firstly, if $C=C^{\prime}$, then $f$ is
an isomorphism, so $\psi$ is an isomorphism locally around
$(C^{\prime},p_{1},\ldots,p_{n+1},L)$. If $C\neq C^{\prime}$, then we have two
cases. Either $p_{n+1}$ belongs in a rational tail and, in this case,
$L=f^{*}f_{*}(L)$ so $\psi$ is again an isomorphism locally around
$(C^{\prime},p_{1},\ldots,p_{n+1},L)$. Or, $p_{n+1}$ is in a bridge $E\subset
C^{\prime}$ such that no other marked points are in $E$. We will now focus on
this case.

If $\deg_{E}(L)=0$, then by Proposition 2.3, we have that $L=f^{*}f_{*}(L)$
and again the map $\psi$ is an isomorphism locally around
$(C^{\prime},p_{1},\ldots,p_{n+1},L)$. We are left with the case where
$\deg_{E}(L)=1$. In this case, we have that $f_{*}(L)$ is not locally free
around $f(p_{n+1})$, which is a node. More so, we have that
$\psi^{-1}(C,p_{1},\ldots,p_{n},f_{*}(L),f(p_{n+1}))$ is isomorphic to
$\mathbb{P}^{1}$. Indeed, every $L^{\prime}$ obtained from gluing $L|_{E^{c}}$
and $\mathcal{O}_{E}(1)$ will have the property that
$\psi(C^{\prime},p_{1},\ldots,p_{n+1},L^{\prime})=(C,p_{1},\ldots,p_{n},f_{*}(L),f(p_{n+1}))$.
The possible gluings are paremeterized by a $\mathbb{P}^{1}$, and we are done.

Secondly we observe that $\deg_{E}(L_{\text{tau}})=1$ for each exceptional
component $E$ contracted by $\psi$. In order to show that, it suffices to
construct a nonconstant map $\delta\colon\mathbb{P}^{1}\to E$ such that
$\delta^{*}(L_{\text{tau}})=\mathcal{O}(1)$. Let $(C,p_{1},\ldots,p_{n},L,p)$
correspond to the point in the universal curve that is contracted by $E$. That
means that $p$ is a node of $C$ and that $L$ fails to be locally free at $p$.
We construct a family $X/\mathbb{P}^{1}$ by gluing two sections on two
families $X_{1}/\mathbb{P}^{1}$ and $X_{2}/\mathbb{P}^{1}$. The family $X_{1}$
is the trivial family $C^{\nu_{p}}\times\mathbb{P}^{1}$ (where $\nu_{p}$
denotes the normalization at $p$) carrying the $n$ trivial sections
$p_{1},\ldots,p_{n}$ and the gluing sections are the points $q_{1},q_{2}$ such
that $\nu_{p}(q_{i})=p$. The family $X_{2}$ is the blowup of the trivial
family $\mathbb{P}^{1}\times\mathbb{P}^{1}$ at $[0:1]\times[0:1]$ and
$[1:0]\times[1:0]$ with a further section $p_{n+1}$ defined as the inverse
image of a constant section (different from $[0:1]$ and $[1:0]$) and the two
gluing sections are the strict transforms of the sections $[0:1]$ and $[1:0]$.
Then we choose any line bundle $F$ on $X$ with the property that
$F|_{X_{1}}=L$ and $F|_{X_{2}}=\mathcal{O}(\widetilde{\Delta})$, for
$\widetilde{\Delta}$ the strict transform of the diagonal in $X_{2}$. Then $F$
is $(\phi,\sigma_{i})$-quasistable, and so the datum of $(X,F)$ defines a
morphism $\delta$. By construction, we have that
$\delta^{*}(L_{\text{tau}})=\sigma_{n+1}^{*}(F)=\sigma_{n+1}^{*}(\mathcal{O}(\widetilde{\Delta}))$,
which equals $\mathcal{O}(1)$ because $\sigma_{n+1}^{*}$ intersects
$\widetilde{\Delta}$ at $1$ reduced point.

Then we prove that the direct image $\psi_{*}L_{\text{tau}}$ of the line
bundle defined in (3.30) equals $F_{\text{tau}}\otimes\pi^{*}(M)$ for some
line bundle $M$. By the previous part combined with Proposition 2.3, we
conclude that $\psi_{*}L_{\text{tau}}$ is rank $1$ and torsion-free. By [KP19,
Appendix 7], it is enough to prove that this equality occurs on an open subset
$U$ of $\overline{\mathcal{C}}_{g,n}(\phi)$ whose complement has codimension
at least $2$. It is easy to show equality over the open set $U$ that is the
universal curve over the line bundle locus in
$\overline{\mathcal{J}}^{d}_{g,n}(\phi)$ (this follows because
$\psi_{|\psi^{-1}(U)}$ is an isomorphism over $U$, and because the restriction
of $F^{\prime}_{\text{tau}}$ to the open set
$\pi^{-1}(\psi^{-1}(U))\subset\overline{\mathcal{C}}_{g,n+1}(\phi,\sigma_{i})$
is a line bundle). This concludes the proof that $\psi$ is a positively
admissible quasistable modification.

Uniqueness follows from Corollary 2.4. ∎

###  3.d. Brill–Noether classes

To the data of a flat family $\pi\colon\mathcal{C}\to S$ of nodal curves of
arithmetic genus $g$ over a nonsingular scheme $S$ and a rank $1$ torsion-free
$\mathcal{F}$ on $\mathcal{C}$ of fiberwise degree $d$, we can associate the
Brill–Noether (or Thom–Porteous) class

(3.31)
$\mathsf{w}_{d}(\mathcal{C}/S,\mathcal{F}):=c_{g-d}(-R^{\bullet}\pi_{\ast}\mathcal{F}).$

This class is supported on the subscheme

$\mathsf{W}_{d}(\mathcal{C}/S,\mathcal{F})=\\{s\in S:\
h^{0}(\mathcal{C}_{s},\mathcal{F}_{s})>0\\}\subset S$

and when the latter is of the expected codimension $g-d$, it coincides with
its fundamental class (with a suitably defined scheme structure– see [Ful98,
Chapter 14]). If $h^{0}(\mathcal{C}_{s},\mathcal{F}_{s})=0$ for all $s\in S$,
then the complex
$-R^{\bullet}\pi_{\ast}\mathcal{F}=R^{1}\pi_{\ast}\mathcal{F}$ is a vector
bundle of rank $g-1-d$, so the Chern class
$\mathsf{w}_{d}(\mathcal{C}/S,\mathcal{F})$ equals zero.

Here are a couple of further basic remarks on these classes.

######  Remark 3.32.

If $I$ is a line bundle on $S$, then
$\mathsf{w}_{d}(\mathcal{C}/S,\mathcal{F})=\mathsf{w}_{d}(\mathcal{C}/S,\mathcal{F}\otimes\pi^{*}I)$.

Indeed, by [Ful98, Example 3.2.2] we have

(3.33) $c_{j}(-R^{\bullet}\pi_{\ast}\mathcal{F}\otimes
I)=\sum_{i=0}^{j}\binom{g-d-1-i}{j-i}c_{i}(-R^{\bullet}\pi_{\ast}\mathcal{F})\cdot
c_{1}(I)^{j-i}$

for all $j\geq 0$. The result follows because, for $j=g-d$, the binomial
coefficient vanishes unless when $i=j$.

######  Remark 3.34.

Let $f\colon\mathcal{C}^{\prime}\to\mathcal{C}$ be a semistable modification
of the family of nodal curves $\pi\colon\mathcal{C}\to S$, and let
$\mathcal{L}$ be a positively admissible line bundle on $\mathcal{C}$ (see
Section 2.c). By Proposition 2.3, we deduce

(3.35) $R^{\bullet}(\pi\circ
f)_{\ast}\mathcal{L}=R^{\bullet}\pi_{\ast}(f_{\ast}\mathcal{L}).$

Conversely, if $\mathcal{F}$ is a rank $1$ torsion free simple sheaf on a
family of stable curves $\mathcal{C}/S$, there exists a quasistable
modification $\mathcal{C}^{\prime}$ of $\mathcal{C}$ and a line bundle
$\mathcal{L}$ on $\mathcal{C}$ such that
$R^{\bullet}f_{\ast}(\mathcal{L})=f_{\ast}\mathcal{L}=\mathcal{F}$ and thus
(3.35) occurs.

The same construction and remarks apply to the case of the semistable
modification
$\pi\colon\overline{\mathcal{C}}_{g,n}^{\prime}(\phi)\to\overline{\mathcal{J}}^{d}_{g,n}(\phi)$
of the universal family, and its tautological line bundle
$\mathcal{L}=\mathcal{L}_{\text{tau}}$ (see [PRvZ20]). We will denote by
$\mathsf{w}_{d}(\phi)$ the corresponding universal class in
$A^{d}(\overline{\mathcal{J}}^{d}_{g,n}(\phi))$ and with
$\mathsf{W}_{d}(\phi)$ the subscheme over which it is supported.

######  Remark 3.36.

When restricted to smooth curves, the scheme
$\mathsf{W}_{d}(\phi)_{|{\mathcal{J}}^{d}_{g,n}}$ is reduced, irreducible, and
of relative codimension $g-d$ (it is the image of the $d$-th symmetric product
via the Abel map). The closure of
$\mathsf{W}_{d}(\phi)_{|{\mathcal{J}}^{d}_{g,n}}$ in
$\overline{\mathcal{J}}^{d}_{g,n}(\phi)$ is contained in
$\mathsf{W}_{d}(\phi)$ and when the two coincide, we have
$\mathsf{w}_{d}(\phi)=[\mathsf{W}_{d}(\phi)]$.

######  Remark 3.37.

In general, the scheme $\mathsf{W}_{d}(\phi)$ fails to be irreducible and of
the expected dimension. For example, if $\phi$ is a stability condition such
that the line bundles of bidegree $(d_{1},d_{2})$ are $\phi$-stable on curves
in the boundary divisor $\Delta_{i,S}$, and either $d_{1}>i$ or $d_{2}>g-i$,
then $\mathsf{W}_{d}(\phi)$ contains the pullback of $\Delta_{i,S}$ in
$\overline{\mathcal{J}}^{d}_{g,n}(\phi)$.

We will discuss more on this matter in Proposition 3.38 and Remark 4.18.

We conclude this section by providing sufficient conditions for the
Brill–Noether class 3.31 defined by the Thom–Porteous Formula to coincide with
the class of the Brill–Noether locus. Some parts of the proof of the next two
propositions will require to employ the fact that ${\mathfrak{C}}_{g,n}(\phi)$
is a stratification of $\overline{\mathcal{J}}^{d}_{g,n}(\phi)$, which we will
discuss in the next section. For this reason, we postpone the proof of the
next result to Section 4.b.1.

As in Section 3.b, we fix coordinates for $V_{g,n}^{d}\cong C_{g,n}^{d}\times
D_{g,n}^{d}$, and let
$V_{g,n}^{d}\ni\phi=((x_{i,1,S})_{(i,S)},(x_{1},\ldots,x_{n}))$ where
$x_{j}=x_{0,2,j}$ for each $1\leq j\leq n$.

######  Proposition 3.38.

We have that $\mathsf{W}_{d}(\phi)$ is the closure of
$\mathsf{W}_{d}(\phi)_{|{\mathcal{J}}^{d}_{g,n}}$ (in particular it is
reduced, irreducible and of the expected codimension, and
$\mathsf{w}_{d}(\phi)=\mathsf{W}_{d}(\phi)$) if and only if
$\phi=((x_{i,1,S})_{(i,S)},(x_{1},\ldots,x_{n}))$ is as follows

  1. (1)

If $d=g-1$, for $i-3/2<x_{i,1,S}<i+1/2$ for all $(i,S)$.

  2. (2)

If $d=g-2$, for $i-3/2<x_{i,1,S}<i-1/2$ for all $(i\geq 1,S)$ and
$-3/2<x_{0,1,S}<1/2$ for all $S$, and

$i-2<\frac{2g-2i-t}{2g-2}\cdot\sum_{j\in S}x_{j}+\Bigg{(}d-\sum_{j\notin
S}x_{j}\Bigg{)}\cdot\frac{2i-2+t}{2g-2}<i+1$

for all vine curve triples $(i,t,S)$ with $t\geq 2$.

  3. (3)

Never if $0<d\leq g-3$.

  4. (4)

If $d<0$, for $d-1/2<x_{i,1,S}<1/2$ for all $(i,S)$, and the coordinates
$x_{1},\ldots,x_{n}$ satisfy

(3.39) $d-1<\frac{2g-2i-t}{2g-2}\cdot\sum_{j\in
S}x_{j}+\Bigg{(}d-\sum_{j\notin S}x_{j}\Bigg{)}\cdot\frac{2i-2+t}{2g-2}<1$

for all vine curve triples $(i,t,S)$ with $t\geq 2$. (3.39).

  5. (5)

If $d=0$, for $-1/2<x_{i,1,S}<1/2$ for all $(i\geq 1,S)$, for
$-3/2<x_{0,1,S}<1/2$ for all $S$, and when the coordinates
$x_{1},\ldots,x_{n}$ satisfy (3.39) for all vine curve triples $(i,t,S)$ with
$t\geq 2$.

We now explicitly define a stability condition that, in each of the cases
(1),(2),(4),(5) listed above, belongs to the ranges that we have identified
for $\mathsf{W}_{d}(\phi)$ to equal the closure of its restriction to the open
part. (This shows, in particular, that these ranges are not empty).

######  Definition 3.40.

For $G\in G_{g,n}$, define the stabilized-canonical divisor $K^{s}_{G}$ to
equal zero at every vertex contained in some rational tail222a rational tail
is a complete subgraph whose genus is $0$ and that is connected to its
complement by exactly $1$ edge, and for every other $v$ to equal
$K^{s}(v)=2g(v)-2+\operatorname{val}^{\prime}(v)$, where
$\operatorname{val}^{\prime}(v)$ is the number of edges at $v$ (counting each
loop twice), except the edges that are contained in some rational tail.

Then define the stabilized canonical element
$\phi^{d}_{\text{scan}}(G)=\frac{d}{2g-2}\cdot K^{s}_{G}\in V_{g,n}^{d}$.

Note that the above is different from the canonical stability
$\phi^{d}_{\text{can}}\in V_{g,n}^{d}$ chosen as the origin in [KP17].

####  3.d.1. Pull-back via Abel–Jacobi sections

Fix integers ${\bf d}=(k;d_{1},\ldots,d_{n})$ such that $d=d_{1}+\ldots+d_{n}$
and integers ${\bf f}=(f_{i,S})_{i,S}$ for every boundary divisor
$G(g-i,1,S)\in G_{g,n}$. Define the universal line bundle

(3.41) $\mathcal{L}=\mathcal{L}_{{\bf d},{\bf
f}}=\omega^{k}_{\overline{\mathcal{C}}_{g,n}/\overline{\mathcal{M}}_{g,n}}\left(\sum_{j=1}^{n}d_{j}x_{j}+\sum_{i,S}f_{i,S^{c}}\cdot
C_{i,S^{c}}\right),$

where $C_{i,S^{c}}\subset\overline{\mathcal{C}}_{g,n}$ is the
component333these unnatural conventions will simplify the formulas in Remark
7.39 and Example 7.40 over the boundary divisor
$\Delta_{g-i,S}=\overline{\mathcal{M}}_{G(g-i,1,S)}\subset\overline{\mathcal{M}}_{g,n}$
that contains the sections in $S^{c}$. Then define $\phi=\phi_{{\bf d},{\bf
f}}\in V_{g,n}^{d}$ to be the multidegree of $\mathcal{L}$.

If $\phi^{+}=\phi^{+}_{{\bf d},{\bf f}}$ is a nondegenerate small perturbation
of $\phi$, then we have that the universal line bundle $\mathcal{L}$ is
$\phi^{+}$-stable, and it defines a (Abel–Jacobi) section
$\sigma=\sigma^{+}_{{\bf d},{\bf
f}}\colon\overline{\mathcal{M}}_{g,n}\to\overline{\mathcal{J}}^{d}_{g,n}(\phi^{+})$.

######  Remark 3.42.

Assume $d<0$, $k=0$, and ${\bf d}$ satisfies $d_{i}\leq 1$ for all $i$, and at
most one of the $d_{i}$’s equals $1$. Assume that ${\bf f}$ satisfies $d\leq
f_{i,S^{c}}+\sum_{j\in S}d_{j}\leq 0$ for all $(i,S)$. Then $\phi_{{\bf
d},{\bf f}}$ satisfies the conditions of Item (4) in Proposition 3.38.

By Proposition 3.38 the pullback

(3.43) $\sigma_{{\bf d},{\bf
f}}^{*}(\mathsf{w}_{d})=0\quad\in\operatorname{A}^{d}(\overline{\mathcal{M}}_{g,n}),$

gives a relation. The LHS of (3.43) can be explicitly written as a linear
combination of standard generators of the tautological ring of
$\overline{\mathcal{M}}_{g,n}$ by means of [PRvZ20, Theorem 1] (see also
Corollary 3.7 and Equation 3.9 of _loc.cit_).

##  4\. Normal crossing stratification categories and blowups

In this section we define the axioms needed for a category of (resolved)
strata of a space stratified by normal crossing divisors that are not
necessarily simple normal crossing. We use this formalism to write some
intersection theoretic formulas (the excess intersection formula and the GRR
formula for the total Chern class) that we will use to derive our main result
Theorem 7.4. Then we define the blow-up category at a stratum with transversal
self-intersection. A construction of such strata categories starting from a
normal crossing divisor, and more generally from a toroidal embedding, is
given in [MMUV22, Definition 3.5].

The main examples we are generalizing are (a) the poset obtained by
intersecting the components of a simple normal crossing divisor and (b) the
stratification of $\overline{\mathcal{M}}_{g,n}$ by topological type, induced
by the boundary divisors
$\Delta=\Delta_{\text{irr}}\cup\bigcup_{i,S}\Delta_{i,S}$ (see Section 2.d).
In the latter case, the relevant category is the category $G_{g,n}$ of stable
$n$-pointed graphs of genus $g$ with morphisms given by graph contractions.

###  4.a. Categories of resolved strata for a normal crossing stratification

Let $\mathfrak{C}$ be a finite skeletal category with a terminal object
$\bullet$ such that every morphism is an epimorphism.

######  Remark 4.1.

In $\mathfrak{C}$ we have that
$\operatorname{Mor}(\alpha,\alpha)=\operatorname{Aut}(\alpha)$. Indeed, if
$f\in\operatorname{Mor}(\alpha,\alpha)$, then there exist natural numbers
$a>b$ such that $f^{a}=f^{b}$, and since $f$ is an epimorphism we have that
$f^{a-b}=\operatorname{Id}_{A}$, which proves that $f$ is an isomorphism.

If $\alpha$ and $\beta$ are distinct elements, we also have that if
$\operatorname{Mor}(\alpha,\beta)\neq\varnothing$ then
$\operatorname{Mor}(\beta,\alpha)=\varnothing$. Indeed, assume that there
exists $f\colon\alpha\to\beta$ and $g\colon\beta\to\alpha$. By the observation
we would have that both $f\circ g$ and $g\circ f$ are automorphisms, which
implies that both $f$ and $g$ are isomorphisms, this contradicts the fact that
$\mathfrak{C}$ is skeletal. This means that the set
$\operatorname{Obj}(\mathfrak{C})$ has a natural poset structure given by
$\alpha\geq\beta$ if $\operatorname{Mor}(\alpha,\beta)\neq\varnothing$.

We say that such $\mathfrak{C}$ is a _(normal crossing) stratification
category_ if its underlying poset is ranked with rank function
$\operatorname{cd}$ with minimum element the terminal object, having
$\operatorname{cd}(\bullet)=0$ and it satisfies the following axiom:

######  Axiom 1.

For each $f\colon\alpha\to\beta$ there exist exactly
$\operatorname{cd}(f):=\operatorname{cd}(\alpha)-\operatorname{cd}(\beta)$
pairs

$(\beta^{\prime},\operatorname{Aut}(\beta^{\prime})g)\ \in\
\operatorname{Obj}(\mathfrak{C})\times(\operatorname{Aut}(\beta^{\prime})\backslash\operatorname{Mor}(\alpha,\beta^{\prime}))$

such that, for each such pair, there exists $i\colon\beta^{\prime}\to\beta$
with $\operatorname{cd}(i)=1$ and $f=i\circ g$. (Note that (a) the existence
of such $i$ is independent of the choice of the representative $g$ in the left
coset $\overline{g}:=\operatorname{Aut}(\beta^{\prime})g$); and (b) since $g$
is an epimorphism, the morphism $i$ is necessarily unique).

From now on we will also fix some notation on $\mathfrak{C}$.

  1. (1)

We write $f_{\alpha}$ for the unique element of
$\operatorname{Mor}(\alpha,\bullet)$.

  2. (2)

If $f_{i}\colon\alpha\to\beta_{i}$ are morphisms for $i=1,\ldots,m$, we define

$\operatorname{Aut}(f_{1},\ldots,f_{m}):=\\{\tau\in\operatorname{Aut}(\alpha);f_{i}\circ\tau=f_{i}\text{
for every }i=1,\ldots,m\\}.$

Note that $\operatorname{Aut}(f_{\alpha})=\operatorname{Aut}(\alpha)$.

  3. (3)

For each morphism $f\colon\beta\to\gamma$ and object
$\alpha\in\operatorname{Obj}(\mathfrak{C})$, we define
$\overline{\operatorname{Mor}}(\alpha,f):=\operatorname{Aut}(f)\backslash\operatorname{Mor}(\alpha,\beta)$.
When $f=f_{\beta}$, we simply write
$\overline{\operatorname{Mor}}(\alpha,\beta):=\overline{\operatorname{Mor}}(\alpha,f_{\beta})=\operatorname{Aut}(\beta)\backslash\operatorname{Mor}(\alpha,\beta)$.

  4. (4)

For each morphism $f\colon\alpha\to\beta$, we let $S_{f}$ denote the set of
all pairs $(\beta^{\prime},\overline{g})$ satisfying the condition in Axiom
(1). Moreover, for each $(\beta^{\prime},\overline{g})\in S_{f}$ we denote by
$i_{\overline{g},f}:=i\colon\beta^{\prime}\to\beta$ the morphism defined in
Axiom 1. We define $S_{\alpha}:=S_{f_{\alpha}}$.

Here are the most relevant examples in this paper.

######  Example 4.2.

(Simple normal crossing). Let $X$ be a nonsingular variety and
$D=D_{1}+\ldots+D_{k}$ be a simple normal crossing divisor. To this we can
associate a category $\mathfrak{C}$ whose objects are the strata and morphisms
are the inclusions. This category $\mathfrak{C}$ is finite, skeletal, has a
terminal element, every morphism is an epimorphism, it is ranked by
codimension, and it satisfies Axiom 1.

The category $\mathfrak{C}$ is _simple normal crossing_ if in addition to
Axiom 1, it satisfies:

######  Axiom 2.

For every $\alpha,\beta\in\operatorname{Obj}(\mathfrak{C})$ the set
$\overline{\operatorname{Mor}}(\alpha,\beta)$ has at most one element.

######  Example 4.3.

The second example is $\mathfrak{C}=G_{g,n}$ introduced in Section 2.d. The
terminal object here is the trivial graph with $1$ vertex of genus $g$
carrying all the markings, and no edges. The rank function is the number of
edges. The set $S_{f}$ of a morphism $f\colon G\to G^{\prime}$ is naturally
identified with the set of edges of $G$ that are contracted by $f$. In
particular, $S_{G}$ equals the edge set $E(G)$.

The rank $1$ objects, the boundary divisors, are either graphs with two
vertices connected by one edge (corresponding to the divisors $\Delta_{i,S}$,
see 2.d), or the graph consisting of $1$ vertex of genus $g-1$ with $1$ loop.

######  Example 4.4.

The main example in this paper is the category
$\mathfrak{C}={\mathfrak{C}}_{g,n}(\phi)$ that we introduced in Definition
3.23, an enhancement of the category $G_{g,n}$ discussed above. The terminal
object is the trivial graph endowed with the unique function that maps its
unique vertex to the integer $d$. The rank of an object $(G,(E_{G},D_{G}))$
equals $|\operatorname{Edges}(G)|+|E_{G}|$. For
$f\colon(G^{\prime},(E_{G}^{\prime},D_{G^{\prime}}))\to(G,(E_{G},D_{G}))$, the
set $S_{f}$ is naturally identified with the set of edges contracted by $f$.

The rank $1$ objects are $(G,(E_{G},D_{G}))$ with $G$ a rank $1$ object of
$G_{g,n}$ and $(E_{G},D_{G})$ a $\phi$-stable pseudodivisor (which implies
$E_{G}=\varnothing$).

######  Example 4.5.

To a nonsingular variety $X$ (or DM stack) endowed with a normal crossing
divisor $D$, [MMUV22, Definition 3.5] associates a stratification category
that respects Axiom 1 above.

Note that the construction of _loc.cit_ in the case
$X=\overline{\mathcal{M}}_{g,n}$ and
$D=\Delta_{irr}+\sum_{(i,S)}\Delta_{i,S}$, which we discussed in Example 4.3,
produces the quotient of the category $G_{g,n}$ of stable graphs where $2$
morphisms are identified whenever they are the same on the corresponding edge
sets. (See [MMUV22, Figure 2] for examples of automorphisms that are
identified to the identity). A similar phenomenon happens for the category of
Example 4.4.

In this paper we prefer instead to work with the usual category of stable
graphs (and its enhancements).

The next proposition is the analogue of the fact that the set of strata that
contain a given stratum is in natural bijection with the subsets of the
divisors that define that stratum.

######  Proposition 4.6.

Given a morphism $f\colon\alpha\to\beta$ and $1\leq
k\leq\operatorname{cd}(\alpha)-\operatorname{cd}(\beta)$, there is a natural
bijection between the set of pairs

$\\{(\gamma,\overline{j})\in\operatorname{Obj}(\mathfrak{C})\times\overline{\operatorname{Mor}}(\alpha,\gamma):\
\operatorname{cd}(\gamma)-\operatorname{cd}(\beta)=k\textrm{ and }\exists
h\colon\gamma\to\beta,f=h\circ j\\}$

(note that $h$ above is unique) and the set $\mathcal{P}(k,S_{f})$ of subsets
of $S_{f}$ containing $k$ elements.

We start by observing the following:

######  Remark 4.7.

Given a factorization
$f\colon\alpha\xrightarrow{j}\gamma\xrightarrow{h}\beta$, there is a natural
inclusion $j^{*}\colon S_{h}\hookrightarrow S_{f}$ given by
$(\beta^{\prime},\overline{g^{\prime}})\mapsto(\beta^{\prime},\overline{g^{\prime}\circ
j})$.

Moreover, we claim that for each
$\overline{j}\in\overline{\operatorname{Mor}}(\alpha,h)$ we have a well-
defined $\overline{j}^{*}(S_{h})$. Indeed, the map $j^{*}$ is the same as
$(j\circ\tau)^{*}$ for every $\tau\in\operatorname{Aut}(\gamma\to\beta)$.

###### Proof.

We first observe that by Remark 4.7, there is a natural map $\lambda_{k,f}$
from the set of pairs, call it $X_{k,f}$, to the set $\mathcal{P}(k,S_{f})$ of
$k$-elements subsets of $S_{f}$, obtained by
$\lambda_{k,f}((\gamma,\overline{j})):=j^{*}(S_{h})$.

Then we prove that the cardinality of $X_{k,f}$ equals
$\binom{\operatorname{cd}(\alpha)-\operatorname{cd}(\beta)}{k}$, which is also
the cardinality of $\mathcal{P}(k,S_{f})$. This is achieved by induction on
$\operatorname{cd}(\alpha)-\operatorname{cd}(\beta)$ and double counting. For
each $c\in S_{f}$, let $X_{k,f,c}$ be the subset of $X_{k,f}$ of elements
whose image via $\lambda_{f}$ contains $c$. By induction hypothesis, we have
that
$|X_{k,f,c}|=\binom{\operatorname{cd}(\alpha)-\operatorname{cd}(\beta)-1}{k-1}$.
By Axiom 1 the number of elements of $\\{(a,b):\ a\in
X_{k,f},b\in\lambda_{k,f}(a)\\}$ equals $k\cdot|X_{k,f}|$ and, it also equals
$(\operatorname{cd}(\alpha)-\operatorname{cd}(\beta))\cdot\binom{\operatorname{cd}(\alpha)-\operatorname{cd}(\beta)-1}{k-1}$.
These two equalities prove that
$|X_{k,f}|=\binom{\operatorname{cd}(\alpha)-\operatorname{cd}(\beta)}{k}$.

Finally we prove that each $\lambda_{k,f}$ is surjective. First we prove that
this is the case for $k=\operatorname{cd}(\alpha)-\operatorname{cd}(\beta)-1$
(or equivalently when
$\operatorname{cd}(\gamma)=\operatorname{cd}(\alpha)-1$). By the previous
paragraph, $\lambda_{k,f}$ is, for this $k$, a function between sets of the
same cardinality, so it is equivalent to prove that it is injective. Let
$a_{1},a_{2}\in X_{k,f}$ be such that
$\lambda_{k,f}(a_{1})=\lambda_{k,f}(a_{2})$ and let $c$ be the only element of
$S_{f}\setminus\lambda_{k,f}(a_{1})$. If $a_{1}\neq a_{2}$, then
$\lambda_{k,f}^{-1}(\\{c\\})$ contains at most
$\operatorname{cd}(\alpha)-\operatorname{cd}(\beta)-2$ elements, but in the
previous paragraph we have established that
$|X_{k,f,c}|=\operatorname{cd}(\alpha)-\operatorname{cd}(\beta)-1$; this
contradicts the assumption $a_{1}\neq a_{2}$.

To prove surjectivity of each $\lambda_{k,f}$ we argue by induction on
$\operatorname{cd}(\alpha)-\operatorname{cd}(\beta)$. Let
$S\in\mathcal{P}(k,S_{f})$. Choose $T\supset S$ with
$T\in\mathcal{P}(\operatorname{cd}(\alpha)-\operatorname{cd}(\beta)-1,S_{f})$.
By the previous paragraph, there exists $d=(\delta,\overline{h})$ such that
$\lambda_{k,f}(d)=T$ and a factorization of $f=h\circ g$ through $\delta$, so
$T=g^{*}(T^{\prime})$ and $S=g^{*}(S^{\prime})$ for some $S^{\prime}\subset
T^{\prime}$ subsets of $S_{h}$. We have
$\operatorname{cd}(\delta)-\operatorname{cd}(\beta)=\operatorname{cd}(\alpha)-\operatorname{cd}(\beta)-1$.
By applying the induction hypothesis to $\lambda_{k,h}$, we find $c\in
X_{k,h}$ such that $\lambda_{k,h}(c)=S^{\prime}$, and so
$\lambda_{k,f}(g^{*}c)=S$. This concludes the proof of surjectivity. ∎

We will now define some important geometric notions in the stratification
category.

######  Definition 4.8.

Fix $f_{i}\colon\alpha_{i}\to\beta$ for $i=1,\ldots,m$, and
$f\colon\gamma\to\beta$. Let $g_{i}\colon\gamma\to\alpha_{i}$ for
$i=1,\ldots,m$ be a collection of morphisms such that $f_{i}\circ g_{i}=f$.

We say that the collection $(g_{i})$ is _generic_ with respect to the tuple
$(f,(f_{i}))$ if $S_{f}=\bigcup_{i=1}^{m}g_{i}^{*}(S_{f_{i}})$.

We say that the collection $(f,(f_{i}))$ is _transversal_ at $(g_{i})$ if
$g_{i}^{*}(S_{f_{i}})\cap g_{j}^{*}(S_{f_{j}})=f^{*}(S_{\beta})$ for very
$i\neq j$.

Following the above definition, we write
$\operatorname{Int}(f_{1},\ldots,f_{m})_{f}$ to denote the set of all generic
tuples $(g_{1},\ldots,g_{m})$.

######  Remark 4.9.

Fix the same data of the above definition. Let $(\tau_{1},\ldots,\tau_{m})$ be
a tuple in $\prod\operatorname{Aut}(f_{i})$ and let $(g_{1},\ldots,g_{m})$ be
a generic collection, then $(\tau_{1}\circ g_{1},\ldots,\tau_{m}\circ g_{m})$
is also generic. A similar result holds for an automorphism
$\tau\in\operatorname{Aut}(f)$. That is: $(g_{i})$ is generic if and only if
$(g_{i}\circ\tau)$ is generic. This gives a natural left action of
$\prod\operatorname{Aut}(f_{i})$ and right action of $\operatorname{Aut}(f)$
on $\operatorname{Int}((f_{i}))$.

Following the remark, we define

$\overline{\operatorname{Int}}(f_{1},\ldots,f_{m})_{f}:=\prod\operatorname{Aut}(f_{i})\backslash\operatorname{Int}(f_{1},\ldots,f_{m})_{f}$

and

$\widetilde{\operatorname{Int}}(f_{1},\ldots,f_{m})_{f}:=\operatorname{Int}(f_{1},\ldots,f_{m})_{f}/\operatorname{Aut}(f).$

Elements of $\overline{\operatorname{Int}}(f_{1},\ldots,f_{m})$ will be
denoted by $(\overline{g}_{1},\ldots,\overline{g}_{m})$, while the elements of
$\widetilde{\operatorname{Int}}(f_{1},\ldots,f_{m})$ will be denoted by
$(g_{1},\ldots,g_{m})\operatorname{Aut}(f)$.

When $f_{1}=\ldots=f_{m}=f^{\prime}$, we write
$\operatorname{SInt}((f^{\prime})^{m})_{f}$ to denote the set of sets (not
tuples) $\\{\overline{g}_{1},\ldots,\overline{g}_{m}\\}$ (here the $g_{i}$
must be pairwise distinct) such that
$(g_{1},\ldots,g_{m})\in\overline{\operatorname{Int}}((f^{\prime})^{m})_{f}$.

###  4.b. Normal crossing stratifications

We say that a category $\mathfrak{C}$ as in the previous section is the
category of strata of a nonsingular DM-stack $X_{\bullet}$ if there exists a
functor

$\displaystyle\mathfrak{C}$ $\displaystyle\to\text{nonsingular DM-stacks}$
$\displaystyle\alpha$ $\displaystyle\mapsto X_{\alpha}$ $\displaystyle
f\colon\alpha\to\beta$ $\displaystyle\mapsto X_{f}\colon X_{\alpha}\to
X_{\beta}$

such that

  1. (1)

The morphisms $X_{f}\colon X_{\alpha}\to X_{\beta}$ are proper and local
complete intersection of codimension $\operatorname{cd}(f)$.

  2. (2)

The quotient stack $\left[\frac{X_{\alpha}}{\operatorname{Aut}(f)}\right]$ is
the normalization of the image of $X_{f}$.

  3. (3)

The normal bundle $N_{f}$ of $X_{f}$ can be written as $N_{f}=\bigoplus_{e\in
S_{f}}\mathbb{L}_{e}$, where, for a pair $e=(\beta^{\prime},\overline{g})\in
S_{f}$, we define $\mathbb{L}_{e}:=g^{*}(N_{i_{g,f}})$.

  4. (4)

If $f_{i}\colon\alpha_{i}\to\beta$ for $i=1,2$ are two morphisms, then the
following diagram

${\displaystyle\bigsqcup_{\begin{subarray}{c}f\colon\gamma\to\beta\\\
(g_{1},g_{2})\operatorname{Aut}(f)\in\widetilde{\operatorname{Int}}(f_{1},f_{2})_{f}\end{subarray}}[X_{G}/\operatorname{Aut}(g_{1},g_{2})]}$${X_{\alpha_{1}}}$${X_{\alpha_{2}}}$${X_{\beta}}$$\scriptstyle{X_{g_{1}}}$$\scriptstyle{X_{g_{2}}}$$\scriptstyle{X_{f_{1}}}$$\scriptstyle{X_{f_{2}}}$

is a fiber diagram.

From now on we will abuse the notation and, for
$f\in\operatorname{Mor}(\alpha,\beta)$, we simply write $f\colon X_{\alpha}\to
X_{\beta}$ in place of $X_{f}\colon X_{\alpha}\to X_{\beta}$.

######  Notation 4.10.

We will use a prime to denote the image of a morphism $f\colon X_{\alpha}\to
X_{\beta}$. In other words, $X^{\prime}_{f}:=\operatorname{Im}(f)\subseteq
X_{\beta}$ and, in particular,
$X^{\prime}_{\alpha}:=\operatorname{Im}(f_{\alpha})\subseteq X_{\bullet}$.

We also say that the objects $X_{\alpha}^{\prime}$ are the embedded strata and
the objects $X_{\alpha}$ are the (resolved) strata.

The two main examples in this paper are that of $\overline{\mathcal{M}}_{g,n}$
and that of $\overline{\mathcal{J}}^{d}_{g,n}(\phi)$:

######  Example 4.11.

The category $G_{g,n}$ is a category of strata of the nonsingular DM-stack
$\overline{\mathcal{M}}_{g,n}$. If $G\in G_{g,n}$, we write
$\overline{\mathcal{M}}_{G}$ for the corresponding stratum and
$\overline{\mathcal{M}}^{\prime}_{G}$ for its image in
$\overline{\mathcal{M}}_{g,n}$ ([ACG11, Chapter XII, Section 10]).

######  Example 4.12.

The category ${\mathfrak{C}}_{g,n}(\phi)$ is a category of strata of the
nonsingular DM-stack $\overline{\mathcal{J}}^{d}_{g,n}(\phi)$ ([MMUV22,
Section 3]). If $(G,(E,D))\in{\mathfrak{C}}_{g,n}(\phi)$, we write
$\mathcal{J}_{G,(E,D)}$ for444each such stratum also depends on $\phi$, but we
do include this dependency to ease the notation the corresponding stratum and
$\mathcal{J}_{G,(E,D)}^{\prime}$ for its image in
$\overline{\mathcal{J}}^{d}_{g,n}(\phi)$.

The point made in Example 4.12 allows us to complete the proof of Proposition
3.38. The next section is devoted to completing that proof.

####  4.b.1. Proof of Proposition 3.38

###### Proof.

  1. (1)

If $d=g-1$, the result of 3.38 follows from [KP17, Theorem 4.1].

  2. (2)

Assume that $d=g-2$. In order to reach our conclusion, we prove that $\phi$ is
in the claimed range if and only if the intersection of $\mathsf{W}_{d}(\phi)$
with the boundary of $\overline{\mathcal{J}}^{d}_{g,n}(\phi)$ has codimension
larger than the expected codimension $2$. Also, the strata that generically
parameterize curves whose irreducible components are singular can be excluded,
because the existence of a nonzero global section is equivalent if those
components are smoothened.

Firstly, we analyse the boundary divisors, which have the form
$\mathcal{J}_{(G(i,1,S),D)}$. The range of $\phi$ in the claim is equivalent
to constraining the divisor $D$ to equal $(i-1,g-i-1)$. It is straightforward
to verify that the locus cut out in $\mathcal{J}_{(G(i,1,S),D)}$ by the
condition of admitting a global section has codimension at least $2$ in
$\mathcal{J}_{(G(i,1,S),D)}$, hence it has codimension at least $3$ in
$\overline{\mathcal{J}}^{d}_{g,n}(\phi)$.

If, on the other hand, the divisor $D$ is of the form $(k-1,g-k-1)$ for $k\neq
i$, then either $\mathcal{J}_{(G(i,1,S),D)}$ is contained in
$\mathsf{W}_{d}(\phi)$, or their intersection has codimension $1$ in
$\mathcal{J}_{(G(i,1,S),D)}$. In both cases, their intersection has
codimension smaller than or equal to $2$ in
$\overline{\mathcal{J}}^{d}_{g,n}(\phi)$.

Then we analyse the codimension $2$ strata $\mathcal{J}_{G,D}$. If $G$ is a
tree, the stability condition is uniquely determined by the stability
condition on the boundary divisors, and so is the stable degree $D$ – the
problem has been resolved in the previous paragraph. We assume therefore that
$G=G(i,2,S)$ is a vine curve with $2$ nodes. Using the change of coordinates
(3.11), the range identified in our statement is equivalent to requesting that
the stable divisor $D$ on $G(i,2,S)$ equals one of $(i-2,g-i)$, $(i-1,g-i-1)$,
$(i,g-i-2)$ or $(i+1,g-i-3)$. In all these cases, one can check that the
generic element of $\mathcal{J}_{G,D}$ does not admit a global section.
Conversely, if $D$ is not one of those $4$ cases, the stratum
$\mathcal{J}_{G,D}$ is contained in $\mathsf{W}_{d}(\phi)$. This concludes our
proof.

  3. (3)

Assume that $1\leq d\leq g-3$. In order to reach our conclusion, it is enough
to prove that for every $\phi$, the intersection of $\mathsf{W}_{d}(\phi)$
with some boundary divisor has codimension smaller than or equal to the
expected codimension $g-d$.

We take $i=\lfloor\frac{g}{2}\rfloor$ and pick any $S\subseteq[n]$, and show
that the intersection of $\mathsf{W}_{d}(\phi)$ with the preimage of
$\Delta_{i,S}$ in $\overline{\mathcal{J}}^{d}_{g,n}(\phi)$ contains a locus of
codimension smaller than or equal to $g-d$ in
$\overline{\mathcal{J}}^{d}_{g,n}(\phi)$ .

The stable bidegree $D$ such that the intersection
$\mathcal{J}_{G(i,1,S),D}\cap\mathsf{W}_{d}(\phi)$ has largest codimension is
$D=(\frac{d}{2},\frac{d}{2})$ for $d$ even (resp
$D=(\frac{d-1}{2},\frac{d+1}{2})$ for $d$ odd). The intersection has
codimension $\lceil\frac{g-d-1}{2}\rceil+2$ in
$\overline{\mathcal{J}}^{d}_{g,n}(\phi)$ and, for $d\leq g-3$, this number is
smaller than or equal to $g-d$.

  4. (4-5)

Assume that $d\leq 0$.

Assume first that $\phi$ is not in the given range. Then, arguing as in the
case $d=g-2$ above, one can check that in some boundary divisor or in some
codimension $2$ vine curve locus (depending on which inequality $\phi$ fails
to satisfy) the intersection with $\mathsf{W}_{d}(\phi)$ has codimension
larger than the expected one (which is $g$ for $d=0$, and by which we mean
that the locus is not empty when $d<0$).

Assume now that $\phi$ is in the given range. We will use the following
result:

######  Proposition 4.13.

If $\phi\in V_{g,n}^{d}$ is nondegenerate and such that the inequality

(4.14) $\phi_{C_{0}}\leq\frac{\left|C_{0}\cap\overline{C_{0}^{c}}\right|}{2}.$

holds for all $(C,p_{1},\ldots,p_{n})\in\overline{\mathcal{M}}_{g,n}$ and for
all subcurves $C_{0}\subseteq C$, then

    1. (a)

if $d=0$, then $F\in\mathsf{W}_{d}(\phi)$ if and only $F$ is the trivial line
bundle.

    2. (b)

if $d<0$, then $\mathsf{W}_{d}(\phi)=\varnothing$.

###### Proof.

Part (a) is [HKP18, Lemma 8, Lemma 9]. Part (b) follows from Lemma 4.15 below.

∎

By applying Proposition 4.13, and observing that both $\phi$ and the
multidegree $D$ of line bundles are stable for graph morphisms, and arguing as
in the proof of Proposition 3.5, we conclude that Inequality (4.14) is
satisfied for all curves $C$ and subcurves $C_{0}$ if and only if it is
satisfied for all vine curves $C$ (and taking $C_{0}$ to be one of its
irreducible components). After applying the change of coordinates (3.11), this
is equivalent to the given range.

The only remaining case to consider is when $d=0$ and $-3/2<x_{0,1,S}<-1/2$
for some $S$. In that case, the intersection of
$\mathcal{J}_{G(0,1,S),(-1,1)}$ with $\mathsf{W}_{d}(\phi)$ has codimension
$g+1$ in $\overline{\mathcal{J}}^{d}_{g,n}(\phi)$, hence the intersection is
in the closure of the restriction of $\mathsf{W}_{d}(\phi)$ to the open part.

∎

In the proof of Proposition 4.13 we used the following.

######  Lemma 4.15.

Assume $d<0$. Let $C$ be a nodal curve, and let $\phi\in
V^{d}_{\textnormal{stab}}(C)$ be such that Inequality (4.14) holds for all
subcurves $C_{0}\subseteq C$.

Then every $\phi$-stable rank-$1$ torsion-free sheaf $F$ on $C$ satisfies
$H^{0}(C,F)=0$.

The lemma generalizes to the case $d<0$ the argument given in in [Dud18, Lemma
3.1] and [HKP18, Lemma 8].

###### Proof.

Let $F$ be one such sheaf. If $F$ is $\phi$-stable, the inequality

(4.16)
$\deg_{C_{0}}(F)<\frac{\left|C_{0}\cap\overline{C_{0}^{c}}\right|}{2}-\delta_{C_{0}}(F)+\phi_{C_{0}}$

holds for all subcurves $\varnothing\neq C_{0}\subsetneqq C$. The latter,
combined with (4.14), implies the inequality

(4.17)
$\deg_{C_{0}}(F)<\left|C_{0}\cap\overline{C_{0}^{c}}\right|-\delta_{C_{0}}(F)$

for all subcurves $\varnothing\neq C_{0}\subsetneqq C$.

The fact that the latter inequality holds on all subcurves $C_{0}$ implies
that $F$ admits no nonzero global sections. Indeed, if such a section $s$
existed denote by $C^{\prime}$ its support. Note that $C^{\prime}\neq C$
because the degree of $F$ is negative. Hence, $C^{\prime}\neq C$ and we have
the inequality

$\deg_{C^{\prime}}(F)\geq\left|C^{\prime}\cap\overline{C^{\prime
c}}\right|-\delta_{C^{\prime}}(F),$

contradicting (4.17). ∎

We conclude this interlude by observing that for all degrees “in the middle”,
the Brill–Noether cycle cannot be of the expected codimension.

######  Remark 4.18.

Assume that $1\leq d\leq g-5$. We claim that there exist no $\phi$ such that
$\mathsf{W}_{d}(\phi)$ has the expected codimension $g-d$. To show this, we
argue in a very similar way to the case $1\leq d\leq g-3$ of the proof of
Proposition 3.38. We let $i=\lfloor\frac{g}{2}\rfloor$ and pick any
$S\subseteq[n]$. In the same way as discussed in loc.cit., for $1\leq d\leq
g-5$, the intersection of $\mathsf{W}_{d}(\phi)$ with the preimage of
$\Delta_{i,S}$ in $\overline{\mathcal{J}}^{d}_{g,n}(\phi)$ contains a locus of
codimension strictly smaller than $g-d$ in
$\overline{\mathcal{J}}^{d}_{g,n}(\phi)$, and this proves our claim.

###  4.c. Intersection theory formulas

In this section we enunciate and prove some results concerning the
intersection theory of this stratification. From now on in this section we fix
$X_{\bullet}$ and its stratification functor.

Since most of our computations are done using Chern classes, we will abuse the
notation as we now explain. Let $p_{1},p_{2},q$ be polynomials in variables
$x_{i,j}$ such that $p_{1}=qp_{2}$. Assume that $L_{i}$ are elements in the
$K$-theory of $X$, and that $A=p_{1}(c_{j}(L_{i}))$ and
$B=p_{2}(c_{j}(L_{i}))$ are formal polynomials in the Chern classes of
$L_{i}$. We will write $\frac{A}{B}$ to mean the class
$q(c_{j}(L_{i}))\cap[X]$ in the Chow group of $X$.

More generally, we will write $\frac{A}{B}\in A^{*}(X)$ to mean that there
exists polynomials $p_{1},p_{2},q$ and $K$-theory elements $L_{i}$ satisfying
the conditions in the previous paragraph.

The main motivation for this is [Ful98], which states that, for a vector
bundle $N$

$\frac{c(L\otimes\wedge^{\bullet}N)}{c_{\operatorname{rk}{N}}(N)}$

is a polynomial in the Chern classes of $L$ and of $N$.

In this language, we have the following excess intersection formula.

######  Proposition 4.19.

Let $f_{i}\colon\alpha_{i}\to\beta$ for $i=1,2$ be two morphisms in
$\mathfrak{C}$ and fix classes
$A_{i}/c_{\operatorname{cd}(f_{i})}(N_{f_{i}})\in A^{*}(X_{\alpha_{i}})$, then

$f_{1*}\Big{(}\frac{A_{1}}{c_{\operatorname{cd}(f_{1})}(N_{f_{1}})}\Big{)}f_{2*}\big{(}\frac{A_{2}}{c_{\operatorname{cd}(f_{2})}(N_{f_{2}})}\Big{)}=\sum_{\begin{subarray}{c}f\colon\gamma\to\beta\\\
(g_{1},g_{2})\operatorname{Aut}(f)\in\widetilde{\operatorname{Int}}(f_{1},f_{2})\end{subarray}_{f}}\frac{f_{*}}{|\operatorname{Aut}(g_{1},g_{2})|}\Big{(}\frac{g_{1}^{*}A_{1}g_{2}^{*}A_{2}}{c_{\operatorname{cd}(f)}(N_{f})}\Big{)}$

where $g_{i}=1,2$ are the base change morphisms (as in Section 4.b, Item (4)).

###### Proof.

This follows directly from Item (4) in Section 4.b and from the excess
intersection formula (see [Ful98, Proposition 17.4.1]). ∎

We will also be using the following corollary of the above formula.

######  Corollary 4.20.

Let $f_{i}\colon\alpha_{i}\to\beta$ be two morphisms in $\mathfrak{C}$ and let
$A_{i}/c_{\operatorname{cd}(f_{i})}(N_{f_{i}})\in A^{*}(X_{\alpha_{i}})$ be
such that $A_{i}$ is invariant under $\operatorname{Aut}(f_{i})$, then

$\frac{f_{1*}(\frac{A_{1}}{c_{\operatorname{cd}(f_{1})}(N_{f_{1}})})}{|\operatorname{Aut}(f_{1})|}\frac{f_{2*}(\frac{A_{2}}{c_{\operatorname{cd}(f_{2})}(N_{f_{2}})})}{|\operatorname{Aut}(f_{2})|}=\sum_{f\colon\gamma\to\beta}\frac{f_{*}}{|\operatorname{Aut}(f)|}\Big{(}\sum_{(\overline{g}_{1},\overline{g}_{2})\in\overline{\operatorname{Int}}(f_{1},f_{2})_{f}}\frac{g_{1}^{*}A_{1}g_{2}^{*}A_{2}}{c_{\operatorname{cd}(f)}(N_{f})}\Big{)}$

###### Proof.

We expand the formula in Proposition 4.19 to obtain

$f_{1*}\Big{(}\frac{A_{1}}{c_{\operatorname{cd}(f_{1})}(N_{f_{1}})}\Big{)}f_{2*}\big{(}\frac{A_{2}}{c_{\operatorname{cd}(f_{2})}(N_{f_{2}})}\Big{)}=\sum_{\begin{subarray}{c}f\colon\gamma\to\beta\\\
(g_{1},g_{2})\in\operatorname{Int}(f_{1},f_{2})_{f}\end{subarray}}\frac{f_{*}}{|\operatorname{Aut}(f)|}\Big{(}\frac{g_{1}^{*}A_{1}g_{2}^{*}A_{2}}{c_{\operatorname{cd}(f)}(N_{f})}\Big{)}$

because
$|(f\colon\gamma\to\beta,(g_{1},g_{2})\operatorname{Aut}(f))|=|\operatorname{Aut}(f)|/|\operatorname{Aut}(g_{1},g_{2})|$.
From there, we have that

$f_{1*}(\frac{A_{1}}{c_{\operatorname{cd}(f_{1})}(N_{f_{1}})})f_{2*}(\frac{A_{2}}{c_{\operatorname{cd}(f_{2})}(N_{f_{2}})})=\\\
=\sum_{\begin{subarray}{c}f\colon\gamma\to\beta\\\
(\overline{g}_{1},\overline{g}_{2})\in\overline{\operatorname{Int}}(f_{1},f_{2})_{f}\end{subarray}}|\operatorname{Aut}(f_{1})||\operatorname{Aut}(f_{2})|\frac{f_{*}}{|\operatorname{Aut}(f)|}(\frac{g_{1}^{*}A_{1}g_{2}^{*}A_{2}}{c_{\operatorname{cd}(f)}(N_{f})})$

and the result follows. ∎

Next, we apply the above to obtain a self-intersection formula.

######  Corollary 4.21.

Let $f\colon\alpha\to\beta$, then

$\bigg{(}\frac{f_{*}}{|\operatorname{Aut}(f)|}\Big{(}\frac{A}{c_{\operatorname{cd}(f)}(N_{f})}\Big{)}\bigg{)}^{k}=\sum_{f^{\prime}\colon\gamma\to\beta}\frac{f^{\prime}_{*}}{|\operatorname{Aut}(f^{\prime})|}(\sum_{(\overline{g}_{1},\ldots,\overline{g}_{k})\in\overline{\operatorname{Int}}(f,\ldots,f)_{f}}\frac{\prod_{j=1}^{k}g_{i}^{*}(A)}{c_{\operatorname{cd}(f^{\prime})}(N_{f^{\prime}})})$

The latter will be used to prove the following GRR formula for the total Chern
class (deduced from the usual one, involving the Chern character).

######  Proposition 4.22 (GRR for the total Chern class).

Let $f\colon\alpha\to\beta$ be a morphism and let $\mathcal{F}$ be an element
in the $K$-theory of $X_{\alpha}$ with rational coefficients. Then

$c(\frac{f_{*}(\mathcal{F})}{|\operatorname{Aut}(f)|})=1+\sum_{\begin{subarray}{c}m\geq
1\\\
f^{\prime}\colon\gamma\to\beta\end{subarray}}\frac{f^{\prime}_{*}}{|\operatorname{Aut}(f^{\prime})|}\bigg{(}\sum_{\\{\overline{g}_{1},\ldots,\overline{g}_{m}\\}\in\operatorname{SInt}((f)^{m})_{f^{\prime}}}\frac{\prod_{j=1}^{m}g_{i}^{*}(c(\bigwedge^{\bullet}N_{f}^{\vee}\otimes\mathcal{F})-1)}{c_{\operatorname{cd}(f^{\prime})}(N_{f^{\prime}})}\bigg{)}$

(This is inspired by [Ful98, Theorem 15.3]).

###### Proof.

We begin with the usual GRR formula

$\operatorname{ch}\big{(}\frac{f_{*}(\mathcal{F})}{|\operatorname{Aut}(f)|}\big{)}=f_{*}\big{(}\frac{\operatorname{ch}(\mathcal{F})}{|\operatorname{Aut}(f)|}\operatorname{td}(N_{f})^{-1}\big{)},$

which, combining with the formula for the Todd class, implies

$\operatorname{ch}_{n}\big{(}\frac{f_{*}(\mathcal{F})}{|\operatorname{Aut}(f)|}\big{)}=\frac{f_{*}}{|\operatorname{Aut}(f)|}\bigg{(}\frac{\operatorname{ch}_{n}(\bigwedge^{\bullet}N_{f}^{\vee}\otimes\mathcal{F})}{c_{\operatorname{cd}(f)}(N_{f})}\bigg{)}.$

By the inversion formula to express the total Chern class in terms of the
Chern character (see e.g. [PRvZ20, Equation 3.9]), we deduce

$c\big{(}\frac{f_{*}(\mathcal{F})}{|\operatorname{Aut}(f)|}\big{)}=\exp\big{(}\frac{f_{*}}{|\operatorname{Aut}(f)|}\bigg{(}\frac{\sum_{n\geq
1}\frac{(-1)^{n-1}}{n}\operatorname{ch}_{n}(\bigwedge^{\bullet}N_{f}^{\vee}\otimes\mathcal{F})}{c_{\operatorname{cd}(f)}(N_{f})}\bigg{)}\big{)}.$

Setting $A:=\sum_{n\geq
1}\frac{(-1)^{n-1}}{n}\operatorname{ch}_{n}(\bigwedge^{\bullet}N_{f}^{\vee}\otimes\mathcal{F})$,
we will then compute

$\star:=\exp\bigg{(}\frac{f_{*}}{|\operatorname{Aut}(f)|}\Big{(}\frac{A}{c_{\operatorname{cd}(f)}(N_{f})}\Big{)}\bigg{)}$

$\displaystyle\star$ $\displaystyle=1+\sum_{k\geq
1}\bigg{(}\frac{f_{*}}{|\operatorname{Aut}(f)|}\Big{(}\frac{A}{c_{\operatorname{cd}(f)}(N_{f})}\Big{)}\bigg{)}^{k}\frac{1}{k!}$
$\displaystyle=1+\sum_{k\geq
1}\sum_{f^{\prime}\colon\gamma\to\beta}\frac{f^{\prime}_{*}}{|\operatorname{Aut}(f^{\prime})|}\Big{(}\sum_{(\overline{g}_{1},\ldots,\overline{g}_{k})\in\overline{\operatorname{Int}}((f)^{k})_{f^{\prime}}}\frac{\prod_{i=1}^{k}g_{i}^{*}(A)}{c_{\operatorname{cd}(f^{\prime})}(N_{f^{\prime}})}\cdot\frac{1}{k!}\Big{)}$
$\displaystyle=1+\sum_{f^{\prime}\colon\gamma\to\beta}\frac{f^{\prime}_{*}}{|\operatorname{Aut}(f^{\prime})|}\Big{(}\sum_{\begin{subarray}{c}m\geq
1\\\
\\{\overline{g}_{1},\ldots,\overline{g}_{m}\\}\in\operatorname{SInt}((f)^{m})_{f^{\prime}}\\\
k_{1},\ldots,k_{m}\geq
1\end{subarray}}\frac{\prod_{i=1}^{m}g_{i}^{*}(A)^{k_{i}}}{c_{\operatorname{cd}(f^{\prime})}(N_{f^{\prime}})}\cdot\frac{1}{(\sum_{i=1}^{m}k_{i})!}\Big{)}$
$\displaystyle=1+\sum_{f^{\prime}\colon\gamma\to\beta}\frac{f^{\prime}_{*}}{|\operatorname{Aut}(f^{\prime})|}\Big{(}\sum_{\begin{subarray}{c}m\geq
1\\\
\\{\overline{g}_{1},\ldots,\overline{g}_{m}\\}\in\operatorname{SInt}((f)^{m})_{f^{\prime}}\end{subarray}}\frac{\prod_{i=1}^{m}(\exp(g_{i}^{*}(A))-1)}{c_{\operatorname{cd}(f^{\prime})}(N_{f^{\prime}})}\Big{)}$

The claim is then obtained by applying again the inversion formula in the form

$\exp(A)=c(\bigwedge^{\bullet}N_{f}\otimes F).$

∎

###  4.d. Blow-up

Starting from a category $\mathfrak{C}$ as in 4.a, here we define the blow-up
category at a stratum with transversal self-intersection. Then for a fixed
stratification functor $X$, we interpret the blow-up category as the
stratification of the blow-up of the nonsingular DM-stack $X$ at that stratum.

######  Definition 4.23.

We say that an object $\delta\in\operatorname{Obj}(\mathfrak{C})$ _has
transversal self-intersection_ if for every pair
$g_{1},g_{2}\colon\gamma\to\delta$, the sets $g_{1}^{*}(S_{\delta})$,
$g_{2}^{*}(S_{\delta})$ are either equal or disjoint.

######  Remark 4.24.

If $g_{1}^{*}(S_{\delta})=g_{2}^{*}(S_{\delta})$, then
$\overline{g}_{1}=\overline{g}_{2}\in\overline{\operatorname{Mor}}(\gamma,\delta)$.
See Proposition 4.6.

######  Example 4.25.

Let $\mathfrak{C}=G_{2k+1,1}$ for some $k\geq 1$. We claim that the vine curve
graph $G=G(k,2,\\{1\\})$ does not have transversal self-intersection.

Indeed, let $G^{\prime}$ be the “triangle” graph with $2$ vertices of genus
$k$ and a third vertex of genus $0$ carrying the marking. There are two
different morphisms $g_{1},g_{2}\colon G^{\prime}\to G$ and
$g_{1}^{*}(S_{G})\cap g_{2}^{*}(S_{G})$ consists of $1$ edge.

######  Definition 4.26.

Let $\delta$ be an object of $\mathfrak{C}$ with transversal self-
intersection. We define _the blowup category
$\operatorname{Bl}_{\delta}\mathfrak{C}$ of $\mathfrak{C}$ at $\delta$_ as
follows.

First consider the following category. Its set of objects consists of pairs
$(\gamma,\mathbf{m})$ where $\gamma$ is an object of $\mathfrak{C}$ and
$\mathbf{m}$ is a function
$\overline{\operatorname{Mor}}(\gamma,\delta)\to\mathcal{P}(S_{\gamma})$ such
that

(4.27) $\varnothing\neq\mathbf{m}(\overline{g})\subseteq g^{*}(S_{\delta})\
\text{ for every }\
\overline{g}\in\overline{\operatorname{Mor}}(\gamma,\delta).$

Its morphisms $(\gamma_{1},\mathbf{m}_{1})\to(\gamma_{2},\mathbf{m}_{2})$ are
morphisms $f\colon\gamma_{1}\to\gamma_{2}$ such that for every
$\overline{g}_{1}\in\overline{\operatorname{Mor}}(\gamma_{1},\delta)$ we have
that one of the conditions hold

  1. (1)

there exists
$\overline{g}_{2}\in\overline{\operatorname{Mor}}(\gamma_{2},\delta)$ such
that $\overline{g}_{1}=\overline{g_{2}\circ f}$ and
$\mathbf{m}_{1}(\overline{g}_{1})\subseteq
f^{*}(\mathbf{m}_{2}(\overline{g}_{2}))$,

  2. (2)

or $\mathbf{m}_{1}(\overline{g}_{1})\cap f^{*}(S_{\gamma_{2}})=\varnothing$.

We then define $\operatorname{Bl}_{\delta}\mathfrak{C}$ as a skeleton of the
above category.

######  Proposition 4.28.

The category $\operatorname{Bl}_{\delta}\mathfrak{C}$ is naturally ranked, and
it satisfies Axiom (1) from Section 4.a.

###### Proof.

Straightforward. ∎

######  Remark 4.29.

The rank of $(\gamma,\mathbf{m})$ is

$\operatorname{rk}(\gamma)-\sum_{\overline{g}\in\overline{\operatorname{Mor}}(\gamma,\delta)}|\mathbf{m}(\overline{g})|.$

Moreover, the set $S_{(\gamma,\mathbf{m})}$ (the codimension $1$ strata that
contain a fixed stratum $(\gamma,\mathbf{m})$) is naturally identified with
$S_{\gamma}\setminus(\bigcup_{\overline{g}\in\overline{\operatorname{Mor}}(\gamma,\delta)}\mathbf{m}(\overline{g}))\cup\overline{\operatorname{Mor}}(\gamma,\delta)$.

Recall Notation 4.10. We define $h\colon\widetilde{X}_{\beta}\to X_{\beta}$ to
be the blow up of $X_{\beta}$ at the union of the images
$X^{\prime}_{g_{1}}\subseteq X_{\beta}$ for every $g_{1}\colon\gamma\to\beta$
such that there exists $g_{2}\colon\gamma\to\delta$ satisfying
$(g_{1},g_{2})\in\operatorname{Int}(f_{\beta},f_{\delta})_{f_{\gamma}}$. We
define $X_{\beta,\mathbf{m}}$ to be

$\prod_{\overline{g}\in\overline{\operatorname{Mor}}(\gamma,\delta)}\mathbb{P}\Big{(}\bigoplus_{e\in\mathbf{m}(\overline{g})}\mathbb{L}_{e}\Big{)}.$

######  Proposition 4.30.

The functor

$\displaystyle\operatorname{Bl}_{\delta}\mathfrak{C}$
$\displaystyle\to\textnormal{nonsingular DM stacks }$
$\displaystyle(\gamma,\mathbf{m})$ $\displaystyle\mapsto
X_{\gamma,\mathbf{m}}$

is a stratification of $\operatorname{Bl}_{X^{\prime}_{\delta}}X_{\bullet}$.

###### Proof.

This follows from [MPS23, Section 4.5] (see also [KKMSD73, Theorem 6, p.90])
where the nonsingular DM stack is constructed as the star subdivision of the
cone stack associated to the stratification. ∎

######  Remark 4.31.

When there exists no morphism $\gamma\to\delta$, there exists a unique
$\mathbf{m}$ such that the pair
$(\gamma,\mathbf{m})\in\operatorname{Bl}_{\delta}\mathfrak{C}$. The latter is
the stratum that corresponds to the strict transform of the image
$X_{\gamma}^{\prime}\subset X_{\bullet}$.

######  Remark 4.32.

Suppose that $\delta$ is a stratum with transversal self intersection and
$f\colon\gamma\to\beta$ is a morphism such that
$\operatorname{Mor}(\beta,\delta)=\varnothing$. Let $(\gamma,\mathbf{m})$ be
an object in $\operatorname{Bl}_{\delta}\mathfrak{C}$. Then the morphism $f$
lifts to a morphism $(\gamma,\mathbf{m})\to(\beta,\varnothing)$ in
$\operatorname{Bl}_{\delta}\mathfrak{C}$ if and only if
$f^{*}S_{\beta}\cap\bigcup_{\overline{g}\in\overline{\operatorname{Mor}}(\gamma,\delta)}\mathbf{m}(g)=\varnothing$.
(That is, when $X^{\prime}_{\gamma,\mathbf{m}}$ is contained in the strict
transform of $X^{\prime}_{\beta}$ in
$\operatorname{Bl}_{X^{\prime}_{\delta}}X_{\bullet}$).

In this paper, the main example of the above construction is going to be the
case where $\mathfrak{C}$ is the category ${\mathfrak{C}}_{g,n}(\phi)$ of
Example 4.12, or a blowup of the latter. We now describe the example of $1$
blowup of ${\mathfrak{C}}_{g,n}(\phi)$ at one of the centers that will be
relevant for our main result.

######  Example 4.33.

Let $\phi\in V_{g,n}^{d}$ and $(G,D)\in{\mathfrak{C}}_{g,n}(\phi)$ be the lift
of a vine curve $G=G(i,t,S)$ by some $\phi$-stable divisor $D$.

Morphisms $f\colon(G^{\prime},(E^{\prime},D^{\prime}))\to(G,D)$ correspond to
subsets $T_{f}\subset V(G^{\prime E^{\prime}})$ such that the complete
subgraphs $G(T_{f}),G(T_{f}^{c})$ in $G^{\prime E^{\prime}}$ are connected and
of genus $i,g-i-t+1$, the markings $S$ are on $G(T_{f})$ and the markings
$S^{c}$ are on $G(T_{f}^{c})$, and $D^{\prime}(G(T_{f}))=D(v_{1})$ and
$D^{\prime}(G(T_{f}^{c}))=D(v_{2})$ (for $v_{1},v_{2}$ the two vertices of
$G$). We let $E(T_{f})\subseteq E(G^{\prime E^{\prime}})$ be the subset of $t$
edges that separate $G(T_{f})$ from $G(T_{f}^{c})$.

Assume that $(G,D)$ is a stratum555because of Lemma 5.27, in this paper we
will never need to blowup any strata of the form $(G,E,D)$ with
$E\neq\varnothing$ with transversal self-intersection. The category
$\operatorname{Bl}_{(G,D)}{\mathfrak{C}}_{g,n}(\phi)$ defined above stratifies
the blowup
$\operatorname{Bl}_{{\mathcal{J}^{\prime}}_{(G,D)}}\overline{\mathcal{J}}^{d}_{g,n}(\phi)$.
We can describe more explicitly its objects as tuples
$(G^{\prime},E^{\prime},D^{\prime},\alpha)$ such that
$(G^{\prime},(E^{\prime},D^{\prime}))\in\operatorname{Obj}({\mathfrak{C}}_{g,n}(\phi))$
and $\alpha$ is a choice, for each morphism
$f\colon(G^{\prime},D^{\prime})\to(G,D)$ (up to automorphisms of $(G,D)$), of
a subset $\varnothing\neq\alpha(\operatorname{Aut}(G,D)f)\subseteq E(T_{f})$.

We now define the psi-classes associated to a given stratum
$\gamma\in\mathfrak{C}$. Recall that each $e\in S_{\gamma}$ corresponds to a
morphism $j_{e}\colon\gamma\to\beta_{e}$ where the latter has rank $1$. Then
define the psi-classes

(4.34)
$\Psi_{\beta_{e}}:=-c_{1}(\mathbb{L}_{e})=-c_{1}(N_{X_{\beta_{e}}}X_{\bullet}),\quad\psi_{\gamma,e}:=j_{e}^{*}\Psi_{\beta_{e}}$

(see Item (3) in the beginning of Section 4.b) for $\mathbb{L}_{e}$).

We will now state and prove a pushforward formula for monomials in psi-classes
under the blowdown morphism. We begin by introducing some notation.

Recall Remark 4.29. For an object $(\gamma,\mathbf{n})$ in
$\operatorname{Bl}_{\delta}\mathfrak{C}$ we define the sets

$\displaystyle S_{\gamma\setminus\delta}$
$\displaystyle:=S_{\gamma}\setminus\bigcup_{j\colon\gamma\to\delta}j^{*}(S_{\delta}),$
$\displaystyle\operatorname{FU}_{\delta}(\gamma,\mathbf{n})$
$\displaystyle:=\bigcup_{j\colon\gamma\to\delta}j^{*}(S_{\delta})\setminus\mathbf{n}(\overline{j}),$
$\displaystyle\operatorname{CU}_{\delta}(\gamma,\mathbf{n})$
$\displaystyle:=\bigcup_{j\colon\gamma\to\delta}\mathbf{n}(\overline{j}).$

(The symbols $\operatorname{FU}$ and $\operatorname{CU}$ will acquire some
meaning in Section 7 as certain collection of edges, see Equation (7.19).)
Note that the unions can equivalently be taken over
$\overline{\operatorname{Mor}}(\gamma,\delta)$ instead of over all morphisms.
(See Remark 4.7).

We define
$H^{\delta}_{\gamma,\mathbf{n}}((g^{\prime}_{e^{\prime}})_{e^{\prime}\in
S_{\gamma,\mathbf{n}}})$ as the set of tuples $((a_{e})_{e\in
S_{\gamma}},(g_{e})_{e\in S_{\gamma}})$ of non-negative integers satisfying
$a_{e}=0$ for every $e\notin\operatorname{FU}_{\delta}(\gamma,\mathbf{n})$,

$\sum_{e\in\mathbf{n}(\overline{j})}(g_{e}+1)=g^{\prime}_{\overline{j}}+1+\sum_{e\in
j^{*}(S_{\delta})\setminus\mathbf{n}(\overline{j})}a_{e}$

for every
$\overline{j}\in\overline{\operatorname{Mor}}(\gamma,\delta)\subseteq
S_{\gamma,\mathbf{n}}$, and $g_{e}=g^{\prime}_{e}$ for every $e\in
S_{\gamma}\setminus\operatorname{CU}_{\delta}(\gamma,\mathbf{n})\subseteq
S_{\gamma,\mathbf{n}}$.

For a morphism $h\colon(\gamma,\mathbf{n})\to(\beta,\mathbf{m})$ and a tuple
$(g^{\prime}_{e^{\prime}})_{e^{\prime}\in S_{\beta,\mathbf{m}}}$ we define
$h^{*}(g^{\prime}_{e^{\prime}})$ as the tuple

$(h^{*}(g^{\prime}_{e^{\prime}}))_{\widetilde{e}}:=\begin{cases}g^{\prime}_{e^{\prime}}&\text{
if }\widetilde{e}=h^{*}(e^{\prime})\text{ for some }e^{\prime}\\\ -1&\text{ if
}\widetilde{e}\in S_{\gamma,\mathbf{n}}\setminus
h^{*}(S_{\beta,\mathbf{m}}).\end{cases}$

We define $M_{\delta}(\gamma)$ to be the set of all function
$\mathbf{m}\colon\overline{\operatorname{Mor}}(\gamma,\delta)\to\mathcal{P}(S_{\gamma})$
satisfying Equation (4.27).

######  Corollary 4.35.

Let $p\colon\operatorname{Bl}_{X_{\delta^{\prime}}}X_{\bullet}\to X_{\bullet}$
be the blowdown morphism, and fix integers $(g^{\prime}_{e^{\prime}}\geq
0)_{e^{\prime}\in S_{\beta,\mathbf{m}}}$. Then the pushforward

$p_{*}\frac{f_{(\beta,\mathbf{m})*}}{|\operatorname{Aut}(\beta,\mathbf{m})|}\Big{(}\prod_{e^{\prime}\in
S_{\beta,\mathbf{m}}}\Psi_{e^{\prime}}^{g^{\prime}_{e}}\Big{)}$

equals

$\sum_{\gamma\in\mathfrak{C}}\frac{f_{\gamma*}}{|\operatorname{Aut}(\gamma)|}\Bigg{(}\sum_{\begin{subarray}{c}\mathbf{n}\in
M_{\delta}(\gamma)\\\
h\in\overline{\operatorname{Mor}}((\gamma,\mathbf{n}),(\beta,\mathbf{m}))\end{subarray}}\sum_{\begin{subarray}{c}(a_{e},g_{e})\in\\\
H^{\delta}_{\gamma,\mathbf{n}}(h^{*}(g^{\prime}_{e^{\prime}}))\end{subarray}}(-1)^{a_{e}}\binom{g_{e}}{a_{e}}\Psi_{e}^{g_{e}-a_{e}}\Bigg{)}.$

###### Proof.

Follows from [BL05, Theorem 4.8] (or [Alu10, Theorem 4.2]). ∎

##  5\. Combinatorial aspects of Wall-Crossing

In this section we fix two stability conditions $\phi^{+}$ and $\phi^{-}$ “on
opposite sides of a stability hyperplane $H$” (Definition 5.1), and give a
description of what will turn out to be the stratification category
$\widetilde{\mathfrak{C}}=\widetilde{\mathfrak{C}}(\phi^{+},\phi^{-})$ of the
resolution $\widetilde{\mathcal{J}}^{d}_{g,n}(\phi^{+},\phi^{-})$, which we
will formally construct in the next section.

Objects of this category are defined in Definition 5.28 as triples
$(G,D,\alpha)$ where $(G,D)$ is an object of
$\overline{\mathcal{J}}^{d}_{g,n}(\phi^{+})$ and $\alpha$ is a certain “vine
function” to be introduced in Definition 5.15. This generalizes the case of
$1$ blowup, described in Example 4.33. Then we study the subcategory
$\widetilde{\mathfrak{C}}_{E}=\widetilde{\mathfrak{C}}(\phi^{+},\phi^{-})$ of
the strata that are in the intersection of the exceptional divisors — this is
the category appearing in our main result Theorem 7.4. The condition that
singles out objects of
$\widetilde{\mathfrak{C}}_{E}\subset\widetilde{\mathfrak{C}}$ is that the
function $\alpha$ should be _full_ , as defined in Definition 5.15. We then
see in Proposition 5.21 that the datum of a full vine function is equivalent
to that of a full forest, and this gives a simpler description of the objects
of $\widetilde{\mathfrak{C}}_{E}$, which is the one that we will use in
Theorem 7.4.

Recall from 3.b that there are $3$ types of hyperplanes. If $H$ is as in
(3.12), then no blowup is required and
${\mathfrak{C}}_{g,n}(\phi)=\widetilde{\mathfrak{C}}$. The second case is when
$H$ is as in (3.13) and $S\neq[n]$. That case will be discussed in 5.c. The
most difficult case is when $H$ is as in (3.13) and $S=[n]$.

######  Definition 5.1.

Let $\phi^{+},\phi^{-}\in V_{g,n}^{d}$ be nondegenerate, and let $H$ be a
stability hyperplane (see Section 3.b). We say that _the polarizations
$\phi^{\pm}$ are on opposite sides of the hyperplane $H$_ if
$\phi^{0}=\frac{\phi^{-}+\phi^{+}}{2}\in H$ is the only degenerate point of
the segment $[\phi^{+},\phi^{-}]\subset V_{g,n}^{d}$.

In other words, $H\ni\phi_{0}$-semistability implies $\phi^{+}$ or $\phi^{-}$
stability, and $\phi^{\pm}$ are small perturbations of $\phi_{0}$. Throughout
we fix $H,\phi^{\pm}$ and $\phi_{0}$ as in the above definition.

###  5.a. Extremal sets, vine functions and full forests

In this section we prove some of our bulk combinatorial results that have to
do with wall-crossing, focusing on the “extremal” multidegree, i.e. those
multidegrees that are $\phi^{+}$-stable but are not $\phi^{-}$-stable. _We
shall fix a $n$ pointed graph $G$ of genus $g$ and a divisor $D$ on $G$
throughout_. (We are not imposing stability conditions here, but the cases we
are interested in are either $G=G^{\prime E^{\prime}}$ for some
$E^{\prime}\subseteq E(G)$ where $G^{\prime}$ is stable, or $G$ is obtained by
forgetting the last marking on a stable $(n+1)$ pointed graph.)

For a subset $V\subseteq V(G)$ we define666We note that the existing
definition of $\beta(V)$ in [Est01] and [AP20] are based on a different sign
convention, however all relevant properties remain the same.

$\beta^{\star}(V):=-\deg(D|_{V})+\phi^{\star}(V)+\frac{\left|E(V,V^{c})\right|}{2},\;\
\ \ \text{ for }\;\star=+,-,0.$

Note that $D$ is $\phi^{\star}$-semistable (Definition 3.17) if and only if
$\beta^{\star}(V)\geq 0$ for every $V\subset V(G)$. Moreover, we have the
following relation for $\beta^{\star}$ (see [AP20, Lemma 4.1])

(5.2) $\beta^{\star}(V)+\beta^{\star}(W)-|E(V\setminus W,W\setminus
V)|=\beta^{\star}(V\cap W)+\beta^{\star}(V\cup W).$

From now on in this chapter we will assume that $D$ is $\phi^{+}$ semistable
on $G$. A subset $V\subsetneqq V(G)$ is called _extremal_ (with respect to
$\phi^{+},\phi^{-}$ and $D$) if

(5.3) $\beta^{+}(V)>0\text{ and }\beta^{-}(V)<0$

In particular, this implies

$\phi^{+}(V)>\phi^{-}(V).$

Note that if $V$ is extremal, then $\beta^{0}(V)=0$.

We are now ready to define the main object of study in this section.

######  Definition 5.4.

We define the poset
$\operatorname{Ext}(G,D)=\operatorname{Ext}_{\phi^{+},\phi^{-}}(G,D)$ as

$\\{V\subseteq V(G);V\text{ is extremal, connected and with connected
complement}\\}$

with the ordering given by inclusion.

######  Remark 5.5.

If $\iota\colon(G,D)\to(G^{\prime},D^{\prime})$ is a specialization and
$V^{\prime}$ is in $\operatorname{Ext}(G^{\prime},D^{\prime})$, we have that
$\iota^{-1}(V^{\prime})$ is in $\operatorname{Ext}(G,D)$.

######  Remark 5.6.

Each element $V$ of $\operatorname{Ext}(G,D)$ corresponds to a morphism from
$G$ to an extremal vine curve stratum $(G^{\prime},D^{\prime})$ (up to
automorphisms of $(G^{\prime},D^{\prime})$) obtained by contracting $E(V,V)$
and $E(V^{c},V^{c})$.

We have the following results for extremal subsets.

######  Proposition 5.7.

Let $V_{1}$ and $V_{2}$ be two extremal subsets. Then $V_{1}\cap V_{2}$ and
$V_{1}\cup V_{2}$ are either empty, extremal or equal to $V(G)$. Moreover, we
have that $E(V_{1}\setminus V_{2},V_{2}\setminus V_{1})=\varnothing$ in all
cases.

###### Proof.

Write $H_{0}=V_{1}\cap V_{2}$, $H_{1}=V_{1}\setminus H_{0}$,
$H_{2}=V_{2}\setminus H_{0}$ and $H_{3}=V_{1}^{c}\cap V_{2}^{c}$ (see Figure
1). Define $\alpha=|E(H_{1},H_{2})|=|E(V_{1}\setminus V_{2},V_{2}\setminus
V_{1})|$.

$H_{0}$$H_{1}$$H_{2}$$H_{3}$$\alpha$ Figure 1.

Since $\beta^{-}(H_{0}\cup H_{1}),\beta^{-}(H_{0}\cup H_{2})<0$ we have that
$\beta^{0}(H_{0}\cup H_{1})=\beta^{0}(H_{0}\cup H_{2})=0$. By (5.2) we have

$\beta^{0}(H_{0}\cup H_{1})+\beta^{0}(H_{0}\cup
H_{2})-\alpha=\beta^{0}(H_{0})+\beta^{0}(H_{0}\cup H_{1}\cup H_{2}),$

and, because $\beta^{0}(H)\geq 0$ for every $H$, we deduce that $\alpha=0$,
and $\beta^{0}(H_{0})=\beta^{0}(H_{0}\cup H_{1}\cup H_{2})=0$.

If $H_{0}\neq\varnothing$, then $\beta^{+}(H_{0})>0$ and if $H_{0}\cup
H_{1}\cup H_{2}\neq V(G)$, then $\beta^{+}(H_{0}\cup H_{1}\cup H_{2})>0$.
Since $\beta^{0}=\frac{\beta^{+}+\beta^{-}}{2}$, we have that
$\beta^{-}(H_{0})<0$ (respectively, $\beta^{-}(H_{0}\cup H_{1}\cup H_{2})<0$)
if $H_{0}\neq\varnothing$ (respectively, if $H_{0}\cup H_{1}\cup H_{2}\neq
V(G)$). This finishes the proof. ∎

######  Proposition 5.8.

Let $V$ be an extremal set.

If $V=V_{1}\sqcup V_{2}$ with $E(V_{1},V_{2})=\varnothing$ and
$V_{1},V_{2}\neq\varnothing$, then $V_{1}$ and $V_{2}$ are extremal.
Similarly, if $V^{c}=W_{1}\sqcup W_{2}$ with $E(W_{1},W_{2})=\varnothing$ and
$W_{1},W_{2}\neq\varnothing$, then $W_{1}^{c}$ and $W_{2}^{c}$ are extremal.

###### Proof.

For the first part, we have that
$\beta^{0}(V)=\beta^{0}(V_{1})+\beta^{0}(V_{2})$, since $\beta^{0}(V)=0$, then
$\beta^{0}(V_{1})=\beta^{0}(V_{2})=0$ as well. So,
$\beta^{-}(V_{i}),\beta^{-}(V_{2})<0$. The second part is proven similarly. ∎

In what follows we will also need an additional hypothesis.

######  Hypothesis 1.

If $V\subseteq V(G)$ is extremal, then $\operatorname{leg}(1)\in V$.

In particular, by Proposition 5.8, we have that if $V\subseteq V(G)$ is
extremal, then $G(V)$ is connected. From now on in this chapter we will always
assume Hypothesis 1.

Hypothesis (1) fixes the following convention on $\phi^{+},\phi^{-}$:

######  Remark 5.9.

If $V$ is an extremal set in $\operatorname{Ext}(G,D)$ then
$\phi^{+}(V)>\phi^{-}(V)$. In particular, if we set
$S:=\operatorname{leg}^{-1}(V)$ and $i:=g(V)$ and $t=|E(V,V^{c})|$, then

$x_{i,t,S}^{+}>x_{i,t,S}^{-}.$

for fixed coordinates $\phi^{\pm}=(x_{i,t,S}^{\pm})_{(i,t,S)}\in V_{g,n}^{d}$
as discussed in Section 3.b. Also, since $\phi^{+}$ and $\phi^{-}$ are on
opposite sides of a hyperplane $H=H(i_{0},t_{0},S_{0})$, Hypothesis 1 is
always satisfied upon possibly switching $\phi^{+}$ and $\phi^{-}$.

######  Remark 5.10.

The results in this section hold more generally for when $\phi^{+}$ and
$\phi^{-}$ lie on opposite sides of a higher codimension stability plane (not
necessarilly a hyperplane, as in Definition 5.1), in which case Hypothesis 1
becomes restrictive.

Here are some important properties of $\operatorname{Ext}(G,D)$ that follow
from Hypothesis 1:

######  Corollary 5.11.

Let $V_{1},V_{2}\in\operatorname{Ext}(G,D)$, then either $V_{1}\cup
V_{2}=V(G)$ or there exists $V\in\operatorname{Ext}(G,D)$ such that $V_{1}\cup
V_{2}\subseteq V$.

###### Proof.

Assume that $V_{1}\cup V_{2}\neq V(G)$. Then, by Proposition 5.7, we have that
$V_{1}\cup V_{2}$ is extremal. By Hypothesis 1, we have that $V_{1}\cup V_{2}$
is also connected. Let $W$ be a connected component of $(V_{1}\cup
V_{2})^{c}$. By Proposition 5.8, we have that $W^{c}$ is extremal. Moreover,
since $V_{1}\cup V_{2}$ is connected (and $G$ is connected), so is $W^{c}$.
This proves that $W^{c}\in\operatorname{Ext}(G,D)$ and $V_{1}\cup
V_{2}\subseteq W^{c}$. ∎

######  Corollary 5.12.

Let $V_{1},V_{2}$ be elements of $\operatorname{Ext}(G,D)$ such that
$V_{1},V_{2}\subseteq V$ for some $V\in\operatorname{Ext}(G,D)$. Then
$V_{1}\cap V_{2}\in\operatorname{Ext}(G,D)$.

###### Proof.

By Proposition 5.7, we have that $V_{1}\cap V_{2}$ is extremal and by
Hypothesis 1 we have that $V_{1}\cap V_{2}$ is nonempty and connected. All
that is left is to prove that $(V_{1}\cap V_{2})^{c}=V^{c}\cup(V\setminus
V_{1})\cup(V\setminus V_{2})$ is connected. But this follows from the fact
that $V^{c}$, $V_{1}^{c}=V^{c}\cup(V\setminus V_{1})$ and
$V_{2}^{c}=V^{c}\cup(V\setminus V_{2})$ are connected. ∎

We are now ready to introduce a key notion to describe the blowup category of
${\mathfrak{C}}_{g,n}(\phi)$ at some vine curve strata.

######  Definition 5.13.

For each $(G,D)$ and lower set $L\subseteq\operatorname{Ext}(G,D)$, we say
that a function $\alpha\colon L\to\mathcal{P}(E(G))$ is a _vine function_ if
the following conditions hold

  1. (1)

$\alpha(V)\subseteq E(V,V^{c})$ for every $V\in L$.

  2. (2)

For all $V\in L$ we have $\alpha(V)=\varnothing$ if and only if there exists
$V^{\prime}\subsetneqq V$ with $V^{\prime}\in\operatorname{Ext}(G,D)$ such
that $\alpha(V^{\prime})\cap E(V,V^{c})\neq\varnothing$.

We usually think that $L$ is part of the datum of $\alpha$, and write
$L_{\alpha}$ for the domain of a vine function $\alpha$. We also define
$|\alpha|=\bigcup_{V\in L_{\alpha}}\alpha(V)\subseteq E(G)$.

######  Definition 5.14.

Given a specialization $\iota\colon(G,D)\to(G^{\prime},D^{\prime})$ we say
that the vine functions $\alpha$ and $\alpha^{\prime}$ are _compatible with
$\iota$_ if

  1. (1)

$\iota^{-1}(V^{\prime})\in L_{\alpha}$ for every $V^{\prime}\in
L_{\alpha^{\prime}}$.

  2. (2)

if $\alpha^{\prime}(V^{\prime})\neq\varnothing$, then
$\alpha(\iota^{-1}(V^{\prime}))\neq\varnothing$.

  3. (3)

if $e^{\prime}\notin|\alpha^{\prime}|$ then $\iota_{E}(e)\notin|\alpha|$.

In that case we write
$\iota\colon(G,D,\alpha)\to(G^{\prime},D^{\prime},\alpha^{\prime})$ and say
that the first triple specializes to the second.

We will also need to introduce some subcategories of the stratification
category of ${\mathfrak{C}}_{g,n}(\phi)$, which will use only some vine
functions which we call “full”. We now introduce those, and then discuss how
this notion is equivalent to the combinatorial notion of a full forest.

######  Definition 5.15.

We say that $\alpha$ is _full_ if $L_{\alpha}=\operatorname{Ext}(G,D)$, and
$|\alpha|=E(G)$.

We will show in Proposition 5.21 how full vine functions are equivalent to the
following notion.

######  Definition 5.16.

A forest $V_{\bullet}\subseteq\operatorname{Ext}(G,D)$ is a _full forest_ in
$\operatorname{Ext}(G,D)$ if

  1. (1)

it contains all maximal elements of $\operatorname{Ext}(G,D)$, and

  2. (2)

the edge set satisfies $E(G)=\bigcup_{V\in V_{\bullet}}E(V,V^{c})$.

We first prove some intermediate results in that direction. For a forest
$V_{\bullet}\subseteq\operatorname{Ext}(G,D)$, and for each
$V^{\prime}\in\operatorname{Ext}(G,D)$, we define

(5.17)
$\operatorname{next}(V^{\prime})=\operatorname{next}_{V_{\bullet}}(V^{\prime}):=\bigcap_{V^{\prime}\subsetneqq
V\in V_{\bullet}}V$

(with the usual convention that the intersection over the empty set equals
$V(G)$).

######  Lemma 5.18.

Let $V_{\bullet}\subseteq\operatorname{Ext}(G,D)$ be a full forest and let
$V_{1}$ and $V_{2}$ be two incomparable elements in $V_{\bullet}$. Then
$V_{1}\cup V_{2}=V(G)$ and $E(V_{1}^{c},V_{2}^{c})=\varnothing$.

###### Proof.

By Corollary 5.11 we have that either $V_{1}\cup V_{2}=V(G)$, or there exists
$V\in\operatorname{Ext}(G,D)$ such that $V_{1},V_{2}\subseteq V$. If the
latter holds, Part (1) of Definition 5.16 implies that $V_{1}$ and $V_{2}$ are
comparable, a contradiction. So $V_{1}\cup V_{2}=V(G)$. The fact that
$E(V_{1}^{c},V_{2}^{c})=\varnothing$ follows from Proposition 5.7. ∎

######  Proposition 5.19.

Let $V_{\bullet}\subseteq\operatorname{Ext}(G,D)$ be a full forest. Let
$V^{\prime}\in\operatorname{Ext}(G,D)$ be a nonmaximal element and let
$V_{1},\ldots,V_{m}$ be the elements of $V_{\bullet}$ that are minimal among
those containing $V^{\prime}$. Then:

  1. (1)

For every $i=1,\ldots,m$,

$E(\operatorname{next}(V^{\prime})\setminus
V^{\prime},V_{i}^{c})\neq\varnothing.$

  2. (2)

$\operatorname{next}(V^{\prime})\neq V^{\prime}$.

  3. (3)

$G(\operatorname{next}(V^{\prime}))$ is connected.

  4. (4)

If $V^{\prime}\in V_{\bullet}$, then
$E(\operatorname{next}(V^{\prime})\setminus
V^{\prime},\operatorname{next}(V^{\prime})\setminus V^{\prime})=\varnothing$

  5. (5)

If $V\in\operatorname{Ext}(G,D)$ is such that
$\operatorname{next}(V^{\prime})\subseteq V$, then there exists
$i\in\\{1,\ldots,m\\}$ such that $V_{i}\subseteq V$.

  6. (6)

all minimal elements of $\operatorname{Ext}(G,D)$ belong to $V_{\bullet}$.

  7. (7)

We have $E(G)=\bigsqcup_{V\in V_{\bullet}}E(V,\operatorname{next}(V)\setminus
V)$.

###### Proof.

For (1), we first notice that $V_{i},V_{j}$ are inconparable if $i\neq j$,
because of the minimality condition. By Lemma 5.18, we have that
$E(V_{i}^{c},V_{j}^{c})=\varnothing$ for all $i\neq j$. Since $V^{\prime
c}=(\operatorname{next}(V^{\prime})\setminus
V^{\prime})\cup\bigcup_{i=1}^{m}V_{i}^{c}$ and $V^{\prime c}$ is connected, we
must have that $E(\operatorname{next}(V^{\prime})\setminus
V^{\prime},V_{i}^{c})\neq\varnothing$. Item (2) follows immediately from (1).

For (3), just notice that $\operatorname{next}(V^{\prime})$ is an extremal
element by Proposition 5.7, then it is connected by Hypothesis 1.

The existence of an edge between vertices of
$\operatorname{next}(V^{\prime})\setminus V^{\prime}$ would witness the
failure for $V_{\bullet}$ to be full. Indeed, let $V\in V_{\bullet}$ be an
element. If $V$ is incomparable with $V^{\prime}$, by Lemma 5.18, we have that
$V\cup V^{\prime}=V(G)$, and that implies that
$\operatorname{next}(V^{\prime})\setminus V^{\prime}\subseteq V$ and hence
$E(\operatorname{next}(V^{\prime})\setminus
V^{\prime},\operatorname{next}(V^{\prime})\setminus V^{\prime}))\cap
E(V,V^{c})=\varnothing$. If $V^{\prime}\subseteq V$, then $V_{i}\subseteq V$
for some $i$, and hence $\operatorname{next}(V^{\prime})\subseteq V$ which
implies that $E(\operatorname{next}(V^{\prime})\setminus
V^{\prime},\operatorname{next}(V^{\prime})\setminus V^{\prime}))\cap
E(V,V^{c})=\varnothing$. If $V\subset V^{\prime}$, then it is clear that
$E(\operatorname{next}(V^{\prime})\setminus
V^{\prime},\operatorname{next}(V^{\prime})\setminus V^{\prime}))\cap
E(V,V^{c})=\varnothing$. By the condition that $V_{\bullet}$ is a full forest,
we have that $E(\operatorname{next}(V^{\prime})\setminus
V^{\prime},\operatorname{next}(V^{\prime})\setminus V^{\prime}))\cap
E(G)=\varnothing$ and this completes the proof of Item (4).

Item (5). This follows from the fact that
$V(G)=\operatorname{next}(V^{\prime})\cup\bigcup_{j=1}^{m}V_{j}^{c}$ and hence
$V^{c}=\bigcup_{j=1}^{m}V^{c}\cap V_{j}^{c}$. Since $G(V^{c})$ is connected
and $E(V_{i}^{c},V_{j}^{c})=\varnothing$ for $j\neq i$ (Lemma 5.18), so we
have that there exists $i\in\\{1,\ldots,m\\}$ such that $V^{c}\cap
V_{j}^{c}=\varnothing$ for every $j\neq i$. This means that $V_{i}\subset V$.

Item (6). Let $V_{0}$ be a minimal element of $\operatorname{Ext}(G,D)$. If
$V_{0}$ is also maximal, there is nothing to do. So we can assume that $V_{0}$
is nonmaximal. Assume by contradiction that $V_{0}\notin V_{\bullet}$. From
Items (2) and (3) we deduce that $E(V_{0},\operatorname{next}(V_{0})\setminus
V_{0})\neq\varnothing$. Let $V\in V_{\bullet}$ we will prove that
$E(V_{0},\operatorname{next}(V_{0})\setminus V_{0})\cap
E(V,V^{c})=\varnothing$ and get a contradiction. If $V_{0}\subseteq V$, then
$\operatorname{next}(V_{0})\subset V$ (recall that $V_{0}\neq V$ because
$V_{0}\notin V_{\bullet}$), then $E(V_{0},\operatorname{next}(V_{0})\setminus
V_{0})\cap E(V,V^{c})=\varnothing$. If $V_{0}\not\subseteq V$, we have that
$V_{0}\cap V\subsetneqq V_{0}$ is extremal by Proposition 5.7; by the
minimality of $V_{0}$ in $\operatorname{Ext}(G,D)$ we have that $(V_{0}\cap
V)^{c}=V_{0}^{c}\cup V^{c}$ is not connected. Since $V_{0}^{c},V^{c}$ are
connected, we have that $V_{0}^{c}\cap V^{c}=\varnothing$ and hence $V_{0}\cup
V=V(G)$. By Proposition 5.7 we have that $E(V_{0}^{c},V^{c})=\varnothing$.
Hence $E(V_{0},V_{0}^{c})\cap E(V,V^{c})=\varnothing$, indeed
$E(V_{0},V_{0}^{c})=E(V_{0}\cap V,V_{0}^{c})$ and $E(V,V^{c})=E(V_{0}\cap
V,V^{c})$. In particular $E(V_{0},\operatorname{next}(V_{0})\setminus
V_{0})\cap E(V,V^{c})=\varnothing$.

Item (7). Let $e$ be an edge of $G$ and let $V^{\prime}$ be an element in
$V_{\bullet}$ that is maximal among those with the property that $e\in
E(V^{\prime},V^{\prime c})$. For each $V^{\prime}\subsetneqq V\in
V_{\bullet}$, we must have that $e\in E(V,V)$, so we have that $e\in
E(V^{\prime},\operatorname{next}(V^{\prime})\setminus V^{\prime})$.

∎

######  Proposition 5.20.

Let $\alpha$ be a vine function and define

$V_{\bullet}:=\\{V\in\operatorname{Ext}(G,D):\
\alpha(V)\neq\varnothing\\}\subseteq\operatorname{Ext}(G,D).$

Then the following hold.

  1. (1)

For every $V_{0}\in\operatorname{Ext}(G,D)$, we have that
$V_{\bullet}\cap\operatorname{Ext}(G,D)_{\subseteq V_{0}}$ is a chain.

  2. (2)

The poset $V_{\bullet}$ is a forest.

  3. (3)

$\alpha(V)\subseteq E(V,\operatorname{next}(V)\setminus V)$ for every $V\in
V_{\bullet}$.

###### Proof.

Let $V_{1},V_{2}\in V_{\bullet}$ be such that $V_{1},V_{2}\subseteq V_{0}$ and
$V_{1}$ and $V_{2}$ are incomparable. By Corollary 5.12, we have that
$V_{1}\cap V_{2}\in\operatorname{Ext}(G,D)$. Moreover, we have that

$E(V_{1}\cap V_{2},(V_{1}\cap V_{2})^{c})\subseteq E(V_{1},V_{1}^{c})\cup
E(V_{2},V_{2}^{c}).$

By the fact that $\alpha$ is a vine function, we have that there exists
$V^{\prime}\subseteq V_{1}\cap V_{2}$ such that $\alpha(V^{\prime})\cap
E(V_{1}\cap V_{2},(V_{1}\cap V_{2})^{c})\neq\varnothing$. This means that
either $\alpha(V^{\prime})\cap E(V_{1},V_{1}^{c})\neq\varnothing$, or
$\alpha(V^{\prime})\cap E(V_{2},V_{2}^{c})\neq\varnothing$, which contradicts
the fact that $\alpha$ is a vine function and
$\alpha(V_{1}),\alpha(V_{2})\neq\varnothing$. This concludes the proof of Item
(1). Item (2) follows directly.

We now prove (3). If $\operatorname{next}(V)=V(G)$ there is nothing to prove.
Otherwise, we have

$E(V,V^{c})\setminus E(V,\operatorname{next}(V)\setminus
V)\subseteq\bigcup_{V\subsetneqq V^{\prime}\in
V_{\bullet}}E(V^{\prime},V^{\prime c}),$

so $\alpha(V)\cap(E(V,V^{c})\setminus E(V,\operatorname{next}(V)\setminus
V))\neq\varnothing$ would imply $\alpha(V)\cap E(V^{\prime},V^{\prime
c})\neq\varnothing$ for some $V^{\prime}\in V_{\bullet}$ with $V\subsetneqq
V^{\prime}$, thus contradicting the assumption that $\alpha$ is a vine
function.

∎

We are now ready to prove the equivalence of full vine functions and full
forests.

######  Proposition 5.21.