# Stable maps of curves and
algebraic equivalence of 1-cycles

János Kollár and Zhiyu Tian<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We show that algebraic equivalence of images of stable maps of curves lifts to
deformation equivalence of the stable maps. The main applications concern
$A_{1}(X)$, the group of 1-cycles modulo algebraic equivalence, for smooth,
separably rationally connected varieties. If $K/k$ is an algebraic extension,
then the kernel of $A_{1}(X_{k})\to A_{1}(X_{K})$ is at most
${\mathbb{Z}}/2{\mathbb{Z}}$. If $k$ is finite, then the image equals the
subgroup of Galois invariant cycles.

For a scheme $X$, let $\operatorname{Mor}(\mbox{Curves},X)$ denote the stack
of morphisms from proper, nodal curves to $X$. If two morphisms
$\pi_{i}:C_{i}\to X$ are deformation equivalent (see Definition 12), then the
image 1-cycles $(\pi_{i})_{*}[C_{i}]$ are algebraically equivalent. The
converse, however, rarely holds.

Nonetheless, we show that if $X$ is smooth, then algebraic equivalence of the
image cycles lifts to deformation equivalence of morphisms. We use
$A\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}B$ to indicate that all
intersection points $A\cap B$ are nodes; see Notation 16.

######  Theorem 1.

Let $X$ be a smooth, projective variety over an algebraically closed field
$K$. Let $\pi_{i}:C_{i}\to X$ (for $i\in I$) be finitely many morphisms of
nodal curves to $X$ such that the $(\pi_{i})_{*}[C_{i}]$ are algebraically
equivalent to each other. Then there is a nodal deformation equivalence (as in
Definition 12)

$None$ $\begin{array}[]{ccccc}C_{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}R_{i}&\subset&S&\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}&B\times
X\\\ \downarrow&&\downarrow&&\downarrow\\\ b_{i}&\in&B&=&B,\end{array}$

such that

  1. (1.2)

$(\pi|_{C_{i}}:C_{i}\to X)\cong(\pi_{i}:C_{i}\to X)$ for $i\in I$, and the

  2. (1.3)

$(\pi|_{R_{i}}:R_{i}\to X)$ are isomorphic to each other for $i\in I$.

That is, the algebraic equivalence is visible at the level of maps, not just
at the level of cycles.

For arithmetic applications, the following version is quite useful.

######  Theorem 2.

Let $k$ be a perfect field with algebraic closure $K$, and $k\subset L\subset
K$. Let $X$ be a smooth, projective, $k$-variety and $\pi_{i}:C_{i}\to X_{L}$
(for $i\in I$) morphisms of nodal curves defined over $L$, such that the
$(\pi_{i})_{*}[C_{i}]$ are algebraically equivalent to each other over $K$.
Assume also that $\chi(C_{i},{\mathcal{O}}_{C_{i}})$ is independent of $i\in
I$.

Then one can choose (1.1) such that, in addition to (1.2–3),

  1. (2.1)

the $\pi|_{C_{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.26582pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}R_{i}}:C_{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}R_{i}\to X_{L}$ are
defined over $L$ for $i\in I$, and the

  2. (2.2)

$(\pi|_{R_{i}}:R_{i}\to X_{L})$ are defined over $k$.

######  Remark 3.

There are some unusual features of Theorem 2.

First, we assume only that the $(\pi_{i})_{*}[C_{i}]$ are algebraically
equivalent over $K$. Correspondingly, the diagram (1.1) itself is defined only
over $K$. It is only the fibers over the points $b_{i}$ that are defined over
$L$.

Second, the $R_{i}$ are defined over the smallest possible field, $k$.

Third, the assumption on $\chi(C_{i},{\mathcal{O}}_{C_{i}})$ is necessary, see
Example 9.

From Theorem 2, we can not conclude that the
$C_{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}R_{i}$ are algebraically
equivalent over $k$ or $L$, since the families $S\to B$ are only defined over
$K$. However, we will be able to prove algebraic equivalence for certain
varieties, which we define next.

######  Definition 4.

Let $X$ be a smooth, proper variety over an algebraically closed field. Given
any $g:{\mathbb{P}}^{1}\to X$, the pull-back $g^{*}T_{X}$ decomposes as
$\sum_{i}{\mathcal{O}}_{{\mathbb{P}}^{1}}(a_{i})$. It is called very free if
$a_{i}>0$ for every $i$, and almost very free if $a_{i}\geq 0$ for every $i$,
with strict inequality for all but one $a_{i}$.

$X$ is separably rationally connected, or src, if there is a very free $g$;
see [Kol96, Sec.IV.3] or [AK03, Sec.7] for many equivalent definitions.

We say that $X$ is separably rationally connected in codimension 1, or src in
codimension 1, if there is an almost very free $g:{\mathbb{P}}^{1}\to X$. Thus
src implies src in codimension 1. The main new examples are varieties that
admit a morphism to a curve $X\to B$ whose general fibers are smooth and
separably rationally connected. These 2 classes give all examples in
characteristic 0.

There are a few more in positive characteristic. General fibers of $X\to B$
could be singular, but contain very free rational curves in their smooth
locus. There are also rationally connected (but not src) examples, studied in
[She10].

Being src (or src in codimension 1) are both open conditions in smooth
families of projective varieties.

For varieties that are src in codimension 1, we turn the deformation
equivalences in Theorem 2 into algebraic equivalences over $L$. This leads to
a series of arithmetic consequences for 1-cycles. Further applications to the
Brauer-Manin obstruction for zero cycles on geometrically rational surfaces
over global function fields and the coniveau filtration are discussed in
[Tia22a, Tia22b].

######  5Algebraic equivalence of 1-cycles.

Let $X$ be a proper scheme over a field $k$. We use $A_{d}(X)$ for the group
of $d$-dimensional cycles modulo algebraic equivalence, the latter denoted by
$Z_{1}\sim_{a}Z_{2}$. (For an introduction, see [Ful98, Sec.10.3], where this
group is denoted by $B_{d}(X)$.)

Let $L/k$ be an algebraic field extension. We have natural maps
$A_{d}(X_{k})\to A_{d}(X_{L})$. The kernel of this is $\deg(L/k)$-torsion.

If $L/k$ is Galois, then the image is contained in the Galois-invariant
subgroup $A_{d}(X_{L})^{\operatorname{Gal}(L/k)}$. Thus our interest is the
kernel and cokernel of the map

$None$ $A_{1}(X_{k})\to A_{1}(X_{L})^{\operatorname{Gal}(L/k)}.$

The following 3 theorems are proved in Section 7. The first one describes the
kernel of (5.1).

######  Theorem 6.

Let $X_{k}$ be a smooth, projective variety over a perfect field $k$ with
algebraic closure $K$. Assume that $X_{K}$ is src in codimension 1. Then the
kernel of the natural map $A_{1}(X_{k})\to A_{1}(X_{K})$ is either trivial or
${\mathbb{Z}}/2{\mathbb{Z}}$. More precisely,

  1. (6.1)

the kernel is trivial if $X_{k}$ contains an odd degree 0-cycle, and

  2. (6.2)

if $Z=\sum d_{i}C_{i}$ and $Z_{K}\sim_{a}0$, then $Z\sim_{a}0$ iff the index
of $X$ (Definition 49) divides
$\chi(Z):=\sum_{i}d_{i}\chi(C_{i},{\mathcal{O}}_{C_{i}})$.

For finite fields (5.1) is an isomorphism.

######  Theorem 7.

Let $k$ be a perfect field with algebraic closure $K$, and $X$ a smooth,
projective $k$-variety that is src in codimension 1. Assume that every
geometrically irreducible $k$-variety has a 0-cycle of degree 1. (For example,
$k$ is finite or pseudo algebraically closed.) Then

$None$ $A_{1}(X_{k})\to A_{1}(X_{K})^{\operatorname{Gal}(K/k)}\quad\mbox{is an
isomorphism.}\quad$

Let $k$ be a field, $C$ a smooth, projective, geometrically irreducible
$k$-curve. Then $X:=C\times{\mathbb{P}}^{1}$ is src in codimension 1. For
$c\in C(K)$, the line $\\{c\\}\times{\mathbb{P}}^{1}$ gives a class in
$A_{1}(X_{K})^{\operatorname{Gal}(K/k)}$. If (7.1) holds then $C$ has a
0-cycle of degree 1. So the assumption on $k$ in Theorem 7 is necessary in all
cases.

The third application concerns specialization of 1-cycles.

######  Theorem 8.

Let $R$ be a Henselian DVR with perfect residue field. Let
$p:X\to\operatorname{Spec}R$ be a smooth, projective morphism with closed
fiber $X_{0}$ and generic fiber $X_{g}$. Assume that $X_{0}$ is src in
codimension 1.

Then the specialization map $A_{1}(X_{g})\to A_{1}(X_{0})$ is an isomorphism.

Next we give some examples concerning Theorems 2 and 6. From now on we write
$\chi(C):=\chi(C,{\mathcal{O}}_{C})$ for proper, 1-dimensional schemes.

######  Example 9.

Take $X={\mathbb{P}}^{3}$, choose $m>0$ and let $C_{1}\subset{\mathbb{P}}^{3}$
be a degree $2m$ curve of genus 1 over ${\mathbb{Q}}$ such that every 0-cycle
has degree divisible by $2m$. Let $C_{2}$ be the union of $m$ disjoint,
conjugate conics, such that again every 0-cycle has degree divisible by $2m$.
Note that $\chi(C_{1})=0$ and $\chi(C_{2})=m$. Let $R$ be any curve and attach
it to $C_{i}$ at $n_{i}$ nodes, where, necessarily, $2m\mid n_{i}$. Since
$\chi(C_{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}R)=\chi(C_{i})+\chi(R)+n_{i}$,
we obtain that

$\chi(C_{1}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}R)-\chi(C_{2}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}R)\equiv m\mod 2m.$

The Euler characteristic is deformation invariant, so the
$C_{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}R$ are not deformation
equivalent.

######  Example 10.

In connection with Theorem 6, we give 3 examples computing the kernel of
$A_{1}(X_{k})\to A_{1}(X_{K})$.

(10.1) Let $Q_{k}\subset{\mathbb{P}}^{4}$ be a quadric without $k$-points. Let
$C_{1}\subset Q_{k}$ be a smooth conic and $C_{2}$ the union of a conjugate
pair of lines. For example, for any $k\subset{\mathbb{R}}$ we can take

$\begin{array}[]{l}Q=(\textstyle{\sum}_{i}x_{i}^{2}=0)\subset{\mathbb{P}}^{4},\
C_{1}=(x_{3}=x_{4}=0),\quad\mbox{and}\\\
C_{2}=(x_{0}=x_{1}-ix_{2}=x_{3}-ix_{4}=0)\cup(x_{0}=x_{1}+ix_{2}=x_{3}+ix_{4}=0).\end{array}$

Then $C_{1}-C_{2}$ is algebraically equivalent to 0 over $K$, but not
algebraically equivalent to 0 over $k$ by (51) since $\chi(C_{1})=1$ and
$\chi(C_{2})=2$.

Thus the kernel of $A_{1}(Q_{k})\to A_{1}(Q_{K})$ is
${\mathbb{Z}}/2{\mathbb{Z}}$.

(10.2) Let $\pi:X\to B$ be a morphism of smooth projective varieties. Assume
that $\dim B=1$ and the generic fiber has odd index. Write the index of $B$ as
$2^{a}(\mbox{odd})$. Then the index of $X$ is also $2^{a}(\mbox{odd})$. Let
$Z_{1},Z_{2}$ be 1-cycles on $X$ that are algebraically equivalent over $K$.
Then the $Z_{i}\to B$ have the same degree, so
$\chi(Z_{1})\equiv\chi(Z_{2})\mod 2^{a}$ by (51.1). Using (49) we get that the
index of $X$ divides $\chi(Z_{1})-\chi(Z_{2})$.

So, if the generic fiber is src, then $A_{1}(X_{k})\to A_{1}(X_{K})$ is an
injection.

(10.3) Let $\pi:X\to S$ be a morphism of smooth projective varieties. Assume
that $\dim S=2$ and the generic fiber has odd index. As before, if
$\operatorname{ind}_{S}=2^{a}(\mbox{odd})$, then
$\operatorname{ind}_{X}=2^{a}(\mbox{odd})$. Let $Z_{1},Z_{2}$ be 1-cycles on
$X$ that are algebraically equivalent over $K$. Then $\pi_{*}Z_{1}$ and
$\pi_{*}Z_{2}$ are also algebraically equivalent over $K$. By the adjunction
formula, they have the same Euler characteristic. This implies that
$\chi(Z_{1})\equiv\chi(Z_{2})\mod 2^{a}$ by (51.1). Using (49) we get that the
index of $X$ divides $\chi(Z_{1})-\chi(Z_{2})$.

So, if $S$ is geometrically rational or ruled, and the generic fiber is src,
then $A_{1}(X_{k})\to A_{1}(X_{K})$ is an injection.

######  11Steps of the proof.

Assume for simplicity that $C_{1},C_{2}\subset X$ are algebraically equivalent
smooth curves of the same genus. By definition, their algebraic equivalence is
certified (15) by a deformation

$None$
$\begin{array}[]{ccccc}S_{i}&\subset&S&\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}&B\times
X\\\ \downarrow&&\downarrow&&\downarrow\\\ b_{i}&\in&B&=&B,\end{array}$

such that $\pi_{*}[S_{1}]=[C_{1}]+Z$ and $\pi_{*}[S_{2}]=[C_{2}]+Z$ for some
1-cycle $Z$. After applying semistable reduction to $S\to B$, we may assume
that all fibers of $S\to B$ are nodal.

Note that in general the $S_{i}$ need not have irreducible components
$C^{\prime}_{i}\subset S_{i}$ that are isomorphic to the $C_{i}$. Even if they
do, and we write $S_{i}=C^{\prime}_{i}\cup R_{i}$, the curves $R_{i}$ and the
maps $\pi_{i}:=\pi|_{R_{i}}:R_{i}\to\operatorname{Supp}Z$ can be very
different from each other.

The moduli space of morphisms of curves to ${\mathbb{P}}^{1}$ was studied by
Hurwitz. Similarly, we view the $R_{i}$ as ‘covers’ of the same
$\operatorname{Supp}Z$, though the maps $R_{i}\to\operatorname{Supp}Z$ need
not even be finite. In Section 2 we give a combinatorial description of going
between various such ‘covers’ by gluing additional components and deformation
equivalences.

Next we want to realize the above combinatorial steps as deformation
equivalences. The general problem is essentially the following:

Question 11.2. Write a reducible curve as $C_{1}\cup C_{2}$ and construct
deformations of each $C_{i}$. Can they be glued to a deformation of $C_{1}\cup
C_{2}$?

If the $C_{i}$ are disjoint then yes, but we run into non-flatness at the
intersection points $C_{1}\cap C_{2}$. However, flatness can be restored by
adding an extra component. The general framework is discussed in Section 3 and
applied to our current setting in Section 4. The main technical result is
(31), which replaces (11.1) with another certificate

$None$ $\begin{array}[]{ccccc}C_{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}R_{i}&\subset&S_{R}&\stackrel{{\scriptstyle\pi_{R}}}{{\longrightarrow}}&B\times
X\\\ \downarrow&&\downarrow&&\downarrow\\\ b_{i}&\in&B&=&B,\end{array}$

such that now $(\pi_{R}:R_{1}\to X)\cong(\pi_{R}:R_{2}\to X)$. (31) also gives
very detailed information on how $S_{R}$ is obtained from $S$.

This completes the geometric part of the argument, proving Theorem 1.

If the $C_{i}$ are defined over some field $k$, then we first construct (11.3)
over the algebraic closure $K$, and then in (36) construct another
certificate, where

$C_{i}+R_{i}\quad\mbox{is replaced by}\quad C_{i}+(\mbox{all Galois conjugates
of }R_{i}).$

This gives us Theorem 2 and completes the part of the proof that holds for all
smooth, projective varieties.

The base curve $B$ of the deformation in (11.3) is geometrically connected,
but usually not definable over $k$, so it says nothing about algebraic
equivalence over $k$.

Now we face the following

Question 11.4. Let $Y$ be a $k$-variety, and $Z_{i}\subset Y$ curves such that
$[Z_{1}]$ and $[Z_{2}]$ are in the same geometrically connected component of
$\operatorname{Hilb}_{1}(Y)$. Are $[Z_{1}]$ and $[Z_{2}]$ algebraically
equivalent over $k$?

The answer is negative in general, and this is where we need that $X$ be src
in codimension 1. We show in (45) that there is a third curve $T$ such that
$[Z_{1}\cup T]$ and $[Z_{2}\cup T]$ are smooth points on the same irreducible
component of $\operatorname{Hilb}_{1}(Y)$ that is also geometrically
irreducible. Thus $[Z_{1}\cup T]$ and $[Z_{2}\cup T]$ are algebraically
equivalent over $k$, and so are $[Z_{1}]$ and $[Z_{2}]$.

This gives us Theorem 6, while Theorems 7–8 require a more careful
understanding of the above procedure.

######  Acknowledgments.

We thank Jean-Louis Colliot-Thélène, Rahul Pandharipande and Olivier
Wittenberg for many helpful and constructive comments. Partial financial
support to JK was provided by the NSF under grant number DMS-1901855, and to
ZT by NSFC grants No.11890660 and No.11890662.

##  1\. Notation and definitions

######  Definition 12.

Let $X$ be a proper scheme over an algebraically closed field $K$. A
deformation equivalence (of curves mapping to $X$) is a diagram

$None$
$\begin{array}[]{ccccc}S_{i}&\subset&S&\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}&B\times
X\\\ \downarrow&&\downarrow&&\downarrow\\\ b_{i}&\in&B&=&B,\end{array}$

such that

  1. (12.2)

$B$ is a connected curve with smooth marked points $b_{i}$,

  2. (12.3)

$S\to B$ is flat, proper, of pure relative dimension 1, and

  3. (12.4)

$S_{i}:=S_{b_{i}}$ is the fiber over $b_{i}$.

The $b_{i}$ and the intersection points of different irreducible components of
$B$ are the pivot points.

Such a deformation is called nodal (or 2-nodal) if all fibers are nodal
curves.

We call a deformation 3-nodal if all but finitely many fibers are nodal
curves, and the pivot fibers have only nodes (like 2 coordinate axes
$(xy=0)\subset{\mathbb{A}}^{2}$) and triple points (like 3 coordinate axes
$(xy=yz=zx=0)\subset{\mathbb{A}}^{3}$).

Our constructions naturally produce 3-nodal deformations, we then make them
2-nodal at the end using (17).

Note on terminology. 3-nodal is not standard. These are sometimes called
ordinary triple points, but that name usually refers to planar triple points
whose tangent cone is 3 distinct lines.

######  13Base change.

Let $(B^{\prime},b^{\prime}_{i})$ be another connected curve with smooth
marked points, and $r:B^{\prime}\to B$ a morphism such that
$r(b^{\prime}_{i})=b_{i}$. By pull-back we get a new deformation equivalence

$B^{\prime}\leftarrow S^{\prime}:=B^{\prime}\times_{B}S\to B^{\prime}\times
X.$

If $S\to B$ is nodal, then so is $S^{\prime}\to B^{\prime}$. If $S^{\prime}\to
B^{\prime}$ is 3-nodal, and the fibers of $S\to B$ over the images of the
pivot points of $B^{\prime}$ are 3-nodal, then $S^{\prime}\to B^{\prime}$ is
also 3-nodal. (For us $B^{\prime}\to B$ will usually come from a general
choice, so the extra condition will not be a problem.)

During the proofs we have to make many base changes, and usually we use $B$ to
denote all base curves.

######  14Improving the singularities of $S\to B$.

Let $p:S\to B$ be a flat, projective morphism of a surface to an irreducible
curves. The semistable reduction theorem [DM69] says that there is a base
change $B^{\rm ss}\to B$, and a resolution of singularities $S^{\rm ss}\to
B^{\rm ss}\times_{B}S$ such that the resulting $S^{\rm ss}\to B^{\rm ss}$ is
smooth over generic points, and the singular fibers are all nodal.

If $B$ is reducible, we could apply this to each irreducible component
$S_{i}\to B_{i}$. However, the fibers of $S^{\rm ss}_{i}\to B^{\rm ss}_{i}$
and of $S^{\rm ss}_{j}\to B^{\rm ss}_{j}$ over $B_{i}\cap B_{j}$ are usually
not isomorphic. So the families $S^{\rm ss}_{i}\to B^{\rm ss}_{i}$ can not be
glued together into a flat family over a connected curve.

Nonetheless, if we have a deformation equivalence as in (12.1) where all but
finitely many fibers are nodal, then, after a suitable base change
$B^{\prime}\to B$, we can get a new family $S^{\prime}\to B^{\prime}$ where
all fibers over smooth points of $B^{\prime}$ are nodal and the fibers over
the pivot points are unchanged.

For this reason, we can and will ignore possible badly singular fibers over
smooth points of $B$.

######  Definition 15.

Let $X$ be a proper scheme over a field $k$ and $Z$ a 1-cycle on $X$. We say
that the deformation (12.1) certifies that $Z\sim_{a}0$ iff
$\pi_{*}[S_{1}]-\pi_{*}[S_{2}]=Z$, as 1-cycles on $X$.

By definition, if $Z\sim_{a}0$, there is always a certificate $B\leftarrow
S\to B\times X$ where $B$ is a smooth, irreducible curve. We can apply stable
reduction to $B\leftarrow S$ and get certificate $B\leftarrow S^{\rm nodal}\to
B\times X$ where the fibers of $S^{\rm nodal}\to B$ are nodal curves.

We can also assume that the general fibers are smooth, but this usually makes
the $S_{i}$ disconnected. For us, connectedness is more important.

######  Notation 16.

Let $X_{i}$ be schemes, $Z_{i}\subset X_{i}$ subschemes and $\sigma:Z_{1}\cong
Z_{2}$ an isomorphism. We use $X_{1}\amalg_{\sigma}X_{2}$ to denote the scheme
obtained by identifying $Z_{1}$ and $Z_{2}$.

Let $C_{i}$ be curves, $Z_{i}\subset C_{i}$ sets of smooth points and
$\sigma:Z_{1}\cong Z_{2}$ an isomorphism. Instead of
$C_{1}\amalg_{\sigma}C_{2}$ we use
$C_{1}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}_{\sigma}C_{2}$ (where the
$\times$ reminds us that we get a node). We use
$C_{1}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}C_{2}$ if the $Z_{i}$ and
$\sigma$ are clear or not important.

######  17From 3-nodes to nodes.

Let $C_{3}\subset{\mathbb{P}}^{3}$ be 3 general lines meeting at point $p$.
Using (38.1–2) we compute that the Hilbert scheme of ${\mathbb{P}}^{3}$ is
smooth of dimension 12 at $[C_{3}]$, and includes the (3-dimensional)
universal deformation of the singularity at the intersection point.

Blow up $p$, let $E\cong{\mathbb{P}}^{2}$ be the exceptional divisor. Let
$\bar{C}_{3}$ be a curve consisting of the birational transform
$C^{\prime}_{3}$ of $C_{3}$, and a conic $C_{0}\subset E$ passing through the
3 points $E\cap C^{\prime}_{3}$. Although the conic is not unique,
$\pi:\bar{C}_{3}\to C_{3}$ is unique as an abstract curve.

The space of morphisms $\operatorname{Mor}(\mbox{Curves},{\mathbb{P}}^{3})$ is
also smooth of dimension 12 at $[\pi:\bar{C}_{3}\to{\mathbb{P}}^{3}]$; cf.
[AK03, 15]. Since $R^{1}\pi_{*}{\mathcal{O}}_{\bar{C}_{3}}=0$, $\pi$ extends
to a morphism

$None$ $\Pi:\operatorname{Mor}(\mbox{Nodal
curves},{\mathbb{P}}^{3})^{\circ}\to\operatorname{Hilb}_{1}({\mathbb{P}}^{3})^{\circ}$

in suitable open neighborhoods of $[\pi:\bar{C}_{3}\to{\mathbb{P}}^{3}]$ and
$[C_{3}]$. Thus $\Pi$ is a local isomorphism. That is, every flat deformation
of the 3-node $C_{3}$ can be lifted to a flat deformation of the morphism
$\pi:\bar{C}_{3}\to C_{3}$. Since $\bar{C}_{3}$ has only nodes, this replaces
3-nodal families with nodal ones. In general, for any $X$ we get

$None$ $\Pi_{X}:\operatorname{Mor}(\mbox{Nodal
curves},X)\dasharrow\operatorname{Hilb}_{1}(X),$

which restricts to an isomorphism over the 3-nodal locus of
$\operatorname{Hilb}_{1}(X)$.

(In fact, nodes and 3-nodes are the only singularities over which $\Pi_{X}$ is
a local isomorphism.)

The following explicit example may be useful. It shows how to ‘jump over’ the
nodes as we slide an auxiliary curve $L$ from one irreducible component of a
nodal curve $C$ to another.

Consider a family of nodes parametrized by a node:

${\mathbb{A}}^{4}_{xyuv}\supset
S:=(xy=uv=0)\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}(xy=0)=:B\subset{\mathbb{A}}^{2}_{xy}.$

Let $D\subset S$ be the image of the section $\sigma(x,y)=(x,y,x,y)$. Thus,
along the $x$-axis we move in the $u$-axis, along the $y$-axis we are in the
$v$-axis. At the origin $(x=y=0)$ we hit the node $(u=v=0)$.

Let $D^{\prime}\subset S$ be the image of the section
$\sigma^{\prime}(x,y)=(x,y,y,x)$. Its equation is $(v-x=u-y=0)$.

Note that $D+D^{\prime}$ is Cartier divisor, with defining equation $x+y=u+v$.

Blow up $D^{\prime}\subset S$ to get $\bar{S}\to S$. The restriction of
$D^{\prime}$ to the $(x=v=0)$ and $(y=u=0)$ planes is a Cartier divisor, so on
these the blow-up is the identity. The restriction of $D^{\prime}$ to the
$(x=u=0)$ and $(y=v=0)$ planes is the origin, so we get their usual blow-up.
The 2 exceptional curves are identified in $\bar{S}$ to get
$\bar{E}\subset\bar{S}$.

The new central fiber consists of $\bar{E}$ and the birational transforms of
$(0,0,u,0)$ and $(0,0,0,v)$, meeting $\bar{E}$ at distinct points
$p_{u},p_{v}$. The birational transform of $D$ is still a section $\bar{D}$,
meeting $\bar{E}$ at a point different from the $p_{u},p_{v}$. Thus in the new
family $\bar{S}\to B$, the section $\bar{D}$ is a Cartier divisor that does
not pass through any of the nodes of the fibers.

##  2\. Maps to curves

Hurwitz proved that for fixed $(g,d)$, all degree $d$ morphisms from genus $g$
smooth curves to ${\mathbb{P}}^{1}$ form an irreducible family. We discuss
what happens when ${\mathbb{P}}^{1}$ is replaced by an arbitrary (possibly
reducible) curve $C$ and we study maps from nodal curves to $C$.

Irreducibility fails already in simple cases. For example, let
$C:=(xyz=0)\subset{\mathbb{P}}^{2}$ be a cycle of 3 lines. Separating one of
its nodes we get 3 morphisms $\pi_{i}:A\to C$ where $A$ is a chain of 3
rational curves, so $\chi(A)=1$. These 3 maps are distinct, rigid and all
their numerical invariants agree.

We aim to prove that by adding more irreducible curves to $A$, we get a
connected moduli space.

######  Definition 18.

Let $k$ be an algebraically closed field, $C$ a proper curve, $A,B$ proper
nodal curves and $\pi_{A}:A\to C,\pi_{B}:B\to C$ morphisms. The following
operations are the elementary changes of these data.

  1. (18.1)

Take a smooth, projective, irreducible curve $\tau:E\to C$ and attach it to
$A$ and $B$ along two subsets of the same cardinality, to get
$\pi^{\prime}_{A}:A^{\prime}:=A\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}E\to C$ and
$\pi^{\prime}_{B}:B^{\prime}:=B\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}E\to C$.

  2. (18.2)

Take constant morphisms $\tau_{A}:{\mathbb{P}}^{1}\to
C,\tau_{B}:{\mathbb{P}}^{1}\to C$ and attach then to $A$ (resp. $B$) along two
subsets of the same cardinality, to get
$\pi^{\prime}_{A}:A^{\prime}:=A\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}{\mathbb{P}}^{1}\to C$ and
$\pi^{\prime}_{B}:B^{\prime}:=B\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}{\mathbb{P}}^{1}\to C$.

  3. (18.3)

Write $A=E\cap F$. Choose $\pi^{\prime}_{F}:F^{\prime}\to C$ to be deformation
equivalent to $\pi|_{F}$ fixing $E\cap F$. Set

$\pi^{\prime}_{A}:(\pi|_{E},\pi^{\prime}_{F}):A^{\prime}:=E\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}F^{\prime}\to
C\quad\mbox{and}\quad\pi^{\prime}_{B}=\pi_{B}:B\to C.$

  4. (18.4)

As in (3) with the roles of $A,B$ reversed.

The main result of this section is the following.

######  Proposition 19.

Let $k$ be an algebraically closed field, $C$ a proper curve, $A,B$ proper
nodal curves and $\pi^{1}_{A}:A^{1}\to C,\pi^{1}_{B}:B^{1}\to C$ morphisms.
The following are equivalent.

  1. (19.1)

$(\pi^{1}_{A})_{*}[A^{1}]=(\pi^{1}_{B})_{*}[B^{1}]$ and
$\chi(A^{1})=\chi(B^{1})$.

  2. (19.2)

There is a sequence of elementary changes (18.1–4)

$\bigl{(}\pi^{1}_{A}:A^{1}\to C,\pi^{1}_{B}:B^{1}\to
C\bigr{)}\dasharrow\cdots\dasharrow\bigl{(}\pi^{r}_{A}:A^{r}\to
C,\pi^{r}_{B}:B^{r}\to C\bigr{)}$

such that $\bigl{(}\pi^{r}_{A}:A^{r}\to
C\bigr{)}\cong\bigl{(}\pi^{r}_{B}:B^{r}\to C\bigr{)}$.

Proof. It is clear that elementary changes preserve both
$(\pi^{i}_{A})_{*}[A^{i}]-(\pi^{i}_{B})_{*}[B^{i}]$ and
$\chi(A^{i})-\chi(B^{i})$, so (2) $\Rightarrow$ (1).

For the converse, we may assume that $\pi^{1}_{A},\pi^{1}_{B}$ are dominant.
The claim is clear if $A^{1}$ or $B^{1}$ are empty.

Let $C_{j}$ be the irreducible components of $C$ with normalizations
$\bar{C}_{j}\to C_{j}$. Take curves $\bar{A}^{1}_{j}\to\bar{C}_{j}$ and
$\bar{B}^{1}_{j}\to\bar{C}_{j}$ such that $A^{1}$ (resp $B^{1}$) is obtained
from $\amalg_{j}\bar{A}^{1}_{j}$ (resp. $\amalg_{j}\bar{B}^{1}_{j}$) by gluing
some point pairs to nodes. These $\bar{A}^{1}_{j},\bar{B}^{1}_{j}$ are almost
unique, except for irreducible components of $A^{1}\cup B^{1}$ that lie over a
singular point of $C$. For these we have finitely many possible choices, we
make any one of them.

Our plan is to show that our claim holds each
$\bigl{(}\pi^{1}_{A}:\bar{A}^{1}_{j}\to\bar{C}_{j},\pi^{1}_{B}:\bar{B}^{1}_{j}\to\bar{C}_{j}\bigr{)}$.
However, $\chi(\bar{A}^{1}_{j})$ can be different from
$\chi(\bar{B}^{1}_{j})$. So first we need a series of elementary changes to
ensure that $\chi(\bar{A}^{i}_{j})=\chi(\bar{B}^{i}_{j})$ for some $i\geq 1$.

It will be convenient to add disjoint copies of $\amalg_{j}\bar{C}_{j}$ to
both $A^{1}$ and $B^{1}$, in order to avoid some trivial special cases.

We have to pay special attention to the nodes of $A^{1}$ and $B^{1}$ that get
separated when we pass to $\amalg_{j}\bar{A}^{1}_{j}$ and
$\amalg_{j}\bar{B}^{1}_{j}$. So let $c\in C$ be a singular point and
$C_{j}(c)$ the local branches of $C$ through $c$. Let
$N^{1}_{A}(c,j_{1},j_{2})\subset A^{1}$ be the set of all nodes of $A^{1}$
over $c$ whose 2 branches end up on $C_{j_{1}}(c)$ and $C_{j_{2}}(c)$ when we
pass to $\amalg_{j}\bar{A}^{1}_{j}$. We call these separating nodes. We start
with some elementary changes to achieve that
$\\#N^{i}_{A}(c,j_{1},j_{2})=\\#N^{i}_{B}(c,j_{1},j_{2})$ for every
$c,j_{1},j_{2}$.

So, let us see how to create a new node on $A^{1}$ by elementary changes. This
is done in 2 steps.

  1. (19.1)

First attach a copy of $E_{1}\cong\bar{C}_{j_{1}}$ to both $A^{1}$ and $B^{1}$
at a point as in (18.1).

  2. (19.2)

Then attach a copy of $E_{2}\cong\bar{C}_{j_{2}}$ to
$A\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}E_{1}$ at a point of $A$
and at the point of $E_{1}$ lying over $c$. Also attach a copy of
$E_{2}\cong\bar{C}_{j_{2}}$ to $B\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}E_{1}$ at 2 points of $B$.

After finitely many such steps, we achieve that
$\\#N^{i}_{A}(c,j_{1},j_{2})=\\#N^{i}_{B}(c,j_{1},j_{2})$ for every
$c,j_{1},j_{2}$.

From now on we will not create more separating nodes.

Next we arrange that $\chi(\bar{A}^{i}_{j})=\chi(\bar{B}^{i}_{j})$ for every
$j$. Fix a curve $C_{1}$. If say $\chi(\bar{A}^{i}_{j})>\chi(\bar{B}^{i}_{j})$
for some $j\neq 1$ we use (18.2) to attach a ${\mathbb{P}}^{1}$ at 2 points of
$\bar{A}^{i}_{j}$ and at 2 points of $\bar{B}^{i}_{1}$. This decreases
$\chi(\bar{A}^{i}_{j})$ by $1$ but leaves $\chi(\bar{B}^{i}_{j})$ unaltered.
Repeating as necessary, and increasing the value of $i$, we achieve that
$\chi(\bar{A}^{i}_{j})=\chi(\bar{B}^{i}_{j})$ for every $j\neq 1$. Next note
that

$\chi(A^{i})=\textstyle{\sum}_{j}\chi(\bar{A}^{i}_{j})-\textstyle{\sum}\\#N^{i}_{A}(c,j_{1},j_{2}),\quad\mbox{and}\quad\chi(B^{i})=\textstyle{\sum}_{j}\chi(\bar{B}^{i}_{j})-\textstyle{\sum}\\#N^{i}_{B}(c,j_{1},j_{2}).$

We arranged that all but 1 of the terms in these identities match up, thus
$\chi(\bar{A}^{i}_{1})=\chi(\bar{B}^{i}_{1})$ also holds.

Next choose embeddings $A^{i}\subset C\times{\mathbb{P}}^{3}$ and
$B^{i}\subset C\times{\mathbb{P}}^{3}$ that have the same bidegree and such
that $N^{i}_{A}(c,j_{1},j_{2})=N^{i}_{B}(c,j_{1},j_{2})$ for every
$c,j_{1},j_{2}$.

From now on we work separately with these
$\bigl{(}\pi^{i}_{A}:\bar{A}^{i}_{j}\to\bar{C}_{j},\pi^{i}_{B}:\bar{B}^{i}_{j}\to\bar{C}_{j}\bigr{)}$.
In (20) we prove a slightly stronger version of our claim, where all the steps
leave the set of separated nodes fixed. The 2-nodal deformation equivalence at
the end fixes all the separated nodes. If this holds, then the 2-nodal
deformation equivalences over the individual $\bar{C}_{j}$ glue together to a
2-nodal deformation equivalence over $C$, as claimed. ∎

######  Lemma 20.

Let $C$ be a smooth, projective, irreducible curve and $A^{1},B^{1}\subset
C\times{\mathbb{P}}^{n}$ two nodal subcurves such that
$\bigl{(}\operatorname{bideg}(A^{1}),\chi(A^{1})\bigr{)}=\bigl{(}\operatorname{bideg}(B^{1}),\chi(B^{1})\bigr{)}$.
Let $P\subset A^{1}\cap B^{1}$ be a finite set of points that are smooth on
both $A^{1}$ and $B^{1}$. Then there is a sequence of elementary changes
(18.1–3) fixing $P$ such that $\bigl{(}\pi^{r}_{A}:A^{r}\to
C\bigr{)}=\bigl{(}\pi^{r}_{B}:B^{r}\to C\bigr{)}$.

Proof. First we achieve that $A^{i}\to C$ is finite and separable. To do this,
choose a smooth complete intersection surface $A^{1}\subset S\subset
C\times{\mathbb{P}}^{3}$ and $m>0$ such that ${\mathcal{O}}_{S}(m)(-A^{1})$ is
very ample. A general section of it gives a smooth curve $F\subset S$. We
attach $F$ to $A^{1}$ along $F\cap A^{1}$ and to $B^{1}$ along the same number
if points as in (18.1). Now $A^{2}=A^{1}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}F$ is a very ample curve
on $S$. It is thus linearly equivalent to a curve $A^{3}$ that is smooth,
separable over $C$, and contains $P$. This is (18.3) with $E=\emptyset$. We
can now do the same for the $B$-curves.

Once $A^{i},B^{i}$ are both finite and separable over $C$, in characteristic 0
we could use [GHS02, 1.1]. It should work in any characteristic, but here is a
weaker version that is good enough for us.

Choose a general projection to $C\times{\mathbb{P}}^{1}$. Let the images be
$A^{\prime},B^{\prime}$, and $P^{\prime}\subset A^{\prime}\cap B^{\prime}$ the
image of $P$. We may assume that $A^{\prime},B^{\prime}$ are smooth at
$P^{\prime}$. Since $A^{\prime},B^{\prime}$ have the same bidegree, they have
the same arithmetic genus. Thus we get the same number of new nodes
$N^{\prime}_{A},N^{\prime}_{B}$. After adding the same number of vertical
lines $L^{\prime}_{A}=\cup_{c}\\{c\\}\times{\mathbb{P}}^{1}$, we may assume
that $A^{\prime}+L^{\prime}_{A}$ and $B^{\prime}+L^{\prime}_{B}$ are linearly
equivalent, and such that $|A^{\prime}+L^{\prime}_{A}|(-P^{\prime})$ is very
ample. This gives the deformation equivalence of $A^{\prime}+L^{\prime}_{A}$
and of $B^{\prime}+L^{\prime}_{B}$. The lines $L^{\prime}_{A},L^{\prime}_{B}$
can be lifted to $L_{A},L_{B}$ but we have a problem at the nodes $N^{\prime}$
that are smoothed in this deformation equivalence. However, these can be
lifted to the required deformation equivalence, after a ramified double cover
and adding lines that connect the preimages of the nodes as in (21). ∎

######  Example 21.

Consider the smoothing of the node $(xy=t)\subset{\mathbb{A}}^{3}_{xyt}$. Take
the ramified double cover $t=s^{2}+s^{3}$. (The cube is needed only in
$\operatorname{char}2$.) Now we have
$(xy=s^{2}+s^{3})\subset{\mathbb{A}}^{3}_{xys}$. Blowing up the origin, the
central fiber is the birational transforms of the $x$ and $y$-axes, connected
by the exceptional curve of the blow up.

##  3\. Gluing of deformations

######  22Gluing deformations.

As in (12.1), consider 2 deformations

$None$ $B\longleftarrow
S^{j}\stackrel{{\scriptstyle\pi^{j}}}{{\longrightarrow}}B\times X$

with sections $\sigma^{j}:B\to S^{j}$.

If $\pi^{1}\circ\sigma^{1}\equiv\pi^{2}\circ\sigma^{2}$ then we can glue the 2
deformations along the isomorphism

$S^{1}\supset\sigma^{1}(B)\cong B\cong\sigma^{2}(B)\subset S^{2}$

to get a new deformation

$None$ $B\longleftarrow
S^{1}\amalg_{\sigma}S^{2}\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}B\times
X,$

whose fiber over $b\in B$ is $S_{b}^{1}\amalg_{\sigma_{b}}S_{b}^{2}$.
Furthermore, if the (22.1) are 3-nodal, the $\sigma^{j}(b)$ is never a 3-node,
and $\sigma^{1}(b),\sigma^{1}(b)$ are never both 2-nodes, then (22.2) is also
3-nodal.

The condition $\pi^{1}\circ\sigma^{1}\equiv\pi^{2}\circ\sigma^{2}$ is,
however, very restrictive, so we are unlikely to be able to glue right away.
However, we can always connect any 2 points of $X$ by some auxiliary curve.
Our plan is to glue both $S^{j}$ to the auxiliary family.

######  23Prelimnary steps.

Creating sections 23.1. Given a deformation as in (12.1) such that $S\to B$
has connected fibers. Let $p_{i}\in S_{i}$ be smooth points. Let $B^{\prime}$
be a general member of a sufficiently ample linear system on $S$ passing
through the $p_{i}$. After base change to $B^{\prime}\to B$, the family
$S^{\prime}\to B^{\prime}$ has a section connecting the points
$p^{\prime}_{i}$. If $S\to B$ is 3-nodal, then so is $S^{\prime}\to
B^{\prime}$. (Note that $B^{\prime}$ needs to avoid the 3-nodes, and that we
pick up a new pivot points where $B^{\prime}$ intersects the double curve of
$S$.)

If the $p_{1},p_{2}$ are on different irreducible components of $S$, then
$B^{\prime}$ is reducible. However, if $B$ is irreducible and $p_{1},p_{2}$
are on the same irreducible component of $S$, then $B^{\prime}$ is
irreducible.

Iterating this we can create any number of sections. We may assume that they
meet only at smooth points and transversally. We can now blow up the
intersection points to get a deformation with disjoint sections.

Creating isomorphic base curves 23.2. Given pointed, connected curves
$(B^{1},b^{1}_{i})$ and $(B^{2},b^{2}_{i})$, let $B$ be a general member of a
sufficiently ample linear system on $B^{1}\times B^{2}$ passing through the
$b_{i}:=(b^{1}_{i},b^{2}_{i})$. The projections give base changes
$\beta^{1}:(B,b_{i})\to(B^{1},b^{1}_{i})$ and
$\beta^{2}:(B,b_{i})\to(B^{2},b^{2}_{i})$.

Thus, given 2 families $S^{j}\to B^{j}$, after a base change we may assume
that $B^{1}=B^{2}$. If the $S^{j}\to B^{j}$ are 3-nodal, so are the new
families over $B$.

######  24Joining 2 points.

Let $\tau_{i}:B\to{\mathbb{P}}^{n}$ be 2 morphisms form a curve $B$ to
${\mathbb{P}}^{n}$. For $b\in B$, let $L_{b}\subset{\mathbb{P}}^{n}$ be the
line joining $\tau_{1}(b)$ and $\tau_{2}(b)$. This gives
$T:{\mathbb{P}}^{1}\times B\to{\mathbb{P}}^{n}$ whose restriction to the $0$
and $\infty$ sections give the $\tau_{i}$. There are, however, some degenerate
cases. First, if $\tau_{1}(b)=\tau_{2}(b)$ for every $b$. Second, if
$\tau_{1}(b)=\tau_{2}(b)$ holds only for finitely many $b_{i}$, we get the
lines over $B\setminus\\{b_{i}\\}$ and then taking the limit should give the
lines over the $b_{i}$. This works if the $b_{i}$ are smooth points, but not
otherwise.

We need a similar systematic way to connect 2 points in any variety $X$.
Instead of lines, we use complete intersection curves. In order to avoid the
above problems with coinciding points, we use complete intersection curves in
$X\times{\mathbb{P}}^{1}$.

######  25$H$-complete intersection curves.

Let $|H|$ be a sufficiently ample linear system on $X$ and $n:=\dim X$. (For
example, any $H\sim mA$ works, where $A$ is very ample and $m\geq 2$.)
Consider $X\times{\mathbb{P}}^{1}$ with coordinate projections $p_{i}$. We
have the linear systems $p_{1}^{*}|H|$ and
$|H^{*}|:=p_{1}^{*}|H|+p_{2}^{*}|{\mathcal{O}}_{{\mathbb{P}}^{1}}(2)|$.

We have a universal family

$None$ $M_{H}\stackrel{{\scriptstyle u}}{{\longleftarrow}}(T_{H}\subset
U_{H})\stackrel{{\scriptstyle\rho}}{{\longrightarrow}}X,$

where $M_{H}$ parametrizes objects $(T\subset L=H_{1}\cap\cdots\cap
H_{n-1}\cap H^{*}_{n}),$ where $T$ is a length 2 subscheme of
$X\times{\mathbb{P}}^{1}$, and $L$ is a smooth, complete intersection of
$H_{1},\dots,H_{n-1}\in p_{1}^{*}|H|$ and of $H^{*}_{n}\in|H^{*}|$. We refer
to this as the $H^{\rm ci}$ family. Its members are double covers of
$H$-complete intersection curves in $X$.

Let $Y$ be a smooth, proper variety. The Hilbert scheme of length 2 subschemes
with a marked point is $\operatorname{Hilb}_{(2)}(Y)\cong B_{\Delta}(Y\times
Y)$, the blow-up of the diagonal of $Y\times Y$.

For the $H^{\rm ci}$ family, $M_{H}$ is an open subset of a fiber bundle over
$\operatorname{Hilb}_{(2)}(X\times{\mathbb{P}}^{1})$ whose fiber is a product
of projective spaces, and the induced map
$M_{H}\to\operatorname{Hilb}_{(2)}(X\times{\mathbb{P}}^{1})$ is surjective.

In particular, we have the following lifting property.

Claim 25.2. Let $B$ be a reduced curve, $b_{j}\in B$ a finite set of points
and $g:B\to\operatorname{Hilb}_{(2)}(X\times{\mathbb{P}}^{1})$ a morphism. For
each point $b_{j}$, choose $m_{j}\in M_{H}$ such that
$\rho(T_{m_{j}})=g(b_{j})$. Then there is a rational map
$g^{\prime}:B\dasharrow M_{H}$ such that

  1. (a)

$\rho\circ g^{\prime}=g$,

  2. (b)

$g^{\prime}$ is defined at the poins $b_{j}$, and $g^{\prime}(b_{j})=m_{j}$. ∎

######  Corollary 26.

Let $X$ be a smooth, projective variety and fix an $H^{\rm ci}$ family as in
(25.1). Let $B$ be a reduced curve, $b_{i}\in B$ a finite set of points and
$\tau^{j}:B\to X$ two morphisms. Choose liftings $\tilde{\tau}^{j}:B\to
X\times{\mathbb{P}}^{1}$ such that $\tilde{\tau}^{1}(b)=\tilde{\tau}^{2}(b)$
happens only at finitely many smooth points. Thus we have
$g:=(\tilde{\tau}^{1},\tilde{\tau}^{2}):B\to\operatorname{Hilb}_{(2)}(X\times{\mathbb{P}}^{1})$.
For each $b_{i}$ choose $H^{\rm ci}$ curves $L_{i}$ through $g(b_{i})$.

Then there is a deformation

$None$
$\begin{array}[]{ccccc}H(\tau)_{i}&\subset&H(\tau)&\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}&B\times
X\\\ \downarrow&&\downarrow&&\downarrow\\\ b_{i}&\in&B&=&B,\end{array}$

with 2 sections $\sigma^{j}:B\to H(\tau)$ such that

  1. (26.2)

$\tau^{j}=\pi\circ\sigma^{j}$ for $j=1,2$,

  2. (26.3)

all but finitely many fibers of $H(\tau)\to B$ are $H^{\rm ci}$ curves, and

  3. (26.4)

$H(\tau)_{i}=L_{i}$ for every $i$.

Proof. The lifting of
$g:B\to\operatorname{Hilb}_{(2)}(X\times{\mathbb{P}}^{1})$ to $g^{\prime}:B\to
M_{H}$ is given by (25.2). The pull-back of the universal family by
$g^{\prime}$ gives $H(\tau)$. ∎

We use $H^{\rm ci}$ families to glue curves together. The situation is clear
for disjoint curves.

######  Definition 27.

Let $\pi_{i}:C_{i}\to X$ be reduced curves and

${\mathbf{L}}:=\bigl{\\{}(\tau_{L}:L\to X,p_{1}\neq p_{2}\in L)\\}$

a set of 2-pointed curves. Let
$\operatorname{Join}(C_{1},r{\mathbf{L}},C_{2})$ denote the set of all curves
of the form

$C_{1}\cup_{\sigma_{1}}(\amalg_{j\in J}L_{j})\cup_{\sigma_{2}}C_{2}$

where $\\#J=r$, the $p_{ij}\in L$ are smooth, and
$\sigma_{i}:\amalg_{j}\\{p_{ij}\\}\lhook\joinrel\to C_{i}^{\rm nodal}$ are
injections such that $\tau_{L_{j}}(p_{ij})=\pi_{i}(\sigma_{i}(p_{ij}))$. Let
$\operatorname{Join}^{\rm nodal}(C_{1},r{\mathbf{L}},C_{2})$ be the subset of
those curves for which the $\sigma_{i}(p_{ij})\in C_{i}$ are smooth.

######  Corollary 28.

Let $C_{i}\to X$ be reduced curves and $H$ an ample divisor class as in (25).
Assume that the nodal loci of the $C_{i}$ are connected. Then

  1. (28.1)

$\operatorname{Join}(C_{1},rH^{\rm ci},C_{2})$ is connected, and

  2. (28.2)

$\operatorname{Join}^{\rm nodal}(C_{1},rH^{\rm ci},C_{2})$ is a dense, open
subset of it. ∎

For families with some intersection points, we need to be more careful.

######  Theorem 29.

Let $X$ be a smooth, projective variety over an algebraically closed field
$K$, and $\pi_{i}:(C_{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}D_{i})\to X$ nodal curves
mapping to $X$ for $i=1,2$. Assume that the $C_{i},D_{i}$ are connected,
$\\#(C_{1}\cap D_{1})=\\#(C_{2}\cap D_{2})$ (call this number $r$) and we have
3-nodal deformation equivalences

$\begin{array}[]{ccccc}C_{i}&\subset&S_{C}&\stackrel{{\scriptstyle\pi_{C}}}{{\longrightarrow}}&B_{C}\times
X\\\ \downarrow&&\downarrow&&\downarrow\\\
b_{i}&\in&B_{C}&=&B_{C},\end{array}\quad\mbox{and}\quad\begin{array}[]{ccccc}D_{i}&\subset&S_{D}&\stackrel{{\scriptstyle\pi_{D}}}{{\longrightarrow}}&B_{D}\times
X\\\ \downarrow&&\downarrow&&\downarrow\\\
b_{i}&\in&B_{D}&=&B_{D}.\end{array}$

Fix an $H^{\rm ci}$ family of curves as in (25.1).

Then, after a base change $\beta_{C}:B\to B_{C}$ and $\beta_{D}:B\to B_{D}$,
there is a 3-nodal deformation equivalence glued from $\beta_{C}^{*}(S_{C})\to
B$, $\beta_{D}^{*}(S_{D})\to B$ and $r$ different $H^{\rm ci}$ families.

Proof. By (23), after a base change $\beta_{C}:B\to B_{C}$ and $\beta_{D}:B\to
B_{D}$ we may assume that there are disjoint sections $\sigma^{C}_{j}:B\to
S_{C}$ and $\sigma^{D}_{j}:B\to S_{D}$ connecting the points in $C_{1}\cap
D_{1}$ with the points in $C_{2}\cap D_{2}$. This gives
$\tau^{C}_{j}:=\pi_{C}\circ\sigma^{C}_{j}:B\to X$ and
$\tau^{D}_{j}:=\pi_{D}\circ\sigma^{D}_{j}:B\to X$.

We apply (26) to the pair of morphisms $\tau^{C}_{j},\tau^{D}_{j}$. We choose
liftings $\tilde{\tau}^{C}_{j}:B\to X\times{\mathbb{P}}^{1}$ and
$\tilde{\tau}^{D}_{j}:B\to X\times{\mathbb{P}}^{1}$ such that
$\tilde{\tau}^{C}_{j}(b_{1})=\tilde{\tau}^{D}_{j}(b_{2})$ for every $j$.

Now use (26) to create $H^{\rm ci}$ families $H^{j}\to B$ with 2 sections
$\sigma^{HC}_{j}:B\to H^{j}$ and $\sigma^{HD}_{j}:B\to H^{j}$.

Now first glue $S_{C}$ to each $H^{j}$ using the $r$ section pairs
$(\sigma^{C}_{j},\sigma^{HC}_{j})$. Then glue the resulting surface to $S_{D}$
using the $r$ section pairs $(\sigma^{D}_{j},\sigma^{HD}_{j})$. ∎

######  Complement 30.

It may be worthwhile to write down explicitly what happens at the points
$C_{i}\cap D_{i}$.

By construction, at each point we glue 3 smooth surfaces. In suitable (formal
or étale) local coordinates the gluing problem becomes

$\begin{array}[]{l}S_{C}\cong{\mathbb{A}}^{2}_{xt}\ni(0,t)\leftrightarrow(g(t),t)\in{\mathbb{A}}^{2}_{zt}\cong
H^{j},\quad\mbox{and}\\\
S_{D}\cong{\mathbb{A}}^{2}_{yt}\ni(0,t)\leftrightarrow(h(t),t)\in{\mathbb{A}}^{2}_{zt}\cong
H^{j},\end{array}$

where $g(0)=h(0)=0$. The $(t=0)$ fiber is the 3 coordinate axes, which can be
given as

$\left(\operatorname{rank}\left(\begin{array}[]{ccc}x&0&z\\\
0&y&z\end{array}\right)\leq 1\right)\subset{\mathbb{A}}^{3}_{xyz}.$

By [Art76, §5], any flat deformation is obtained by varying the entries of the
above matrix. In our case, the glued surface can be written as

$\left(\operatorname{rank}\left(\begin{array}[]{ccc}x&0&z{-}g(t)\\\
0&y&z{-}h(t)\end{array}\right)\leq 1\right)\subset{\mathbb{A}}^{4}_{xyzt}.$

##  4\. Modifying deformation equivalences

######  Theorem 31.

Let $X$ be a smooth, projective variety over an algebraically closed field
$k$, and

$\begin{array}[]{ccccc}C_{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}D_{i}&\subset&S&\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}&B\times
X\\\ \downarrow&&\downarrow&&\downarrow\\\ b_{i}&\in&B&=&B\end{array}$

a 3-nodal deformation equivalence. Assume that the $C_{i},D_{i}$ are
connected, $\chi(D_{1})=\chi(D_{2})$ and $\pi_{*}[D_{1}]=\pi_{*}[D_{2}]$.

Then, there is a 3-nodal deformation equivalence

$\begin{array}[]{ccccc}C_{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}R_{i}&\subset&S_{R}&\stackrel{{\scriptstyle\pi_{R}}}{{\longrightarrow}}&B_{R}\times
X\\\ \downarrow&&\downarrow&&\downarrow\\\
b_{i}&\in&B_{R}&=&B_{R},\end{array}$

such that

  1. (31.1)

$(\pi_{R}:R_{1}\to X)\cong(\pi_{R}:R_{2}\to X)$,

  2. (31.2)

$R_{i}\cap C_{i}=D_{i}\cap C_{i}$, and

  3. (31.3)

$S_{R}$ is glued from 4 types of pieces: pull-backs of $S$, deformation
equivalences over
$\operatorname{Supp}\pi(D_{1})=\operatorname{Supp}\pi(D_{2})$, trivial
families $C_{i}\times B^{\prime}\to B^{\prime}$, and $H^{\rm ci}$ families.

Proof. We apply (19) to

$\pi|_{D_{1}}:D_{1}\to\operatorname{Supp}\pi(D_{1})\quad\mbox{and}\quad\pi|_{D_{2}}:D_{2}\to\operatorname{Supp}\pi(D_{2})=\operatorname{Supp}\pi(D_{1}).$

Then we show that each of the steps (18.1–4) can be lifted to 3-nodal
deformation equivalences.

For (18.1) this is achieved using (29): we glue $S\to B$ to the trivial family
$E\times B\to B$ using additional $H^{\rm ci}$ families. For (18.2) we glue
$S\to B$ to the trivial family ${\mathbb{P}}^{1}\times D_{1}\to D_{1}$, again
using additional $H^{\rm ci}$ families.

Steps (18.3–4) are already deformation equivalences.

This is almost what we need, except that in (29) we added a number of $H^{\rm
ci}$ families. Thus the fibers over $b_{1}$ and $b_{2}$ are of the form

$\bigl{(}C_{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}R_{i}\bigr{)}\amalg_{\sigma}\bigl{(}\cup_{j\in
J}L_{ij}\bigr{)}$

where the $L_{ij}$ are $H^{\rm ci}$ curves attached to nodes of
$C_{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}R_{i}$ by $\sigma$. By
construction, we have the same number of $H^{\rm ci}$ curves for $i=1,2$.

We can now slide these $H^{\rm ci}$ curves away from the nodes using (28), to
get the same $H^{\rm ci}$ curves for $i=1,2$. Thus fibers over $b_{1}$ and
$b_{2}$ are now

$\bigl{(}C_{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}R_{i}\bigr{)}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}\bigl{(}\cup_{j\in
J}L^{\prime}_{j}\bigr{)},$

which is what we wanted. ∎

######  Corollary 32.

Let $X$ be a smooth, projective variety over an algebraically closed field
$K$, and $\pi_{i}:A_{i}\to X$ connected, nodal curves mapping to $X$. Assume
that $\chi(A_{1})=\chi(A_{2})$ and
$(\pi_{1})_{*}[A_{1}]=(\pi_{2})_{*}[A_{2}]$. Then there is a 3-nodal
deformation equivalence

$None$ $\begin{array}[]{ccccc}A_{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}L_{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}R_{i}&\subset&S&\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}&B\times
X\\\ \downarrow&&\downarrow&&\downarrow\\\ b_{i}&\in&B&=&B,\end{array}$

such that

  1. (32.2)

$(\pi:R_{1}\to X)\cong(\pi:R_{2}\to X)$,

  2. (32.3)

$L_{i}$ is an $H^{\rm ci}$ curve, meeting both $A_{i}$ and $R_{i}$ at a single
smooth point, and

  3. (32.4)

$S$ is glued from 4 types of pieces as in (31.3), (with $A_{i}$ replacing
$D_{i}$).

Proof. Pick any $H^{\rm ci}$ curve $L$ meeting both $A_{1}$ and $A_{2}$ at a
single smooth point. There is a trivial deformation equivalence
$(A_{1}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}L\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}A_{2})\times{\mathbb{P}}^{1}\to{\mathbb{P}}^{1}$.
Apply (31) with $D_{1}=A_{1}$ and $D_{2}=A_{2}$. ∎

It is worth noting that the fibers of $S\to B$ move very little in
$\operatorname{Chow}_{1}(X)$. To state this, let $|H^{\rm
ci}|\subset\operatorname{Chow}_{1}(X)$ denote the closure of the locus of
images of $H^{\rm ci}$ curves.

######  Complement 33.

We can choose (32.1) with the following property. There is an effective
1-cycle $Z$ and morphisms $\tau_{i}:B\to|H^{\rm ci}|$ such that

$None$ $\pi_{*}[S_{b}]=Z+\textstyle{\sum}_{i}L(\tau_{i}(b))$

for every $b\in B$. (We stress that (33.1) is an actual equality of cycles.) ∎

##  5\. Proof of Theorems 1–2

We start with a key special case, where we have a more precise answer.

######  Proposition 34.

Let $X$ be a smooth, projective variety over an algebraically closed field
$K$, and $\pi_{i}:C_{i}\to X$ connected, nodal curves mapping to $X$. Assume
that $\chi(C_{1})=\chi(C_{2})$ and
$(\pi_{1})_{*}[C_{1}]\sim_{a}(\pi_{2})_{*}[C_{2}]$. Then there is a 3-nodal
deformation equivalence

$None$ $\begin{array}[]{ccccc}C_{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}L_{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}R_{i}&\subset&S&\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}&B\times
X\\\ \downarrow&&\downarrow&&\downarrow\\\ b_{i}&\in&B&=&B,\end{array}$

such that

  1. (34.2)

$(\pi:L_{1}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}R_{1}\to
X)\cong(\pi:L_{2}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}R_{2}\to X)$, and

  2. (34.3)

$L_{i}$ is an $H^{\rm ci}$ curve, meeting both $C_{i}$ and $R_{i}$ at a single
smooth point.

Furthermore, if the $C_{i}$ have a common irreducible component $D$, then we
can also achieve that

  1. (34.4)

$L_{i}$ meets $C_{i}$ at a point on $D$.

Proof. Let $B\leftarrow T\to T\times X$ be a nodal deformation equivalence
certifying $(\pi_{1})_{*}[C_{1}]\sim_{a}(\pi_{2})_{*}[C_{2}]$. We may assume
that $T\to B$ has connected fibers. We also have the trivial deformation
equivalences $C_{i}\times B\to B$. By (29) we can glue these 3 together, using
2 different $H^{\rm ci}$ families to get a 3-nodal deformation equivalence

$None$
$\begin{array}[]{ccccc}T^{\prime}_{i}&\subset&T^{\prime}&\stackrel{{\scriptstyle\pi^{\prime}}}{{\longrightarrow}}&B\times
X\\\ \downarrow&&\downarrow&&\downarrow\\\ b_{i}&\in&B&=&B,\end{array}$

such that $T^{\prime}_{i}=C_{1}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}L_{1}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}T_{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}L_{2}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}C_{2}$.

Apply (31) with $D_{1}=L_{1}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}T_{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}L_{2}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}C_{2}$ and
$D_{2}=C_{1}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}L_{1}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}T_{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}L_{2}$. ∎

######  35Proof of Theorem 1.

First apply (37) to get morphisms of nodal curves
$\bar{\pi}_{i}:C_{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}D\to X$ such that, the
$C_{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}D$ are geometrically
connected, and $\chi(C_{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}D)$ is independent of $i$.

Now we are in the situation of (34). Thus, for any two $i,j$ we have
deformation equivalences between curves of the form

$(C_{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}D)\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}L_{ij}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}R_{ij}\quad\mbox{and}\quad(C_{j}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}D)\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}L_{ij}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}R_{ij}.$

We can choose the $L_{ij}$ to meet the
$C_{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}D$ and
$C_{j}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}D$ at a point of $D$ by
(34.4). Finally, for any $\ell$, take

$None$ $(C_{\ell}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}D)\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}\bigl{(}\amalg_{ij}L_{ij}\bigr{)}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}\bigl{(}\amalg_{ij}R_{ij}\bigr{)}.$

We claim that they are 3-nodal deformation equivalent to each other. By
symmetry consider $\ell=1,2$. Then we have

$None$ $(C_{\ell}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}D)\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}\bigl{(}\amalg_{ij}L_{ij}\bigr{)}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}\bigl{(}\amalg_{ij}R_{ij}\bigr{)}=R_{12}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}L_{12}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}(C_{\ell}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}D)\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}\bigl{(}\amalg^{\prime}_{ij}L_{ij}\bigr{)}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}\bigl{(}\amalg^{\prime}_{ij}R_{ij}\bigr{)}$

where $\amalg^{\prime}$ means that we omit the pair $ij=12$. Here

$R_{12}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}L_{12}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}(C_{1}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}D)\quad\mbox{and}\quad
R_{12}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}L_{12}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}(C_{2}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}D)$

are 3-nodal deformation equivalent by construction. We can now use (28) to add
the $L_{ij}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}R_{ij}$ one at a time (for
$ij\neq 12$).

This gives a deformation equivalence that has some 3-nodal fibers, they can be
eliminated using (17). ∎

######  36Proof of Theorem 2.

We follow the arguments in (35). First apply (37) to get morphisms of nodal
$k$-curves $\bar{\pi}_{i}:C_{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}D\to X$ such that the
$C_{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}D$ are geometrically
connected and $\chi(C_{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}D)$ is independent of $i$.

We are in the situation of (34–35). Thus, working over the algebraic closure
$K\supset k$, we have the curves

$None$ $(C_{\ell}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}D)\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}\bigl{(}\amalg_{ij}L_{ij}\bigr{)}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}\bigl{(}\amalg_{ij}R_{ij}\bigr{)}.$

as in (35.1). Set $p_{ij}=D\cap L_{ij}$.

Next fix a pair $ij$. The plan is to replace
$L_{ij}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}R_{ij}$ by all of its
Galois conjugates $(L_{ij}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}R_{ij})^{\sigma}$ and
attach them to $D$ at the point $p_{ij}^{\sigma}$. In order to keep the new
curve nodal, we need the various $p_{ij}^{\sigma}$ to be different from each
other and from the points in $C_{\ell}\cap D$.

Here it is important that we are free to move the curves $L_{ij}$, and use
(61) to guarantee a choice for which
$L_{ij}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}R_{ij}$ is defined over
$k(p_{ij})$, while avoiding any given finite subset of points of $D$.

We can now do this for all the pairs $ij$ in some order. Of course we need to
make sure that a new $p_{ij}$ is distinct form all nodes created earlier. ∎

In this construction, the degrees $k(p_{ij})/k$ end up different. With a more
careful choice in (61) we can achieve that these degrees are the same. It is,
however, unlikely that one can choose the fields $k(p_{ij})$ the same.

######  Lemma 37.

Let $k$ be a perfect field and $L/k$ a finite field extension. Let $X$ be a
smooth, projective, $k$-variety and $\pi_{i}:C_{i}\to X_{L}$ (for $i\in I$)
finitely many morphisms of reduced curves defined over $L$. Assume that either
$\operatorname{ind}_{C_{i}}=1$ for every $i\in I$, or $\chi(C_{i})$ is
independent of $i$. Then there is a smooth, geometrically irreducible
$k$-curve $\pi_{D}:D\to X$ and morphisms of curves
$\bar{\pi}_{i}:C_{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}D\to X_{L}$ such that

  1. (37.1)

the $(C_{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}D)^{\rm nodal}$ are
geometrically connected,

  2. (37.2)

$\chi(C_{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}D)$ is independent of $i$,
and

  3. (37.3)

$(\bar{\pi}_{i}|_{D}:D\to X)\cong(\pi_{D}:D\to X)$.

Note that the assumption on $\chi(C_{i})$ is necessary by Example 9.

Proof. Working in $Y:=X\times{\mathbb{P}}^{3}$, we may assume that the
$C_{i}\subset Y$ are disjoint. Let $Z_{i}\subset C_{i}^{\rm sm}$ be reduced
subschemes that have at least 1 point on each irreducible component. Let
$D\subset Y$ be a smooth, complete intersection curve, defined over $k$, that
contains the $Z_{i}$. Now attach $D$ to $C_{i}$ at $Z_{i}$ to get
$C_{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}D$. Note that

$\chi(C_{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}D)=\chi(C_{i})+\chi(D)-\deg
Z_{i}.$

If $\operatorname{ind}_{C_{i}}=1$ for every $i$ then we can choose the $Z_{i}$
such that $\chi(C_{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}D)$ is independent of $i$.
We can always choose the $\deg Z_{i}$ to be the same. Then
$\chi(C_{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}D)$ is independent of $i$,
and the other conditions are clear. ∎

##  6\. Smoothing combs

Let $X$ be a smooth projective variety and $C\subset X$ a smooth curve with
normal bundle $N_{C}$. If $N_{C}$ is generated by global sections and
$H^{1}(C,N_{C})=0$, then $C$ can be deformed in all directions inside $X$, but
otherwise $C$ may have no deformations at all. The technique of combs,
originated in [KMM92, GHS03], relies on the observation that if we attach
enough smooth rational curves $R_{i}$ to $C$, and the $N_{R_{i}}$ are
generated by global sections, then the resulting nodal curve
$C\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}\amalg_{i}R_{i}$ has a
better deformation theory than $C$ itself. We call $C$ the handle and the
$R_{i}$ the teeth of the comb $C\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}\amalg_{i}R_{i}$. (See
also [AK03, Sec.6] for an introductory treatment.) The proof uses 2 properties
of the curves $R$. At a general point $c\in C$,

  * •

the tangent space $T_{c}X$ is spanned by $T_{c}C$ and the $T_{c}R$ for all
$R\ni c$, and

  * •

$H^{1}(R,N_{R}(-c))=0$ for all $R\ni c$.

If $X$ is src, then there are rational curves $R$ through every point and in
every direction with ample $N_{R}$, as needed. If $X\to B$ has smooth src
general fibers, then we get the needed rational curves in the smooth fibers,
but not in the singular fibers. So the method only works for handles that
dominate $B$.

Thus, in this case, we proceed in 2 steps. First we attach rational curves to
$H^{\rm ci}$ curves to achieve the needed properties. Then use these
(reducible, high genus) curves as teeth for handles that are contained in the
singular fibers. To accommodate both steps, we work out the details in a
general setting.

Each projective family of nodal curves $B_{R}\leftarrow S_{R}\to B\times X$
can be realized as an embedded surface in $B\times Y$ where
$Y:=X\times{\mathbb{P}}^{3}$. We can thus view the deformation as a
geometrically connected curve
$B_{R}\lhook\joinrel\to\operatorname{Hilb}_{1}(Y)$ that parametrizes nodal
curves.

######  38Hilbert scheme.

For a projective $k$-scheme $Y$, let $\operatorname{Hilb}_{1}(Y)$ denote the
Hilbert scheme of 1-dimensional, closed subschemes. Let
$\operatorname{Hilb}_{1}^{\rm nodal}(Y),\operatorname{Hilb}_{1}^{\rm
3-nodal}(Y)$ denote the subschemes parametrizing nodal (resp. 3-nodal) curves.
We use $\operatorname{Hilb}_{1}^{\rm unobst}(Y)$ to denote the open subscheme
parametrizing reduced curves, all of whose singularities have unobstructed
deformations.

For a curve $[C]\in\operatorname{Hilb}_{1}^{\rm unobst}(Y)$, let
$N_{C}:=\operatorname{\mathcal{H}\\!\it{om}}_{C}(I_{C}/I_{C}^{(2)},{\mathcal{O}}_{C})$
denote its normal sheaf. (It is locally free for nodes, but not for 3-nodes.)
Then

  1. (38.1)

$H^{0}(C,N_{C})$ is the tangent space of $\operatorname{Hilb}_{1}(Y)$ at
$[C]$, and

  2. (38.2)

$H^{1}(C,N_{C})$ is the obstruction space for deforming $C\subset Y$;

see, for example, [Kol96, I.2.14].

We say that $C$ is free if $N_{C}$ is generated by global sections and
$H^{1}(C,N_{C})=0$. Let $C=C_{1}+C_{2}$. Restriction gives a natural injection
$I_{C}/I_{C}^{(2)}\lhook\joinrel\to
I_{C_{1}}/I_{C_{1}}^{(2)}+I_{C_{2}}/I_{C_{2}}^{(2)}$, hence we get

$0\to N_{C_{1}}+N_{C_{2}}\to N_{C_{1}+C_{2}}\to T\to 0,$

where $T$ is supported at $C_{1}\cap C_{2}$. In particular, we get the
following.

Claim 38.3. If $C_{1},C_{2}$ are free and $C_{1}+C_{2}$ is locally
unobstructed, then $C_{1}+C_{2}$ is also free. ∎

The open subscheme parametrizing free curves is denoted by
$\operatorname{Hilb}_{1}^{\rm free}(Y)$. Note that
$\operatorname{Hilb}_{1}^{\rm free}(Y)$ is smooth. It has the following
general properties.

Claim 38.4. Every irreducible component of $\operatorname{Hilb}_{1}^{\rm
free}(Y)$ is a connected component. ∎

Claim 38.5. Assume that $[Z_{1}]$ and $[Z_{2}]$ are smooth $k$-points on the
same irreducible component of $\operatorname{Hilb}_{1}(Y)$. Then
$[Z_{1}]\sim_{a}[Z_{2}]$ over $k$. ∎

######  39Families of teeth.

Let $X$ be a proper variety. A family of pointed nodal curves over $X$ is a
diagram

$None$
$M\stackrel{{\scriptstyle\pi}}{{\longleftarrow}}C_{M}\stackrel{{\scriptstyle\tau}}{{\longrightarrow}}X$

where $\pi:C_{M}\to M$ is a proper, flat morphism with nodal curve fibers,
plus a section $s:M\to C_{M}$ such that $\pi$ is smooth along $s(M)$.

For $p\in M$ let $\tau_{p}:C_{p}\to X$ denote the restriction of $\tau$ to the
fiber over $p$. If every $\tau_{p}$ is an embedding, we have a flat family of
nodal curves in $X$.

For $x\in X$, the condition $\tau\circ s(p)=x$ defines $M_{x}\subset M$ and
the the subfamily $C_{M_{x}}\to M_{x}$.

If $C_{p}\to X$ is an immersion at $s(p)$ for every $p\in M$, then the
derivative of $\tau_{p}$ at $s(p)$ gives $d\tau:M\to{\mathbb{P}}_{X}(T_{X})$.
If $\tau_{p}:C_{p}\to X$ is an embedding, then we have the normal bundle
$N_{C_{p}}$.

The following properties are various versions of saying that the curves
$\tau_{p}:C_{p}\to X$ ‘move freely’ in $X$.

  1. (39.2)

For every $x\in X$, $\tau|_{M_{x}}:C_{M_{x}}\setminus s(M_{x})\to X$ is
equidimensional.

  2. (39.3)

$d\tau:M\to{\mathbb{P}}_{X}(T_{X})$ is smooth and surjective.

  3. (39.4)

$d\tau$ has geometrically irreducible fibers.

  4. (39.5)

$H^{1}\bigl{(}C_{p},N_{C_{p}}(-s(p))\bigr{)}=0$ for every $p\in M$.

Our discussions in (25) show that the family of pointed, smooth $H^{\rm ci}$
curves satisfy (2–4), but they almost never satisfy (5).

However, if $X$ is separably rationally connected, then there is a way to
create a family that also satisfies (5),

######  40Combs.

Let $X$ be a proper variety over an algebraically closed field $K$. Let
${\mathbf{W}}$ be a set of curves on $X$ and

$M\stackrel{{\scriptstyle\pi}}{{\longleftarrow}}T_{M}\stackrel{{\scriptstyle\tau}}{{\longrightarrow}}X\quad\mbox{with}\quad
s:M\to T_{M}$

a family of pointed curves as in (39.1); called teeth from now on, and denoted
by ${\mathbf{T}}$. A comb with handle in ${\mathbf{W}}$ and $r$ teeth in
${\mathbf{T}}$ is a curve $C\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}(\amalg_{i\in
I}T_{p_{i}})$, where $C\in{\mathbf{W}}$, $p_{i}\in M$ and
$\sigma:\amalg_{i}s(p_{i})\lhook\joinrel\to C^{\rm sm}$ is a gluing morphism
such that $\tau(s(p_{i}))=\sigma(s(p_{i}))$ for every $i$. If we allow
$\sigma:\amalg_{i}s(p_{i})\lhook\joinrel\to C^{\rm nodal}$, we get a 3-nodal
comb. It is an embedded comb in the natural map to $X$ is an embedding. Let
$\operatorname{EComb}({\mathbf{W}},r{\mathbf{T}})$ denote the set of all such
subschemes.

A comb is called balanced if each irreducible component of $C$ has at least
$r/(2c)$ teeth attached at smooth points, where $c$ is the number of
irreducible components of $C$. (2 is pretty arbitrary choice, any larger
constant would work for us.)

These form a subset $\operatorname{EComb}^{\rm
bal}({\mathbf{W}},r{\mathbf{T}})\subset\operatorname{EComb}({\mathbf{W}},r{\mathbf{T}})$.

If ${\mathbf{W}},{\mathbf{T}}$ are constructible subsets of
$\operatorname{Hilb}_{1}(X)$, then so are
$\operatorname{EComb}({\mathbf{W}},r{\mathbf{T}})$ and
$\operatorname{EComb}^{\rm bal}({\mathbf{W}},r{\mathbf{T}})$.

The following two theorems are slight generalizations of [GHS03, Sec.2], see
also [AK03, Sec.6].

######  Theorem 41.

Let $X$ be a smooth, projective variety over a perfect field $k$. Assume that
either $X$ is src and $\dim X\geq 3$, or src in codimension 1 and $\dim X\geq
4$. Let ${\mathbf{R}}$ be an irreducible component of the space of almost very
free morphisms ${\mathbb{P}}^{1}\to X$.

  1. (41.1)

Let ${\mathbf{W}}$ be any $H^{\rm ci}$ family. Then there is an $r>0$ and a
geometrically irreducible open subset
${\mathbf{T}}\subset\operatorname{EComb}({\mathbf{W}},r{\mathbf{R}})$ that
satisfies (39.2–5).

Proof. As we noted, the $H^{\rm ci}$ family satisfies (39.2–3), which implies
the same for $\operatorname{EComb}({\mathbf{W}},r{\mathbf{R}})$. The key
property (39.5) is proved in [GHS03, Sec.2].

One needs to pay some attention to (39.4). ${\mathbf{R}}$ parametrizes certain
maps $g:{\mathbb{P}}^{1}\to X$. Their lifts $\bar{g}:{\mathbb{P}}^{1}\to
X\times{\mathbb{P}}^{1}$ are parametrized by
$\bar{\mathbf{R}}:={\mathbf{R}}\times\mathrm{PGL}_{2}$. We have the universal
rational curve map $\rho:{\mathbb{P}}^{1}_{\bar{\mathbf{R}}}\to
X\times{\mathbb{P}}^{1}$. Given an $H^{\rm ci}$ curve $L\subset
X\times{\mathbb{P}}^{1}$, the curves $R\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}L$ are parametrized by
$\rho^{-1}(L)$. By Bertini’s connectedness theorem, $\rho^{-1}(L)$ is
geometrically irreducible for general $L$; see [Jou83, 4.10 and 6.10].

If ${\mathbf{R}}$ itself is geometrically irreducible, then
$\operatorname{EComb}({\mathbf{W}},r{\mathbf{R}})$ is geometrically
irreducible, and we are done. (In fact, one can always choose such an
${\mathbf{R}}$.) In general, if ${\mathbf{R}}$ decomposes into geometrically
irreducible components ${\mathbf{R}}_{i}$, then ${\mathbf{T}}$ consist of
those combs that have the same number of teeth from each ${\mathbf{R}}_{i}$. ∎

Next we use ${\mathbf{T}}$ obtained in (41) as teeth of the next comb
construction, but first a definition.

######  Definition 42.

A comb $C\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}(\amalg_{i}T_{i})\subset
X$ is $d$-free along the handle if for every subcurve $C_{1}\subset C$,

$None$ $H^{1}\bigl{(}C_{1}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}(\amalg_{i}T_{i}),N_{C_{1}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.26582pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}(\amalg_{i}T_{i})}(-D)\bigr{)}=0$

for every effective Cartier divisor $D\subset C_{1}^{\rm sm}$ of degree $\leq
d$.

Given ${\mathbf{W}},{\mathbf{T}}$, let $\operatorname{EComb}^{\rm
d-free}({\mathbf{W}},r{\mathbf{T}})$ denote the set of all balanced, 3-nodal
combs that are free and $d$-free along their handle.

Comments. Requiring (42.1) for all $C_{1}\subset C$ will be important in (55).

It would have been simpler to require (42.1) for every
$D\subset\bigl{(}C\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}(\amalg
T_{i})\bigr{)}^{\rm sm}$. However, the teeth $T_{i}$ that we use contain the
rational curves $R_{i}$ of (41) as irreducible components. If $X$ is src, we
can choose these $R_{i}$ to have arbitrarily positive normal bundle. If $X$ is
only src in codimension 1, then the $R_{i}$ are contained in the fibers of
$X\to B$, thus their normal bundle always has a trivial summand. So
$H^{1}\bigl{(}R_{i},N_{R_{i}}(-2)\bigr{)}$ is never 0. We could fix this by
replacing the reducible curves ${\mathbf{T}}$ obtained in (41) by their
general smooth deformations. This would necessitate some extra steps to work
over nonclosed fields.

######  Theorem 43.

Let $Y$ be a smooth, projective variety and
${\mathbf{W}}\subset\operatorname{Hilb}_{1}^{\rm unobst}(Y)$ a bounded set.
Let ${\mathbf{T}}$ be a family of teeth satisfying (39.2–5) and fix $d>0$.

Then there is an $r>0$ such that, for every $C\in{\mathbf{W}}$,
$\operatorname{EComb}^{\rm d-free}(C,r{\mathbf{T}})$ is open and dense in
$\operatorname{EComb}^{\rm bal}(C,r{\mathbf{T}})$.

Proof. Choose $C\in{\mathbf{W}}$ and let $D\subset C^{\rm sm}$ be an effective
Cartier divisor of degree $\leq d$. $H^{1}\bigl{(}C,N_{C}(-D)\bigr{)}$ is dual
to $\operatorname{Hom}_{C}\bigl{(}N_{C}(-D),\omega_{C}\bigr{)}$.

The computation of [GHS03, Sec.2], which is done at smooth points of $C$,
shows that if
$\eta\in\operatorname{Hom}_{C}\bigl{(}N_{C}(-D),\omega_{C}\bigr{)}$, then
attaching a tooth $T_{c}$ at a point of $c\in\operatorname{Supp}(\eta)$ kills
$\eta$, provided $\eta$ is nonzero on the tangent space of $T_{c}$ at $c$.
Thus there is an $r_{0}$ such that if we attach at least $r_{0}$ general teeth
at general points to all irreducible components on $C$, then
$H^{1}\bigl{(}C,N_{C}(-D)\bigr{)}$ gets killed. Now take $r\geq 2cr_{0}$,
where $c$ is the number of irreducible components of $C$.

Finally note that both the dimension of $H^{1}\bigl{(}C,N_{C}(-D)\bigr{)}$ and
the number of irreducible components are uniformly bounded in a bounded set of
curves. ∎

The main technical result is the following.

######  Theorem 44.

Let $Y$ be a smooth, projective variety of dimension $\geq 3$ over a perfect
field $k$. Let ${\mathbf{W}}\subset\operatorname{Hilb}_{1}^{\rm nodal}(Y)$ be
a bounded, locally closed subset, and ${\mathbf{T}}$ a family of pointed
curves satisfying (39.2–5). Fix $d>0$.

Then there is an $r>0$ such that the forgetful map

$\Pi:\operatorname{EComb}^{\rm
d-free}({\mathbf{W}},r{\mathbf{T}})\longrightarrow{\mathbf{W}}$

  1. (44.1)

has connected fibers,

  2. (44.2)

satisfies the curve lifting property (48), and

  3. (44.3)

surjective on $L$-points for every $k\subset L\subset K$.

######  Corollary 45.

Using the notation of (44), assume in addition that ${\mathbf{W}}$ is
geometrically connected. Then $\operatorname{EComb}^{\rm
d-free}({\mathbf{W}},r{\mathbf{T}})$ is contained in a single geometrically
irreducible component $\operatorname{Env}({\mathbf{W}},r{\mathbf{T}})$ of
$\operatorname{Hilb}_{1}(Y)$; called its envelope.

Proof. $\operatorname{EComb}^{\rm d-free}({\mathbf{W}},r{\mathbf{T}})$ is
geometrically connected by (48). Thus the unique connected component of
$\operatorname{Hilb}_{1}^{\rm free}(Y)$ containing it is geometrically
irreducible by (38.4). ∎

######  Corollary 46.

Using the notation of (44), assume in addition that ${\mathbf{W}}$ is
geometrically connected. If $[C_{1}],[C_{2}]\in{\mathbf{W}}(L)$, then $C_{1}$
and $C_{2}$ are algebraically equivalent over $L$.

Proof. By (44.3) there are free combs
$C_{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}(\amalg_{j}T_{ij})$
defined over $L$. By (38.5), the $C_{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}(\amalg_{j}T_{ij})$ are
algebraically equivalent over $L$. The $\amalg_{j}T_{ij}$ are also smooth
$L$-points of the $r$th symmetric power of ${\mathbf{T}}$, hence the
$\amalg_{j}T_{ij}$ are also algebraically equivalent to each other over $L$. ∎

######  47Proof of 44.

Let $C\in{\mathbf{W}}$ with irreducible components $\\{C_{i}:i\in I\\}$. The
combs where we attach $m_{i}$ teeth to $C_{i}$ form an irreducible family. By
(43) there is an $r_{0}$ such that general such combs are $d$-free if
$m_{i}\geq r_{0}$ for every $i$. If $m_{i}>r_{0}$ for every $i$, then, as we
slide one of the teeth across a node of $C$, we get a 3-nodal comb, which is
still $d$-free be (38.3). Thus, if $r\geq(r_{0}+1)\cdot\\#I$, then
$\operatorname{EComb}^{\rm d-free}(C,r{\mathbf{T}})$ is connected.

In order to check the curve lifting property (48) let $(B,b_{1},b_{2})$ be an
irreducible, smooth, 2-pointed curve, and
$\tau:(B,b_{1},b_{2})\to{\mathbf{W}}$ a morphism. By pull-back we get a family
$C_{B}\to B$. Attach $m$ general teeth to each irreducible component of the
$C_{i}$ at the points $c_{ij}\in C_{i}$. After a further base change, we may
assume to have sections $\sigma_{ij}:B^{\prime}\to C^{\prime}_{B}$, such that
$\sigma_{ij}(b^{\prime}_{i})=c_{ij}$. We keep $B^{\prime}$ irreducible, thus
we can not control on which irreducible components of $C_{3-i}$ the points
$\sigma_{3-i,j}(b_{i})$ are on. However, after attaching further teeth, we
have a family in $\operatorname{EComb}^{\rm d-free}(C,r{\mathbf{T}})$.

Assume next that $C$ is defined over a field $L$ and
$C\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}(\amalg_{j}T_{j})$ is
$d$-free and defined over $K$. Adding all the
$\operatorname{Gal}(K/L)$-conjugate teeth, we get a comb
$C\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}(\amalg_{j,\sigma}T_{j}^{\sigma})$
which is $d$-free and defined over $L$. We need to make sure that the
$T_{j}^{\sigma}$ are attached to $C$ at distinct points and that their number
is uniformly bounded. Such a choice of the $T_{j}$ is possible by (60). ∎

######  48Curve lifting property.

Let $K$ be algebraically closed and $g:Y\to X$ a morphism of $K$-schemes of
finite type. If $g$ is not proper, it can happen that $X$ is connected, the
fibers of $g$ are nonempty and connected, but $Y$ is not. The implication,
however, holds, if $g$ has the following property.

Let $(B,b_{1},b_{2})$ be a 2-pointed, irreducible (not necessarily proper)
curve and $\tau:(B,b_{1},b_{2})\to X$ a morphism. Then there is 2-pointed,
irreducible curve $(B^{\prime},b^{\prime}_{1},b^{\prime}_{2})$ and a
commutative diagram

$\begin{array}[]{ccc}(B^{\prime},b^{\prime}_{1},b^{\prime}_{2})&\stackrel{{\scriptstyle\tau^{\prime}}}{{\to}}&Y\\\
\downarrow&&\hphantom{g}\downarrow g\\\
(B,b_{1},b_{2})&\stackrel{{\scriptstyle\tau}}{{\to}}&X.\end{array}$

##  7\. Arithmetic applications

First we discuss the index and the genera of curves.

######  49Euler characteristic of 1-cycles.

Let $k$ be a field, $X$ a proper $k$-scheme and $\operatorname{ind}_{X}$ the
index of $X$, that is, the gcd of the degrees of all 0-cycles on $X$.

Let $C\subset X$ be an irreducible curve. We check in (50) that
$\operatorname{ind}_{X}\mid 2\chi(C)$ and set

$\chi^{*}(C):=\tfrac{2\chi(C)}{\operatorname{ind}_{X}}\mod 2.$

Thus, if $\operatorname{ind}_{X}$ is odd then $\chi^{*}(C)=0$ for every $C$.

We extend $\chi^{*}$ to reducible curves and 1-cycles
$Z=\textstyle{\sum}_{i}d_{i}C_{i}$ by linearity.

We check in (51) that if $Z$ is algebraically equivalent to 0 then
$\chi^{*}(Z)=0$. Thus $\chi^{*}$ descends to a homomorphism
$\chi^{*}:A_{1}(X)\to{\mathbb{Z}}/2{\mathbb{Z}}$.

######  50.

Let $k$ be a field, $X$ a proper $k$-scheme and $\operatorname{ind}_{X}$ the
index of $X$.

Let $p:C\to X$ be a proper, reduced curve mapping to $X$ and $\pi:\bar{C}\to
C$ the normalization. Then
$p_{*}(\pi_{*}{\mathcal{O}}_{\bar{C}}/{\mathcal{O}}_{C})$ is a 0-cycle on $X$,
hence its degree is divisible by $\operatorname{ind}_{X}$. Thus
$\chi(\bar{C})\equiv\chi(C)\mod\operatorname{ind}_{X}$.

The canonical class of $\bar{C}$ is represented by a 0-cycle, and $\deg
K_{\bar{C}}=-2\chi(\bar{C})$. Thus $\operatorname{ind}_{X}\mid
2\chi(\bar{C})$. We have 2 possibilities:

  1. (50.1)

if $\operatorname{ind}_{X}$ is odd then $\chi(C)\equiv
0\mod\operatorname{ind}_{X}$, and

  2. (50.2)

if $\operatorname{ind}_{X}$ is even then
$\chi(C)\equiv\chi^{*}(C)(\operatorname{ind}_{X}/2)\mod\operatorname{ind}_{X}$.

(This is related to the elw indices defined in [Kol13]. For $d$-cycles, we get
a similar invariant modulo
$\gcd\bigl{(}\operatorname{elw}_{d-1},\mu(\operatorname{Td}_{d})\bigr{)}$
where $\mu(\operatorname{Td}_{d})$ is the denominator appearing in the Todd
class in dimension $d$.)

######  Proposition 51.

(cf. [Kol13, 3–5]) Let $X$ be a proper $k$-scheme. Then the function on
1-cycles $Z\mapsto\chi^{*}(Z)\in{\mathbb{Z}}/2{\mathbb{Z}}$ is preserved by
algebraic equivalence.

Proof. Let $p:C\to X$ be a proper morphism from a nonsingular curve to $X$
such that $p_{*}[C]=[Z]$. The Euler characteristic of fibers of flat morphisms
is a locally constant function on the base, so it is enough to prove that

$None$
$\chi(C)\equiv\textstyle{\sum}_{i}d_{i}\chi(Z_{i})\mod\operatorname{ind}_{X}.$

By linearity, it is enough to prove this for a morphism $\tau:C\to B$ of
nonsingular, irreducible curves over $X$.

If $\tau(C)$ has dimension 0, then $\chi(C)=\deg\tau_{*}{\mathcal{O}}_{C}-\deg
R^{1}\tau_{*}{\mathcal{O}}_{C}$, and both terms on the right hand side are
divisible by $\operatorname{ind}_{X}$. If $\tau$ is finite then

$\chi(C)=\chi(\tau_{*}{\mathcal{O}}_{C})=\deg\tau_{*}{\mathcal{O}}_{C}+\deg\tau\cdot\chi({\mathcal{O}}_{B}).$

Here $\deg\tau_{*}{\mathcal{O}}_{C}$ is divisible by $\operatorname{ind}_{X}$.
∎

######  Lemma 52.

Let $X$ be a smooth, projective variety over a perfect field $k$ with 1-cycles
$Z_{1},Z_{2}$ such that $\chi^{*}(Z_{1})=\chi^{*}(Z_{2})$. Let $A\subset X$ be
a smooth, curve meeting all irreducible components of $Z_{1}\cup Z_{2}$ at
smooth points and such that $\operatorname{ind}_{X}=\operatorname{ind}_{A}$.

Then there are connected, nodal $k$-curves $\pi_{i}:C_{i}\to X$ such that
$(\pi_{i})_{*}[C_{i}]=Z_{i}+2A$ and $\chi(C_{1})=\chi(C_{2})$.

Proof. We have connected, nodal $k$-curves $\rho_{i}:D_{i}\to X$ such that
$(\rho_{i})_{*}[D_{i}]=\operatorname{red}(Z_{i})+A$. If $D_{ij}\subset D_{i}$
is an irreducible component, and $(\rho_{i})_{*}[D_{ij}]$ appears in $Z_{i}$
with multiplicity $d_{ij}$, then we replace $D_{ij}$ with a degree $d_{ij}$
cyclic cover $C_{ij}\to D_{ij}$ that ramifies along all the nodes. Now we have
connected, nodal $k$-curves $C^{\prime}_{i}=\cup C_{ij}$ such that
$(\pi^{\prime}_{i})_{*}[C^{\prime}_{i}]=Z_{i}$.

Note that $\operatorname{ind}_{X}$ divides
$\chi(C^{\prime}_{1})-\chi(C^{\prime}_{2})$. Also, since
$\operatorname{ind}_{X}=\operatorname{ind}_{A}$, the curve $A$ has double
covers $A_{i}\to A$ such that
$\chi(A_{2})-\chi(A_{1})=\chi(C^{\prime}_{1})-\chi(C^{\prime}_{2})$. We now
get $C_{i}$ by gluing $C^{\prime}_{i}$ to $A_{i}$. ∎

The stronger form of Theorem 6 is the following.

######  Theorem 53.

Let $X_{k}$ be a smooth, projective variety over a perfect field $k$ with
algebraic closure $K$. Assume that $X_{K}$ is src in codimension 1. Then

$A_{1}(X_{k})\to A_{1}(X_{K})+{\mathbb{Z}}/2{\mathbb{Z}}\quad\mbox{given
by}\quad[Z_{k}]\mapsto\bigl{(}[Z_{K}],\chi^{*}(Z_{k})\bigr{)}$

is injective.

Proof. If $Z_{k}\sim_{a}0$ then also $Z_{K}\sim_{a}0$, and $\chi^{*}(Z)=0$ by
(51).

Conversely, assume that $\chi^{*}(Z)=0$. By (52) there are connected, nodal
$k$-curves $\pi_{i}:C_{i}\to X$ such that
$Z\sim_{a}(\pi_{1})_{*}[C_{1}]-(\pi_{2})_{*}[C_{2}]$ and
$\chi(C_{1})=\chi(C_{2})$.

Then, by Theorem 2, there is a nodal deformation equivalence

$\begin{array}[]{ccccc}C_{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}R&\subset&S&\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}&B\times
X\\\ \downarrow&&\downarrow&&\downarrow\\\ b_{i}&\in&B&=&B,\end{array}$

where the $\pi|_{C_{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.26582pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}R}:C_{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}R\to X$ are defined over
$k$. Note that

$(\pi_{1})_{*}[C_{1}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}R]-(\pi_{2})_{*}[C_{2}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}R]=(\pi_{1})_{*}[C_{1}]-(\pi_{2})_{*}[C_{2}]\sim_{a}Z.$

Finally $(\pi_{1})_{*}[C_{1}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}R]-(\pi_{2})_{*}[C_{2}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}R]\sim_{a}0$ by (46)
applied to $C_{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}R$. ∎

######  54Proof of Theorem 7.

Injectivity follows from (6.1).

Let $Z$ be a 1-cycle on $X_{K}$ such that $Z\sim_{a}Z^{\sigma}$ for every
$\sigma\in\operatorname{Gal}(K/k)$. By Theorem 2 we may certify
$Z\sim_{a}Z^{\sigma}$ by a nodal curve
$C\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}R\subset
X\times{\mathbb{P}}^{3}$ that is nodal deformation equivalent to
$C^{\sigma}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}R$ for every
$\sigma\in\operatorname{Gal}(K/k)$. The union of these nodal equivalences and
their $\operatorname{Gal}(K/k)$-conjugates gives a geometrically connected
${\mathbf{W}}\subset\operatorname{Hilb}_{1}(X\times{\mathbb{P}}^{3})$ which
contains all the $C^{\sigma}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}R$.

Choose a geometrically irreducible family of teeth ${\mathbf{T}}$, and apply
(45). We get the envelope $\operatorname{Env}({\mathbf{W}},r{\mathbf{T}})$
that contains a free comb $(C\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}R)\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}(\amalg_{j}T_{j})$ and all
of its $\operatorname{Gal}(K/k)$-conjugates
$\bigl{(}(C\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}R)\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}(\amalg_{j}T_{j})\bigr{)}^{\sigma}$.
Thus $\operatorname{Env}({\mathbf{W}},r{\mathbf{T}})$ is geometrically
irreducible.

By assumption there are 0-cycles $P$ on ${\mathbf{T}}$ and $Q$ on
$\operatorname{Env}({\mathbf{W}},r{\mathbf{T}})$ of degree 1. Let $T_{P}$ and
$E_{Q}$ be the corresponding 1-cycles on $X\times{\mathbb{P}}^{3}$. Then
$[E_{Q}]-r[T_{P}]$ is a 1-cycle on $X\times{\mathbb{P}}^{3}$ that is
algebraically equivalent to $C\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}R$ over $K$. Thus
$[E_{Q}]-r[T_{P}]-[R]$ is algebraically equivalent to $C$ over $K$. Projecting
it to $X$ gives a 1-cycle on $X$ that is algebraically equivalent to $[Z]$
over $K$. ∎

######  55Proof of Theorem 8.

Pick $[Z_{0}]\in A_{1}(X_{0})$. We can represent it as the projection of a
nodal curve $C_{0}\subset Y:=X\times{\mathbb{P}}^{3}$. By (43) there is a free
comb $C_{0}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}(\amalg_{j}T_{j})$ where
$\amalg_{j}T_{j}$ is defined over $k$. Thus
$\operatorname{Hilb}_{1}(Y/R)\to\operatorname{Spec}R$ is smooth at
$[C_{0}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}(\amalg_{j}T_{j})]$ and
also at $[\amalg_{j}T_{j}]$. Since $R$ is Henselian, these lift to
$[E_{g}],[T_{g}]\in\operatorname{Hilb}_{1}(Y_{g})$. Thus $[E_{g}]-[T_{g}]$
shows that $A_{1}(X_{g})\to A_{1}(X_{0})$ is surjective.

For injectivity, pick $Z^{1}_{g},Z^{2}_{g}$ with specializations
$Z^{1}_{0}\sim_{a}Z^{2}_{0}$. Then
$\operatorname{ind}_{X_{0}}\mid\chi(Z^{1}_{0})-\chi(Z^{2}_{0})$ by (6.2). Note
that $\operatorname{ind}_{X_{0}}=\operatorname{ind}_{X_{g}}$, so
$\operatorname{ind}_{X_{g}}\mid\chi(Z^{1}_{g})-\chi(Z^{2}_{g})$. Thus, by
(6.2), it is enough to show that $Z^{1}_{g}$ and $Z^{2}_{g}$ become
algebraically equivalent after a field extension of $k(g)$.

By (37) we can choose nodal representatives $C^{i}_{g}\to Z^{i}_{g}$ such that
$\chi(C^{1}_{g})=\chi(C^{2}_{g})$. After semistable reduction, we may replace
the $C^{i}_{g}$ by flat families of nodal curves $C^{i}_{R}\subset
X_{R}\times_{R}{\mathbb{P}}^{3}_{R}$. We apply Theorem 1 to get

$\begin{array}[]{ccccc}C^{i}_{0}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}E&\subset&S&\stackrel{{\scriptstyle}}{{\hookrightarrow}}&B\times
X_{0}\times{\mathbb{P}}^{3}\\\ \downarrow&&\downarrow&&\downarrow\\\
b_{i}&\in&B&=&B.\end{array}$

Now apply (45) to ${\mathbf{W}}:=B$ with $d:=\max\\{\\#(C^{1}_{0}\cap
E),\\#(C^{2}_{0}\cap E)\\}$. We get $d$-free combs
$(C^{i}_{0}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}E)\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}(\amalg_{j}T^{i}_{j})$
that are in the same irreducible component of $\operatorname{Hilb}_{1}^{\rm
free}(Y_{0})$.

The $C^{i}_{0}\cap E$ are smooth points on $C^{i}_{0}$, so can be lifted to a
union of sections $\Sigma^{i}\subset C^{i}_{R}$. We can also view
$C^{i}_{0}\cap E$, as a Cartier divisor on $E$. Since the comb is $d$-free,

$H^{1}\bigl{(}E\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}(\amalg_{j}T^{i}_{j}),N_{E\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.26582pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}(\amalg_{j}T^{i}_{j})}(-(C^{i}_{0}\cap
E))\bigr{)}=0,$

hence every deformation of $C^{i}_{0}\cap E$ extends to a deformation of
$E\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}(\amalg_{j}T^{i}_{j})$. In
particular, each $E\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}(\amalg_{j}T^{i}_{j})$ can
be lifted to a flat family of curves $V^{i}_{R}$ containing $\Sigma^{i}$. Then
$C^{i}_{R}\cup V^{i}_{R}$ is flat over $R$.

Since the $E\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}(\amalg_{j}T^{i}_{j})=V^{i}_{0}$
are smooth points on same irreducible component of
$\operatorname{Hilb}_{1}^{\rm free}(Y_{0})$, the same holds for the general
fibers $V^{i}_{g}$. Hence $V^{1}_{g}\sim_{a}V^{2}_{g}$.

The same argument shows that $C^{1}_{g}\cup V^{1}_{g}\sim_{a}C^{2}_{g}\cup
V^{2}_{g}$. Therefore $C^{1}_{g}\sim_{a}C^{2}_{g}$. ∎

##  8\. Loops in $\operatorname{Chow}_{1}$

######  Definition 56.

Let $X$ be a scheme. A length $m$ chain in $X$ is a collection of morphisms

${\mathcal{L}}:=\bigl{\\{}\pi^{i}:(W^{i},Z^{i}_{1},Z^{i}_{2})\to X\colon
i=1,\dots,m\bigr{\\}},$

where the $W^{i}$ are irreducible schemes, $Z^{i}_{1},Z^{i}_{2}$ are
irreducible, closed subsets and $\pi^{i}(Z^{i}_{2})=\pi^{i+1}(Z^{i+1}_{1})$
(as sets).

${\mathcal{L}}$ is a loop if $\pi^{m}(Z^{m}_{2})=\pi^{1}(Z^{1}_{1})$ also
holds. A loop is contractible if all the $\pi^{i}(W^{i})$ are contained in the
same irreducible component of $X$.

A connected, pointed curve
$\pi_{B}^{i}:(B^{i},p^{i}_{1},p^{i}_{2})\to(W^{i},Z^{i}_{1},Z^{i}_{2})$ is a
representative. Assume that we also have connected, pointed curves
$\pi_{C}^{i}:(C^{i},q^{i}_{1},q^{i}_{2})\to p^{i}(Z^{i}_{2})$ such that
$\pi^{i}\circ\pi_{B}^{i}(p^{i}_{2})=\pi_{C}^{i}(q^{i}_{1})$ and
$\pi^{i+1}\circ\pi_{B}^{i+1}(p^{i}_{1})=\pi_{C}^{i}(q^{i}_{2})$ (indexing
modulo $m$). Alternating the $B^{i}$ with the $C^{i}$ gives a loop in $X$. We
call it a representative of ${\mathcal{L}}$.

######  Definition 57.

Let $X$ be a proper scheme over a field and

${\mathcal{L}}:=\bigl{\\{}\pi^{i}:(W^{i},Z^{i}_{1},Z^{i}_{2})\to X\colon
i=1,\dots,m\bigr{\\}},$

a loop in $\operatorname{Chow}_{1}(X)$. Let $Z_{V}\subset
V\subset\operatorname{Chow}_{1}(X)$ be irreducible subvarieties. The
translation of ${\mathcal{L}}$ by $(V\supset Z_{V})$ is the loop made from the

$(W^{i}\oplus V,Z^{i}_{1}\oplus Z_{V},Z^{i}_{2}\oplus
Z_{V})\to\operatorname{Chow}_{1}(X),$

where for $U_{1},U_{2}\subset\operatorname{Chow}_{1}(X)$, we set $U_{1}\oplus
U_{2}:=\\{Z_{1}+Z_{2}:Z_{i}\in U_{i}\\}$.

######  Theorem 58.

Let $X$ be a smooth, projective variety, and ${\mathcal{L}}$ a loop in
$\operatorname{Chow}_{1}(X)$. Then there is a 1-cycle $Z$ and $r>0$ such that
the translation of ${\mathcal{L}}$ by $[Z]\oplus r|H^{\rm ci}|$ is
representable by a loop in $\operatorname{Hilb}_{1}^{\rm
nodal}(X\times{\mathbb{P}}^{3})$.

Proof. Choose points $z^{i}_{j}\in Z^{i}_{j}$. By (15), there are 2-nodal
deformation equivalences

$(B^{i},b^{i}_{1},b^{i}_{2})\stackrel{{\scriptstyle}}{{\longleftarrow}}S^{i}\stackrel{{\scriptstyle\pi^{i}}}{{\longrightarrow}}B^{i}\times
X$

representing $(W^{i},z^{i}_{1},z^{i}_{2})$. After adding a suitable 1-cycle
$D$ and using (37), we may assume that the $S^{i}\to B^{i}$ have connected
fibers and their Euler characteristic is independent of $i$.

The problem is that, while $S^{i}_{2}\to X$ and $S^{i+1}_{1}\to X$ give the
same 1-cycles on $X$, they are not isomorphic as maps.

Next we apply (32–33) to each $\pi^{i}:S^{i}_{2}\to X$ and
$\pi^{i+1}:S^{i+1}_{1}\to X$. We get nodal curves $R_{i}$ and 3-nodal
deformation equivalences

$(C^{i},c^{i}_{1},c^{i}_{2})\stackrel{{\scriptstyle}}{{\longleftarrow}}T^{i}\stackrel{{\scriptstyle\tau^{i}}}{{\longrightarrow}}C^{i}\times
X$

such that

$T^{i}_{1}\cong S^{i}_{2}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}L_{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}R_{i}\quad\mbox{and}\quad
T^{i}_{2}\cong S^{i+1}_{1}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}L_{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}R_{i}.$

We can now assemble a 3-nodal representative loop from the pieces

$S^{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}_{j}\bigl{(}L_{j}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}R_{j})\quad\mbox{and}\quad
T^{i}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}_{j\neq
i}\bigl{(}L_{j}\mathbin{\ooalign{$\cup$\cr\hfil\raise
1.80832pt\hbox{$\scriptscriptstyle\times$}\hfil\cr}}R_{j}).$

Now take $[Z]=[D]+\sum_{j}[\tau^{j}_{*}(R_{j})]$. We can turn it nodal using
(17). ∎

Combining this with (45) we get the following.

######  Corollary 59.

Let $X$ be a smooth, projective variety, and ${\mathcal{L}}$ a loop in
$\operatorname{Chow}_{1}(X)$. Assume that $X$ is src in codimension 1.

Then a suitable translate of ${\mathcal{L}}$ is contractible. ∎

##  9\. Controlling residue fields

In Sections 5–6 we need the following to control residue fields.

######  Lemma 60.

Let $k$ be a perfect field and $\\{p_{i}:X_{i}\to Y_{i}:i\in I\\}$ finitely
many nonconstant morphisms of $k$-schemes of finite type. Then there are
closed points $x_{i}\in X_{i}$ with images $y_{i}=p_{i}(x_{i})$ such that the
residue fields $k(x_{i}),k(y_{i})$ are all isomorphic. Moreover, the following
hold.

  1. (60.1)

If $k$ is infinite, then there is an infinite set $D\subset{\mathbb{N}}$ such
that, for every $d\in D$, there are infinitely many such points with
$\deg(k(x_{i})/k)=d$.

  2. (60.2)

If $k$ is finite, then there is an infinite set $D\subset{\mathbb{N}}$ such
that, for every $d\in D$, there are $\geq\frac{1}{2}|k|^{d}$ such points with
$\deg(k(x_{i})/k)=d$.

  3. (60.3)

If the $\\{p_{i}:X_{i}\to Y_{i}:i\in I\\}$ are chosen from a bounded family,
and $\\#I$ is bounded, then a fixed $D\subset{\mathbb{N}}$ works for all of
them.

Proof. We may assume that the $X_{i},Y_{i}$ are affine and the $p_{i}$ are
flat. Choose dominant morphisms $Y_{i}\to{\mathbb{A}}^{1}$ and let
$q:Z\to{\mathbb{A}}^{1}$ be the fiber product of all $X_{i}\to
Y_{i}\to{\mathbb{A}}^{1}$.

Apply [Poo01] to $q:Z\to{\mathbb{A}}^{1}$. We get a dense set of closed points
$z\in Z$ such that $k(z)=k(q(z))$. Let $x_{i}\in X_{i}$ be the coordinate
prjections of $z$. Then $k(z)\supset k(x_{i})\supset k(y_{i})\supset k(q(z))$,
so they are all equal.

The claims (1–3) may be clearer from [KLOS23, 7.2.1]. ∎

A typical application is the following.

######  Corollary 61.

Let $k$ be a perfect field, $X$ a $k$-scheme of finite type and
$Z_{1},Z_{2}\subset X$ positive dimensional, irreducible subsets. Let

$M\leftarrow C_{M}\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}X$

be an irreducible family of curves joining $Z_{1}$ and $Z_{2}$. That is,
$M\leftarrow C_{M}$ is flat, and there are 2 sections $\sigma_{i}:M\to C_{M}$
such that the $\pi\circ\sigma_{i}$ give dominant morphisms form $M$ to
$Z_{i}$. Then there is a dense set of closed points $m\in M$ such that
$k(m)=k\bigl{(}\pi\circ\sigma_{1}(m)\bigr{)}=k\bigl{(}\pi\circ\sigma_{2}(m)\bigr{)}$.∎

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Princeton University, Princeton NJ 08544-1000,

Beijing International Center for Mathematical Research,

Peking University, 100871, Beijing, China