Node Polynomials for Curves on Surfaces

Node Polynomials for Curves on Surfaces††This paper is a contribution to the
Special Issue on Enumerative and Gauge-Theoretic Invariants in honor of Lothar
Göttsche on the occasion of his 60th birthday. The full collection is
available at https://www.emis.de/journals/SIGMA/Gottsche.html

Steven KLEIMAN a and Ragni PIENE b

S. Kleiman and R. Piene

a) Room 2-172, Department of Mathematics, MIT,
a) 77 Massachusetts Avenue, Cambridge, MA 02139, USA<EMAIL_ADDRESS>
b) Department of Mathematics, University of Oslo,
b) PO Box 1053, Blindern, NO-0316 Oslo, Norway<EMAIL_ADDRESS>
Received February 24, 2022, in final form July 28, 2022; Published online
August 02, 2022

We complete the proof of a theorem we announced and partly proved in [Math.
Nachr. 271 (2004), 69–90, math.AG/0111299]. The theorem concerns a family of
curves on a family of surfaces. It has two parts. The first was proved in that
paper. It describes a natural cycle that enumerates the curves in the family
with precisely $r$ ordinary nodes. The second part is proved here. It asserts
that, for $r\leq 8$, the class of this cycle is given by a computable
universal polynomial in the pushdowns to the parameter space of products of
the Chern classes of the family.

enumerative geometry; nodal curves; nodal polynomials; Bell polynomials;
Enriques diagrams; Hilbert schemes

14N10; 14C20; 14H40; 14K05

Happy 60th, Lothar

##  1 Introduction

This paper is the fourth in a series about enumerating nodal curves on smooth
complex surfaces. Here we complete the proof of Theorem 2.5 on p. 74 in [16].
It has two parts. The first was proved in [16]. It describes a natural cycle
$U(r)$ on the parameter space of a family of pairs of a surface and of a curve
on it; $U(r)$ enumerates the curves with precisely $r$ ordinary nodes. The
second part is proved here. It asserts that, for $r\leq 8$, the class $[U(r)]$
is given by a computable universal polynomial in the pushdowns of products of
the Chern classes of the family.

The second part was not proved in [16], because we believed our approach,
inspired by Vainsencher’s paper [27], would eventually yield an algorithm for
computing the entire polynomial for $[U(r)]$ not only for $r\leq 7$, but also
for $r=8$ and perhaps for all $r$. So we chose to publish only the
construction of $U(r)$ and to postpone the rest. Unfortunately, we were too
optimistic. Thus here we work out an ad hoc determination of the polynomial
for $r=8$; specifically, we show that the “correction term” is independent of
the family, and so can be found by working out a particular example, such as
we did in [16, Example 3.8, p. 80].

In [16, Remark 2.7, p. 74] we conjectured that the class $[U(r)]$ is given for
all $r$ by a universal polynomial in certain classes $y(a,b,c)$, defined here
in Section 4.4, which are pushdowns of products of the (relative) Chern
classes of the family. Moreover, this polynomial should be of the form
$P_{r}(a_{1},\dots,a_{r})/r!$, where $P_{r}(a_{1},\dots,a_{r})$ is the $r$th
(complete) Bell polynomial and the $a_{i}$ are linear polynomials in the
$y(a,b,c)$.

Göttsche [9] had already conjectured the special case where the pairs consist
of a fixed surface and of the divisors in a linear system. This celebrated
conjecture was proved independently by Tzeng [26] and Kool–Shende–Thomas [18].
For the history of the case of plane curves, see [16, Remark 3.7, p. 78] and
the more recent [4].

A part of our conjecture has now been proved by Laarakker [19, Theorem A, p.
4921]. He defined a cycle $\gamma(r)$ which, under suitable genericity
assumptions on the family, is supported on $U(r)$, and its class $[\gamma(r)]$
is given by a universal polynomial in the $y(a,b,c)$. Although he did not
prove that the polynomial is Bell (see his footnote [19, p. 4918]), he did
prove that $[\gamma(r)]$ is “multiplicative” when the family of surfaces is a
direct sum of families over the same base (see [19, Lemma 5.5 and Remark 5.6,
p. 4936]). When the family is trivial, Göttsche had observed that this
multiplicative property implies the polynomial is Bell. However, when the
family is nontrivial, the multiplicative property is insufficient.

In [16] we applied our theorem in several enumerations involving nontrivial
families of surfaces, including the family of all planes in $\mathbb{P}^{4}$.
In [19, Theorem B, p. 4922] Laarakker proved that the number of $r$-nodal
plane curves of degree $d$ in $\mathbb{P}^{3}$ meeting the appropriate number
of general lines, is given by a universal polynomial in $d$ of degree $\leq
9+2r$. Moreover, he explicitly computed the polynomial for $r\leq 12$. In [21]
Mukherjee, Paul, and Singh did the same; they obtained a recursive formula,
and verified that their results agree with Laarakker’s. In [5] Das and
Mukherjee treated the case where the curves may have one additional nonnodal
singularity. In [22] Mukherjee and Singh did the same for rational curves.

In [16, Remark 2.7, p. 74] we conjectured that universal polynomials also
enumerate curves with any given equisingularity type. In [13, Theorem 10.1, p.
713] Kazaryan gave a “topological justification” of our conjecture, but gave
no algebraic proof. He worked with a linear system on a fixed surface, and
found several explicit formulas for curves with singularities of codimension
$\leq 7$. A few of these formulas had been given in [15, Theorem 1.2, p. 210].
In [2] Basu and Mukherjee gave recursive formulas for the number of curves in
a linear system on a fixed surface that have $r$ nodes and one additional
singularity of codimension $\leq 8-r$. In particular, their formula for
$8$-nodal curves recovers ours in this case; see [15, Theorem 1.1, p. 210].

In [20] Li and Tzeng and, independently in [23], Rennemo proved the existence
of universal polynomials enumerating divisors with isolated singularities of
given topological or analytical types in a trivial family of varieties of
arbitrary dimension.

In short, we work here over an algebraically closed field of characteristic
$0$ with pairs $(F/Y,D)$, where $Y$ is a Cohen–Macaulay algebraic scheme,
$F/Y$ is a smooth projective family of surfaces, and $D$ is a relative, or
$Y$-flat, effective divisor on $F$. We let $\pi\colon F\to Y$ denote the
structure map.

In Section 2, given a pair $(F/Y,D)$, we recall from [15, pp. 226–227] the
construction and elementary properties of its induced pairs
$(F_{i}/X_{i},D_{i})$. Then we prove some further properties. Intuitively,
$(F_{i}/X_{i},D_{i})$ represents a family of curves that sit on blowups of the
surfaces of $F/Y$ and that have one less $i$-fold point.

In Section 3, the main results are Lemmas 3.2 and 3.3, which concern
properties of certain subschemes of the relative Hilbert scheme
$\operatorname{Hilb}_{D/Y}^{3r}$. In Section 4, we develop some results of
bivariant intersection theory for use in the subsequent sections. Our
treatment here generalizes and improves our shorter one in [15].

In Section 5, we state the main theorem, Theorem 5.4. Then we prove a key
recursion relation; we prove the theorem for $r\leq 7$; and we explain what
more is needed for $r=8$. The difficulty is that the induced pair
$(F_{2}/X_{2},D_{2})$ does not satisfy the hypotheses of the theorem, as
$D_{2}/X_{2}$ has nonreduced fibers in codimension $7$ above the relative
quadruple-point locus $X_{4}$ of $D/Y$.

Therefore, the recursion that works for $r\leq 7$ must be corrected
accordingly. In Section 6 we find an expression for the correction term, and
in Section 7 we prove that the correction term is equal to $C[X_{4}]$ for some
integer $C$ that is independent of the given $(F/Y,D)$. Our proof illustrates
the advantage of developing intersection theory over any universally catenary
Noetherian base. Thus, to complete the proof of the theorem, it suffices to
compute the integer $C$ in a particular case, such as that of $8$-nodal
quintic plane curves, which we did in [16, Example 3.8, p. 80].

However, our proof requires an additional genericity hypothesis: the analytic
type of a fiber of $D$ at an ordinary quadruple point must not remain constant
along any irreducible component of $X_{4}$. This hypothesis comes into play at
just one spot in the proof of Lemma 7.4 to ensure a certain map is flat. We
believe that Lemma 7.4 and Theorem 5.4 hold without this hypothesis. At any
rate, the hypothesis is usually fulfilled in practice.

##  2 The induced pairs

The induced pairs $(F_{i}/X_{i},D_{i})$ of a given pair $(F/Y,D)$ play a
central role in the present work. So, in this section, we recall the theory
and develop it further. Here $F$ and $Y$ need only be Noetherian, and $F/Y$
need only be of finite type.

######  2.1.

The induced pairs. From [15, pp. 226–227], let’s recall the construction and
elementary properties of the induced pairs, but make a few minor changes
appropriate for the present work.

Denote by $p_{j}\colon F\times_{Y}F\to F$ the $j$th projection, by
$\Delta\subset F\times_{Y}F$ the diagonal subscheme, and by
$\mathcal{I}_{\Delta}$ its ideal. Say $D$ is defined by the global section
$\sigma$ of the invertible sheaf $\mathcal{O}_{F}(D)$. Then $\sigma$ induces a
section $\sigma_{i}$ of the sheaf of relative twisted principal parts,

$\mathcal{P}_{F/Y}^{i-1}(D):=p_{1*}\bigl{(}p_{2}^{*}\mathcal{O}_{F}(D)\big{/}\mathcal{I}_{\Delta}^{i}\bigr{)}\qquad\text{for}\quad
i\geq 1.$ (2.1)

Take the scheme of zeros of $\sigma_{i}$ to be $X_{i}$, and set $X_{0}:=F$.

Then $X_{1}=D$. Further, a geometric point of $X_{i}$, that is, a map
$\xi\colon\operatorname{Spec}(K)\to X_{i}$, where $K$ is an algebraically
closed field, is just a geometric point $\xi$ of $F$ at which the fiber
$D_{\pi(\xi)}$ has multiplicity at least $i$. Also, as $i$ varies, the $X_{i}$
form a descending chain of closed subschemes.

The sheaf $\mathcal{P}_{F/Y}^{i-1}(D)$ fits into the exact sequence,

$0\to\mathop{\rm
Sym}\nolimits^{i-1}\Omega^{1}_{F/Y}(D)\to\mathcal{P}_{F/Y}^{i-1}(D)\to\mathcal{P}_{F/Y}^{i-2}(D)\to
0,$

where the first term is the symmetric power of the sheaf of relative
differentials, twisted by $\mathcal{O}_{F}(D)$. Hence
$\mathcal{P}_{F/Y}^{i-1}(D)$ is locally free of rank $\binom{i+1}{2}$ by
induction on $i$. Therefore, at each scheme point $x\in X_{i}$, we have

$\operatorname{cod}_{x}(X_{i},F)\leq\binom{i+1}{2},$ (2.2)

where, as usual, $\operatorname{cod}_{x}(X_{i},F)$ stands for the minimum
$\min(\dim\mathcal{O}_{F,\eta})$ as $\eta$ ranges over the generizations of
$x$ in $X_{i}$. If $\operatorname{cod}_{x}X_{i}=\binom{i+1}{2}$ and if $Y$ is
Cohen–Macaulay at $\pi(x)$, then, since $F/Y$ is smooth, $X_{i}$ is a local
complete intersection in $F$ at $x$, and is Cohen–Macaulay at $x$.

Denote by $\beta\colon F^{\prime}\to F\times_{Y}F$ the blowup along $\Delta$,
and by $E$ the exceptional divisor. Set $\varphi^{\prime}:=p_{1}\beta$ and
$\pi^{\prime}:=p_{2}\beta$. Then $\pi^{\prime}\colon F^{\prime}\to F$ is again
a smooth family of surfaces, and projective if $\pi$ is; in fact, over a point
$\xi$ of $F$, the fiber $F^{\prime}_{\xi}:=\pi^{\prime-1}(\xi)$ is just the
blowup (via $p_{1})$ of the fiber $F_{\pi(\xi)}:=\pi^{-1}\pi(\xi)$ at $\xi$.
For each $i$, set $F_{i}:={\pi^{\prime}}^{-1}(X_{i})$, and denote by
$\pi_{i}\colon F_{i}\to X_{i}$ the restriction of $\pi^{\prime}$. In sum, we
have this diagram:

$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
F\times_{Y}F}$$\scriptstyle{\\!\\!p_{1}}$$\scriptstyle{p_{2}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
F^{\prime}}$$\scriptstyle{\quad\
\beta}$$\scriptstyle{\pi^{\prime}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces
F_{i}}$$\scriptstyle{\
\beta^{\prime}_{i}}$$\scriptstyle{\pi_{i}}$$\textstyle{Y}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
F}$$\scriptstyle{\pi}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
F}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
X_{i}.}$$\scriptstyle{\ \beta_{i}}$

In addition, given $r\geq 1$, set

$r_{i}:=r-{\binom{i+1}{2}}+2,\qquad
D^{\prime}_{i}:={\varphi^{\prime}}^{-1}D-iE\qquad\text{and}\qquad
D_{i}:=D^{\prime}_{i}\big{|}_{F_{i}}.$ (2.3)

As $F^{\prime}$ has no associated points on $E$, the subscheme
${\varphi^{\prime}}^{-1}D$ is an effective divisor; so $D^{\prime}_{i}$ is a
divisor on $F^{\prime}$. If $i\geq 1$, then

$D^{\prime}_{i}=D^{\prime}_{i-1}-E.$

In [15, p. 227], we proved the second assertion of the next lemma. Taking a
little more care, we now prove the first too. Later, in Lemma 2.8, we relate
$X_{i}$ and $r_{i}$.

######  Lemma 2.2.

For each $i\geq 1$, the subscheme $X_{i}$ of $F$ is the largest subscheme over
which $D^{\prime}_{i}$ is effective. Furthermore,
$D_{i}:=D^{\prime}_{i}\big{|}_{F_{i}}$ is relative effective on $F_{i}/X_{i}$.

###### Proof.

By definition of $X_{i}$, a $Y$-map $t\colon T\to F$ factors through $X_{i}$
iff $t^{*}\sigma_{i}=0$. Now, $\mathcal{P}_{F/Y}^{i-1}(D)$ is locally free on
$F$, so flat over $Y$; hence, $(1\times
t)^{*}\mathcal{I}_{\Delta}^{i}\mathcal{O}_{F\times_{Y}F}(p_{1}^{*}D)$ is a
subsheaf of $(1\times t)^{*}\mathcal{O}_{F\times_{Y}F}(p_{1}^{*}D)$ owing to
display (2.1). Therefore, $t^{*}\sigma_{i}=0$ iff $(1\times
t)^{*}p_{1}^{*}\sigma\colon\mathcal{O}_{F\times_{Y}T}\to(1\times
t)^{*}\mathcal{O}_{F\times_{Y}F}(p_{1}^{*}D)$ factors through that subsheaf.

Let $q\colon F\times_{Y}T\to F$ denote the projection. Then $q=p_{1}(1\times
t)$. So

$(1\times
t)^{*}\mathcal{O}_{F\times_{Y}F}(p_{1}^{*}D)=\mathcal{O}_{F\times_{Y}T}(q^{*}D).$

Let $\Gamma\subset F\times_{Y}T$ be the graph subscheme of $t$, and
$\mathcal{I}_{\Gamma}$ its ideal. Then $(1\times t)^{-1}\Delta=\Gamma$. Hence
$(1\times t)^{*}\mathcal{I}_{\Delta}^{i}=\mathcal{I}_{\Gamma}^{i}$. Therefore,
$t^{*}\sigma_{i}=0$ iff
$q^{*}\sigma\colon\mathcal{O}_{F\times_{Y}T}\to\mathcal{O}_{F\times_{Y}T}(q^{*}D)$
factors through $\mathcal{I}_{\Gamma}^{i}\mathcal{O}_{F\times_{Y}T}(q^{*}D)$.

Set $F^{\prime}_{T}:=F^{\prime}\times_{F}T$ and $\beta_{T}:=\beta\times_{F}T$.
Then $\beta_{T}\colon F^{\prime}_{T}\to F\times_{Y}T$ is the blowup of
$F\times_{Y}T$ along $\Gamma$ as $(1\times
t)^{*}\mathcal{I}_{\Delta}^{i}=\mathcal{I}_{\Gamma}^{i}$. Set
$E_{T}:=E\times_{F}T$. Then $E_{T}$ is the exceptional divisor. Trivially,
$I_{\Gamma}^{i}\mathcal{O}_{F^{\prime}_{T}}=\mathcal{O}_{F^{\prime}_{T}}(-iE_{T})$.
However,
$I_{\Gamma}^{i}\buildrel\sim\over{\hbox{$\longrightarrow$}}(\beta_{T})_{*}\mathcal{O}_{F^{\prime}_{T}}(-iE_{T})$
since $\Gamma$ is a local complete intersection; see [7, display (6), p. 601];
so the projection formula yields

$I_{\Gamma}^{i}\mathcal{O}_{F\times_{Y}T}(q^{*}D)=\beta_{T*}\mathcal{O}_{F^{\prime}_{T}}(\beta_{T}^{*}q^{*}D-iE_{T}).$

Set $\varphi^{\prime}_{T}:=q\beta_{T}$. Then, therefore, $t^{*}\sigma_{i}=0$
iff
$\varphi_{T}^{\prime*}\sigma\colon\mathcal{O}_{F^{\prime}_{T}}\to\mathcal{O}_{F^{\prime}_{T}}(\varphi_{T}^{\prime*}D)$
factors through $\mathcal{O}_{F^{\prime}_{T}}(\varphi_{T}^{\prime*}D-iE_{T})$.

Let $\tau\colon F^{\prime}_{T}\to F^{\prime}$ denote the projection. Then
$\varphi_{T}^{\prime*}D-iE_{T}=\tau^{*}D^{\prime}_{i}$. Therefore,
$t^{*}\sigma_{i}=0$ iff $\tau^{*}D^{\prime}_{i}$ is effective. Thus $X_{i}$ is
the largest subscheme of $F$ over which $D^{\prime}_{i}$ is effective.

In particular, on every fiber of $\pi_{i}$, the restriction of $D_{i}$ is
effective. Furthermore, $\pi_{i}$ is flat. Hence, $D_{i}$ is relative
effective. Thus the lemma holds. ∎

######  Lemma 2.3.

Let $(F/Y,D)$ be a pair. Then forming all of the induced pairs
$(F_{i}/X_{i},D_{i})$ commutes with arbitrary base change $g\colon
Y^{\prime}\to Y$.

###### Proof.

It follows from [17, Proposition 3.4, p. 422] that the formation of $F^{(1)}$
and $E^{(1)}$ commutes with base change. Set $g^{\prime}\colon
F\times_{Y}Y^{\prime}\to F$. By [11, Proposition 16.4.5, p. 19], we have
${g^{\prime}}^{*}\mathcal{P}^{i-1}_{F/Y}(D)=\mathcal{P}^{i-1}_{F\times_{Y}Y^{\prime}/Y^{\prime}}\big{(}{g^{\prime}}^{-1}(D)\big{)}$,
and the section $\sigma_{i}$ pulls back to the corresponding section
$\sigma^{\prime}_{i}$. Hence the zero scheme of $\sigma^{\prime}_{i}$ is equal
to $X_{i}\times_{Y}Y^{\prime}$. ∎

######  Definition 2.4.

Let $Y(\infty)$ denote the subset of $Y$ whose geometric points are those
$\eta$ of $Y$ whose fiber $D_{\eta}$ is not reduced.

Fix a minimal Enriques diagram $\mathbf{D}$; see [15, Section 2, p. 213].
Denote by $Y(\mathbf{D})$ the subset of $Y$ whose geometric points are those
$\eta$ whose fiber $D_{\eta}$ has diagram $\mathbf{D}$.

######  2.5.

Arbitrarily near points. Recall the following notions, notation, and results.
First, as in [17, Definition 3.1, p. 421], for $j\geq 0$, iterate the
construction of $\pi^{\prime}\colon F^{\prime}\to F$ from $\pi\colon F\to Y$
to obtain $\pi^{(j)}\colon F^{(j)}\to F^{(j-1)}$ with $\pi^{(0)}:=\pi$, with
$\pi^{(1)}:=\pi^{\prime}$, and so forth. By [17, Proposition 3.4, p. 422], the
$Y$-schemes $F^{(j)}$ represent the functors of arbitrarily near points of
$F/Y$; the latter are defined in [17, Definition 3.3, p. 422]. As in [17,
Definition 3.1, p. 421], we denote by $\varphi^{(j)}\colon F^{(j)}\to
F^{(j-1)}$ the map equal to the composition of the blowup and the first
projection, and by $E^{(j)}\subset F^{(j)}$ the exceptional divisor.

Given a minimal Enriques diagram $\mathbf{D}$ on $j+1$ vertices, fix an
ordering $\theta$ of these vertices. Also, let $\mathbf{U}$ be the unweighted
diagram underlying $\mathbf{D}$. By [17, Theorem 3.10, p. 425], the functor of
arbitrarily near points with $(\mathbf{U},\theta)$ as associated diagram is
representable by a $Y$-smooth subscheme $F(\mathbf{U},\theta)$ of $F^{(j)}$.

By [17, Corollary 4.4, p. 430], the group of automorphisms
$\operatorname{Aut}(\mathbf{U})$ acts freely on $F(\mathbf{U},\theta)$. So its
subgroup $\operatorname{Aut}(\mathbf{D})$, of automorphisms of $\mathbf{D}$,
does too. Set

$Q(\mathbf{D}):=F(\mathbf{U},\theta)\big{/}{\operatorname{Aut}}(\mathbf{D});$

it is independent of the choice of $\theta$ by [17, Theorem 5.7, p. 438]. Set
$d:=\deg\mathbf{D}$.

Form the structure map and the universal injection of [17, Theorem 5.7, p.
438]:

$q\colon\ Q(\mathbf{D})\to Y\qquad\text{and}\qquad\Psi\colon\
Q(\mathbf{D})\to\operatorname{Hilb}^{d}_{F/Y};$

in fact, $\Psi$ is a an embedding in characteristic 0. The construction and
study of $\Psi$ is based on the modern theory of complete ideals. Finally, set

$G(\mathbf{D}):=\operatorname{Hilb}^{d}_{D/Y}\times_{\operatorname{Hilb}^{d}_{F/Y}}Q(\mathbf{D}).$
(2.4)

######  Lemma 2.6.

The sets $Y(\mathbf{D})$ and $Y(\infty)$ are constructible; in fact,
$Y(\infty)$ is closed if $F/Y$ is proper. Furthermore, for all $z\in
G(\mathbf{D})$ and $y\in Y(\mathbf{D})$, we have

$\operatorname{cod}_{z}(G(\mathbf{D}),Q(\mathbf{D}))\leq
d\qquad\text{and}\qquad\operatorname{cod}_{y}(Y(\mathbf{D}),Y)\leq\operatorname{cod}\mathbf{D}.$
(2.5)

Finally, for only finitely many $\mathbf{D}$, is either
$G(\mathbf{D})\smallsetminus q^{-1}Y(\infty)$ or $Y(\mathbf{D})$ nonempty.

###### Proof.

Note that $Y(\infty)$ is just the image in $Y$ of the set of $x\in X_{2}$ at
which the fiber of $X_{2}/Y$ is of dimension at least 1. This set is closed in
$X_{2}$, so in $F$. Hence $Y(\infty)$ is constructible; in fact, $Y(\infty)$
is closed if $\pi$ is proper.

Only finitely many $\mathbf{D}$ arise from the fibers of $D/Y$; indeed, this
statement is proved in [16, Lemma 2.4, p. 73] without making use of its
blanket hypothesis that $Y$ is Cohen–Macaulay and of finite type over the
complex numbers; that proof just requires $Y$ to be Noetherian. Thus there are
only finitely many $\mathbf{D}$ such that $Y(\mathbf{D})$ is nonempty; denote
the set of these $\mathbf{D}$ by $\Sigma$.

The subscheme $\operatorname{Hilb}^{d}_{D/Y}$ of
$\operatorname{Hilb}^{d}_{F/Y}$ is locally cut out by $d$ equations by [1,
Proposition 4, p. 5]. Therefore, the first bound holds in (2.5).

The definitions yield $q(G(\mathbf{D}))\supset Y(\mathbf{D})$. Further, take
any $y\in q(G(\mathbf{D}))\smallsetminus Y(\infty)$, and let
$\mathbf{D}^{\prime}$ be the diagram of $D_{K}$, where $K$ is the algebraic
closure of $k(y)$. Then the definitions yield a natural injection
$\alpha\colon\mathbf{D}\hookrightarrow\mathbf{D}^{\prime}$ such that each
$V\in\mathbf{D}$ has weight at most that of $\alpha(V)$. So
$\deg\mathbf{D}^{\prime}>d$ if $y\notin Y(\mathbf{D})$. Hence

$Y(\mathbf{D})=q(G(\mathbf{D}))\smallsetminus\bigl{(}Y(\infty)\cup\big{(}{\textstyle\bigcup}\\{q(G(\mathbf{D}^{\prime}))\mid\mathbf{D}^{\prime}\in\Sigma\text{
and }\deg\mathbf{D}^{\prime}>d\\}\big{)}\bigr{)}.$

But $G(\mathbf{D})$ and the $G(\mathbf{D}^{\prime})$ are locally closed. Thus
$Y(\mathbf{D})$ is constructible.

To prove the second bound in (2.5), note that $G(\mathbf{D})$ has a unique
point, $z$ say, lying over the given $y$. Now, $Q(\mathbf{D})/Y$ is smooth of
relative dimension $\dim\mathbf{D}$ by [17, Theorem 3.10, p. 425]. Thus, as
desired,

$\operatorname{cod}_{y}(Y(\mathbf{D}),Y)=\operatorname{cod}_{z}(G(\mathbf{D}),Q(\mathbf{D}))-\dim\mathbf{D}\leq
d-\dim\mathbf{D}=\operatorname{cod}\mathbf{D}.$

Finally, suppose $G(\mathbf{D})\smallsetminus h^{-1}Y(\infty)$ is nonempty.
Then, as we have just seen, there is an injection
$\alpha\colon\mathbf{D}\hookrightarrow\mathbf{D}^{\prime}$, where
$\mathbf{D}^{\prime}\in\Sigma$, and each $V\in\mathbf{D}$ has weight at most
that of $\alpha(V)$. But there are only finitely many such $\mathbf{D}$, as
desired. ∎

######  Definition 2.7.

We say that $(F/Y,D)$ is _$r$ -generic_ if for every minimal Enriques diagram
$\mathbf{D}$ and for every $y\in Y(\mathbf{D})$, we have

$\operatorname{cod}_{y}(Y(\mathbf{D}),Y)\geq\min(r+1,\operatorname{cod}\mathbf{D}).$
(2.6)

We say that $(F/Y,D)$ is _strongly $8$-generic_ if it is 8-generic and if the
analytic type of $D_{\pi(x)}$ at an ordinary quadruple point $x\in X_{4}$ is
not constant along any irreducible component $Z$ of $X_{4}$; that is, the
cross ratio of the four tangents at $x$ is not the same for all $x\in Z$.

######  Proposition 2.8.

Fix $r$. Assume that $Y$ is universally catenary and that $(F/Y,D)$ is
$r$-generic. Then, for each $i\geq 2$, the induced pair $(F_{i}/X_{i},D_{i})$
is $r_{i}$-generic.

###### Proof.

Fix $i$. Let $\mathbf{D}^{\prime}$ be a minimal Enriques diagram. Let $x$ be a
generic point of the closure of $X_{i}(\mathbf{D}^{\prime})$. Then $x\in
X_{i}(\mathbf{D}^{\prime})$ as $X_{i}(\mathbf{D}^{\prime})$ is constructible
by Lemma 2.6 applied with $(F_{i}/X_{i},D_{i})$ and $\mathbf{D}^{\prime}$ for
$(F/Y,D)$ and $\mathbf{D}$. Set $y:=\pi(x)$. Let $K$ be an algebraically
closed field containing $k(x)$; then $K$ contains $k(y)$ too.

Consider the curves $D_{K}$ and $(D_{i})_{K}$. Note $x\in
X_{i}(\mathbf{D}^{\prime})$. So the curve $(D_{i})_{K}$ is reduced, and is
obtained from $D_{K}$ as follows: blow up $F_{K}$, at the $K$-point, $x_{K}$
say, defined by $x$; take the preimage of $D_{K}$; and subtract $i$ times the
exceptional divisor. Hence $D_{K}$ is reduced and of multiplicity either $i$
or $i+1$ at $x_{K}$. In the latter case, $(D_{i})_{K}$ contains the
exceptional divisor; in the former, it doesn’t. In either case, let
$\mathbf{D}$ be the diagram of $D_{K}$. Then [17, Proposition 2.8, p. 420]
yields

$\operatorname{cod}(\mathbf{D})\geq\operatorname{cod}(\mathbf{D}^{\prime})+\binom{i+1}{2}-2.$
(2.7)

Since $F/Y$ is flat, the dimension formula yields

$\dim\mathcal{O}_{F,x}=\dim\mathcal{O}_{Y,y}+\dim\mathcal{O}_{F_{y},x}.$

However, $x$ is the generic point of a component, $X$ say, of the closure of
$X_{i}(\mathbf{D}^{\prime})$; hence,
$\dim\mathcal{O}_{F,x}=\operatorname{cod}_{x}(X,F)$. So $y=\pi(x)$. So
$\dim\mathcal{O}_{Y,y}=\operatorname{cod}_{y}(\pi(X),Y)$. Further, $F/Y$ is of
relative dimension 2; so $\dim\mathcal{O}_{F_{y},x}=2$. Thus

$\operatorname{cod}_{x}(X,F)-2=\operatorname{cod}_{y}(\pi(X),Y).$ (2.8)

However, $y:=\pi(x)\in Y(\mathbf{D})$. Hence,

$\operatorname{cod}_{y}(\pi(X),Y)\geq\operatorname{cod}_{y}(Y(\mathbf{D}),Y).$

Combine the last two displays; then (2.6) yields

$\operatorname{cod}_{x}(X,F)-2\geq\min(r+1,\operatorname{cod}\mathbf{D}).$
(2.9)

Since $Y$ is universally catenary, $F$ is catenary; hence,

$\operatorname{cod}_{x}(X,X_{i})=\operatorname{cod}_{x}(X,F)-\operatorname{cod}_{x}(X_{i},F).$
(2.10)

Hence (2.9) and (2.2) yield

$\operatorname{cod}_{x}(X,X_{i})\geq\min(r+1,\operatorname{cod}\mathbf{D})+2-\binom{i+1}{2}.$

Therefore, (2.3) and (2.7) yield the desired lower bound:

$\operatorname{cod}_{x}(X,X_{i})\geq\min(r_{i}+1,\operatorname{cod}\mathbf{D}^{\prime}).$
∎

######  Corollary 2.9.

Fix $r$. Assume $(F/Y,D)$ is $r$-generic. Fix $i\geq 2$, let $X$ be a
component of $X_{i}$, take $x\in X\smallsetminus Y(\infty)$, and set
$y:=\pi(x)$. Then

$\displaystyle\operatorname{cod}_{x}(X,F)=\binom{i+1}{2},\qquad\operatorname{cod}_{y}(\pi(X),Y)=\binom{i+1}{2}-2\qquad\text{if}\quad
r_{i}\geq-1,$ (2.11) $\displaystyle\operatorname{cod}_{x}(X,F)\geq
r+3,\qquad\operatorname{cod}_{y}(\pi(X),Y)\geq r+1\qquad\text{if}\quad
r_{i}\leq-1.$ (2.12)

###### Proof.

Plainly we may assume $x$ is the generic point of $X$. Let $K$ be an
algebraically closed field containing $k(x)$, so $k(y)$. Then $D_{K}$ is
reduced as $x\notin Y(\infty)$. Let $\mathbf{D}$ be the diagram of $D_{K}$,
and $\mathbf{D}^{\prime}$ that of $D_{i}$. Then $X$ is a component of the
closure of $X_{i}(\mathbf{D}^{\prime})$. So we may appeal to the proof of
Proposition 2.8. Note that equation (2.10) is trivial here, and we do not need
$Y$ to be universally catenary.

Since $x\in X_{i}$, at the corresponding $K$-point, $D_{K}$ is of multiplicity
at least $i$. Hence $\mathbf{D}$ has a root of weight at least $i$. So
$\operatorname{cod}\mathbf{D}\geq\binom{i+1}{2}-2$.

Suppose $r_{i}\geq-1$. Then (2.3) yields $r+1\geq\binom{i+1}{2}-2$. So (2.9)
yields

$\operatorname{cod}_{x}(X,F)\geq\binom{i+1}{2}.$

But the opposite inequality is (2.2), which always holds. So equality holds.
Thus, in (2.11), the first equation holds. The second follows from it and
(2.8).

Suppose $r_{i}\leq-1$ instead. Then $\binom{i+1}{2}-2\geq r+1$. So
$\operatorname{cod}\mathbf{D}\geq r+1$. Hence (2.9) and (2.8) yield (2.12).
Thus the corollary is proved. ∎

##  3 Virtual double points

The minimal Enriques diagram $r\mathbf{A}_{1}$ consists of $r$ roots of weight
2 and no other vertices. The corresponding scheme $G(r\mathbf{A}_{1})$ is
particularly important, as it is equal to the subscheme of the Hilbert scheme
$\operatorname{Hilb}^{3r}_{D/Y}$ associated to the geometric fibers of $D/Y$
with at least $r$ distinct singular points. Moreover, we need to consider it
for various $(F/Y,D)$ and $r$. So, for clarity, we set
$G(F/Y,D;r):=G(r\mathbf{A}_{1})$.

In this section, we first recall the basic properties of $G(F/Y,D;r)$, which
were treated in [17, Proposition 5.9, p. 439]. Then we fix $r\geq 1$, and
assume $(F/Y,D)$ is $r$-generic. For each $i\geq 1$, we find a natural large
open subscheme of

$H_{i}:=G(F_{i}/X_{i},D_{i};r_{i})$

such that the associated geometric fibers of $D_{i}/X_{i}$ have exactly
$r_{i}$ nodes. None lies on the exceptional divisor of a fiber of $F_{i}$.
Further, adding the exceptional divisor to the fiber of $D_{i}/X_{i}$ yields a
fiber of $D_{i-1}/X_{i-1}$, and thus establishes an isomorphism from the
preceding open subscheme to a natural open subscheme of $H_{i-1}$, which is
dense in the preimage of $X_{i}$. These results are treated in Lemmas 3.2 and
3.3 below for later use.

######  3.1.

Subschemes of the Hilbert scheme. Fix $r$. If $r\geq 1$, let $H(r)$ denote the
open subscheme of $\operatorname{Hilb}^{r}_{F/Y}$ over which the universal
family is smooth; in other words, $H(r)$ parameterizes the unions of $r$
distinct reduced points in the geometric fibers of $F/Y$. By convention, if
$r=0$, then $H(r)$ and $\operatorname{Hilb}^{r}_{F/Y}$ are both equal to $Y$;
if $r\leq-1$, then both are empty.

If $r\geq 1$, then Proposition 5.9 on p. 439 in [17] asserts that
$H(r)=Q(r\mathbf{A}_{1})$ and that the map $\Psi\colon
H(r)\to\operatorname{Hilb}^{3r}_{F/Y}$ is given on $T$-points, where $T$ is a
$Y$-scheme, by sending a subscheme $W$ of $F_{T}$, say with ideal
$\mathcal{I}$, to the subscheme $W^{\prime}$ with ideal $\mathcal{I}^{2}$
(note that $W^{\prime}$ is flat, because the standard sequence

$0\to\mathcal{I}/\mathcal{I}^{2}\to\mathcal{O}_{W^{\prime}}\to\mathcal{O}_{W}\to
0$

is exact and because $\mathcal{I}/\mathcal{I}^{2}$ and $\mathcal{O}_{W}$ are
flat); furthermore, $\Psi$ is always an embedding.

Consequently, we may view $G(r\mathbf{A}_{1})$ as a subscheme of
$\operatorname{Hilb}^{3r}_{D/Y}$. Set

$G(F/Y,D;r):=G(r\mathbf{A}_{1})\subset\operatorname{Hilb}^{3r}_{D/Y}$

to avoid confusion. Furthermore, set $G(F/Y,D;0):=Y$, and for $r\leq-1$, set
$G(F/Y,D;r)\allowbreak:=\varnothing$. Finally, for an arbitrary fixed $r$ and
for $i\geq 0$, set

$H_{i}:=G(F_{i}/X_{i},D_{i};r_{i}).$

######  Lemma 3.2.

Fix $r\geq 1$. Assume that $(F/Y,D)$ is $r$-generic and that $Y(\infty)$ is
empty. Then there is an open subscheme $U\subset Y$ such that $(1)$ for every
$y\in Y\smallsetminus U$,

$\vspace{-.5ex}\operatorname{cod}_{y}(Y\smallsetminus U,Y)\geq r+1$ (3.1)

and $(2)$ for every $i\geq 1$ with $r_{i}\geq 0$, if we set

$U_{i}:=\big{(}\pi^{-1}U\big{)}\cap(X_{i}\smallsetminus
X_{i+1})\qquad\text{and}\qquad
V_{i}:=\big{(}\pi^{-1}U\big{)}\cap(X_{i-1}\smallsetminus X_{i+1}),$

then $U_{i}$ is dense in $X_{i}$, and there is a natural isomorphism of
$F$-schemes

$\gamma_{i}\colon\
H_{i}\times_{F}U_{i}\buildrel\sim\over{\hbox{$\longrightarrow$}}H_{i-1}\times_{F}U_{i}.$

###### Proof.

Let $U$ be the complement in $Y$ of the union of the closures of those
$Y(\mathbf{D})$ with $\operatorname{cod}\mathbf{D}\geq r+1$. Then $U$ is open
since there are only finitely many nonempty $Y(\mathbf{D})$ by Lemma 2.6. By
the same token, (2.6) implies (3.1). Thus (1) holds.

Fix $i\geq 1$ such that $r_{i}\geq 0$. Then $r+1>\binom{i+1}{2}-2$. Let $X$ be
a component of $X_{i}$; let $x\in X$, set $y:=\pi(x)$. Then (2.11) yields that
$r+1>\operatorname{cod}_{y}(\pi(X),Y)$; moreover, if $x\in X_{i+1}$, then
$\operatorname{cod}_{y}(\pi(X_{i+1}),Y)>\operatorname{cod}_{y}(\pi(X),Y)$. So
(3.1) implies $\pi(X)\smallsetminus\pi(X_{i+1})\not\subset Y\smallsetminus U$.
Hence $\pi(X_{i}\smallsetminus X_{i+1})$ meets $U$. Thus $U_{i}$ is dense in
$X_{i}\smallsetminus X_{i+1}$.

Let $z\in H_{i}\times_{F}U_{i}$; let $x$ be its image in $U_{i}$, and set
$y:=\pi(x)$. Let $K$ be an algebraically closed field containing $k(z)$, so
$k(x)$ and $k(y)$ too. Then $D_{K}$ is reduced since $Y(\infty)$ is empty, and
$D_{K}$ has multiplicity exactly $i$ at the $K$-point $x_{K}$ defined by $x$
since $x\in U_{i}$, so $x\notin X_{i+1}$. Hence $(D_{i})_{K}$ is reduced, and
does not contain the exceptional divisor $E_{K}$.

Let $\mathbf{D}$ be the diagram of $D_{K}$, and $\mathbf{D}^{\prime}$ that of
$(D_{i})_{K}$. By [17, Proposition 2.8, p. 420], we have

$\operatorname{cod}(\mathbf{D})\geq\operatorname{cod}(\mathbf{D}^{\prime})+\binom{i+1}{2}-2,$
(3.2)

with equality if and only if $D_{K}$ has an ordinary $i$-fold point at
$x_{K}$. Now, $(D_{i})_{K}$ has at least $r_{i}$ singular points since $z\in
H_{i}$; hence, formula (2.6.2) in [17, p. 419] yields
$\operatorname{cod}(\mathbf{D}^{\prime})\geq r_{i}$ since
$\mathbf{D}^{\prime}$ has at least $r_{i}$ roots, each root has multiplicity
at least $2$, and the summands in that formula corresponding to the other
vertices of $\mathbf{D}^{\prime}$ are nonnegative. So the right-hand side of
(3.2) is at least $r$. However, $r\geq\operatorname{cod}(\mathbf{D})$ since
$y\in U$. So equality obtains everywhere. Hence $D_{K}$ has an ordinary
$i$-fold point at $x_{K}$. Furthermore, $(D_{i})_{K}$ has exactly $r_{i}$
singular points, each is an ordinary double point, and none lies on $E_{K}$;
also, $(D_{i})_{K}$ and $E_{K}$ meet transversally in $i$ points.

We define $\gamma_{i}$ as follows. A $T$-point of its source
$H_{i}\times_{F}U_{i}$ is given by a map $T\to U_{i}$ and a $T$-smooth
subscheme $W\subset F^{\prime}_{T}$ of relative length $r_{i}$ whose squared
ideal defines a subscheme $W^{\prime}\subset F^{\prime}_{T}$ contained in
$(D_{i})_{T}$. Owing to the discussion above, in every geometric fiber of
$F^{\prime}_{T}/T$, the fibers of $(D_{i})_{T}$ and $E_{T}$ meet transversally
in $i$ points. Hence, since $(D_{i})_{T}$ and $E_{T}$ are relative effective
divisors, their intersection is a $T$-smooth subscheme $Z\subset
F^{\prime}_{T}$ of relative length $i$.

Let $Z^{\prime}$ be the subscheme of $F^{\prime}_{T}$ defined by the squared
ideal of $Z$. Then $Z^{\prime}$ is contained in the sum $(D_{i})_{T}+E_{T}$,
which is equal to $(D_{i-1})_{T}$. So $W\cup Z$ is a $T$-smooth subscheme of
$F^{\prime}_{T}$ of relative length $r_{i}+i$, or $r_{i-1}$. And its squared
ideal defines a subscheme of $F^{\prime}_{T}$, namely $W^{\prime}\cup
Z^{\prime}$, which is contained in $(D_{i-1})_{T}$. So $W\cup Z$ determines a
$T$-point of $H_{i-1}\times_{F}U_{i}$, and the latter scheme is to be the
target of $\gamma_{i}$. We define $\gamma_{i}$ by sending $W$ to $W\cup Z$.
Plainly, $\gamma_{i}$ is injective on $T$-points since $W$ is determined by
$W\cup Z$ as the part off $E_{T}$.

To prove $\gamma_{i}$ is surjective on $T$-points, fix a $T$-point of
$H_{i-1}\times_{F}U_{i}$. It is given by a map $T\to U_{i}$ and a $T$-smooth
subscheme $S\subset F^{\prime}_{T}$ of relative length $r_{i-1}$ such that its
squared ideal defines a subscheme $S^{\prime}\subset F^{\prime}_{T}$ contained
in $(D_{i-1})_{T}$. Then $(D_{i-1})_{T}=(D_{i})_{T}+E_{T}$ by (2.3), and
$(D_{i})_{T}$ is relative effective by Lemma 2.2 since $U_{i}\subset X_{i}$.
Let $W$ be the part of $S$ off $E_{T}$. Plainly, $W$ is a $T$-smooth subscheme
of $F^{\prime}_{T}$, and its squared ideal defines a subscheme contained in
$(D_{i})_{T}$.

Consider a geometric point of $T$, say with (algebraically closed) field $K$.
Then $D_{K}$ is reduced since $Y(\infty)$ is empty, and $D_{K}$ has
multiplicity exactly $i$ at the center of $K$ since $T$ maps into $U_{i}$.
Hence $(D_{i})_{K}$ is reduced, and $(D_{i})_{K}\cap E_{K}$ is a scheme of
length $i$. Now, $S_{K}$ is $K$-smooth of length $r_{i-1}$. Hence $S_{K}$
consists of $r_{i-1}$ distinct reduced points, of which at most $i$ lie on
$E_{K}$. So $W_{K}$ consists of at least $r_{i-1}-i$, or $r_{i}$, distinct
reduced points. By choosing any $r_{i}$ of them, we obtain a $K$-point of
$H_{i}\times_{F}U_{i}$. But then, by the discussion of such points right after
(3.2), there was no choice: $(D_{i})_{K}$ has exactly $r_{i}$ singular points,
and all are ordinary nodes. Hence $W_{K}$ consists exactly of $r_{i}$ distinct
reduced points. Thus $W$ is of relative length $r_{i}$.

Therefore, $W$ defines a $T$-point of $H_{i}\times_{F}U_{i}$. According to the
discussion above, this $T$-point is carried by $\gamma_{i}$ to the $T$-point
of $H_{i-1}\times_{F}U_{i}$ that is given by $R$, where $R:=W\cup Z$ and
$Z:=(D_{i})_{T}\cap E_{T}$. To prove that $\gamma_{i}$ is surjective on
$T$-points, so bijective on $T$-points, so an isomorphism, it remains to prove
that $R=S$.

The equation $R=S$ may be checked locally over $T$ and locally on $F$. So we
may replace $T$ and $F$ by affine open subsets $\operatorname{Spec}(A)$ and
$\operatorname{Spec}(B)$. Then $B$ is étale over a polynomial subring
$A[x,y]$. Let $I\subset B$ denote the ideal of $S$. Shrinking $F$ further if
necessary, we can find an $f\in B$ that generates the ideal of
$(D_{i-1})_{T}$. Then $f\in I^{2}$ as $R$ determines a $T$-point of $H_{i-1}$.
Hence $f$, $\partial f/\partial x$, $\partial f/\partial y\in I$. But those
three elements generate the ideal of $Z:=(D_{i})_{T}\cap E_{T}$ on a
neighborhood $N$ of $E_{T}$. Hence $Z\supset S\cap N$. But both $Z$ and $S\cap
N$ are $T$-flat of relative length $i$. Hence $Z=S\cap N$. But $R$ and $S$ are
equal off $E_{T}$. Thus $R=S$, as desired. ∎

######  Lemma 3.3.

Under the conditions of Lemma 3.2, the closed subscheme
$H_{i-1}\times_{F}U_{i}$ of $H_{i-1}\times_{F}V_{i}$ is also open.

###### Proof.

Consider any $T$-point of $H_{i-1}\times_{F}V_{i}$. Let $T^{\prime}$ be the
preimage of $H_{i-1}\times_{F}U_{i}$. It suffices to prove $T^{\prime}$ is an
open subscheme, as we may take $T=H_{i-1}\times_{F}V_{i}$.

Let $\mathcal{I}\subset\mathcal{O}_{T}$ denote the ideal of $T^{\prime}$. Then
it suffices to prove that the stalk $\mathcal{I}_{t}$ vanishes for all $t\in
T^{\prime}$ for the following reason. Since $\mathcal{I}$ is coherent, the
$t\in T$, where $I_{t}$ vanishes form an open subset $T^{\prime\prime}$. By
hypothesis, $T^{\prime}\subset T^{\prime\prime}$. But, if $t\notin
T^{\prime}$, then $I_{t}=\mathcal{O}_{T,t}$; whence, $T^{\prime}\supset
T^{\prime\prime}$. So $T^{\prime}=T^{\prime\prime}$. Give $T^{\prime\prime}$
the induced structure as an open subscheme of $T$. Then $T^{\prime}$ is the
closed subscheme of $T^{\prime\prime}$ with ideal
$\mathcal{I}\mid_{T^{\prime\prime}}$. But
$\mathcal{I}\mid_{T^{\prime\prime}}=0$. Thus $T^{\prime}$ is equal to the open
subscheme $T^{\prime\prime}$.

Given $t\in T^{\prime}$, to check if $\mathcal{I}_{t}$ vanishes, we may
replace $T$ by $\operatorname{Spec}(\mathcal{O}_{T,t})$. Thus we may assume
that $T$ is of the form $\operatorname{Spec}(A)$, where $A$ is local and that
$T^{\prime}$ is nonempty. Then it suffices to prove the ideal $I\subset A$ of
$T^{\prime}$ vanishes, or equivalently, $T=T^{\prime}$.

There exists a flat local homomorphism $A\to B$ such that $B$ is complete and
its residue class field is algebraically closed. Then $A\to B$ is faithfully
flat. So $I$ vanishes if $I\otimes_{A}B$ does. Thus we may replace $A$ by $B$,
and so assume that $A$ is complete and its residue class field is
algebraically closed.

Consider the composition $T\to H_{i-1}\times_{F}V_{i}\to V_{i}$. Via it,
$T^{\prime}$ is the preimage of $U_{i}$. Hence $T=T^{\prime}$ if and only if
$(D_{i}^{\prime})_{T}$ is effective, owing to Lemma 2.2. By the same token,
$(D_{i-1}^{\prime})_{T}$ is effective.

Consider the local ring $C$ of $F_{T}$ at the closed point of the center of
the blowing up $F^{\prime}_{T}\to F_{T}$. Let $\widehat{C}$ be its completion.
Since $C\to\widehat{C}$ is faithfully flat, $(D_{i}^{\prime})_{T}$ is
effective if and only if $(D_{i}^{\prime})_{T}\otimes_{C}\widehat{C}$ is
effective.

As $F/Y$ is a smooth family of surfaces and as $\widehat{C}$ is complete with
algebraically closed residue class field, $\widehat{C}$ is a power series
ring; say $\widehat{C}=A[[u,v]]$. Say that the section $T\to F_{T}$ is defined
by mapping $u$, $v$ to $a,b\in A$. Replacing $u$, $v$ by $u-a$, $v-b$, we may
assume that $T\to F$ is defined by mapping $u$, $v$ to $0$, $0$.

Let $f$ in $A[[u,v]]$ define the pullback of $D$. Write
$f=f_{1}+f_{2}+\dotsb$, where $f_{j}$ is homogeneous of degree $j$ in $u$ and
$v$. Then $f_{j}=0$ for $1\leq j\leq i-2$ since
$(D_{i-1}^{\prime})_{T}\otimes_{C}\widehat{C}$ is effective. It remains to
prove that $f_{i-1}=0$.

To prove that $f_{i-1}=0$, denote the maximal ideal of $A$ by $\mathbf{m}$,
and write

$f_{i-1}(u,v)=a_{1}u^{i-1}+a_{2}u^{i-2}v+\dotsb+a_{i}v^{i-1}\qquad\text{with}\quad
a_{j}\in A.$ (3.3)

Then it suffices to prove that $a_{j}\in\mathbf{m}^{n}$ for all $j$ and $n\geq
0$.

Since $T$ maps into $H_{i-1}$, there is a $T$-smooth subscheme $S\subset
F^{\prime}_{T}$ of relative length $r_{i-1}$ whose squared ideal defines a
subscheme $S^{\prime}\subset F^{\prime}_{T}$ contained in $(D_{i-1})_{T}$.
Since $A$ has an algebraically closed residue class field $K$, the fiber
$S_{K}$ consists of $r_{i-1}$ distinct points. Of them, exactly $i$ lie on
$E_{K}$ according to our discussion above. Further, the fiber $S_{K}$ is the
singular locus of $(D_{i-1})_{K}$, which consists of $r_{i-1}$ ordinary double
points, and $i$ of them constitute $(D_{i})_{K}\cap E_{K}$, which is a
transverse intersection.

By replacing $u$ and $v$ with suitable linear combinations of themselves, we
may assume that the $u$-axis is not tangent to $D_{K}$ at the center of the
blowing up. Now, $A$ is complete; so by Hensel’s lemma, $S$ decomposes into
the disjoint sum of $r_{i-1}$ sections. Of them, $i$ sections meet $E_{T}$.
Hence, they correspond to $A$-algebra maps

$s_{j}\colon\ A[[u,v]][w]\big{/}(u-vw)\to A\qquad\text{for}\quad 1\leq j\leq
i.$

Set $b_{j}:=s_{j}(v)$ and $c_{j}:=s_{j}(w)$. Then, for all $j$, let’s check
that

$b_{j}\in\mathbf{m}\qquad\text{and}\qquad f_{i}(c_{j},1)\in\mathbf{m}.$

The first relation holds as the closed points of the sections lie on $E_{K}$.
The second holds because these same points lie on $(D_{i})_{K}\cap E_{K}$.
Further, the points are distinct; so mod $\mathbf{m}$, the $c_{j}$ are
distinct elements of $K$.

Proceeding by induction on $n\geq 1$, suppose that $b_{j}\in\mathbf{m}^{n}$
for all $j$. Set

$\bar{f}(v,w):=f(vw,v)/v^{i-1}.$

Then $\bar{f}$ defines the pullback of $(D_{i-1})_{T}$. Now,
$S\subset(D_{i-1})_{T}$. Hence, for each $j$,

$0=\bar{f}(b_{j},c_{j})=f_{i-1}(c_{j},1)+b_{j}f_{i}(c_{j},1)+b_{j}^{2}d_{j}$

for some $d_{j}\in A$. But $f_{i}(c_{j},1)\in\mathbf{m}$. Thus
$f_{i-1}(c_{j},1)\in\mathbf{m}^{n+1}$.

From (3.3), we obtain the following linear system of equations for the
$a_{j}$:

$f_{i-1}(c_{j},1)=a_{1}c_{j}^{i-1}+a_{2}c_{j}^{i-2}+\dotsb+a_{i}\qquad\text{for}\quad
1\leq j\leq i.$

The coefficient matrix is Vandermonde. Its determinant is invertible in $A$,
as the $c_{j}$ are distinct mod $\mathbf{m}$. As
$f_{i-1}(c_{j},1)\in\mathbf{m}^{n+1}$ for all $j$, solving yields
$a_{j}\in\mathbf{m}^{n+1}$.

To complete the proof, we must show $b_{j}\in\mathbf{m}^{n+1}$ for each $j$.
Set $I_{j}:=\operatorname{Ker}(s_{j})$. Then $\bar{f}\in I_{j}^{2}$ as
$S^{\prime}\subset(D_{i-1})_{T}$. Hence $\partial\bar{f}/\partial w\in I_{j}$.
Therefore,

$0=(\partial\bar{f}/\partial w)(b_{j},c_{j})=(\partial f_{i-1}/\partial
u)(c_{j},1)+b_{j}(\partial f_{i}/\partial u)(c_{j},1)+b_{j}^{2}d_{j}^{\prime}$
(3.4)

for some $d_{j}^{\prime}\in A$. Now, $a_{j}\in\mathbf{m}^{n+1}$; so (3.3)
yields

$(\partial f_{i-1}/\partial
u)(c_{j},1)=(i-1)a_{1}c_{j}^{i-2}+(i-2)a_{2}c_{j}^{i-3}+\dotsb+a_{i-1}\in\mathbf{m}^{n+1}.$

But $b_{j}\in\mathbf{m}^{n}$. So (3.4) yields $b_{j}(\partial f_{i}/\partial
u)(c_{j},1)\in\mathbf{m}^{n+1}$. But $(c_{j},1)$ is, mod $\mathbf{m}$, a
simple root of $f_{i}$; so $(\partial f_{i}/\partial
u)(c_{j},1)\notin\mathbf{m}$. Thus $b_{j}\in\mathbf{m}^{n+1}$, as desired. ∎

##  4 Intersection theory

For use in the remaining sections, we extend the intersection theory of
bivariant classes developed in [8, Chapter 17] and generalized over any
universally catenary base in [14, Sections 2 and 3], in [25], and in [24,
Chapter 42]. However, only [8] is cited below.

######  4.1.

Push down. Assume that $f\colon X\to Y$ is a map of schemes such that its
orientation class $[f]$ is defined [8, Section 17.4, p. 326]. If $f$ is also
proper, define an additive map

$f_{\\#}\colon\ A^{*}(X)\to A^{*}(Y)\qquad\text{by}\quad
f_{\\#}a:=f_{*}(a\cdot[f]),$

where $f_{*}\colon A^{*}(f)\to A^{*}(Y)$ is the proper push-forward operation
discussed in $({\bf P}_{2})$ on p. 322 of [8].

######  Proposition 4.2.

Let $a,b\in A^{*}(X)$. Assume that $a$ is a polynomial in Chern classes of
vector bundles on $X$. Then $f_{\\#}a\cdot f_{\\#}b=f_{\\#}b\cdot f_{\\#}a$.

###### Proof.

By $({\bf A}_{12})$ on p. 323 of [8], product and push-forward commute; so

$f_{\\#}a\cdot f_{\\#}b=f_{*}(a\cdot[f]\cdot f_{*}(b\cdot[f])).$

Let $p_{i}\colon X\times_{Y}X\to X$ denote the $i$th projection. Form the
diagram

$\begin{CD}X\times_{Y}X@>{1}>{}>X\times_{Y}X@>{p_{2}}>{}>X\\\
@V{}V{p_{1}}V@V{}V{p_{1}}V@V{}V{f}V\\\
X@>{1}>{}>X<EMAIL_ADDRESS>
Apply the projection formula of $({\bf A}_{123})$ of [8, p. 323] with $f:=f$,
with $g:=f$, with $h:=1_{Y}$, with $c:=[f]$ and with $d:=b\cdot[f]$. The
result is

$[f]\cdot f_{*}(b\cdot[f])=p_{1*}\bigl{(}f^{*}([f])\cdot b\cdot[f]\bigr{)}.$

The definitions of $f^{*}$, of $[f]$, and of $p_{2}$ yield
$f^{*}([f])=[p_{2}]$. But Axiom $({\bf C}_{2})$ on p. 320 of [8] yields
$[p_{2}]\cdot b=p_{2}^{*}(b)\cdot[p_{2}]$. Thus $[f]\cdot
f_{*}(b\cdot[f])=p_{1*}(p_{2}^{*}(b)\cdot[p_{2}]\cdot[f])$.

Apply this projection formula again, but now with $f:=1_{X}$, with $g:=p_{1}$,
with $h:=f$, with $c:=a$ and with $d:=p_{2}^{*}(b)\cdot[p_{2}]\cdot[f]$. The
result is

$a\cdot
p_{1*}\bigl{(}p^{*}_{2}(b)\cdot[p_{2}]\cdot[f]\bigr{)}=p_{1*}\bigl{(}p_{1}^{*}(a)\cdot(p_{2}^{*}(b)\cdot[p_{2}]\cdot[f])\bigr{)}.$

The formula just before Proposition 17.4.1 on p. 327 of [8] yields
$[p_{2}]\cdot[f]=[fp_{2}]$. So the functoriality of pushforwards, stated in
$({\bf A}_{2})$ on p. 323 of [8], yields

$f_{*}\bigl{(}p_{1*}\bigl{(}p_{1}^{*}(a)\cdot
p_{2}^{*}(b)\cdot[p_{2}]\cdot[f]\bigr{)}\bigr{)}=(fp_{1})_{*}\bigl{(}p_{1}^{*}(a)\cdot
p_{2}^{*}(b)\cdot[fp_{2}])\bigr{)}.$

Putting it all together yields

$f_{\\#}a\cdot f_{\\#}b=(fp_{1})_{*}\bigl{(}p_{1}^{*}(a)\cdot
p_{2}^{*}(b)\cdot[fp_{2}]\bigr{)}.$ (4.1)

By hypothesis, $a$ is a polynomial in Chern classes of vector bundles on $F$.
But product and pullback commute by property $({\bf A}_{13})$ on p. 323 of
[8]. So $p_{1}^{*}(a)$ is the same polynomial in the same Chern classes of the
pullbacks under $p_{1}$ of those vector bundles. But, as stated just before
Proposition 17.3.2 on p. 325 of [8], Chern classes commute with all bivariant
classes. Thus $p_{1}^{*}(a)\cdot p_{2}^{*}(b)=p_{2}^{*}(b)\cdot p_{1}^{*}(a)$.

Of course, $fp_{1}=fp_{2}$. Thus

$f_{\\#}a\cdot f_{\\#}b=(fp_{2})_{*}\bigl{(}p_{2}^{*}(b)\cdot
p_{1}^{*}(a)\cdot[fp_{1}]\bigr{)}.$

But $a$ and $b$ are arbitrary in (4.1); moreover, $p_{1}$ and $p_{2}$ may be
interchanged. So

$(fp_{2})_{*}\bigl{(}p_{2}^{*}(b)\cdot
p_{1}^{*}(a)\cdot[fp_{1}]\bigr{)}=f_{\\#}b\cdot f_{\\#}a.$

Thus $f_{\\#}a\cdot f_{\\#}b=f_{\\#}b\cdot f_{\\#}a$, as asserted. ∎

Assume $g\colon Y^{\prime}\to Y$, and consider $f^{\prime}\colon
X^{\prime}:=X\times_{Y}Y^{\prime}\to Y^{\prime}$. Then,

$g^{*}f_{\\#}=f^{\prime}_{\\#}g^{*}\colon\ A^{*}(X)\to A^{*}(Y^{\prime}).$
(4.2)

This property results from property $({\bf A}_{23})$ on p. 323 of [8] as
follows:

$\displaystyle g^{*}f_{\\#}(a)$
$\displaystyle=g^{*}f_{*}(a\cdot[f])=f^{\prime}_{*}g^{*}(a\cdot[f])=f^{\prime}_{*}(g^{*}(a)\cdot
g^{*}[f])$
$\displaystyle=f^{\prime}_{*}(g^{*}(a)\cdot[f^{\prime}])=f^{\prime}_{\\#}(g^{*}(a)).$

Assume that $f\colon X\to Y$ and $g\colon Y\to Z$ are proper and that $[f]$,
$[g]$, and $[gf]$ exist. Then

$(gf)_{\\#}=g_{\\#}f_{\\#},$

since, by [8, Section 17.4, p. 327],

$(gf)_{\\#}(a)=g_{*}f_{*}(a\cdot[gf])=g_{*}f_{*}(a\cdot[f]\cdot[g])=g_{*}(f_{*}(a\cdot[f])\cdot[g])=g_{\\#}(f_{\\#}(a)).$

######  4.3.

Blowups. Let $\iota\colon W\to V$ be a closed, regular embedding of
codimension $d$. Then the orientation class $[\iota]$ is defined. Let
$V^{\prime}$ denote the blowup of $V$ along $W$, with exceptional divisor $E$.
Then $E=\mathbb{P}\big{(}\nu^{\vee}\big{)}$, where $\nu$ is the normal bundle
of $W$ in $V$. Set $\xi:=c_{1}(\mathcal{O}_{V^{\prime}}(-E))$. The map
$f\colon E\to W$ is flat, hence has an orientation class $[f]$. Then, by [8,
Corollary 4.2.2, p. 75], for $k\geq 1$,

$f_{\\#}\xi^{k}=-s_{k-d}\big{(}\nu^{\vee}\big{)},$ (4.3)

where $s\big{(}\nu^{\vee}\big{)}=c(\nu)^{-1}$ denotes the Segre class of
$\nu^{\vee}$.

######  4.4.

Derived classes. Consider the setup of Section 2.1. We have the Chern classes
$v:=c_{1}(\mathcal{O}_{F}(D))$, $w_{j}:=c_{j}\big{(}\Omega^{1}_{F/Y}\big{)}$
(for $j=1,2$), $e:=c_{1}(\mathcal{O}_{F^{\prime}}(E))\in A^{*}(F^{\prime})$,
where $E$ is the exceptional divisor on $F^{\prime}$. Let $\beta\colon
E\to\Delta\cong F$ also denote the restriction of $\beta\colon F^{\prime}\to
F\times_{Y}F$. Recall that $\beta_{i}\colon X_{i}\to F$ and
$\beta^{\prime}_{i}\colon F_{i}\to F^{\prime}$ denote the inclusions. Let
$v,w_{j},e\in A^{*}(F^{\prime})$ also denote their own pullbacks via
$\varphi^{\prime}=p_{1}\beta$. As in [15, 16], we have

$c_{1}(\mathcal{O}_{F_{i}}(D_{i}))=(\beta^{\prime}_{i})^{*}(v-ie),\qquad
c_{1}\big{(}\Omega^{1}_{F^{\prime}/F}\big{)}=w_{1}+e,\qquad\text{and}\qquad
c_{2}\big{(}\Omega^{1}_{F^{\prime}/F}\big{)}=w_{2}-e^{2}.$

We also have the following relations:

$e^{3}+w_{1}e^{2}+w_{2}e=0,\qquad\beta_{\\#}e=0,\qquad\text{and}\qquad\beta_{\\#}e^{2}=-s_{0}\big{(}\nu^{\vee}\big{)}.$

The first relation results from the relation $e^{2}+w_{1}e+w_{2}=0$ on $E$,
cf. [8, Remark 3.2.4, p. 55]; the second and third, from equation (4.3).

Let $\iota\colon\Delta\to F\times_{Y}F$ denote the embedding, which is regular
of codimension $2$, since $\pi\colon F\to Y$ is smooth of relative dimension
$2$. As a class on $F\times_{Y}F$, we can write $\beta_{\\#}e^{2}=-\iota_{\\#}
s_{0}\big{(}\nu^{\vee}\big{)}$.

Set $y(a,b,c):=\pi_{\\#}\big{(}v^{a}w_{1}^{b}w_{2}^{c}\big{)}\in A^{*}(Y)$.
(Note, by Proposition 4.2, multiplying these $y(a,b,c)$ is commutative.) The
corresponding classes for the map $\pi_{i}\colon F_{i}\to X_{i}$ (defined in
Section 2.1) are

$y_{i}(a,b,c):={\pi_{i}}_{\\#}(\beta^{\prime}_{i})^{*}\big{(}(v-ie)^{a}(w_{1}+e)^{b}\big{(}w_{2}-e^{2}\big{)}^{c}\big{)}.$

By (4.2),
$\beta_{i}^{*}\pi^{\prime}_{\\#}=\pi_{i\\#}{\beta^{\prime}_{i}}^{*}$; hence,

$y_{i}(a,b,c)=\beta_{i}^{*}\pi^{\prime}_{\\#}\big{(}(v-ie)^{a}(w_{1}+e)^{b}\big{(}w_{2}-e^{2}\big{)}^{c}\big{)}.$

We have $\pi^{\prime}=p_{2}\beta$, and

$\displaystyle\beta_{\\#}\big{(}(v-ie)^{a}(w_{1}+e)^{b}\big{(}w_{2}-e^{2}\big{)}^{c}\big{)}$
$\displaystyle=\beta_{\\#}\big{(}v^{a}w_{1}^{b}w_{2}^{c}-Q(i;a,b,c)e^{2}\big{)}$
$\displaystyle=p_{1}^{*}\big{(}v^{a}w_{1}^{b}w_{2}^{c}\big{)}+p_{1}^{*}Q(i;a,b,c)\cdot\iota_{\\#}s_{0}\big{(}\nu^{\vee}\big{)},$

where $Q(i;a,b,c)$ is the (weighted homogeneous, degree $a+b+2c-2$) polynomial
in $v$, $w_{1}$, $w_{2}$ returned by the function in Algorithm 2.3 in [16, p.
72]. Hence, since $[\pi^{\prime}]=[p_{2}\beta]=[\beta]\cdot[p_{2}]$ by [8,
Section 17.4, p. 327]), we get

$\pi^{\prime}_{\\#}\big{(}(v-ie)^{a}(w_{1}+e)^{b}\big{(}w_{2}-e^{2}\big{)}^{c}\big{)}=p_{2\\#}p_{1}^{*}\big{(}v^{a}w_{1}^{b}w_{2}^{c}\big{)}+p_{2\\#}\bigl{(}p_{1}^{*}Q(i;a,b,c)\cdot\iota_{\\#}s_{0}\big{(}\nu^{\vee}\big{)}\bigr{)}.$

Since $p_{1}\iota=p_{2}\iota$, we have

$p_{1}^{*}Q(i;a,b,c)\cdot\iota_{\\#}s_{0}\big{(}\nu^{\vee}\big{)}=\iota_{\\#}\big{(}\iota^{*}p_{1}^{*}Q(i;a,b,c)\cdot
s_{0}\big{(}\nu^{\vee}\big{)}\big{)}=p_{2}^{*}Q(i;a,b,c)\cdot\iota_{\\#}s_{0}\big{(}\nu^{\vee}\big{)}.$

Since $s_{0}(\nu^{\vee})=c_{0}(\nu)=:1_{\Delta}\in A^{*}(\Delta)$ is the class
that acts as identy on $A_{*}(\Delta)$ and since $p_{2}\iota$ is an
isomorphism, we get $p_{2\\#}\iota_{\\#}s_{0}\big{(}\nu^{\vee}\big{)}=1_{F}$.
Since $p_{2\\#}p_{1}^{*}=\pi^{*}\pi_{\\#}$, we obtain

$\pi^{\prime}_{\\#}\big{(}(v-ie)^{a}(w_{1}+e)^{b}\big{(}w_{2}-e^{2}\big{)}^{c}\big{)}=\pi^{*}\pi_{\\#}v^{a}w_{1}^{b}w_{2}^{c}+Q(i;a,b,c).$

Thus we have

$y_{i}(a,b,c)=\beta^{*}_{i}(\pi^{*}y(a,b,c)+Q(i;a,b,c)).$ (4.4)

##  5 The main theorem

Fix a smooth projective family of surfaces $\pi\colon F\to Y$, and a relative
effective divisor $D$ on $F/Y$. For each $r\geq 1$, we introduce a natural
cycle $U(D,r)$ on $Y$ that enumerates the fibers $D_{y}$ with $r$ nodes. Our
first goal is to prove Proposition 5.3, which gives a recursive relation for
the class $u(D,r):=[U(D,r)]$ in terms of the classes $u(D_{i},r_{i})$ of the
induced pairs $(F_{i}/X_{i},D_{i})$ introduced in Section 2.1. This relation
is the key to the proof of our main theorem, Theorem 5.4.

######  Definition 5.1.

Fix $r\geq 1$. Form the direct image on $Y$ of the fundamental cycle
$[G(F/Y,D;r)]$, remove the part supported in $Y(\infty)$, and denote the
result by $U(D,r)$. In other words, $U(D,r)$ is obtained as follows. For each
generic point $z$ of $G(F/Y,D;r)$, let $n_{z}$ be the length of
$\mathcal{O}_{z}$ over $\mathcal{O}_{q(z)}$ provided this length is finite and
$q(z)\notin Y(\infty)$; otherwise, let $n_{z}$ be $0$. Let
$\overline{\\{q(z)\\}}$ be the closure of $\\{q(z)\\}$.

$U(D,r):=\sum_{z}n_{z}\overline{\\{q(z)\\}}.$

In addition, let $U(D,0)$ denote the fundamental cycle of $Y$, and set
$U(D,-r):=0$.

Finally, set $u(D,r):=[U(D,r)]$. It’s a class on $Y$.

######  Proposition 5.2.

Fix $r\geq 1$. Assume that the pair $(F/Y,D)$ is $r$-generic. Then $U(D,r)$
has pure codimension $r$, and its support is just the closure of
$Y(r\mathbf{A}_{1})$.

###### Proof.

This follows from the second part of Lemma 2.6, with
$\mathbf{D}=r\mathbf{A}_{1}$: Let $z$ be a generic point of $G(F/Y,D;r)$. Then
$\operatorname{cod}_{z}(G(F/Y,D;r),H(r))\leq 3r$ because
$\operatorname{Hilb}^{3r}_{D/Y}$ is the zero scheme of a section of a locally
free sheaf of rank $3r$ on $\operatorname{Hilb}^{3r}_{F/Y}$ by [1, Proposition
4, p. 5]. ∎

######  Proposition 5.3.

Fix $r\geq 1$. Then the following formula holds:

$u(D,r)=\frac{1}{r}\pi_{\\#}\sum_{i\geq
2}(-1)^{i}{\beta_{i}}_{\\#}u(D_{i},r_{i}),$ (5.1)

where $\beta_{i}\colon X_{i}\to F$ denote the inclusions.

###### Proof.

Notice that the sum in (5.1) is finite, as $r_{i}=r-\binom{i+1}{2}+2$ by (2.3)
and as $U(D,s)=0$ for $s<0$ by Definition 5.1.

Let’s first explain set-theoretically why the formula should hold. Consider a
closed point $y\in Y(r)$. The curve $D_{y}$ has precisely $r$ nodes, and if we
blow up one of the nodes, $x\in X_{2}$ say, the strict transform $(D_{2})_{x}$
has $r-1$ nodes. Hence, above $y\in Y(r)$, we get $r$ points $x\in
X_{2}(r-1)$. But not all $r-1$-nodal curves of $D_{2}/X_{2}$ arise in this
way: if $x\in X_{3}$ is an (ordinary) triple point of $D_{\pi(x)}$, then
$(D_{2})_{x}=(D_{3})_{x}+E_{x}$, hence it has the 3 nodes $(D_{3})_{x}\cap
E_{x}$. Therefore, set-theoretically, $X_{2}(r-1)$ consists of two parts, one
mapping $r:1$ to $Y(r)$, the other mapping $1:1$ to
$Y(\mathbf{D}_{4}+(r-4)\mathbf{A}_{1})$. The second part is equal to the part
of $X_{3}(r-4)$ not contained in $X_{4}$. The part contained in $X_{4}$ is
equal to $X_{4}(r-8)$ minus the part contained in $X_{5}$, and so on.

The fact that this reasoning is valid on the cycle level is precisely what
Lemma 3.3 shows. It therefore only remains to show that there is a natural map

$G(F_{2}/X_{2},D_{2};r-1)\smallsetminus G(F_{3}/X_{3},D_{3};r-4)\to
G(F/Y,D;r)$

which is $r:1$. If $\mathcal{Z}^{\prime}\to T$ is a $T$-point of
$G(F_{2}/X_{2},D_{2};r-1)\smallsetminus G(F_{3}/X_{3},D_{3};r-4)$ (i.e., a
family of $r-1$ double points in $D_{2}\subset F_{2}$ over $T$, none of which
lie on $E$), we send it to the image
${\varphi^{\prime}}(\mathcal{Z}^{\prime}\cup 2E)$ in $F$ to get a family over
$T$ of $r$ double points in $D$. This map induces an $r:1$ map from the
components of the cycle $U(D_{2},r-1)$ that are supported on
$X_{2}\smallsetminus X_{3}$ to $U(D,r)$. ∎

Notice that the case $r=8$ of (5.1) looks different from (4.6) on p. 230 of
[15]. Indeed, in [15], we used $u(D_{2},7)$ to denote what we denote by
$\frac{1}{7!}P_{7}(a_{\bullet}(D_{2}))\cap[X_{2}]$ here. However, the
mathematics is consistent.

Our main result is Theorem 5.4, the first part of which was proved in [16]. We
now prove the last part of the theorem, namely the part concerning the
expression for $u(D,r)=[U(D,r)]$, where $U(D,r)$ is the cycle introduced in
Definition 5.1.

Recall from Section 2 that for a given pair $(F/Y,D)$, the subscheme
$X_{i}\subset F$ denotes the scheme of zeros of the natural section
$\sigma_{i}$ of $\mathcal{P}^{i-1}_{F/Y}(D)$. After making some
simplifications, we prove the theorem when $X_{4}=\varnothing$. This proof is
easy, and it yields the case $r\leq 7$. We then consider the case $r=8$, which
is more difficult due to the presence of nonreduced fibers in codimension
$r_{2}=7$ in the family of curves of the induced pair $(F_{2}/X_{2},D_{2})$.

######  Theorem 5.4 (Main).

Let $\pi\colon F\to Y$ be a smooth projective family of surfaces, and $D$ a
relative effective divisor. Assume $Y$ is Cohen–Macaulay and equidimensional.
Fix an integer $r\geq 0$, and assume

  1. $(i)$

if $Y(\infty)\neq\varnothing$, we have $\operatorname{cod}Y(\infty)\geq r+1$,

  2. $(ii)$

the pair $(F/Y,D)$ is $r$-generic.

Then either $Y(r\mathbf{A}_{1})$ is empty, or it has pure codimension $r$; in
either case, its closure $\overline{Y(r\mathbf{A}_{1})}$ is the support of a
natural nonnegative cycle $U(D,r)$.

Let $b_{s}(D)$ be the polynomial in $v$, $w_{1}$, $w_{2}$ output by Algorithm
2.3 in [16], set $a_{s}(D):=\pi_{\\#}b_{s}(D)$, and let $P_{r}$ be the $r$th
Bell polynomial. Assume $r\leq 8$, and if $r=8$, then $(F/Y,D)$ is strongly
$8$-generic. Then the rational equivalence class $u(D,r):=[U(D,r)]$ is given
by the formula

$u(D,r)=\frac{1}{r!}P_{r}(a_{1}(D),\ldots,a_{r}(D))\cap[Y].$

###### Proof.

First of all, we may assume $Y(\infty)=\varnothing$. Indeed,
$\operatorname{cod}Y(\infty)\geq r+1$ by hypothesis. Hence we may replace $Y$
by $Y\smallsetminus Y(\infty)$, and thus assume that all the fibres of
$\pi|_{D}\colon D\to Y$ are reduced.

Second, $\operatorname{cod}X_{i}=\binom{i+1}{2}$ for $i=2,3,4$ by Corollary
2.9. Therefore, $X_{i}$ is a local complete intersection in $F$, and $F$ is
smooth over the Cohen–Macaulay scheme $Y$, hence is Cohen–Macaulay, and so
$X_{i}$ is too. Since $Y$ is equidimensional, so is $F$, and hence so is
$X_{i}$.

By [16, Lemma 2.4, p. 73] there are at most finitely many $\mathbf{D}$ such
that $Y(\mathbf{D})$ is nonempty; hence, we may remove all $Y(\mathbf{D})$
with $\operatorname{cod}Y(\mathbf{D})\geq r+1$. If $x\in X_{i}$ is a closed
point, $i\geq 5$, then $D_{\pi(x)}$ contains a point of multiplicity at least
$i$; hence, the minimal Enriques diagram $\mathbf{D}$ of $D_{\pi(x)}$
satisfies $\operatorname{cod}\mathbf{D}\geq\binom{i+1}{2}-2\geq 13$ by the
formula for $\operatorname{cod}\mathbf{D}$ in [15, p. 217]. Therefore,
$\operatorname{cod}Y(\mathbf{D})\geq 13>r$, since $r\leq 8$. Hence
$X_{i}=\varnothing$ for $i\geq 5$. If $x\in X_{4}$, then $x\in D_{\pi(x)}$ has
multiplicity $\geq 4$; hence, $\operatorname{cod}\mathbf{D}\geq 8$. If
$\operatorname{cod}\mathbf{D}\geq 9$, then
$\operatorname{cod}Y(\mathbf{D})\geq 9$; so $Y(\mathbf{D})=\varnothing$. Hence
we have $\operatorname{cod}\mathbf{D}=8$. But the only diagram with a root of
multiplicity 4 and codimension 8 is the diagram $X_{1,0}$ corresponding to an
ordinary quadruple point; see [15, Figures 2–6, p. 218].

The recursive formula of Proposition 5.3 applies. It gives, for $r\leq 8$,

$ru(D,r)=\pi_{\\#}\sum_{i=2}^{4}(-1)^{i}\beta_{i\\#}u(D_{i},r_{i}),$

where $r_{2}=r-1$, $r_{3}=r-4$ and $r_{4}=r-8$.

######  Proposition 5.5.

The theorem holds if $X_{4}=\varnothing$ and $Y(\infty)=\varnothing$.

###### Proof.

The proof is by induction on $r$. For $r=1$, we have

$\displaystyle u(D,1)$
$\displaystyle={\pi}_{\\#}{\beta_{2}}_{\\#}u(D_{2},0)=\pi_{\\#}[X_{2}]=\pi_{\\#}x_{2}\cap[Y]$
$\displaystyle=\pi_{\\#}b_{1}(D)\cap[Y]=a_{1}(D)\cap[Y]=P_{1}(a_{1}(D))\cap[Y].$

Assume next that $r\geq 2$ and that the theorem holds for all families
verifying the hypotheses of the theorem with $r$ replaced by $r^{\prime}<r$.
In particular, the statement then holds for the induced pairs
$(F_{i}/X_{i},D_{i})$, for $i=2,3$, defined in Section 2.1. Indeed,
$X_{i}(\infty)=\varnothing$, and $(F_{i}/X_{i},D_{i})$ is $r_{i}$-generic by
Proposition 2.8; that is, $(ii)$ of the theorem holds with $r$ replaced by
$r_{i}$.

To simplify the notation, let us write

$P_{m}(z_{\bullet}):=P_{m}(z_{1},\dots,z_{m}),$ (5.2)

where $P_{m}$ is the $m$th Bell polynomial and $z_{1},\dotsc,z_{m}$ are
variables. Then we get

$r!u(D,r)=\pi_{\\#}\bigl{(}\beta_{2\\#}P_{r-1}(a_{\bullet}(D_{2}))-(r-1)!/(r-4)!\beta_{3\\#}P_{r-4}(a_{\bullet}(D_{3}))\bigr{)}\cap[Y].$

By definition, $a_{s}(D_{i})=\pi_{i\\#}b_{s}(D_{i})$. By applying (4.4) to the
polynomials $b_{s}(D)$ (cf. [16, Algorithm 2.3]),

$\beta_{i\\#}P_{m}(a_{\bullet}(D_{i}))=P_{m}(\pi^{*}a_{\bullet}(D)+Q(i,b_{\bullet}(D)))\cdot
x_{i}(D).$

By the binomial property of the Bell polynomials [3, equation (4.9), p. 265],
we have

$P_{m}(\pi^{*}a_{\bullet}(D)+Q(i,b_{\bullet}(D)))=\sum_{k=0}^{m}\binom{m}{k}P_{m-k}(\pi^{*}a_{\bullet}(D))P_{k}(Q(i,b_{\bullet}(D))).$

Plugging this in and using the definition of $b_{s}(D)$ and $a_{s}(D)$, we get

$r!u(D,r)=\sum_{k=0}^{r-1}\binom{r-1}{k}P_{r-1-k}(a_{\bullet}(D))a_{k+1}(D)\cap[Y]=P_{r}(a_{\bullet}(D))\cap[Y],$

where the last equality follows from the recursive property of the Bell
polynomials [3, equation (4.2), p. 263]. ∎

When $r\leq 7$, we can remove all $Y(\mathbf{D})$ with
$\operatorname{cod}\mathbf{D}\geq 8$. If $\mathbf{D}$ contains a root of
multiplicity $\geq 4$, then
$\operatorname{cod}\mathbf{D}\geq\binom{4+1}{2}-2=8$, hence we may assume
$X_{4}=\varnothing$. This proves the theorem for $r\leq 7$.

Assume $r=8$. By Proposition 5.3, we have

$8u(D,8)=\pi_{\\#}\bigl{(}\beta_{2\\#}u(D_{2},7)-\beta_{3\\#}u(D_{3},4)+\beta_{4\\#}u(D_{4},0)\bigr{)}.$

The induced pairs $(F_{i}/X_{i},D_{i})$, $i=3,4$ satisfy the conditions of the
theorem, with $r$ replaced by $r_{i}$, hence, by the case $r\leq 7$ of the
theorem:

$u(D_{3},4)=\frac{1}{4!}P_{4}(a_{\bullet}(D_{3}))\cap[X_{3}]\qquad\text{and}\qquad
u(D_{4},0)=[X_{4}].$

Note that, since $F$ is Cohen–Macaulay, $[X_{i}]=x_{i}\cap[F]$ with $x_{i}$ as
in Algorithm 2.3 in [16]. The induced pair $(F_{2}/X_{2},D_{2})$ does not
satisfy the conditions for $r$ replaced by $r_{2}$. Indeed, note that
$D_{2}|_{F_{4}}=(D-2E)|_{F_{4}}=(D-4E+2E)|_{F_{4}}=D_{4}+2E|_{F_{4}}$ and that
$D_{4}$ is relative effective on $F_{4}/X_{4}$ by Lemma 2.2; hence, $D_{2}$
has nonreduced fibers above $X_{4}$. So $X_{2}(\infty)=X_{4}$, and hence has
codimension $r_{2}=7$ in $X_{2}$.

However, from what we have seen above, if we restrict the family $F_{2}\to
X_{2}$ to $X_{2}\smallsetminus X_{4}$, then
$\frac{1}{7!}P_{7}(a_{\bullet}(D_{2}))\cap[X_{2}\smallsetminus X_{4}]$ is the
class of the 7-nodal curves of that family. So the difference

$\frac{1}{7!}P_{7}(a_{\bullet}(D_{2}))\cap[X_{2}]-u(D_{2},7)$ (5.3)

is the _correction term_ we are looking for. It is the class of a cycle of
codimension 7, supported on the codimension 7 subscheme $X_{4}$ of $X_{2}$. As
Theorem 7.5 shows, (5.3) is equal to $C[X_{4}]$, where the constant $C$ is an
integer, which is independent of the given pair $(F/Y,D)$. Hence it suffices
to compute $C$ in any particular case; for example, in [16, Example 3.8, p.
80], we worked out the case of $8$-nodal quintic plane curves, and found
$C=3280$. Thus Theorem 5.4 is proved. ∎

######  Remark 5.6.

Assume $(F/Y,D)$ is the direct sum of two pairs $(F^{\prime}/Y,D^{\prime})$
and $(F^{\prime\prime}/Y,D^{\prime\prime})$ over the same base. Then the
$r$-nodal curves of $D\to Y$ consists of the unions of the $(r-i)$-nodal
curves of $D^{\prime}\to Y$ and the $i$-nodal curves of $D^{\prime\prime}\to
Y$ for $i=0,\dots,r$. Hence, the existence of a universal polynomial for
$r$-nodal curves implies that the generating series for $(F/Y,D)$ is equal to
the product of the generating series for $(F^{\prime}/Y,D^{\prime})$ and
$(F^{\prime\prime}/Y,D^{\prime\prime})$. This fact was observed by Göttsche in
the case of a trivial family [9, p. 525], and by Laarakker in the general case
[19, Section 5.1, p. 4935]. In the case of a trivial family, Göttsche observed
that this multiplicativity implies that the universal polynomials are Bell
polynomials. However, as observed by Laarakker, this conclusion does not
follow in the case of a nontrivial family.

Let $a_{j}(D)$, $a_{j}(D^{\prime})$, $a_{j}(D^{\prime\prime})$ be the classes,
introduced in Theorem 5.4, for the three pairs. Clearly,
$a_{j}(D)=a_{j}(D^{\prime})+a_{j}(D^{\prime\prime})$. So (for $r\leq 8$), in
the notation of (5.2),

$u(D,r)=\frac{1}{r!}P_{r}(a_{\bullet}(D))\cap[Y]=\frac{1}{r!}P_{r}(a_{\bullet}(D^{\prime})+a_{\bullet}(D^{\prime\prime}))\cap[Y].$

By the binomial property of the Bell polynomials, the right-hand side is equal
to

$\sum_{i=0}^{r}\frac{1}{(r-i)!}P_{r-i}(a_{\bullet}(D^{\prime}))\frac{1}{i!}P_{i}(a_{\bullet}(D^{\prime\prime}))\cap[Y].$

Hence the Bell polynomial shape of the universal polynomials is in agreement
with the multiplicative property of the generating series of $(F/Y,D)$.

##  6 An expression for the correction term

We now find an expression for the correction term (5.3). First, in Section 6.1
we define some useful schemes. Then in Lemma 6.2, we give an expression for
$u(D_{2},7)$, obtained via repeated use of the recursion formula of
Proposition 5.3. Then in Section 6.3, we introduce classes $e(W_{i})$ on
$X_{2}$ of cycles on $X_{4}$. Finally, in Proposition 6.4, we express (5.3) as
a linear combination of the $e(W_{i})$.

######  6.1.

Some important schemes. Let
$\big{(}F_{2}^{(1)}/X_{2}^{(0)},D_{2}^{(1)}\big{)}:=(F_{2}/X_{2},D_{2})$ be
the induced pair of $(F/Y,D)$. Define recursively
$\big{(}F_{2}^{(j+1)}/X_{2}^{(j)},D_{2}^{(j+1)}\big{)}$ as the induced pair of
$\big{(}F_{2}^{(j)}/X_{2}^{(j-1)},D_{2}^{(j)}\big{)}$. Let
$\big{(}F_{3}^{(j+1)}/X_{3}^{(j)},D_{3}^{(j+1)}\big{)}$ be the induced pair of
$\big{(}F_{2}^{(j)}/X_{2}^{(j-1)},D_{2}^{(j)}\big{)}$. Let
$X_{2}\big{(}D_{3}^{(j+1)}\big{)}\subset F_{3}^{(j+1)}$ be the zero scheme of
the section of $\mathcal{P}^{1}_{\pi_{3}^{(j+1)}}\big{(}D_{3}^{(j+1)}\big{)}$
induced by that defining the divisor $D_{3}^{(j+1)}$.

Let $D^{(j+1)}-E^{(j)}$ denote the restriction of the divisor
$D^{(j+1)}-{\varphi^{(j+1)}}^{-1}E^{(j)}$ to
$F^{(j+1)}|_{X_{2}(D_{3}^{(j)})}$. This divisor is effective; so it induces a
section of the restriction of
$\mathcal{P}^{1}_{\pi^{(j+1)}}\big{(}D^{(j+1)}-{\varphi^{(j+1)}}^{-1}E^{(j)}\big{)}$.
Let $X_{2}\big{(}D^{(j+1)}-E^{(j)}\big{)}$ denote its scheme of zeros. Let
$D^{(j+2)}-E^{(j)}$ denote the restriction of
$D^{(j+2)}-{\varphi^{(j+2)}}^{-1}{\varphi^{(j+1)}}^{-1}E^{(j)}$ to
$F^{(j+2)}|_{X_{2}(D^{(j+1)}-E^{(j)})}$, and
$X_{2}\big{(}D^{(j+2)}-E^{(j)}\big{)}$ the scheme of zeros of the induced
section of the restriction of
$\mathcal{P}^{1}_{\pi^{(j+2)}}\big{(}D^{(j+2)}-{\varphi^{(j+2)}}^{-1}{\varphi^{(j+1)}}^{-1}E^{(j)}\big{)}$.

Form these five equidimensional schemes of dimension $\dim X_{2}-7$, or $\dim
X_{4}$:

$\displaystyle\mathfrak{X}_{2}^{(7)}:=\overline{X_{2}^{(7)}\smallsetminus
X_{2}^{(7)}|_{X_{4}}},$
$\displaystyle\mathfrak{X}_{3}^{(4)}:=\overline{X_{3}^{(4)}\smallsetminus
X_{3}^{(4)}|_{X_{4}}},$
$\displaystyle\mathfrak{X}_{2}\big{(}D_{3}^{(4)}\big{)}:=\overline{X_{2}\big{(}D_{3}^{(4)}\big{)}\smallsetminus
X_{2}\big{(}D_{3}^{(4)}\big{)}|_{X_{4}}},$
$\displaystyle\mathfrak{X}_{2}\big{(}D^{(4)}-E^{(3)}\big{)}:=\overline{X_{2}\big{(}D^{(4)}-E^{(3)}\big{)}\smallsetminus
X_{2}\big{(}D^{(4)}-E^{(3)}\big{)}|_{X_{4}}},$
$\displaystyle\mathfrak{X}_{2}\big{(}D^{(4)}-E^{(2)}\big{)}:=\overline{X_{2}\big{(}D^{(4)}-E^{(2)}\big{)}\smallsetminus
X_{2}\big{(}D^{(4)}-E^{(2)}\big{)}|_{X_{4}}}.$

For $j=2,\dots,7$, consider the composed map
$\pi^{(j)}\circ\dotsb\circ\pi^{(1)}\colon F^{(j)}\to F$, and let
$\pi_{j}\colon F^{(j)}|_{X_{2}}\to X_{2}$ denote its restriction.

######  Lemma 6.2.

Then

$\displaystyle u(D_{2},7)={}$
$\displaystyle\frac{1}{7!}\pi_{7\\#}\big{[}\mathfrak{X}_{2}^{(7)}\big{]}-\frac{3!}{7!}\pi_{4\\#}\big{[}\mathfrak{X}_{3}^{(4)}\big{]}-\frac{4!}{7!}\pi_{4\\#}\big{[}\mathfrak{X}_{2}\big{(}D_{3}^{(4)}\big{)}\big{]}-\frac{5!}{7!2!}\pi_{4\\#}\big{[}\mathfrak{X}_{2}\big{(}D^{(4)}-E^{(3)}\big{)}\big{]}$
$\displaystyle-\frac{6!}{7!3!}\pi_{4\\#}\big{[}\mathfrak{X}_{2}\big{(}D^{(4)}-E^{(2)}\big{)}\big{]}.$

###### Proof.

As $(F/Y,D)$ is 8-generic, the successive induced pairs are $j$-generic (for
appropriate $j$) by Proposition 2.8. Thus the lemma follows from repeated use
of the recursion formula of Proposition 5.3. ∎

######  6.3.

The classes $\boldsymbol{e(W_{i})}$. Next we find an expression for each term
appearing in the formula for $u(D_{2},7)$ in Lemma 6.2. We just consider
$\mathfrak{X}_{2}^{(7)}$, since the other schemes can be studied in a similar
way and their classes have similar expressions.

By definition, $X_{2}^{(j)}$ is the scheme of zeros of the section
$\sigma_{2}^{(j)}$ of
$\mathcal{P}^{1}_{\pi_{2}^{(j)}}\big{(}D_{2}^{(j)}\big{)}$ induced by the
section $\sigma^{(j)}$ defining $D_{2}^{(j)}$. For $j=1,\dots,6$, set

$\displaystyle\mathfrak{X}_{2}^{(j)}:=\overline{X_{2}^{(j)}\smallsetminus
X_{2}^{(j)}|_{X_{4}}},\qquad\mathfrak{F}_{2}^{(j+1)}:=F_{2}^{(j+1)}|_{\mathfrak{X}_{2}^{(j)}},$
(6.1) $\displaystyle
W_{1}^{(j)}:=\big{(}X_{2}^{(j)}|_{\mathfrak{X}_{2}^{(j-1)}}\big{)}|_{X_{4}}.$
(6.2)

Then $X_{2}^{(j)}|_{\mathfrak{X}_{2}^{(j-1)}}=\mathfrak{X}_{2}^{(j)}\cup
W_{1}^{(j)}$. Notice the $\mathfrak{F}_{2}^{(j)}$ are equidimensional of
dimension $\dim X_{2}-j+3$ and that
$\operatorname{cod}\big{(}\mathfrak{X}_{2}^{(j)},\mathfrak{F}_{2}^{(j)}\big{)}=3$.
Thus $\dim\mathfrak{X}_{2}^{(j)}=\dim X_{2}-j$.

To determine the dimensions of the “excess schemes” $W_{1}^{(j)}$, consider
the fibers of $W_{1}^{(j)}\to W_{1}^{(j-1)}\cap\mathfrak{X}_{2}^{(j-1)}$.
Starting with $x\in X_{4}$, we have
$\big{(}D_{2}^{(1)}\big{)}_{x}=\Gamma_{x}\cup 2E^{(1)}_{x}$, where
$\Gamma_{x}$ is the strict transform of $D_{\pi(x)}$ under the blowup of
$F_{\pi(x)}$ at $x$. Hence, a local calculation yields
$\big{(}W_{1}^{(1)}\big{)}_{x}$, which is $\big{(}X_{2}^{(1)}\big{)}_{x}$, is
equal to $E^{(1)}_{x}$ with four embedded points at the intersections of
$\Gamma_{x}$ with $E^{(1)}_{x}$. Thus $\dim W_{1}^{(1)}=\dim X_{4}+1$.

Next take $z\in\big{(}W_{1}^{(1)}\big{)}_{x}$, but $z\notin\Gamma_{x}$. Then
the fiber $\big{(}W_{1}^{(2)}\big{)}_{z}$ is the strict transform of
$E^{(1)}_{x}$ plus four embedded points. If $z\in\Gamma_{x}\cap E^{(1)}_{x}$,
then $\big{(}W_{1}^{(2)}\big{)}_{z}$ has an additional point, namely, the
intersection of the strict transform of $\Gamma_{x}$ with $E^{(2)}_{z}$. Hence
$\dim W_{1}^{(2)}=\dim X_{4}+2$. Continuing, we get

$\dim W_{1}^{(3)}=\dim X_{4}+3=\dim X_{2}-4=\dim X_{2}^{(3)}-1.$

Thus $W_{1}^{(j)}\subset X_{2}^{(j)}$ and $\mathfrak{X}_{2}^{(j)}=X_{2}^{(j)}$
for $j\leq 3$.

For $j=4$ we get $\dim W_{1}^{(4)}=\dim X_{4}+4=\dim X_{2}-3=\dim
X_{2}^{(4)}+1$. Then $\dim W_{1}^{(4)}\cap\mathfrak{X}_{2}^{(4)}\leq\dim
X_{2}^{(4)}-1$. Hence $\dim W_{1}^{(5)}\leq\dim X_{2}^{(4)}-1+1=\dim
X_{2}^{(5)}+1$. Continuing, we get $\dim W_{1}^{(j)}\leq\dim X_{2}^{(j)}+1$
for $j\geq 4$.

To simplify notation, set
$\mathcal{P}^{(j)}:=\mathcal{P}^{1}_{\pi^{(j)}}\big{(}D^{(j)}\big{)}$.
Consider $\mathcal{P}^{(7)}$ restricted to $\mathfrak{F}_{2}^{(7)}$. The
scheme of zeros of its section $\sigma_{2}^{(7)}$ is equal to
$\mathfrak{X}_{2}^{(7)}\cup W_{1}^{(7)}$. Blow up $\mathfrak{F}_{2}^{(7)}$
along $W_{1}^{(7)}$ and apply the residual formula for top Chern classes [8,
Example 14.1.4, p. 245]. After pushing down to $\mathfrak{F}_{2}^{(7)}$, we
find

$\big{[}\mathfrak{X}_{2}^{(7)}\big{]}=c_{3}\big{(}\mathcal{P}^{(7)}\big{)}\cap\big{[}\mathfrak{F}_{2}^{(7)}\big{]}+\big{\\{}c\big{(}\mathcal{P}^{(7)}\big{)}\cap
s\big{(}W_{1}^{(7)},\mathfrak{F}_{2}^{(7)}\big{)}\big{\\}}_{\dim X_{4}}.$
(6.3)

Here is why (6.3) holds.

Let $\sigma^{\prime}$ denote the induced section of $\mathcal{P}^{(7)}$
twisted by the ideal sheaf of the exceptional divisor on the blowup of
$\mathfrak{F}_{2}^{(7)}$. Let $\mathbb{Z}(\sigma^{\prime})$ denote the
localized top Chern class of the pullback of $\mathcal{P}^{(7)}$ with respect
to $\sigma^{\prime}$. The zero scheme $Z(\sigma^{\prime})$ is equal to the
strict transform of $\mathfrak{X}_{2}^{(7)}$, hence has codimension 3 in the
blowup of $\mathfrak{F}_{2}^{(7)}$. It follows from [8, Proposition 14.1(b),
p. 244] that $\mathbb{Z}(\sigma^{\prime})$ is the class of a positive cycle
with support $Z(\sigma^{\prime})$. Since the blowup of
$\mathfrak{F}_{2}^{(7)}$ need not be Cohen–Macaulay, we cannot immediately
conclude that $\mathbb{Z}(\sigma^{\prime})=[Z(\sigma^{\prime})]$. However,
since $\mathfrak{F}_{2}^{(7)}\smallsetminus\mathfrak{F}_{2}^{(7)}|_{X_{4}}$ is
Cohen–Macaulay, the restrictions of $\mathbb{Z}(\sigma^{\prime})$ and
$[Z(\sigma^{\prime})]$ agree above
$\mathfrak{X}_{2}^{(7)}\smallsetminus\mathfrak{X}_{2}^{(7)}|_{X_{4}}$; hence
they are equal.

Since
$\big{[}\mathfrak{F}^{(7)}_{2}\big{]}=\big{[}F^{(7)}_{2}|_{\mathfrak{X}^{(6)}_{2}}\big{]}=\pi^{(7)*}\big{[}\mathfrak{X}_{2}^{(6)}\big{]}$,
the pushdown of the first term under $\pi^{(7)}$ gives
$\pi^{(7)}_{\\#}c_{3}\big{(}\mathcal{P}^{(7)}\big{)}\cap\big{[}\mathfrak{X}_{2}^{(6)}\big{]}$
by the projection formula. We then replace
$\big{[}\mathfrak{X}_{2}^{(6)}\big{]}$ by the analogue of equation (6.3) and
push down the resulting terms by $\pi^{(6)}$.

Continuing this way, we get a formula for
$\pi_{7\\#}\big{[}\mathfrak{X}_{2}^{(7)}\big{]}$. To simplify notation, set

$\displaystyle d_{j}(W_{1}):=\big{\\{}c\big{(}\mathcal{P}^{(7-j)}\big{)}\cap
s\big{(}W_{1}^{(7-j)},\mathfrak{F}_{2}^{(7-j)}\big{)}\big{\\}}_{\dim
X_{4}+j}\qquad\text{for}\quad j=0,\dotsc,3,$ $\displaystyle
e(W_{1}):=\pi^{(1)}_{\\#}\cdots\pi^{(7)}_{\\#}d_{0}(W_{1})+\pi^{(1)}_{\\#}\cdots\pi^{(6)}_{\\#}\big{(}\pi^{(7)}_{\\#}c_{3}\big{(}\mathcal{P}^{(7)}\big{)}d_{1}(W_{1})\big{)}$
$\displaystyle\hphantom{e(W_{1}):=}{}+\pi^{(1)}_{\\#}\cdots\pi^{(5)}_{\\#}\big{(}\pi^{(6)}_{\\#}\big{(}\pi^{(7)}_{\\#}c_{3}\big{(}\mathcal{P}^{(7)}\big{)}c_{3}\big{(}\mathcal{P}^{(6)}\big{)}d_{2}(W_{1})\big{)}\big{)}$
$\displaystyle\hphantom{e(W_{1}):=}{}+\pi^{(1)}_{\\#}\cdots\pi^{(4)}_{\\#}\big{(}\pi^{(5)}_{\\#}\big{(}\pi^{(6)}_{\\#}\big{(}\pi^{(7)}_{\\#}c_{3}\big{(}\mathcal{P}^{(7)}\big{)}c_{3}\big{(}\mathcal{P}^{(6)}\big{)}c_{3}\big{(}\mathcal{P}^{(5)}\big{)}\big{)}d_{3}(W_{1})\big{)}\big{)}.$

Note that the $d_{j}(W_{1})$ are classes in
$A^{*}\big{(}W_{1}^{(7-j)}\big{)}$, and that restricting the map $\pi_{7-j}$
gives a proper map $W_{1}^{(7-j)}\to X_{4}$.

Then the resulting formula for $\pi_{7\\#}[\mathfrak{X}_{2}^{(7)}]$ is the
following:

$\pi_{7\\#}\big{[}\mathfrak{X}_{2}^{(7)}\big{]}=\pi^{(1)}_{\\#}\big{(}\pi^{(2)}_{\\#}\big{(}\dotsb\big{(}\pi^{(7)}_{\\#}c_{3}\big{(}\mathcal{P}^{(7)}\big{)}\dotsb\big{)}c_{3}\big{(}\mathcal{P}^{(2)}\big{)}\big{)}c_{3}\big{(}\mathcal{P}^{(1)}\big{)}\big{)}\cap[X_{2}]+e(W_{1}).$

Similarly, we obtain formulas for the classes
$\pi_{4\\#}\big{[}\mathfrak{X}_{3}^{(4)}\big{]}$ and
$\pi_{4\\#}\big{[}\mathfrak{X}_{2}\big{(}D_{3}^{(4)}\big{)}\big{]}$ and
$\pi_{4\\#}\big{[}\mathfrak{X}_{2}\big{(}D^{(4)}-E^{(3)}\big{)}\big{]}$ and
$\pi_{4\\#}\big{[}\mathfrak{X}_{2}\big{(}D^{(4)}-E^{(2)}\big{)}\big{]}$. For
$i=2,\dots,5$, define the classes $e(W_{i})$ on $X_{2}$ correspondingly.

######  Proposition 6.4.

The correction term (5.3) is equal to

$\displaystyle\frac{1}{7!}P_{7}(a_{\bullet}(D_{2}))\cap[X_{2}]\\!-u(D_{2},7)=\frac{1}{7!}e(W_{1})\\!-\frac{3!}{7!}e(W_{2})\\!-\frac{4!}{7!}e(W_{3})\\!-\frac{5!}{7!2!}e(W_{4})\\!-\frac{6!}{7!3!}e(W_{5}).$

###### Proof.

Recall that the classes $a_{s}(D_{2})$ on $X_{2}$ are obtained by pushing down
the classes $b_{s}(D_{2})$ on $F_{2}$ obtained by applying Algorithm 2.3 of
[16, p. 72] to the pair $(F_{2}/X_{2},D_{2})$. In the case that
$X_{4}=\varnothing$, the Algorithm would have produced the formula
$u(D_{2},7)=\frac{1}{7!}P_{7}(a_{\bullet}(D_{2}))\cap[X_{2}]$. Removing the
classes $e(W_{i})$, we get

$\displaystyle\frac{1}{7!}P_{7}(a_{\bullet}(D_{2}))\cap[X_{2}]={}$
$\displaystyle\frac{1}{7!}\bigl{(}\pi_{7\\#}\big{[}\mathfrak{X}_{2}^{(7)}\big{]}-e(W_{1})\bigr{)}-\frac{3!}{7!}\bigl{(}\pi_{4\\#}\big{[}\mathfrak{X}_{3}^{(4)}\big{]}-e(W_{2})\bigr{)}$
$\displaystyle-\frac{4!}{7!}\bigl{(}\pi_{4\\#}\big{[}\mathfrak{X}_{2}\big{(}D_{3}^{(4)}\big{)}\big{]}-e(W_{3})\bigr{)}$
$\displaystyle-\frac{5!}{7!2!}\bigl{(}\pi_{4\\#}\big{[}\mathfrak{X}_{2}\big{(}D^{(4)}-E^{(3)}\big{)}\big{]}-e(W_{4})\bigr{)}$
$\displaystyle-\frac{6!}{7!3!}\bigl{(}\pi_{4\\#}\big{[}\mathfrak{X}_{2}\big{(}D^{(4)}-E^{(2)}\big{)}\big{]}-e(W_{5})\bigr{)}.$

Lemma 6.2 now yields the asserted formula. ∎

##  7 Independence of the correction term

In this section, we prove Theorem 7.5, which asserts that the correction term
(5.3) is equal to $C[X_{4}]$, where $C$ is independent of the strongly
8-generic pair $(F/Y,D)$ with $Y(\infty)=\varnothing$. We work locally
analytically on $F$ at a general closed point $x$ in $X_{4}$. Section 7.1
describes the local setup. Lemma 7.2 asserts that locally we have the
properness we need to pushdown classes. Lemma 7.3 asserts that the key classes
$e(W_{i})$ pull back to their local counterparts
$e\big{(}\widehat{W}_{i}\big{)}$.

Lemma 7.4 asserts that the coefficient in $e\big{(}\widehat{W}_{i}\big{)}$ of
$\big{[}\widehat{X}_{4}\big{]}$ depends only on the analytic type of the
ordinary quadruple point $x\in D_{\pi(x)}$; namely, on the cross ratio of the
four tangents at $x$. Its proof requires $(F/Y,D)$ to be strongly 8-generic.
Finally, we prove Theorem 7.5 by exhibiting a pair $(F/Y,D)$, where $X_{4}$ is
irreducible and where any given value of the cross ratio appears at some $x\in
X_{4}$.

######  7.1.

The local setup. Fix an 8-generic pair $(F/Y,D)$ with $Y(\infty)=\varnothing$,
and a general closed point $x\in X_{4}$. By general, we mean that $x$ lies on
a single irreducible component $Z$ of $X_{4}$ and that $x$ is an ordinary
quadruple point of $D_{\pi(x)}$. Let us arrange for every $x\in X_{4}$ to be
general as follows. First, if two components $Z^{\prime}$ and
$Z^{\prime\prime}$ of $X_{4}$ meet, then $\dim(Z^{\prime}\cap
Z^{\prime\prime})<\dim X_{4}$. So $\operatorname{cod}\pi(Z^{\prime}\cap
Z^{\prime\prime})>\operatorname{cod}\pi(X_{4})$. But
$\operatorname{cod}(\pi(X_{4}),Y)=8$ by (2.11) as $(F/Y,D)$ is 8-generic.
Hence we may discard $Z^{\prime}\cap Z^{\prime\prime}$. Second we may discard
the locus of $y\in Y$, where $D_{y}$ has a singularity $x$ worse than an
ordinary quadruple point, again because $(F/Y,D)$ is 8-generic.

Set $\widetilde{F}:=\operatorname{Spec}\mathcal{O}_{F,x}$ and
$\widetilde{Y}:=\operatorname{Spec}\mathcal{O}_{Y,\pi(x)}$ and
$\widetilde{D}:=\operatorname{Spec}\mathcal{O}_{D,x}$. Denote the induced pair
of $\big{(}\widetilde{F}/\widetilde{Y},\widetilde{D}\big{)}$ by
$\big{(}\widetilde{F}_{2}/\widetilde{X}_{2},\widetilde{D}_{2}\big{)}$. The
bundles of relative principal parts are compatible not only with the base
change $\widetilde{Y}\to Y$, but also with the maps $\widetilde{F}\to F$ and
$\widetilde{F}^{(j)}\to F^{(j)}$; cf. [11, Proposition 16.4.14, p. 22]. So
although the $\widetilde{X}_{i}$ for $i\geq 2$ are defined in terms of
$\big{(}\widetilde{F}/\widetilde{Y},\widetilde{D}\big{)}$, we have
$\widetilde{X}_{i}=\operatorname{Spec}\widetilde{\mathcal{O}}_{X_{i},x}$.
Similarly, the schemes constructed in Section 6 for $(F_{2}/X_{2},D_{2})$
induce the corresponding schemes for
$\big{(}\widetilde{F}_{2}/\widetilde{X}_{2},\widetilde{D}_{2}\big{)}$. Denote
by $\widetilde{W}^{(j)}_{i}$ the scheme corresponding to $W^{(j)}_{i}$; see
(6.2).

Next consider the completions of the local rings, giving us a pair
$\big{(}\widehat{F}/\widehat{Y},\widehat{D}\big{)}$. Replacing the principal
parts bundles by their completions, cf. [6, Example 16.14, p. 416], construct
the $\widehat{X}_{i}$, the induced pair
$\big{(}\widehat{F}_{2}/\widehat{X}_{2},\widehat{D}_{2}\big{)}$, and the
corresponding schemes of Section 6. Since the complete principal parts bundles
are pullbacks,
$\widehat{X}_{i}=\operatorname{Spec}\widehat{\mathcal{O}}_{X_{i},x}$, and all
the schemes of Section 6 for $(F_{2}/X_{2},D_{2})$ pull back to the
corresponding schemes for
$\big{(}\widehat{F}_{2}/\widehat{X}_{2},\widehat{D}_{2}\big{)}$. Denote by
$\widehat{W}^{(j)}_{i}$ the scheme corresponding to $\widetilde{W}^{(j)}_{i}$,
so to $W^{(j)}_{i}$.

The classes $e(W_{i})$ of Section 6.3 are sums of pushdowns of classes on the
$W^{(j)}_{i}$. By Lemma 7.2 below, the $\widehat{W}^{(j)}_{i}$ are proper over
$\widehat{X}_{2}$; hence, we may form the corresponding classes
$e(\widehat{W}_{i})$ for the pair
$\big{(}\widehat{F}_{2}/\widehat{X}_{2},\widehat{D}_{2}\big{)}$. Denote them
by $e(\widehat{W}_{i})$.

Let $\epsilon\colon\widehat{X}_{2}\to X_{2}$ denote the composition of the
flat maps $\widehat{X}_{2}\to\widetilde{X}_{2}$ and $\widetilde{X}_{2}\to
X_{2}$. Then $\big{[}\widehat{X}_{4}\big{]}=\epsilon^{*}[X_{4}]$, and Lemma
7.3 asserts $e(\widehat{W}_{i})=\epsilon^{*}e(W_{i})$.

Each $e(W_{i})$ is the class of a cycle $U_{i}$ on $X_{4}$ of dimension $\dim
X_{4}$. Say that the component $Z$ of $X_{4}$ containing $x$ appears in
$U_{i}$ with coefficient $C^{\prime}_{i}$ and in the fundamental cycle
$|X_{4}|$ with coefficient $C^{\prime\prime}_{i}$. Set
$C_{i}:=C_{i}^{\prime}/C_{i}^{\prime\prime}$. Then the cycles $U_{i}$ and
$C_{i}|X_{4}|$ become equal after restriction to a neighborhood of $Z$, so the
classes $e(W_{i})$ and $C_{i}[X_{4}]$ do too. Thus

$e\big{(}\widehat{W}_{i}\big{)}=C_{i}\big{[}\widehat{X}_{4}\big{]};$ (7.1)

furthermore, $C_{i}$ is independent of the choice of $x$ in $Z$.

######  Lemma 7.2.

The schemes $\widehat{W}^{(j)}_{i}$ are proper over $\widehat{X}_{2}$.

###### Proof.

Let us first show that the schemes $\widetilde{W}^{(j)}_{i}$ and
$W^{(j)}_{i}|_{\widetilde{X}_{2}}$ have the same support. It suffices to
consider only the $W^{(j)}_{1}$ as the other cases are similar.

Let $E_{j}\subset F^{(j)}$ be the union of the strict transforms of the
exceptional divisors $E^{(1)},\dots,E^{(j)}$; see [17, Definition 3.5, p.
423]. It follows from the description in Section 6 that $W^{(j)}_{1}$ is
supported in $E_{j}$. The fibers of the exceptional divisor of
$\widetilde{F}_{2}$ are the same as the corresponding fibers of the
exceptional divisor of $F_{2}$. Hence, above $x\in\widetilde{X}_{2}$, the
fiber of $E^{(j)}$ lies in $\widetilde{F}_{2}^{(j)}$. Thus
$W^{(j)}_{1}|_{\widetilde{X}_{2}}$ has support in $\widetilde{F}^{(j)}_{2}$,
and it is the same as the support of $\widetilde{W}^{(j)}_{i}$; moreover, this
support is proper over $\widetilde{X}_{2}$.

For $w\in W^{(j)}_{i}|_{\widetilde{X}_{2}}$, the map
$\mathcal{O}_{F^{(j)}_{2},w}\twoheadrightarrow\mathcal{O}_{W^{(j)},w}$ pulls
back to
$\mathcal{O}_{\widetilde{F}^{(j)}_{2},w}\twoheadrightarrow\mathcal{O}_{\widetilde{W}^{(j)},w}$.
But clearly
$\mathcal{O}_{F^{(j)}_{2},w}=\mathcal{O}_{\widetilde{F}^{(j)}_{2},w}$; hence
$\mathcal{O}_{W^{(j)}_{i},w}=\mathcal{O}_{\widetilde{W}^{(j)_{1}},w}$. Hence
$\widetilde{W}^{(j)}_{i}=W^{(j)}_{i}|_{\widetilde{X}_{2}}$. Therefore,
$\widetilde{W}^{(j)}_{i}$ is proper over $\widetilde{X}_{2}$. Thus the
pullback $\widehat{W}^{(j)}_{i}$ is proper over $\widehat{X}_{2}$. ∎

######  Lemma 7.3.

We have $e\big{(}\widehat{W}_{i}\big{)}=\epsilon^{*}e(W_{i})$.

###### Proof.

It suffices to check that each summand of $e(W_{i})$ pulls back to the
corresponding summand of $e\big{(}\widehat{W}_{i}\big{)}$. Here we only
consider the first summand of $e(W_{1})$, since the other cases are similar.

In (6.1), we defined the schemes $\overline{F}_{2}^{(j)}$. Denote by
$\epsilon_{2}^{(j)}\colon\overline{\widehat{F}}_{2}^{(j)}\to\overline{F}_{2}^{(j)}$
the map induced by the map $\epsilon$ defined in Section 7.1. Notice that, as
$\epsilon_{2}^{(7)}$ is flat,

$s\big{(}\widehat{W}_{1}^{(7)},\overline{\widehat{F}}_{2}^{(7)}\big{)}=s\big{(}\epsilon_{2}^{(7)-1}W_{1}^{(7)},\epsilon_{2}^{(7)-1}\overline{F}_{2}^{(7)}\big{)}=\epsilon_{2}^{(7)*}s\big{(}W_{1}^{(7)},\overline{F}_{2}^{(7)}\big{)};$

cf. [8, Proposition 4.2(b), p. 74]. But
$\widehat{\pi}_{\\#}^{(j)}\epsilon_{2}^{(j)*}=\epsilon_{2}^{(j-1)*}\pi_{\\#}^{(j)}$
by (4.2). Thus

$\displaystyle e\big{(}\widehat{W}_{i}\big{)}$
$\displaystyle=\widehat{\pi}^{(1)}_{\\#}\dotsb\widehat{\pi}^{(7)}_{\\#}\big{\\{}c\big{(}\widehat{\mathcal{P}}^{(7)}\big{)}\cap
s\big{(}\widehat{W}_{1}^{(7)},\overline{\widehat{F}}_{2}^{(7)}\big{)}\big{\\}}_{\dim
X_{4}}$
$\displaystyle=\widehat{\pi}^{(1)}_{\\#}\cdots\widehat{\pi}^{(7)}_{\\#}\big{\\{}\epsilon_{2}^{(7)*}c\big{(}\mathcal{P}^{(7)}\big{)}\cap\epsilon_{2}^{(7)*}s\big{(}W_{1}^{(7)},\overline{F}_{2}^{(7)}\big{)}\big{\\}}_{\dim
X_{4}}$
$\displaystyle=\widehat{\pi}^{(1)}_{\\#}\cdots\widehat{\pi}^{(6)}_{\\#}\epsilon_{2}^{(6)*}\pi_{\\#}^{(7)}\big{\\{}c\big{(}\mathcal{P}^{(7)}\big{)}\cap
s\big{(}W_{1}^{(7)},\overline{F}_{2}^{(7)}\big{)}\big{\\}}_{\dim X_{4}}$
$\displaystyle=\dotsb=\epsilon^{*}\pi^{(1)}_{\\#}\cdots\pi^{(7)}_{\\#}\big{\\{}c\big{(}\mathcal{P}^{(7)}\big{)}\cap
s\big{(}W_{1}^{(7)},\overline{F}_{2}^{(7)}\big{)}\big{\\}}_{\dim X_{4}}$
$\displaystyle=\epsilon^{*}e(W_{i}),$

as desired. ∎

######  Lemma 7.4.

Assume $(F/Y,D)$ is strongly $8$-generic. Then $C_{i}$ depends just on the
analytic type of $D_{\pi(x)}$ at $x$, but is otherwise inpenendent of the
choice of $(F/Y,D)$.

###### Proof.

Recall that $x$ is an ordinary quadruple point of
$\widehat{D}_{\widehat{\pi}(x)}$. Let $(\mathbb{V}/\mathbb{B},\mathbb{D})$ be
its versal deformation; see [12, Example 14.0.1, p. 101 and Theorem 14.1, p.
103]. Recall how $(\mathbb{V}/\mathbb{B},\mathbb{D})$ is constructed. Take
variables $t_{1},\dots,t_{9},u,v$. Identify $\widehat{F}_{\pi(x)}$ with
$\operatorname{Spec}k[[u,v]]$. Say $\widehat{D}_{\pi(x)}$ is defined by
$f(u,v)$ in $k[[u,v]]$, and choose $g_{1},\dots,g_{9}$ in $k[u,v]$ whose
classes in $k[[u,v]]/(f,f_{u},f_{v})$ form a basis of that vector space. Then

$\mathbb{B}:=\operatorname{Spec}k[t_{1},\dots,t_{9}]\qquad\text{and}\qquad\mathbb{D}:=\operatorname{Spec}B[[u,v]]\big{/}\Bigl{(}f+\sum
t_{i}g_{i}\Bigr{)}.$

Note that $(\mathbb{V}/\mathbb{B},\mathbb{D})$ depends just on the analytic
type of $D_{\pi(x)}$ at $x$.

Since $x\in\widehat{D}_{\pi(x)}$ is an ordinary quadruple point,
$f=f_{4}+f_{5}+\cdots$, where $f_{4}$ is a product of independent linear
forms. Choose the $g_{i}$ so that only $g_{9}\in(u,v)^{4}$. Define
$\mathbb{B}_{4}$ by the vanishing of $t_{1},\dotsc,t_{8}$. Then
$b\in\mathbb{B}$ lies in $\mathbb{B}_{4}$ iff the fiber $\mathbb{D}_{b}$ has a
quadruple point.

Recall from [12, Theorem 14.1, p. 103] that there exists a map
$\delta\colon\widehat{Y}\to\mathbb{B}$ such that $\widehat{D}$ and
$\mathbb{D}\times_{\mathbb{B}}\widehat{Y}$ become isomorphic after completion
along their fibers over $\widehat{\pi}(x)$. Since $\widehat{D}$ is complete at
$x$, it is already complete along its fiber. Form the completions
$\widehat{\mathbb{V}}$, $\widehat{\mathbb{B}}$, $\widehat{\mathbb{D}}$,
$\widehat{\mathbb{B}}_{4}$ at the origin $b_{0}$ of $\mathbb{B}$. Then
$\delta\colon\widehat{Y}\to\mathbb{B}$ factors through a map
$\widehat{\delta}\colon\widehat{Y}\to\widehat{\mathbb{B}}$, and $\widehat{D}$
is isomorphic to the completion of
$\widehat{\mathbb{D}}\times_{\widehat{\mathbb{B}}}\widehat{Y}$ along its fiber
over $\widehat{\pi}(x)$. Each subscheme $\widehat{X}_{i}$ of $\widehat{D}$ is
the pullback of the corresponding subscheme of
$\widehat{\mathbb{D}}\times_{\widehat{\mathbb{B}}}\widehat{Y}$, which, in
turn, is the pullback of the corresponding subscheme
$\widehat{\mathbb{X}}_{i}$ of $\widehat{\mathbb{D}}$.

Let us show $\widehat{\delta}\colon\widehat{Y}\to\widehat{\mathbb{B}}$ is
flat. Notice
$\widehat{\delta}\big{(}\widehat{\pi}\big{(}\widehat{X}_{4}\big{)}\big{)}\subset\widehat{\mathbb{B}}_{4}$.
But
$\widehat{\delta}\big{(}\widehat{\pi}\big{(}\widehat{X}_{4}\big{)}\big{)}\neq\\{b_{0}\\}$
because $(F/Y,D)$ is strongly 8-generic. However,
$\dim\widehat{\mathbb{B}}_{4}=1$. It follows that
$\operatorname{cod}\big{(}\delta^{-1}(b_{0}),\widehat{\pi}\big{(}\widehat{X}_{4}\big{)}\big{)}=1$.
Now,
$\operatorname{cod}\big{(}\widehat{\pi}(\widehat{X}_{4}),\widehat{Y}\big{)}=8$
by (2.11) since $(F/Y,D)$ is 8-generic. So
$\operatorname{cod}\big{(}\widehat{\delta}^{-1}(b_{0}),\widehat{Y}\big{)}=9$.
But $\widehat{Y}$ is Cohen–Macaulay, and $\widehat{\mathbb{B}}$ is smooth.
Thus $\widehat{\delta}$ is flat by [10, Proposition 15.4.2, p. 230].

Form the class $e\big{(}\widehat{\mathbb{W}}_{i}\big{)}$ for
$\big{(}\widehat{\mathbb{V}}/\widehat{\mathbb{B}},\widehat{\mathbb{D}}\big{)}$
analogous to the class $e\big{(}\widehat{W}_{i}\big{)}$ for
$\big{(}\widehat{F}/\widehat{Y},\widehat{D}\big{)}$. The map
$\widehat{\mathbb{X}}_{2}\times_{\widehat{\mathbb{B}}}\widehat{Y}\to\widehat{\mathbb{X}}_{2}$
is flat, as it is induced by $\widehat{\delta}$. Since the map
$\widehat{X}_{2}\to\mathbb{X}_{2}\times_{\mathbb{B}}\widehat{Y}$ is also flat,
we can argue as in the proof of Lemma 7.3 to conclude that
$e\big{(}\widehat{\mathbb{W}}_{i}\big{)}$ pulls back to
$e\big{(}\widehat{W}_{i}\big{)}$. Owing to the same flatness, the fundamental
class $\big{[}\widehat{\mathbb{X}}_{4}\big{]}$ on $\widehat{\mathbb{X}}_{2}$
pulls back to the fundamental class $\big{[}\widehat{X}_{4}\big{]}$ on
$\widehat{X}_{2}$.

Form the equation
$e\big{(}\widehat{\mathbb{W}}_{i}\big{)}=\mathbb{C}_{i}\big{[}\widehat{\mathbb{X}}_{4}\big{]}$
on $\widehat{\mathbb{X}}_{2}$ analogous to
$e\big{(}\widehat{W}_{i}\big{)}=C_{i}\big{[}\widehat{X}_{4}\big{]}$ on
$\widehat{X}_{2}$, see (7.1). The former equation pulls back to the latter
owing to the preceding paragraph. Thus $C_{i}=\mathbb{C}_{i}$. But
$\mathbb{C}_{i}$ depends just on $(\mathbb{V}/\mathbb{B},\mathbb{D})$, so just
on the analytic type of $D_{\pi(x)}$ at $x$. Thus $C_{i}$ depends just on the
analytic type of $D_{\pi(x)}$ at $x$. ∎

######  Theorem 7.5.

Assume $(F/Y,D)$ is strongly $8$-generic. Then the correction term (5.3) is
equal to $C[X_{4}]$, where $C$ is independent of the choice of $(F/Y,D)$.

###### Proof.

By Lemma 7.4, each $C_{i}$ depends just on the analytic type of $D_{\pi(x)}$
at $x$; that is, on the cross ratio of the four tangents at $x$. By the last
line in Section 7.1, furthermore, $C_{i}$ is independent of the choice of $x$
in $Z$. Below, we exhibit a pair where $X_{4}$ is irreducible and where any
given value of the cross ratio appears at some $x\in X_{4}$. It follows that
$C_{i}$ is independent of the choice of $(F/Y,D)$. Finally, Proposition 6.4
now implies that $C$ is independent too, as desired

To build the pair, say $k$ is the base field, take variables
$t_{1},\dotsc,t_{8},t,u,v$, and set

$\displaystyle\mathcal{A}:=k\biggl{[}t_{1},\dotsc,t_{8},t,\frac{1}{t},\frac{1}{t-1}\biggr{]},\qquad\mathcal{B}:=A[u,v],\qquad\mathcal{C}:=\mathcal{B}/(g),$

where

$\displaystyle
g:=t_{1}+t_{2}u+t_{3}v+t_{4}u^{2}+t_{5}uv+t_{6}v^{2}+t_{7}u^{2}v+t_{8}uv^{2}+uv(u-v)(u-tv).$

Set $Y:=\operatorname{Spec}\mathcal{A}$ and $F:=\mathbb{P}^{2}_{k}\times Y$
and $D:=\overline{\operatorname{Spec}\mathcal{C}}$.

Then $X_{4}\subset\operatorname{Spec}\mathcal{B}$. Its ideal $\mathcal{I}$ is
generated by the partial derivatives with respect to $u$ and $v$ of $g$ up to
order three; so $I=(t_{1},\dots,t_{8},u,v)$. It follows that
$X_{4}=\operatorname{Spec}(\mathcal{B}/\mathcal{I})=\operatorname{Spec}k\big{[}t,\frac{1}{t},\frac{1}{t-1}\big{]}$.
Thus $X_{4}$ is irreducible.

Given $c\in k$, let $x\in X_{4}$ be the point with $t=c$. Then the fiber
$D_{\pi(x)}\subset\mathbb{P}^{2}_{k}$ is equal to the four lines
$uv(u-v)(u-cv)$ through $(0:0:1)$ with cross ratio equal to $c$. Thus all
cross ratios appear in this family, as desired. ∎

### Acknowledgements

Thanks are due to the referee for pointing out our inadvertent change of
notation from [15], which is discussed immediately after the proof of
Proposition 5.2.

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