perspective, the geometric analogues of the arithmetic results established in
this memoir specifically concern analytic and algebraic varieties fibered over
a projective curve. However, in this “fibered” context, the simple framework
of Subsections 1.3.1, 1.3.2, and 1.4.1 excludes significant examples. Indeed
the self-intersection of sections of surfaces fibered over a projective curve
is in general negative, as established by Arakelov [Ara71] and Szpiro [Szp81,
Chapitre III].

It turns out that the results in Subsections 1.3.1, 1.3.2, and 1.4.1, dealing
with a smooth connected projective curve $C$ embedded in a smooth analytic
surface ${\mathcal{V}}$, may be extended to situations where $C$ is replaced
by a general connected projective curve, possibly non-reduced, satisfying a
suitable positivity property that generalizes the positivity (1.4.1) of the
self-intersection of $C$.

These generalizations of our previous results will allow us to establish
proper analogues of our later arithmetic results in Sections 2.2 and 2.3.
These are established by some amplifications of our earlier arguments in
Subsections 1.3.1, 1.3.2, and 1.4.1, which still only relies on basic
properties of algebraic curves and surfaces.

The content of this chapter is more technical than the one of Chapter 1, and
could be skipped by a reader mainly interested by the arithmetic results of
this memoir.

The reader with an interest in complex analytic geometry will observe that the
analytic discussion in Subsection 1.4.3, where the positivity of
$\deg_{C}N_{C}{\mathcal{V}}$ is related to the pseudoconcavity properties of
the (germ of) analytic surface ${\mathcal{V}}$, is not generalized to the more
general framework of this chapter, although it seems likely that it could be.

For instance one expects that, if an effective divisor $D$ in a connected
smooth complex analytic surface ${\mathcal{V}}$ satisfies the condition
$\mathbf{CNB}$ introduced in Subsection 2.1.1 below, then its support $|D|$
admits a basis of open neighborhoods in ${\mathcal{V}}$ that are
pseudoconcave. This would follow from a suitable generalization of the simple
argument on Levi forms in Proposition 1.4.4.

However, at this stage, we do not know whether such a basis of pseudoconcave
open neighborhoods exists for a general $\mathbf{CNB}$ divisor. This
difficulty with the analytic approach confers a special interest to the
algebraic approach in this chapter when dealing with general $\mathbf{CNB}$
divisors.

####  2.1. Compact, connected, nef and big effective divisors in analytic
surfaces

#####  2.1.1. The condition $\mathbf{CNB}$ and its consequences

In this section, we denote by ${\mathcal{V}}$ a connected smooth complex
analytic surface. For every effective divisor $D$ in ${\mathcal{V}}$, we may
introduce the following conditions:

$\mathbf{CNB}$: _The support $|D|$ of $D$ is compact and connected._ _If we
denote by $(D_{i})_{i\in I}$ the irreducible components of $D$, then we have:_

(2.1.1) $\mbox{\emph{for every }$i\in I$, }D_{i}\cdot D\geq 0.$

_Moreover:_

(2.1.2) $D\cdot D>0.$

When $|D|$ is compact and connected, the non-negativity condition (2.1.1) is
equivalent to the following one:

(2.1.3) $\mbox{\emph{for every effective divisor $\widetilde{D}$ in
${\mathcal{V}}$, } }\widetilde{D}\cdot D\geq 0.$

When the first two conditions in $\mathbf{CNB}$ are satisfied, the validity of
(2.1.2) is equivalent to the existence of $i\in I$ such that the inequality
(2.1.1) is strict.

Observe also that, when the surface ${\mathcal{V}}$ is projective, an
effective divisor $D$ satisfies the conditions $\mathbf{CNB}$ if and only if
$D$, or equivalently the line bundle ${\mathcal{O}}_{\mathcal{V}}(D)$, is nef
and big.111The connectedness of $|D|$ when ${\mathcal{O}}_{\mathcal{V}}(D)$ is
nef and big follows from the Hodge index inequality, by an argument of C.P.
Ramanujam which actually establishes that $D$ is numerically connected. See
for instance [Szp81, Chapter II] or [Bos99, Proposition 2.2]. The name
$\mathbf{CNB}$ for the above condition stands for compact, connected, nef and
big, although the general formalism of big line bundles and divisors does not
really makes sense on a general analytic surface.

From now on, we suppose that $D$ is an effective divisor in ${\mathcal{V}}$
that satisfies the conditions $\mathbf{CNB}$, we denote by $C$ an irreducible
component of $D$ such that:

(2.1.4) $C\cdot D>0,$

and we denote by $\mu$ the multiplicity of $C$ in $D$. We may decompose $D$ as
a sum:

(2.1.5) $D=\mu\,C+E,$

where $E$ is an effective divisor with compact support not containing $C$.

######  Proposition 2.1.1.

For every complex analytic line bundle $M$ over ${\mathcal{V}}$, the
$\mathbb{C}$-vector space of its analytic sections $\Gamma({\mathcal{V}},M)$
is finite dimensional. Moreover:

(2.1.6) $\dim_{\mathbb{C}}\Gamma({\mathcal{V}},M)\leq C(c_{1}(M)\cdot D)$

where, for every $n\in\mathbb{Z}$:

(2.1.7) $C(n):=\sum_{0\leq i\leq\lfloor n/C\cdot
D\rfloor}\left(1+\lfloor(n-i\,C\cdot D)/\mu\rfloor\right)^{+}.$

Observe that we still have $C(n)=0$ if $n<0,$ and $C(0)=1$, and that:

$C(n)\sim\frac{n^{2}}{2\mu\,C\cdot D},\quad\mbox{when
$n\longrightarrow+\infty$}.$

###### Proof.

Let us choose a point $P$ in $C_{\mathrm{reg}}.$

As in the proof of Proposition 1.4.1, we may consider the decreasing
filtration $(E^{i})_{i\in\mathbb{N}}$ of $\Gamma({\mathcal{V}},M)$ defined by:

$E^{i}:=\Gamma({\mathcal{V}},M\otimes{\mathcal{O}}_{\mathcal{V}}(-iC)).$

For every $i\in\mathbb{N},$ we may also consider the restriction map:

$\eta^{i}:E^{i}=\Gamma({\mathcal{V}},M\otimes{\mathcal{O}}_{\mathcal{V}}(-iC))\longrightarrow\Gamma(C,(M\otimes{\mathcal{O}}_{\mathcal{V}}(-iC))_{|C})=\Gamma(C,M_{|C}\otimes
N_{C}{\mathcal{V}}^{\otimes(-i)})$

defined by:

$\eta^{i}(s):=s_{|C}.$

Its kernel is $E^{i+1}$.

For every $k\in\mathbb{N}$, we may introduce the closed subscheme of $C$,
supported by $P$, defined by the effective Cartier divisor $kP$ in $C$. We
shall denote it by $kP$, and consider the composite map:

${\eta}^{i}_{k}:E^{i}/E^{i+1}\longrightarrow\Gamma(C,M_{|C}\otimes
N_{C}{\mathcal{V}}^{\otimes(-i)})\longrightarrow(M_{|C}\otimes
N_{C}{\mathcal{V}}^{\otimes(-i)})_{|kP},$

defined by:

${\eta}^{i}_{k}([s]):=(\eta^{i}(s))_{|kP}.$

######  Lemma 2.1.2.

The map $\eta^{i}_{k}$ is injective if the following inequality holds:

$i\,C\cdot D+\mu k>c_{1}(M)\cdot D.$

###### Proof of Lemma 2.1.2.

Let us consider a non-zero class $[s]$ in $E^{i}/E^{i+1}$ that lies in
$\ker{\eta}^{i}_{k}$. The divisor of the analytic section $s$ of $M$ may be
written:

(2.1.8) ${\mathrm{div}\,}s=iC+\widetilde{D}+R,$

where $\widetilde{D}$ and $R$ are effective divisor, where $\widetilde{D}$ is
supported on $|E|$ and $R$ meets $D$ properly. The scheme-theoretic
intersection $R\cap C$ contains $kP$ as a subscheme, and consequently:

$R\cdot C\geq k.$

Consequently:

$R\cdot D=\mu\,R\cdot C+R\cdot E\geq\mu k.$

Moreover, as already observed, the non-negativity condition (2.1.1) on $D$
implies:

$\widetilde{D}\cdot D\geq 0.$

Together with (2.1.8), this implies:

$c_{1}(M)\cdot D={\mathrm{div}\,}s\cdot D\geq i\,C\cdot D+\mu k.\qed$

According to Lemma 2.1.2, we have:

$i\geq\lfloor(c_{1}(M)\cdot D)/(C\cdot D)\rfloor+1\Longrightarrow
E^{i}/E^{i+1}=\\{0\\},$

and:

$i\leq\lfloor(c_{1}(M)\cdot D)/(C\cdot
D)\rfloor\Longrightarrow\dim_{\mathbb{C}}E^{i}/E^{i+1}\leq
1+\lfloor(c_{1}(M)\cdot D-i\,C\cdot D)/\mu\rfloor.$

Since the filtration $(E^{i})_{i\in\mathbb{N}}$ satisfies
$E^{0}=\Gamma({\mathcal{V}},M)$ and $\bigcap_{i\in\mathbb{N}}E^{i}=\\{0\\}$,
this implies the finite dimensionality of $\Gamma({\mathcal{V}},M)$ and the
upper bound (2.1.6) on its dimension. ∎

Proposition 2.1.1 notably implies the validity of the “weak dimension
estimates” (1.4.11). In turn, it leads to the following generalization of the
algebraicity result in Proposition 1.4.3 by a straightforward modification of
its proof:

######  Proposition 2.1.3.

Let ${\mathcal{V}}$ be a smooth connected complex analytic surface containing
an effective divisor $D$ satisfying condition $\mathbf{CNB}$. If
$\gamma:{\mathcal{V}}\rightarrow Z$ is a complex analytic map from
${\mathcal{V}}$ to some complex quasi-projective variety $Z$, then either $f$
is constant, or $f({\mathcal{V}})$ coincides with $f(|D|)$ and is an
irreducible projective curve, or $f({\mathcal{V}})$ is contained, as a Zariski
dense subset, in some irreducible closed algebraic surface in $Z$.

The computations of intersection numbers in Subsection 1.3.1 also may be
generalized to the present setting.

######  Proposition 2.1.4.

Let ${\mathcal{V}}$ be a smooth connected complex analytic surface containing
an effective divisor $D$ satisfying condition $\mathbf{CNB}$, and let us keep
the notation introduced in (2.1.4) and (2.1.5) above. For every complex
analytic map $\gamma:{\mathcal{V}}\rightarrow Z$ with Zariski dense image from
${\mathcal{V}}$ to a connected smooth complex algebraic surface $Z$, such that
$\gamma_{\mid C}:C\rightarrow\gamma(C)$ is birational, the following estimates
are satisfied:

(2.1.9) $\gamma_{\ast}D\cdot\gamma_{\ast}D\geq e(\gamma)\,\mu\,C\cdot
D\geq\mu\,C\cdot D,$

where $e(\gamma)$ denotes the ramification index of $\gamma$ along $C$.

###### Proof.

According to the projection formula, the following equality of intersection
numbers holds:

$\gamma_{\ast}D\cdot\gamma_{\ast}D=\gamma^{\ast}\gamma_{\ast}D\cdot D.$

Moreover we have:

$\gamma_{\ast}C=\gamma(C),$

and, according to the very definition of $e(\gamma),$ the irreducible curve
$C$ occurs with multiplicity $e(\gamma)$ in the effective divisor
$\gamma^{\ast}\gamma(C).$ This implies the following inequality of divisors on
${\mathcal{V}}$:

$\gamma^{\ast}\gamma_{\ast}D=\mu\,\gamma^{\ast}\gamma_{\ast}C+\gamma^{\ast}\gamma_{\ast}E\geq\mu\,\gamma^{\ast}\gamma_{\ast}C\geq
e(\gamma)\,\mu\,C.$

Together with the non-negativity condition (2.1.3) on $D$, this implies:

$\gamma^{\ast}\gamma_{\ast}D\cdot D\geq e(\gamma)\,\mu\,C\cdot D,$

and (2.1.9) follows. ∎

Using Proposition 2.1.4 together with the general degree bound in Proposition
1.1.2, we may derive a generalization of the degree bound (1.3.7) in
Proposition 1.3.1.

Indeed let us assume that $f:V\rightarrow U$ is a morphism of complex
algebraic varieties, where $U$ and $V$ are two smooth connected quasi-
projective complex surfaces, and that we are given complex analytic maps
$\alpha$ and $\beta$ from ${\mathcal{V}}$ to $X$ and $Y$ respectively, that
fit into a commutative diagram:

(2.1.10) $\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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Let us assume that _the image of $\alpha$ is Zariski dense in $U$_ — this
implies that the image of $\beta$ is Zariski dense in $V$ and that $f$ is
dominant — and that _the restriction of $\alpha$ to $C$_:

$\alpha_{\mid C}:C\longrightarrow\alpha(C)$

_is birational_. Then $\beta_{\mid C}:C\longrightarrow\beta(C)$ also is
birational, and Proposition 2.1.4 applied to $\gamma:=\beta$ implies:

$\beta_{\ast}D\cdot\beta_{\ast}D\geq C\cdot D>0.$

Therefore Proposition 1.1.2 applies with $A:=\alpha_{\ast}(D)$ and
$B=\beta_{\ast}(D)$. This proves that $\deg f$ _satisfies the following upper
bound_ :

(2.1.11) $\deg
f\leq\frac{\alpha_{\ast}D\cdot\alpha_{\ast}D}{\beta_{\ast}D\cdot\beta_{\ast}D}\leq\frac{\alpha_{\ast}D\cdot\alpha_{\ast}D}{\mu\,C\cdot
D}.$

This generalizes Proposition 1.3.1. In turn the upper bound (2.1.11) implies a
generalization of the results _à la_ Lefschetz-Nori concerning étale
fundamental groups in Proposition 1.3.2, where now the conclusion reads:

(2.1.12)
$[\pi_{1}^{\mathrm{\acute{e}t}}(U,x):\alpha_{\mid|D|\ast}(\pi_{1}^{\mathrm{\acute{e}t}}(|D|,c))]\leq\frac{\alpha_{\ast}D\cdot\alpha_{\ast}D}{\mu\,C\cdot
D}.$

#####  2.1.2. Compatibility of $\mathbf{CNB}$ divisors with modifications.
Applications to meromorphic maps

Condition $\mathbf{CNB}$ turns out to be preserved by modifications:

######  Proposition 2.1.5.

Let $D$ be an effective divisor in some connected smooth analytic surface
${\mathcal{V}}$, and let $\nu:{\mathcal{V}}^{\prime}\rightarrow{\mathcal{V}}$
be a modification222 Recall that a modification
$\nu:{\mathcal{V}}^{\prime}\rightarrow{\mathcal{V}}$ of ${\mathcal{V}}$ is a
proper complex analytic map of degree 1 from a connected smooth complex
analytic surface ${\mathcal{V}}^{\prime}$ onto ${\mathcal{V}}$. Then there
exists a closed discrete subset $Z$ of ${\mathcal{V}}$ such that:
$c_{\mid{\mathcal{V}}^{\prime}\setminus\nu^{-1}(Z)}:{\mathcal{V}}^{\prime}\setminus\nu^{-1}(Z)\rightarrow{\mathcal{V}}\setminus
Z$ is an isomorphism, and $\nu^{-1}(P)$ is a compact connected analytic curve
in ${\mathcal{V}}^{\prime}$ for every $P\in{\mathcal{V}}$.  of
${\mathcal{V}}$. If $D$ satisfies $\mathbf{CNB}$, then the divisor
$D^{\prime}:=\nu^{\ast}(D)$ in ${\mathcal{V}}^{\prime}$ also satisfies
$\mathbf{CNB}$.

###### Proof.

The compactness and the connectedness of $|D|$ imply the compactness and the
connectedness of $|D^{\prime}|=\nu^{-1}(|D|)$ since $\nu$ is proper with
connected fibers. The validity of (2.1.2) and (2.1.3) for the divisor $D$,
together with the projection formula, implies its validity for $D^{\prime}$. ∎

Observe also that, with the notation of Proposition 2.1.5, the projection
formula also implies the equality:

(2.1.13) $D^{\prime}\cdot D^{\prime}=D\cdot D.$

Moreover, if $C$ is an irreducible component of $D$ of mutiplicity $\mu$, then
its strict transform $C^{\prime}$ in ${\mathcal{V}}^{\prime}$ is an
irreducible component of multiplicity $\mu$ in $D^{\prime}$, and we have:

(2.1.14) $C^{\prime}\cdot D^{\prime}=C\cdot D.$

Proposition 2.1.5 and these observations allow one to generalize our previous
results to situations, where instead of complex analytic maps from the
analytic surface ${\mathcal{V}}$ to a smooth algebraic surface, one considers
meromorphic maps.

To formulate this generalization, recall that, for every meromorphic map:

$\alpha:{\mathcal{V}}\dashrightarrow X$

of range a smooth projective complex surface $X$, there exists a modification:

$\nu^{\prime}:{\mathcal{V}}^{\prime}\longrightarrow{\mathcal{V}}$

of ${\mathcal{V}}$ that is “adapted to $\alpha$”, namely such that, composed
with $\nu$, the meromorphic map $\alpha$ defines an actual complex analytic
map:

$\alpha^{\prime}:=\alpha\circ\nu:{\mathcal{V}}\longrightarrow X.$

The meromorphic map $\alpha$ defines a correspondence $\Gamma_{\alpha}$ from
${\mathcal{V}}$ to $X$, proper over ${\mathcal{V}}$, and for every divisor $D$
with compact support in ${\mathcal{V}}$, we may consider the image
$\Gamma_{\alpha\ast}D$ of $D$ by this correspondence. It is a divisor in $X$,
effective when $D$ is effective, and in terms of a modification $\nu$ adapted
to $\alpha$ as above, it may defined by the equality:

$\Gamma_{\alpha\ast}D=\alpha^{\prime}_{\ast}\nu^{\ast}D.$

######  Proposition 2.1.6.

Let ${\mathcal{V}}$ be a connected smooth complex analytic surface, and let
$D$ be an effective divisor in ${\mathcal{V}}$ satisfying $\mathbf{CNB}$. Let
$C$ be an irreducible component of $D$ such that:

$C\cdot D>0,$

and let $\mu$ be its multiplicity in $D$. Consider a commutative diagram::

(2.1.15) $\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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0.0pt\hbox{$\textstyle{X,}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{gathered}$

where $X$ and $Y$ are connected smooth projective complex surfaces, $f$ is a
dominant morphism, and $\alpha$ and $\beta$ are meromorphic maps.

If the image of $\alpha$ is Zariski dense in $X$, and if the restriction
$\alpha_{\mid C}$ of $\alpha$ to $C$ is birational onto its image, then the
degree of $f$ satisfies the following upper bound:

(2.1.16) $\deg
f\leq\frac{\Gamma_{\alpha\ast}D\cdot\Gamma_{\alpha\ast}D}{\Gamma_{\beta\ast}D\cdot\Gamma_{\beta\ast}D}\leq\frac{\Gamma_{\alpha\ast}D\cdot\Gamma_{\alpha\ast}D}{\mu\,C\cdot
D}.$

###### Proof.

The degree bound (2.1.16) follows from the estimates (2.1.11) — which
generalizes the estimates in Proposition 1.3.1 to general $\mathbf{CNB}$
divisors — applied to the commutative diagram:

$\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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defined by some modification ${\mathcal{V}}^{\prime}$ of ${\mathcal{V}}$
adapted both to $\alpha$ and $\beta$, and to the $\mathbf{CNB}$ divisor
$D^{\prime}$ in ${\mathcal{V}}$. ∎

#####  2.1.3. Generalization to $\mathbb{Q}$-divisors

Observe finally that, if $D$ is an effective divisor in a connected smooth
analytic complex surface ${\mathcal{V}}$ and $n$ a positive integer, $D$
satisfies $\mathbf{CNB}$ if an only if $nD$ does. Consequently the condition
$\mathbf{CNB}$ makes sense, not only for effective divisors in
${\mathcal{V}}$, but for any effective $\mathbb{Q}$-divisor.

Moreover the results of 2.1.1 and 2.1.2 extend to this more general setting,
notably the degree bound (2.1.11), the index bound _Ã la_ Nori (2.1.12), the
compatibility with modifications, and Proposition 2.1.6. This is a formal
consequence of the definitions and of the fact that the quotients of
intersection numbers in the right-hand side of the estimates (2.1.11),
(2.1.12), or (2.1.16) are unchanged when the divisor $D$ is replaced by some
positive multiple.

####  2.2. Application to analytic and algebraic varieties fibered over a
projective curve

#####  2.2.1. Complex analytic surfaces fibered over a projective curve:
$\mathbf{CNB}$ divisors and equilibrium divisors

In this section we consider the following data: a connected smooth projective
complex curve $C$, a connected smooth complex analytic surface
${\mathcal{V}}$, a surjective (necessarily flat) complex analytic map, with
connected fibers:

$\pi_{{\mathcal{V}}}:{\mathcal{V}}\longrightarrow C,$

a complex analytic section of $\pi_{{\mathcal{V}}}$:

$\varepsilon:C\longrightarrow{\mathcal{V}},$

and its image:

$C_{\mathcal{V}}:=\varepsilon(C).$

Let also $\Sigma$ be a non-empty finite subset of $C$, and let :

$\mathring{C}:=C\setminus\Sigma$

be its complement. It defines a smooth connected affine complex algebraic
curve. We shall assume that, _for every $x\in\Sigma$, the (possibly non-
reduced) connected curve $\pi_{\mathcal{V}}^{-1}(x)$ is non-compact_ and that
we are given _a compact connected curve
$F_{x}\subset|\pi_{\mathcal{V}}^{-1}(x)|$ such that:_

$F_{x}\cap C_{\mathcal{V}}\neq\emptyset.$

######

From the data of the section $\varepsilon_{\mathcal{V}}$ and of the “vertical
curves” $(F_{x})_{x\in\Sigma}$, we may carry out the following construction.

Recall that our assumptions on the curves $\pi^{-1}(x)$ and $F_{x}$ imply that
the intersection form restricted to divisors in ${\mathcal{V}}$ supported on

$F:=\bigcup_{x\in\Sigma}F_{x}$

is negative definite. In turns, this implies that a divisor $\Delta$ supported
on $F$ is effective if $\Delta\cdot W\leq 0$ for every irreducible component
of $F$.333See for instance [MB89, Lemme 3.7], where $(D.C)\geq 0$ should read
$(D.C)\leq 0$.

This implies that there exists a unique $\mathbb{Q}$-divisor $V$ supported on
$F$ such that, for every irreducible component $W$ of $F$:

(2.2.1) $(C_{\mathcal{V}}+V)\cdot W=0,$

and moreover that $V$ is effective. The support $|D|$ of the divisor

$D:=C_{\mathcal{V}}+V$

is easily seen to be connected, again as a consequence of the negativity of
the intersection form on divisors supported by $F$. In turn, this implies the
equality:

$|V|=F.$

Moreover we have:

$D\cdot D=C_{\mathcal{V}}\cdot D,$

and consequently $D$ satisfies condition $\mathbf{CNB}$ if and only if $D\cdot
D$ is positive.

The construction of the divisor $D$ starting from the section $\varepsilon$
and the vertical curves $(F_{x})_{x\in\Sigma}$ is a geometric counterpart of
the construction in Subsection 6.2.2 below of the Arakelov divisor:

$(P,(g_{V_{\sigma},P_{\sigma}})_{\sigma:K\hookrightarrow\mathbb{C}})$

attached to some formal-analytic surface:

${\widetilde{\mathcal{V}}}:=({\widehat{\mathcal{V}}},(V_{\sigma},P_{\sigma},\iota_{\sigma})_{\sigma\hookrightarrow\mathbb{C}})$

by means of the equilibrium potentials $g_{V_{\sigma},P_{\sigma}}$ associated
to the pointed compact Riemann surfaces with boundary
$(V_{\sigma},P_{\sigma})$. Accordingly the divisor $D$ could be referred to as
the _equilibrium divisor_ associated to $\varepsilon$ and
$(F_{x})_{x\in\Sigma}$.

######

Among the $\mathbf{CNB}$ $\mathbb{Q}$-divisors of the form
$C_{{\mathcal{V}}}+V$ with $V$ supported on $F$, when they exist, the
equilibrium divisor $D$ satisfies the following extremal property:

######  Proposition 2.2.1.

If there exists an effective $\mathbb{Q}$-divisor $\widetilde{V}$ supported on
$F$ such that the divisor

(2.2.2) $\widetilde{D}:=C_{\mathcal{V}}+\widetilde{V}$

satisfies $\mathbf{CNB}$, then the divisor $D$ satisfies $\mathbf{CNB}$.
Moreover, we have:

(2.2.3) $V\geq\widetilde{V},$ (2.2.4) $C_{\mathcal{V}}\cdot D\geq
C_{\mathcal{V}}\cdot\widetilde{D},$

and:

(2.2.5) $D\cdot D\geq\widetilde{D}\cdot\widetilde{D},$

and equality holds in (2.2.5) if and only if $\widetilde{D}=D$.

###### Proof.

Let $\widetilde{V}$ an effective $\mathbb{Q}$-divisor supported by $F$ such
that the divisor $\widetilde{D}$ defined by (2.2.2) satisfies $\mathbf{CNB}$,
and let:

$\Delta:=V-\widetilde{V}.$

Then, for every irreducible component $W$ of $F$, we have:

$\Delta\cdot W=-(\widetilde{D}-D)\cdot W=-\widetilde{D}.W\leq 0.$

Since $\Delta$ is supported by $F$, it is therefore effective. This implies
(2.2.3), and the inequality (2.2.4) immediately follows. In particular, we
have:

(2.2.6) $C_{\mathcal{V}}\cdot D\geq 0.$

Moreover, according to (2.2.1), we also have:

$D\cdot\Delta=0.$

Consequently:

$\widetilde{D}\cdot\widetilde{D}=(D-\Delta)\cdot(D-\Delta)=D\cdot
D+\Delta\cdot\Delta.$

Since $\Delta\cdot\Delta$ is non-positive and vanishes if and only if
$\Delta=0,$ this establishes the inequality (2.2.5) and the last assertion of
Proposition 2.2.1.

Finally the relations (2.2.1), (2.2.6), (2.2.5), and the positivity of
$\widetilde{D}\cdot\widetilde{D}$ imply that $D$ satisfies $\mathbf{CNB}$.∎

######  Corollary 2.2.2.

The following conditions are equivalent:

  1. (i)

$C_{\mathcal{V}}\cup F$ is the support of a $\mathbb{Q}$-divisor satisfying
$\mathbf{CNB}$;

  2. (ii)

$D\cdot D>0;$

  3. (iii)

$D$ satisfies $\mathbf{CNB}$.

The analogue of Corollary 2.2.2 in the arithmetic setting will be established
in Section 6.3.

#####  2.2.2. Compatibility with modifications

In the present “fibered” situation, the compatibility of condition
$\mathbf{CNB}$ with modifications discussed in 2.1.2 may be complemented by
the following remarks.

Let us keep the notation of 2.2.1, and let us consider a modification
$\nu:{\mathcal{V}}^{\prime}\rightarrow{\mathcal{V}}$ of ${\mathcal{V}}$. Then
the morphism:

$\pi_{{\mathcal{V}}^{\prime}}:=\pi_{\mathcal{V}}\circ\nu:{\mathcal{V}}^{\prime}\longrightarrow
C$

is surjective with connected fibers, and the strict transform
$C_{{\mathcal{V}}^{\prime}}$ of $C_{{\mathcal{V}}}$ in
${\mathcal{V}}^{\prime}$ is the image of a section:

$\varepsilon^{\prime}:C\longrightarrow{\mathcal{V}}^{\prime}$

of $\pi_{{\mathcal{V}}^{\prime}}$.

The indetermination locus of $\nu^{-1}$ meets
$\mathring{C}_{\mid{\mathcal{V}}}:=\varepsilon(\mathring{C})$ in a finite
subset $\varepsilon(\widetilde{\Sigma})$. Let us define:

$\Sigma^{\prime}:=\Sigma\cup\widetilde{\Sigma},$

and for every $x$ in $\Sigma$ (resp. in $\widetilde{\Sigma}$):

$F^{\prime}_{x}:=\nu^{-1}(F_{x})\quad\mbox{(resp.
$F^{\prime}_{x}:=\nu^{-1}(\varepsilon(x))$).}$

The $F^{\prime}_{x},$ $x\in\Sigma^{\prime},$ are compact connected curves in
$|\pi_{{\mathcal{V}}^{\prime}}^{-1}(x)|$ such that:

$F^{\prime}_{x}\cap C_{{\mathcal{V}}^{\prime}}\neq\emptyset.$

The effective $\mathbb{Q}$-divisor $D^{\prime}:=\nu^{\ast}(D)$ in
${\mathcal{V}}^{\prime}$ may be written:

$D^{\prime}=C_{{\mathcal{V}}^{\prime}}+V^{\prime}$

where $V^{\prime}$ is a divisor with support:

$F^{\prime}:=\bigcup_{x\in\Sigma^{\prime}}F^{\prime}_{x}.$

Moreover the projection formula together with (2.2.1) imply that, for every
irreducible component $W^{\prime}$ of $F^{\prime}$, we have:

$(C_{{\mathcal{V}}^{\prime}}+V^{\prime})\cdot W^{\prime}=0.$

This shows that the divisor $D^{\prime}$ in ${\mathcal{V}}^{\prime}$ coincides
with the equilibrium divisor associated to the section $\varepsilon^{\prime}$
and the family of vertical curves $(F^{\prime}_{x})_{x\in\Delta^{\prime}}$ in
${\mathcal{V}}^{\prime}$.

In particular, we have:

$D^{\prime}\cdot D^{\prime}=C_{{\mathcal{V}}^{\prime}}\cdot D^{\prime}.$

Moreover, as already observed:

$D^{\prime}\cdot D^{\prime}=D\cdot D,$

and therefore $D^{\prime}$ satisfies $\mathbf{CNB}$ if and only if $D$
satisfies it.

#####  2.2.3. Applications: morphisms from fibered analytic surfaces to
fibered algebraic surfaces

In this subsection and in the next one, we keep the notation of Subsection
2.2.1 and we assume that _the $\mathbb{Q}$-divisor $D$ has positive self-
intersection_:

$D\cdot D=C\cdot D>0,$

and therefore satisfies $\mathbf{CNB}$.

Observe that, if $X$ is a connected smooth projective complex surface and if

$\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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31.53471pt\raise-36.83331pt\hbox{\hbox{\kern 0.0pt\raise
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is a commutative diagram where $\alpha$ is a meromorphic map and $\pi_{X}$ is
a morphism of complex algebraic varieties,444or equivalently, by GAGA, of
complex analytic manifolds. The morphism $\pi_{X}$ is necessarily non-
constant, hence surjective and flat. then $\alpha_{\mid C_{\mathcal{V}}}$ is
trivially birational onto its image. Moreover the image of $\alpha$ is easily
seen to be contained in some algebraic curve in $X$ if and only if $\alpha$
factorizes through $\pi_{{\mathcal{V}}}$. In other words, the image of
$\alpha$ is Zariski dense in $X$ if and only if $\alpha$ is not constant on
the fibers of $\pi_{\mathcal{V}}$.

Applied to the present setting, Proposition 2.1.6 for $\mathbb{Q}$-divisors
takes the following form:

######  Theorem 2.2.3.

Consider a commutative diagram:

(2.2.7) $\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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where $X$ and $Y$ are two projective complex surfaces, with $X$ smooth and
connected and $Y$ integral, where $\alpha$ and $\beta$ are meromorphic maps,
and where $f$ and $\pi_{X}$ are morphisms of complex algebraic varieties.

If $\alpha$ is not constant on the fibers of $\pi_{\mathcal{V}}$, then $f$ is
dominant and its degree satisfies:

(2.2.8) $\deg
f\leq\frac{\Gamma_{\alpha\ast}D\cdot\Gamma_{\alpha\ast}D}{C_{\mathcal{V}}\cdot
D}=\frac{\Gamma_{\alpha\ast}D\cdot\Gamma_{\alpha\ast}D}{D\cdot D}.$

Actually Proposition 2.1.6 immediately establishes Theorem 2.2.3 when the
surface $Y$ is smooth. By considering a resolution of $Y$, this implies
Theorem 2.2.3 in general.

Similarly, the amplification of Proposition 1.3.2 concerning $\mathbf{CNB}$
divisors discussed at the end of Section 2.1.1, implies the following result
_à la_ Lefschetz-Nori concerning étale fundamental groups:

######  Theorem 2.2.4.

Consider a commutative diagram:

(2.2.9) $\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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30.93124pt\raise-37.80553pt\hbox{\hbox{\kern 0.0pt\raise
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30.93124pt\raise-37.80553pt\hbox{\hbox{\kern 0.0pt\raise
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where $U$ is a connected smooth quasi-projective complex surface, $\alpha$ a
complex analytic map, and $\pi_{V}$ a morphism of complex algebraic
varieties.555The morphism $\pi_{U}$ is necessarily surjective and flat.

If $\alpha$ is non constant on the fibers of $\pi_{\mathcal{V}},$ then the
image of the morphism between étale fundamental groups:

(2.2.10) $\alpha_{\mid C_{\mathcal{V}}\cup
F\ast}:\pi_{1}^{\mathrm{\acute{e}t}}(C_{\mathcal{V}}\cup
F,c)\longrightarrow\pi_{1}^{\mathrm{\acute{e}t}}(U,x)$

induced by the map:

$\alpha_{\mid C_{\mathcal{V}}\cup F}:C_{\mathcal{V}}\cup F\rightarrow U$

is a subgroup of finite index; moreover:

(2.2.11) $[\pi_{1}^{\mathrm{\acute{e}t}}(U,x):\alpha_{\mid C_{\mathcal{V}}\cup
F\ast}(\pi_{1}^{\mathrm{\acute{e}t}}(C_{\mathcal{V}}\cup
F,c))]\leq\frac{\alpha_{\ast}D\cdot\alpha_{\ast}D}{C_{\mathcal{V}}\cdot
D}=\frac{\alpha_{\ast}D\cdot\alpha_{\ast}D}{D\cdot D}.$

In (2.2.10) and (2.2.11), $c$ denotes a complex point of $C$, and
$x:=\alpha\circ\varepsilon(c)$ its image in ${U}$.

#####  2.2.4. The field of meromorphic functions
${\mathcal{M}}({\mathcal{V}})$

The map between fields of meromorphic functions:

$\pi_{\mathcal{V}}^{\ast}:{\mathcal{M}}(C)\longrightarrow{\mathcal{M}}({\mathcal{V}}),\quad
f\longmapsto f\circ\pi_{\mathcal{V}}$

allows one to identify the field ${\mathcal{M}}(C)=\mathbb{C}(C)$ of rational
functions over $C$ with a subfield of the field ${\mathcal{M}}({\mathcal{V}})$
of meromorphic functions over ${\mathcal{V}}$. Then $\mathbb{C}(C)$ is easily
seen to be algebraically closed in ${\mathcal{M}}({\mathcal{V}})$. Moreover
any meromorphic function $f\in{\mathcal{M}}({\mathcal{V}})$ defines a
meromorphic map “fibered over $C$,” namely:

$f_{C}:=(\pi_{\mathcal{V}},f):{\mathcal{V}}\dashrightarrow\mathbb{P}^{1}_{C}:=C\times\mathbb{P}^{1}_{C}.$

######  Theorem 2.2.5.

The following alternative holds: either
${\mathcal{M}}({\mathcal{V}})=\mathbb{C}(C),$ or
${\mathcal{M}}({\mathcal{V}})$ is an extension of finite type and of
transcendence degree one of $\mathbb{C}(C)$.

Moreover, for every $f$ in
${\mathcal{M}}({\mathcal{V}})\setminus\mathbb{C}(C),$
${\mathcal{M}}({\mathcal{V}})$ is a finite extension of the purely
transcendental extension $\mathbb{C}(C)(f)$ of $\mathbb{C}(C)$, and its degree
satisfies the following upper bound:

(2.2.12)
$[{\mathcal{M}}({\mathcal{V}}):\mathbb{C}(C)(f)]\leq\frac{\Gamma_{f_{C}\ast}D\cdot\Gamma_{f_{C}\ast}D}{C_{\mathcal{V}}\cdot
D}=\frac{\Gamma_{f_{C}\ast}D\cdot\Gamma_{f_{C}\ast}D}{D\cdot D}.$

The right-hand side of (2.2.12) is clearly unchanged when ${\mathcal{V}}$ is
replaced by some connected open neighborhood ${\mathcal{V}}^{\prime}$ of
$C_{\mathcal{V}}\cup F$. This immediately implies that, in Theorem 2.2.5, we
may replace the complex analytic surface ${\mathcal{V}}$ by its germ
${\mathcal{V}}^{{\rm an}}_{C_{\mathcal{V}}\cup F}$ along
$C_{{\mathcal{V}}}\cup F$:

######  Corollary 2.2.6.

Theorem 2.2.5 still holds when the field ${\mathcal{M}}({\mathcal{V}})$ is
replaced by the field ${\mathcal{M}}({\mathcal{V}}^{{\rm
an}}_{C_{\mathcal{V}}\cup F})$ of germs of meromorphic functions on
${\mathcal{V}}$ along $C_{\mathcal{V}}\cup F$. Moreover there exists a
connected open neighborhood ${\mathcal{V}}^{\prime}$ of $C_{\mathcal{V}}\cup
F$ in ${\mathcal{V}}$ such that any germ of meromorphic functions in
${\mathcal{M}}({\mathcal{V}}^{{\rm an}}_{C_{\mathcal{V}}\cup F})$ extends to a
meromorphic function in ${\mathcal{V}}^{\prime}$.

As mentioned in 1.4.2 above, the results of the first part of this monograph
admit variants in formal geometry. We shall leave this to the interested
reader, and will only mention that Theorem 2.2.5 still holds when the field
${\mathcal{M}}({\mathcal{V}})$ is replaced by the field
${\mathcal{M}}(\widehat{{\mathcal{V}}}_{C_{\mathcal{V}}\cup F})$ of formal
meromorphic functions, in the sense of [HM68], on the formal completion
$\widehat{{\mathcal{V}}}_{C_{\mathcal{V}}\cup F}$ of ${\mathcal{V}}$ along the
projective curve $C_{\mathcal{V}}\cup F$. The field
${\mathcal{M}}({\mathcal{V}}^{{\rm an}}_{C_{\mathcal{V}}\cup F})$ naturally
embeds into the field
${\mathcal{M}}(\widehat{{\mathcal{V}}}_{C_{\mathcal{V}}\cup F})$, and an
intriguing question is whether these two fields always coincide.666These
fields are easily seen to coincide when ${\mathcal{M}}({\mathcal{V}}^{{\rm
an}}_{C_{\mathcal{V}}\cup F})\neq\mathbb{C}(C).$ Accordingly the question is
equivalent to the following one: _does the equality
${\mathcal{M}}({\mathcal{V}}^{{\rm an}}_{C_{\mathcal{V}}\cup
F})=\mathbb{C}(C)$ imply the equality_
${\mathcal{M}}(\widehat{{\mathcal{V}}}_{C_{\mathcal{V}}\cup F})=\mathbb{C}(C)$
? We refer the reader to [CG81], [Hir81], [Ste86], and [GPR94, Section VII.4]
for results and references concerning the validity of such a “formal
principle” in similar situations.

###### Proof of Theorem 2.2.5.

It is enough to show that, if $f\in{\mathcal{M}}({\mathcal{V}})$ does not
belong to $\mathbb{C}(C)$ — or equivalently is not constant on the fibers of
$\pi_{{\mathcal{V}}}$ — then, for every $g\in{\mathcal{M}}({\mathcal{V}})$,
the subfield $\mathbb{C}(C)(f,g)$ of ${\mathcal{M}}({\mathcal{V}})$ is a
finite extension of $\mathbb{C}(C)(f)$, of degree at most:

(2.2.13)
$\frac{\Gamma_{f_{C}\ast}D\cdot\Gamma_{f_{C}\ast}D}{C_{\mathcal{V}}\cdot
D}=\frac{\Gamma_{f_{C}\ast}D\cdot\Gamma_{f_{C}\ast}D}{D\cdot D}.$

Indeed this will implies that any finitely generated extension of
$\mathbb{C}(C)(f)$ in ${\mathcal{M}}({\mathcal{V}})$ is a finite extension of
$\mathbb{C}(C)(f)$ of degree at most (2.2.13), and consequently that
${\mathcal{M}}({\mathcal{V}})$ itself is a finite extension of
$\mathbb{C}(C)(f)$ of degree at most (2.2.13).

Let us consider $f$ and $g$ in ${\mathcal{M}}({\mathcal{V}})$ as above. There
exists a modification of ${\mathcal{V}},$

$\nu:{\mathcal{V}}^{\prime}\longrightarrow{\mathcal{V}},$

that is adapted both to $f$ and $g$, namely such that the meromorphic
functions $f^{\prime}:=f\circ\nu$ and $g^{\prime}:=g\circ\nu$ on
${\mathcal{V}}^{\prime}$ define actual analytic maps:

$f^{\prime}:{\mathcal{V}}^{\prime}\longrightarrow\mathbb{P}^{1}(\mathbb{C})\quad\mbox{and}\quad
g^{\prime}:{\mathcal{V}}^{\prime}\longrightarrow\mathbb{P}^{1}(\mathbb{C}).$

We may consider the complex analytic map:

(2.2.14)
$\gamma^{\prime}:=(\pi_{{\mathcal{V}}^{\prime}},f^{\prime},g^{\prime}):{\mathcal{V}}^{\prime}\longrightarrow
C\times\mathbb{P}^{1}(\mathbb{C})\times\mathbb{P}^{1}(\mathbb{C}),$

where $\pi_{{\mathcal{V}}^{\prime}}:=\pi_{{\mathcal{V}}}\circ\nu.$ Since the
divisor $D^{\prime}:=\nu^{\ast}(D)$ in ${\mathcal{V}}^{\prime}$ satisfies
$\mathbf{CNB}$, we may apply Proposition 2.1.3 to $\gamma^{\prime}$. Moreover,
since $f^{\prime}$ is not constant on the fibers of
$\pi_{{\mathcal{V}}^{\prime}}$, the image of

$f^{\prime}_{C}:=(\pi_{{\mathcal{V}}^{\prime}},f^{\prime}):{\mathcal{V}}^{\prime}\longrightarrow
C\times\mathbb{P}^{1}(\mathbb{C})=:\mathbb{P}^{1}_{C}$

is Zariski dense. Consequently, we are in the last case in the conclusion of
Proposition 2.1.3, and there exists an irreducible closed surface $H$ in
$C\times\mathbb{P}^{1}(\mathbb{C})^{2}$ which contains
$\gamma^{\prime}({\mathcal{V}})$ as a Zariski dense subset.

If we denote by:

$p_{12}:C\times\mathbb{P}^{1}(\mathbb{C})^{2}\longrightarrow
C\times\mathbb{P}^{1}(\mathbb{C})=:\mathbb{P}^{1}_{C}$

the projection on the first two factors, we may consider the commutative
diagrams:

$\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
7.52pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 33.1492pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
40.71169pt\raise-19.35387pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.66028pt\hbox{$\scriptstyle{p_{12\mid H}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
40.71169pt\raise-27.06999pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathcal{V}}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
13.69305pt\raise-32.0383pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.30278pt\hbox{$\scriptstyle{f_{C}^{\prime}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 31.52pt\raise-38.70775pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
9.89355pt\raise-13.61082pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.38196pt\hbox{$\scriptstyle{\gamma^{\prime}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 37.55641pt\raise-3.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
31.52pt\raise-38.70775pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathbb{P}^{1}_{C},}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{gathered}$

and:

(2.2.15) $\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
6.75pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 32.3792pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{H\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
39.94168pt\raise-19.35387pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.66028pt\hbox{$\scriptstyle{p_{12\mid H}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
39.94168pt\raise-27.06999pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathcal{V}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
13.30806pt\raise-32.59386pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.74722pt\hbox{$\scriptstyle{f_{C}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 30.75pt\raise-38.70775pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
10.49712pt\raise-14.16638pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8264pt\hbox{$\scriptstyle{\gamma}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 36.845pt\raise-3.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
30.75pt\raise-38.70775pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{$\textstyle{\mathbb{P}^{1}_{C},}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{gathered}$

where $\gamma:=\gamma^{\prime}\circ\nu^{-1}.$ Then, by construction, the maps
$f_{C}$ and $\gamma$ induce isomorphisms of $\mathbb{C}$-algebras:

$f_{C}^{\ast}:\mathbb{C}(\mathbb{P}^{1}_{C})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathbb{C}(C)(f)$

and:

$\gamma^{\ast}:\mathbb{C}(H)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathbb{C}(C)(f,g),$

and we have:

$[\mathbb{C}(C)(f,g):\mathbb{C}(C)(f)]=\deg p_{12\mid
H}:H\longrightarrow\mathbb{P}^{1}_{C}.$

The degree bound (2.2.8) in Theorem 2.2.3 applied to the diagram (2.2.15)
shows that this degree is bounded from above by (2.2.13). ∎

####  2.3. Examples and complements – The universal meromorphic map
$\varphi:{\mathcal{V}}\dashrightarrow{\mathcal{V}}^{\mathrm{alg}}$

In this section, we complete the results of the previous section concerning
analytic surfaces fibered over a projective curves endowed with a
$\mathbf{CNB}$ divisor. Notably in Subsection 2.3.1, we discuss the
construction of such analytic surfaces by blowing up points, starting from an
arbitrary algebraic surface fibered over a projective curve, equipped with a
section. We also pursue the study of the field of meromorphic functions on
these analytic surfaces.

#####  2.3.1. Deleting infinitely near points and $\mathbf{CNB}$ divisors

######

In this subsection, we consider the following analytic data: a connected
smooth projective complex curve $C$, a connected smooth projective complex
surface $X$, a surjective (necessarily flat) morphism of complex (algebraic or
analytic) varieties:

$\pi_{X}:X\longrightarrow C,$

an (algebraic or analytic) section of $\pi_{X}$:

$\varepsilon:C\longrightarrow X,$

and its image:

$C_{X}:=\varepsilon(C).$

The connectedness of the projective variety $X$ and the existence of the
section $\varepsilon$ imply the connectedness of the fibers of $\pi_{X}$.

Moreover we choose a point $x$ of $C$ whose fiber $X_{x}:=\pi_{X}^{-1}(x)$ is
smooth.

######

We may construct a system of modifications of $X$:

$X=:X^{0}\stackrel{{\scriptstyle\;\;\nu^{1}}}{{\longleftarrow}}X^{1}\stackrel{{\scriptstyle\;\;\nu^{2}}}{{\longleftarrow}}X^{2}\longleftarrow\cdots{\longleftarrow}X^{n-1}\stackrel{{\scriptstyle\;\;\nu^{n}}}{{\longleftarrow}}X^{n}\longleftarrow\cdots$

by defining:

$\nu^{n}:X^{n}\longrightarrow X^{n-1}$

to be the blowing-up of $X^{n-1}$ at some point $x_{n-1}$, for every $n\geq
1,$ where the points $(x_{n})_{n\geq 0}$ are chosen as follows:

  * •

$x_{0}$ is a point in $X_{x}$ distinct of the intersection $\varepsilon(x)$ of
$X_{x}$ with $C_{X}$;

  * •

for every $n\geq 1,$ $x_{n}$ is a point of the exceptional divisor of
$\nu^{n}$:

$Z_{n}:=(\nu^{n})^{-1}(x_{n-1})$

distinct from the intersection of $Z^{n}$ with the proper transform
$\widetilde{Z}^{n-1}$ of $Z^{n-1}$ by $\nu^{n}$, where by convention
$Z^{0}:=X_{x}$.

The section $\varepsilon$ of $\pi_{X}$ lifts to a section:

$\varepsilon^{n}:C\longrightarrow X^{n}$

of the morphism:

$\pi_{X^{n}}:=\pi_{X}\circ\nu^{1}\circ\cdots\circ\nu^{n}:X^{n}\longrightarrow
C.$

Let us denote by $\widetilde{X}^{n}_{x}$ the proper transform of $X_{x}$ by:

$\nu^{1}\circ\cdots\circ\nu^{n}:X^{n}\longrightarrow X.$

The divisor $X^{n}_{x}:=\pi_{X^{n}}^{-1}(x)$ may be written:

$X^{n}_{x}=\widetilde{X}^{n}_{x}+\sum_{1\leq i\leq n}E^{n}_{i},$

where $E^{n}_{n}=Z^{n}$ is the exceptional divisor of $\nu^{n}$, and where
$E^{n}_{i}$ is the proper transform of $E^{n-1}_{i}$ by $\nu^{n}$ when $i<n$.
(Observe that $E^{n}_{n-1}=\widetilde{Z}^{n-1}$, and that when $i<n-1$,
$\nu^{n}$ is actually an isomorphism over $E^{n-1}_{i}$.)

By construction, the $E^{n}_{i}$ are smooth rational curves in $X^{n}_{x}$.
Moreover the possibly non-vanishing intersection numbers between the divisors
$C_{X^{n}}:=\varepsilon_{X^{n}}(C)$, $\widetilde{X}^{n}_{x}$, and $E^{n}_{i}$,
$1\leq i\leq n$, are as follows:

$C_{X^{n}}\cdot C_{X^{n}}=C_{X}\cdot C_{X},\quad
C_{X^{n}}\cdot\widetilde{X}^{n}_{x}=1,\quad\widetilde{X}^{n}_{x}\cdot\widetilde{X}^{n}_{x}=-1,\quad\widetilde{X}^{n}_{x}\cdot
E^{n}_{1}=1,$ $E^{n}_{i}\cdot E^{n}_{i+1}=1\quad\mbox{and}\quad E^{n}_{i}\cdot
E^{n}_{i}=-2\quad\mbox{if $1\leq i\leq n-1,$}$

and:

$E^{n}_{n}\cdot E^{n}_{n}=-1.$

See Figure 2. This is easily established by induction on $n$.

$\times$$x_{n}$$E^{n}_{n}$$0$$-1$$E^{n}_{n-1}$$1$$-2$$E^{n}_{2}$$n-2$$-2$$E^{n}_{1}$$n-1$$-2$$\widetilde{X}^{n}_{x}$$n$$-1$$C_{X^{n}}$$1$$C_{X}\cdot
C_{X}$ Figure 2. The divisors $C_{X^{n}}$ and
$X^{n}_{x}=\tilde{X}^{n}_{x}+\sum_{i=1}^{n}E_{i}^{n}$ inside $X^{n}$. In red,
the self-intersections of their components; in blue, their multiplicity in the
divisor $D$.

######

For every $n\geq 1$, we may consider the compact connected curve in
$X_{x}^{n}$:

$F_{x}:=\widetilde{X}_{x}^{n}\cup\bigcup_{1\leq i\leq n-1}E^{n}_{i}.$

If ${\mathcal{V}}_{n}$ is a connected open neighborhood, in the analytic
topology, of $C_{X^{n}}\cup F_{x}$ in $X^{n}$, we may apply the construction
in 2.2.1 to ${\mathcal{V}}:={\mathcal{V}}_{n},$
$\pi_{\mathcal{V}}:=\pi_{X\mid{\mathcal{V}}_{n}}$, $\Sigma:=\\{x\\}$, and
$\varepsilon:=\varepsilon^{n}$.

A straightforward computation shows that the “equilibrium divisor” $D=:D^{n}$
defined by this construction is:

$D^{n}=C_{X^{n}}+n\widetilde{X}^{n}_{x}+\sum_{1\leq i\leq n-1}(n-i)E^{n}_{i}.$

Its self-intersection satisfies:

$D^{n}\cdot D^{n}=C_{X^{n}}\cdot D^{n}=C_{X}\cdot C_{X}+n.$

Consequently, if $n$ satisfies the inequality:

$n\geq 1+(-C_{X}\cdot C_{X})^{+},$

then the self-intersection $D^{n}\cdot D^{n}$ is positive, and $D^{n}$ is a
$\mathbf{CNB}$ divisor in ${\mathcal{V}}_{n}$. This contrasts with the fact
that the self-intersection $C_{X}\cdot C_{X}$ is in general negative. Indeed
it follows from the work of Arakelov [Ara71] and Szpiro [Szp81, Chapitre III]
that, if $C_{X}\cdot C_{X}$ is positive (respectively vanishes), then the
smooth fibers of $\pi_{X}:X\rightarrow C$ are rational curves (resp. $\pi_{X}$
is isotrivial).

#####  2.3.2. Non-constant morphisms to fibered algebraic surfaces

In this subsection, we use the notation of 2.2.3 and 2.2.4. So we denote by
$\pi_{\mathcal{V}}:{\mathcal{V}}\rightarrow C$ a connected smooth complex
analytic surface fibered over $C,$ and by $D:=C_{\mathcal{V}}+V$ its
“equilibrium divisor”, and we assume that $D$ satisfies $\mathbf{CNB}$.

A variant of the proof of Theorem 2.2.5 allows one to derive the following
result from the degree bound in Theorem 2.2.3.

######  Theorem 2.3.1.

Consider a commutative diagram:

$\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
6.93124pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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14.24785pt\raise 4.50694pt\hbox{{}\hbox{\kern 0.0pt\raise
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 31.43889pt\raise 0.0pt\hbox{\hbox{\kern
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.82361pt\hbox{$\scriptstyle{\pi_{\mathcal{V}}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-27.97221pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
31.43889pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
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38.9736pt\raise-18.90276pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern
38.9736pt\raise-27.97221pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
11.70598pt\raise-31.69164pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.74722pt\hbox{$\scriptstyle{\mathrm{Id}_{C}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
30.93124pt\raise-37.80553pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
30.93124pt\raise-37.80553pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{C,}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{gathered}$

where $X$ is a smooth connected projective complex surface, $\alpha$ is a
meromorphic map, and $\pi_{X}$ is a morphism of complex algebraic varieties.

If $\alpha$ is not constant on the fibers of $\pi_{\mathcal{V}}$, then the map
between fields of meromorphic functions:

$\alpha^{\ast}:\mathbb{C}(X)\longrightarrow{\mathcal{M}}({\mathcal{V}})$

is a finite extension of fields, and its degree satisfies the following upper
bound:

(2.3.1)
$[{\mathcal{M}}({\mathcal{V}}):\alpha^{\ast}\mathbb{C}(X)]\leq\frac{\Gamma_{\alpha\ast}D\cdot\Gamma_{\alpha\ast}D}{C_{\mathcal{V}}\cdot
D}=\frac{\Gamma_{\alpha\ast}D\cdot\Gamma_{\alpha\ast}D}{D\cdot D}.$

###### Proof.

The field $\mathbb{C}(X)$ is clearly an extension of transcendence degree one
of $\mathbb{C}(C)$. According to Theorem 2.2.5, the field
${\mathcal{M}}({\mathcal{V}})$, which contains $\alpha^{\ast}\mathbb{C}(X)$,
is therefore an extension of finite type and of transcendence degree one of
$\mathbb{C}(C)$. This implies that ${\mathcal{M}}({\mathcal{V}})$ is a finite
extension of $\alpha^{\ast}\mathbb{C}(X)$. To complete the proof of (2.3.1),
we shall show that, for every $\varphi\in{\mathcal{M}}({\mathcal{V}}),$ the
degree of the extension
$\alpha^{\ast}\mathbb{C}(X)(\varphi)=\alpha^{\ast}\mathbb{C}(X)[\varphi]$ of
$\alpha^{\ast}\mathbb{C}(X)$ satisfies the following inequality:

(2.3.2)
$[\alpha^{\ast}\mathbb{C}(X)(\varphi):\alpha^{\ast}\mathbb{C}(X)]\leq\frac{\Gamma_{\alpha\ast}D\cdot\Gamma_{\alpha\ast}D}{C_{\mathcal{V}}\cdot
D}=\frac{\Gamma_{\alpha\ast}D\cdot\Gamma_{\alpha\ast}D}{D\cdot D}.$

To achieve this, we consider an integral complex projective surface $Y$ whose
field of rational functions $\mathbb{C}(Y)$, as a $\mathbb{C}$-algebra, is
isomorphic to $\alpha^{\ast}\mathbb{C}(X)(\varphi)$. We may assume that the
composite map:

$\mathbb{C}(X)\stackrel{{\scriptstyle\alpha^{\ast}}}{{\longrightarrow}}\alpha^{\ast}\mathbb{C}(X)(\varphi)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\mathbb{C}(Y)$

is induced by a morphism $f:Y\rightarrow X$ of complex algebraic varieties.
Moreover the map:

$\mathbb{C}(Y)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\alpha^{\ast}\mathbb{C}(X)(\varphi)\,{\lhook\joinrel\longrightarrow}\,{\mathcal{M}}({\mathcal{V}})$

is induced by some meromorphic map:

$\beta:{\mathcal{V}}\dashrightarrow Y.$

The maps $\alpha,$ $\beta,$ and $f$ fit into a commutative diagram (2.2.7),
and (2.3.2) follows from the conclusion (2.2.8) of Theorem 2.2.3, since:

$\deg f=[\alpha^{\ast}\mathbb{C}(X)(\varphi):\alpha^{\ast}\mathbb{C}(X)].\qed$

For later reference, we state the following straightforward consequence of
Theorem 2.3.1:

######  Corollary 2.3.2.

Consider a commutative diagram:

$\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
6.93124pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathcal{V}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
14.24785pt\raise 4.50694pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{\alpha}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 32.01459pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
0.0pt\raise-18.90276pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.82361pt\hbox{$\scriptstyle{\pi_{\mathcal{V}}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-27.97221pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
32.01459pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
38.9736pt\raise-18.90276pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.82361pt\hbox{$\scriptstyle{\pi_{U}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
38.9736pt\raise-27.97221pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
11.70598pt\raise-31.69164pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.74722pt\hbox{$\scriptstyle{\mathrm{Id}_{C}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
30.93124pt\raise-37.80553pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
30.93124pt\raise-37.80553pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{C,}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{gathered}$

where $U$ is a smooth connected complex surface, $\alpha$ is a complex
analytic map, and $\pi_{U}$ is a morphism of complex algebraic varieties.

If $\alpha$ is an open immersion,777or equivalently, an injective complex
analytic local diffeomorphism. then the pull-back by $\alpha$ induces an
isomorphism:

$\alpha^{\ast}:\mathbb{C}(U)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathcal{M}}({\mathcal{V}}).$

#####  2.3.3. The universal meromorphic map
$\varphi:{\mathcal{V}}\dashrightarrow{\mathcal{V}}^{\mathrm{alg}}$

In this subsection, as in 2.3.2, we use the notation of 2.2.3-4. Moreover we
suppose that the field ${\mathcal{M}}({\mathcal{V}})$ is not reduced to
$\mathbb{C}(C)$.

######

Under the above assumption, according to Theorem 2.2.5, the field
${\mathcal{M}}({\mathcal{V}})$ is an extension of finite type and of
transcendence degree one of $\mathbb{C}(C)$. Moreover the field
$\mathbb{C}(C)$ is easily seen to be algebraically closed in
${\mathcal{M}}({\mathcal{V}})$, and the restriction of meromorphic functions
on ${\mathcal{V}}$ to the section $\varepsilon_{\mathcal{V}}$ defines a place:

$\varepsilon_{\mathcal{V}}^{\ast}:{\mathcal{M}}({\mathcal{V}})\dashrightarrow\mathbb{C}(C)$

of the field ${\mathcal{M}}({\mathcal{V}})$ with values in $\mathbb{C}(C)$
that restricts to the identity on $\mathbb{C}(C)$.

Consequently there exists a smooth projective geometrically irreducible curve
${\mathcal{C}}$ over $\mathbb{C}(C)$, a $\mathbb{C}(C)$-rational point $P$ of
${\mathcal{C}}$, and an isomorphism of field extensions of $\mathbb{C}(C)$:

$i:\kappa({\mathcal{C}})=\mathbb{C}(C)({\mathcal{C}})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathcal{M}}({\mathcal{V}})$

such that the place $\varepsilon_{\mathcal{V}}^{\ast}$ of
${\mathcal{M}}({\mathcal{V}})$ coincides, via the isomorphism $i$, with the
place of $\kappa({\mathcal{C}})$ defined by the rational point $P$; in other
words:

$\varepsilon^{\ast}\circ i=P^{\ast}.$

Moreover the triple $({\mathcal{C}},P,i)$ is uniquely determined, up to a
unique isomorphism, by the above conditions.

We may choose a smooth projective model over $C$ of the curve ${\mathcal{C}}$
over $\mathbb{C}(C)$ — namely a smooth connected projective complex surface
$S$, equipped with a surjective morphism of complex varieties:

$\pi_{S}:S\longrightarrow C,$

such that the generic fiber of $\pi_{X}$ is isomorphic to ${\mathcal{C}}$.
Then the $\mathbb{C}(C)$-rational point $P$ of ${\mathcal{C}}\simeq
S_{\mathbb{C}(C)}$ defines a section of $\pi_{S}$:

$\varepsilon_{S}:C\longrightarrow X.$

Moreover the isomorphism of field extensions of $\mathbb{C}(C)$:

$i:\mathbb{C}(S)\simeq\kappa({\mathcal{C}})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathcal{M}}({\mathcal{V}})$

is induced by a meromorphic map $\varphi:{\mathcal{V}}\dashrightarrow S$ which
fits into a commutative diagram:

$\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
6.93124pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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0.0pt\raise-0.8264pt\hbox{$\scriptstyle{\varphi}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 32.61945pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.82361pt\hbox{$\scriptstyle{\pi_{\mathcal{V}}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-27.97221pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
32.61945pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
38.9736pt\raise-18.90276pt\hbox{{}\hbox{\kern 0.0pt\raise
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3.0pt}}}}}}\ignorespaces{\hbox{\kern
38.9736pt\raise-27.97221pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
11.70598pt\raise-31.69164pt\hbox{{}\hbox{\kern 0.0pt\raise
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3.0pt}}}}}}\ignorespaces{\hbox{\kern
30.93124pt\raise-37.80553pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
30.93124pt\raise-37.80553pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{C,}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{gathered}$

The map $\varphi$ is non-constant on the fibers of $\pi_{\mathcal{V}}$, and
satisfies:

$\varepsilon_{S}=\varphi\circ\varepsilon_{\mathcal{V}}.$

The family $(S,\pi_{S},\varepsilon_{S},\varphi)$ so constructed is unique up
to a birational isomorphism of $S$ fibered over $C$.

We shall denote by:

${\mathcal{V}}^{\mathrm{alg}}:=S$

for this projective variety over $C$ — which is well defined up to fibered
birational isomorphism – and the meromorphic map:

$\varphi:{\mathcal{V}}\dashrightarrow{\mathcal{V}}^{\mathrm{alg}}$

will be called the _universal meromorphic map_ from ${\mathcal{V}}$ to a
projective complex variety fibered over $C$. It is readily seen to satisfy the
universal property indicated by this terminology.

######

Let us consider a commutative diagram:

$\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
6.93124pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\mathcal{V}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
14.24785pt\raise 4.50694pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{\alpha}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 31.43889pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
0.0pt\raise-18.90276pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.82361pt\hbox{$\scriptstyle{\pi_{\mathcal{V}}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 0.0pt\raise-27.97221pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
31.43889pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
38.9736pt\raise-18.90276pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.82361pt\hbox{$\scriptstyle{\pi_{X}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
38.9736pt\raise-27.97221pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
11.70598pt\raise-31.69164pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern
30.93124pt\raise-37.80553pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
30.93124pt\raise-37.80553pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{C,}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{gathered}$

where $X$ is a smooth connected projective complex surface, $\alpha$ is a
complex analytic map, and $\pi_{X}$ is a morphism of complex algebraic
varieties.

According to Corollary 2.3.2, if $\alpha$ is an open immersion, we may choose
$X$ for ${\mathcal{V}}^{\mathrm{alg}}$, and the map:

$\alpha:{\mathcal{V}}\longrightarrow X=:{\mathcal{V}}^{\mathrm{alg}}$

is the universal meromorphic map. However we will show in Subsection 2.3.4
that in general the universal meromorphic map
$\varphi:{\mathcal{V}}\dashrightarrow{\mathcal{V}}^{\mathrm{alg}}$ may be
ramified along the section $\varepsilon_{\mathcal{V}}$.

######

Once a model ${\mathcal{V}}^{\mathrm{alg}}$ has been chosen, we may introduce
a modification:

$\nu:{\mathcal{V}}^{\prime}\longrightarrow{\mathcal{V}}$

that is adapted to the meromorphic map
$\varphi:{\mathcal{V}}\dashrightarrow{\mathcal{V}}^{\mathrm{alg}}$, and denote
by:

$\varphi^{\prime}:=\varphi\circ\nu:{\mathcal{V}}^{\prime}\longrightarrow{\mathcal{V}}^{\mathrm{alg}}$

the associated complex analytic map. Then we may consider the following
$\mathbb{Q}$-divisor in ${\mathcal{V}}^{\mathrm{alg}}$:

$\Gamma_{\varphi\ast}D:=\varphi^{\prime}_{\ast}\nu^{\ast}D.$

It is an effective $\mathbb{Q}$-divisor which may be written:

$\Gamma_{\varphi\ast}D=C_{{\mathcal{V}}^{\mathrm{alg}}}+W,$

where $C_{{\mathcal{V}}^{\mathrm{alg}}}$ is the image of the canonical section
$\varepsilon_{{\mathcal{V}}^{\mathrm{alg}}}$ of
$\pi_{{\mathcal{V}}^{\mathrm{alg}}}$, and where $W$ is a vertical
$\mathbb{Q}$-divisor in ${{\mathcal{V}}^{\mathrm{alg}}}$. Using that $D$ and
therefore $\nu^{\prime\ast}D$ satisfy the condition $\mathbf{CNB}$ and the
projection formula, one easily sees that the $\mathbb{Q}$-divisor
$\Gamma_{\varphi\ast}D$ is actually big and nef.

This implies that the set of closed integral curves in
${\mathcal{V}}^{\mathrm{alg}}$ disjoint from its support
$|\Gamma_{\varphi\ast}D|$ is finite. We shall denote by $E$ the union of these
curves; it is the largest closed reduced subscheme of
${\mathcal{V}}^{\mathrm{alg}}$ of pure dimension one that is disjoint from
$|\Gamma_{\varphi\ast}D|$.

######

The previous discussion admits a straightforward variant where the complex
manifold ${\mathcal{V}}$ (resp. the field ${\mathcal{M}}({\mathcal{V}})$) is
replaced by the germ of analytic manifold ${\mathcal{V}}^{\mathrm{an}}_{C\cup
F}$ of ${\mathcal{V}}$ along $C\cup F$ (resp. by the field
${\mathcal{M}}({\mathcal{V}}^{\mathrm{an}}_{C\cup F})$ of germs of meromorphic
functions along $C\cup F$).

In this way, we define a universal (germ of) meromorphic map:

(2.3.3) $\varphi:{\mathcal{V}}^{\mathrm{an}}_{C\cup
F}\dashrightarrow[{\mathcal{V}}^{\mathrm{an}}_{C\cup F}]^{\mathrm{alg}}$

from ${\mathcal{V}}^{\mathrm{an}}_{C\cup F}$ to a smooth projective complex
surface fibered over $C$, which induces an isomorphism of field extensions of
$\mathbb{C}(C)$:

$\varphi^{\ast}:\mathbb{C}([{\mathcal{V}}^{\mathrm{an}}_{C\cup
F}]^{\mathrm{an}})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathcal{M}}({\mathcal{V}}^{\mathrm{alg}}_{C\cup
F}).$

The existence of this universal meromorphic map and the fact that the divisor
$\Gamma_{\varphi\ast}D$ in $[{\mathcal{V}}^{\mathrm{an}}_{C\cup
F}]^{\mathrm{alg}}$ is big and nef allow one to establish some finiteness
result concerning the subalgebra ${\mathcal{A}}$ of
${\mathcal{M}}({\mathcal{V}}^{\mathrm{an}}_{C\cup F})$ consisting in the germs
of meromorphic function on ${\mathcal{V}}^{\mathrm{an}}_{C\cup F}$ that are
holomorphic on ${\mathcal{V}}^{\mathrm{an}}_{C\cup
F}\setminus\pi_{{\mathcal{V}}}^{-1}(\Sigma).$ The analogue of this result in
the arithmetic setting will be the content of Theorem 9.1.1.

From now on, for simplicity we shall denote:

${\widetilde{\mathcal{V}}}:={\mathcal{V}}^{\mathrm{an}}_{C\cup F}.$

Moreover we shall denote by $E$ the largest closed reduced subscheme of
${\widetilde{\mathcal{V}}}^{\mathrm{alg}}$ of pure dimension one that is
disjoint from $|\Gamma_{\varphi\ast}D|$.

Recall that the complement of the non-empty finite set $\Sigma$ in the
projective curve $C$ defines an affine complex curve:

$\mathring{C}:=C\setminus\Sigma.$

We shall denote its ring of regular function by:

$\mathbb{C}[\mathring{C}]:=\Gamma(\mathring{C},{\mathcal{O}}^{\rm{alg}}_{\mathring{C}})=\varinjlim_{n}\Gamma(C,{\mathcal{O}}_{C}(n\Sigma)).$

Clearly ${\mathcal{A}}$ is a $\mathbb{C}[\mathring{C}]$-algebra.

Let us denote by $E$ the (finite) union of the closed integral curves in
${{\widetilde{\mathcal{V}}}^{\mathrm{alg}}}$ disjoint from
$|\Gamma_{\varphi\ast}D|.$ The following proposition is established by
analyzing when the divisor of the pull-back $\varphi^{\ast}f$ in
${\mathcal{M}}({\widetilde{\mathcal{V}}})$ of a rational function
$f\in\mathbb{C}({\widetilde{\mathcal{V}}}^{\mathrm{alg}})$ is contained in
$\pi^{-1}(\Sigma)$. The details of its proof will be left to the reader.

######  Proposition 2.3.3.

The isomorphism of field extensions of $\mathbb{C}(C)$:

$\varphi^{\ast}:\mathbb{C}({{\widetilde{\mathcal{V}}}^{\mathrm{alg}}})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathcal{M}}({\widetilde{\mathcal{V}}})$

restricts to an isomorphism of $\mathbb{C}[\mathring{C}]$-algebras:

$\varphi^{\ast}:\Gamma\big{(}{{\widetilde{\mathcal{V}}}^{\mathrm{alg}}}\setminus{(E\cup\pi_{{\widetilde{\mathcal{V}}}^{\mathrm{alg}}}}^{-1}(\Sigma)\big{)},{\mathcal{O}}_{{\widetilde{\mathcal{V}}}^{\mathrm{alg}}})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathcal{A}}.$

Recall that Zariski [Zar54] has shown that, for any smooth projective complex
surface $X$ and any effective divisor $C$ in $X$, the $\mathbb{C}$-algebra:

$\Gamma(X\setminus|C|,{\mathcal{O}}_{X})\simeq\varinjlim_{n}\Gamma(X,{\mathcal{O}}_{X}(nC))$

is finitely generated.888The paper [Zar54] is reproduced in [Zar73, p.
261-274]. The proof of this finiteness generation result, incomplete in the
original publication [Zar54], is completed in [Zar73, p. 275].

Applied to $X:={\widetilde{\mathcal{V}}}^{\mathrm{alg}}$ and
$C:=E\cup\pi_{{\widetilde{\mathcal{V}}}^{\mathrm{alg}}}^{-1}(\Sigma)$, this
implies:

######  Corollary 2.3.4.

${\mathcal{A}}$ is a $\mathbb{C}[\mathring{C}]$-algebra of finite type.

#####  2.3.4. The universal meromorphic map
$\varphi:{\mathcal{V}}\dashrightarrow{\mathcal{V}}^{\mathrm{alg}}$ may be
ramified

In this subsection, we construct some examples that show that the “universal
meromorphic map”

$\varphi_{\mathcal{V}}:{\mathcal{V}}\dashrightarrow{\mathcal{V}}^{\mathrm{alg}}$

introduced in Subsection 2.3.3 may be ramified along the section
$C_{\mathcal{V}}$ of $\pi_{\mathcal{V}}$.

###### Algebraic data

As in 2.3.1, let us consider a connected smooth projective complex curve $C$,
a connected smooth projective complex surface $X$, a surjective (necessarily
flat) morphism of complex (algebraic or analytic) varieties:

$\pi_{X}:X\longrightarrow C,$

an (algebraic or analytic) section of $\pi_{X}$:

$\varepsilon:C\longrightarrow X,$

and its image:

$C_{X}:=\varepsilon(C).$

Let moreover $\Sigma$ be a non-empty finite subset of $C$, and for every
$x\in\Sigma$, let $F_{x}$ be a compact connected reduced curve $F_{x}$
strictly contained in $|\pi_{X}^{-1}(x)|$ such that:

$F_{x}\cap C_{X}\neq\emptyset,$

and let:

$F:=\bigcup_{x\in\Sigma}F_{x}.$

To these data, we may associate an equilibrium $\mathbb{Q}$-divisor $D$
supported by $C_{X}\cup F$ by the construction of Subsection 2.2.1999applied
to the complex surface ${\mathcal{V}}$ defined as an open neighborhood in $X$
of $C_{X}\cup F$ that does not contain any of the fibers $\pi^{-1}(x)$ for
$x\in\Sigma.$, namely the unique $\mathbb{Q}$-divisor of the form:

$D=C_{X}+V$

with $V$ supported on $F$, such that, for every component $W$ of $F$, we have:

(2.3.4) $D\cdot W=0.$

We shall make the following further assumptions:

  * •

the $\mathbb{Q}$-divisor $D$ has integral coefficients; in other words, $D$ is
a divisor in $X$;

  * •

the divisor $D$ satisfies $\mathbf{CNB}$; equivalently, the intersection
number $D\cdot D=C_{X}\cdot D$ is positive.

The construction in 2.3.1 allows one to construct instances of such objects
from any projective surface fibered over $C$ endowed with a section.

###### Constructing analytic surfaces fibered over $C$

We consider the following additional data:

  * •

a positive integer $e$;

  * •

a connected open neighborhood ${\mathcal{V}}$, in the analytic topology, of
$C_{X}\cup F$ in $X$, such that ${\mathcal{V}}$ contains no fiber
$\pi^{-1}(x)$, $x\in C$;

  * •

a smooth effective analytic divisor101010or equivalently, a closed complex
submanifold of dimension $1$ in ${\mathcal{V}}$. $R$ in ${\mathcal{V}}$,
meeting $C_{X}$ transversally, such that the following two conditions are
satisfied:

(2.3.5) $R\subset X\setminus F,$

and:

(2.3.6) $C_{X}\cdot R\equiv-\,C_{X}\cdot D\quad\mod e.$

For every positive integer $e$, a neighborhood ${\mathcal{V}}$ of $C_{X}\cup
F$ and a divisor $R$ in ${\mathcal{V}}$ satisfying the conditions above
clearly exist.

We shall denote by:

$\iota:{\mathcal{V}}\longrightarrow X$

the inclusion morphism.

Consider the complex analytic line bundle
${\mathcal{O}}_{\mathcal{V}}(R+C_{X}+V)$ over ${\mathcal{V}}$. According to
(2.3.4), (2.3.5) and (2.3.6), the degree of its restriction to every component
of $C_{X}\cup F$ is divisible by $e$. Consequently, after possibly replacing
${\mathcal{V}}$ by a smaller neighborhood of $C\cup F$ in $X$, we may assume
that there exists a complex analytic line bundle ${\mathcal{L}}$ over
${\mathcal{V}}$ and an isomorphism of complex analytic line bundles over
${\mathcal{V}}$:

(2.3.7)
$s:{\mathcal{O}}_{\mathcal{V}}(R+C_{X}+V)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathcal{L}}^{\otimes
e}.$

Using ${\mathcal{L}}$ and $s$, we may construct the cyclic cover:

$c:{\mathcal{W}}\longrightarrow{\mathcal{V}}$

obtained by taking the $e$-th root out of $s$; see for instance [EV92,
Sections 3.3-14]. By construction, ${\mathcal{W}}$ is a normal complex
analytic surface, equipped with an action of the cyclic group
$G:=\mathbb{Z}/e\mathbb{Z},$ and $c$ defines an isomorphism of complex
analytic spaces:

(2.3.8)
${\mathcal{W}}/G\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathcal{V}}.$

We shall denote by
$\sigma:{\mathcal{W}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathcal{W}}$
the automorphism of ${\mathcal{W}}$ defined by the action of the generator
$[1]$ of $\mathbb{Z}/e\mathbb{Z}$.

The cover $c$ is totally ramified over $C_{X}$, and therefore establishes an
isomorphism:

$c_{\mid
C_{\mathcal{W}}}:C_{\mathcal{W}}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}C_{X}$

between $C_{X}$ and the image $C_{\mathcal{W}}$ of some complex analytic
section $\varepsilon_{\mathcal{W}}$ of the complex analytic morphism:

$\pi_{\mathcal{W}}:=\pi_{X}\circ\iota\circ c:{\mathcal{W}}\longrightarrow C.$

We may finally consider a resolution111111The normal analytic surface
${\mathcal{W}}$ has a finite set of singular points, included in
$c^{-1}((R\cup C\cup F)_{\mathrm{sing}})$, where $(R\cup C\cup
F)_{\mathrm{sing}}$ denotes the finite set of singular points of $R\cup C\cup
F$.:

$\nu:\widetilde{{\mathcal{W}}}\longrightarrow{\mathcal{W}}$

of ${\mathcal{W}}$ and the morphism:

$\pi_{\widetilde{{\mathcal{W}}}}:=\pi_{W}\circ
r:\widetilde{{\mathcal{W}}}\longrightarrow C.$

The proper transform $C_{\widetilde{W}}$ of $C_{W}$ by $\nu$ is the image of
an analytic section of $\pi_{\widetilde{{\mathcal{W}}}}:$

$\varepsilon_{\widetilde{W}}:C\longrightarrow\widetilde{W}.$

Let $\Sigma^{\prime}$ be the finite subset of $C\setminus\Sigma$ such that:

$R\cap C_{X}=\varepsilon_{X}(\Sigma^{\prime}),$

and let:

$\widetilde{\Sigma}:=\Sigma\sqcup{\Sigma^{\prime}}.$

For every $x\in\widetilde{\Sigma},$ we define:

$\displaystyle\widetilde{F}_{x}$
$\displaystyle:=(c\circ\nu)^{-1}(F_{x})\quad\quad\,\mbox{if $x\in\Sigma$}$
$\displaystyle:=\nu^{-1}(\varepsilon_{X}(x))\quad\quad\;\;\;\mbox{if
$x\in\Sigma^{\prime}$.}$

Then, for every $x\in\widetilde{\Sigma},$ $\widetilde{F}_{x}$ is a compact
connected curve in $\pi_{{\mathcal{W}}}^{-1}(x)$, which itself is non-compact,
and contains $\varepsilon_{\widetilde{{\mathcal{W}}}}(x)$. Moreover the
$\mathbb{Q}$-divisor in $\widetilde{{\mathcal{W}}}$:

$\widetilde{D}:=\frac{1}{e^{\prime}}\,(c\circ\nu)^{\ast}D$

is easily seen to be the equilibrium divisor associated to the the section
$\varepsilon_{\widetilde{{\mathcal{W}}}}$ and to the family
$(\widetilde{F}_{x})_{x\in\widetilde{\Sigma}},$ and to satisfy $\mathbf{CNB}$.

######

We have constructed the following morphisms of complex analytic or algebraic
varieties fibered over $C$:

(2.3.9) $\widetilde{{\mathcal{W}}}\stackrel{{\scriptstyle
r}}{{\longrightarrow}}{\mathcal{W}}\stackrel{{\scriptstyle
c}}{{\longrightarrow}}{\mathcal{V}}\stackrel{{\scriptstyle\iota}}{{\longrightarrow}}X,$

and we may consider the induced maps between fields of meromorphic functions:

(2.3.10)
$\mathbb{C}(X)\stackrel{{\scriptstyle\iota^{\ast}}}{{\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}}}{\mathcal{M}}({\mathcal{V}})\stackrel{{\scriptstyle
c^{\ast}}}{{\longrightarrow}}{\mathcal{M}}({\mathcal{W}})\stackrel{{\scriptstyle\nu^{\ast}}}{{\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}}}{\mathcal{M}}(\widetilde{{\mathcal{W}}}).$

The map $\iota^{\ast}$ is indeed an isomorphism as a consequence of Corollary
2.3.2 applied to $\alpha:=\iota.$

The main result of this subsection is the following proposition.

######  Proposition 2.3.5.

If the analytic divisor $R$ is Zariski dense in $X$, then the cover $c$
induces an isomorphism of fields of meromorphic functions:

(2.3.11)
$c^{\ast}:{\mathcal{M}}({\mathcal{V}})\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{\mathcal{M}}({\mathcal{W}}).$

###### Proof.

The action of $G$ on the complex analytic surface ${\mathcal{W}}$ induces an
action of $G$ on the field ${\mathcal{M}}({\mathcal{W}})$. Moreover, according
to the isomorphism (2.3.8), its fixed field coincides with the image of
${\mathcal{M}}({\mathcal{V}})$ by $c^{\ast}$:

${\mathcal{M}}({\mathcal{W}})^{G}=c^{\ast}{\mathcal{M}}({\mathcal{V}}).$

This shows that ${\mathcal{M}}({\mathcal{W}})$ is a cyclic extension of
$c^{\ast}{\mathcal{M}}({\mathcal{V}})$ of degree $e^{\prime}$ dividing $e$, of
Galois group:

$\big{\\{}\sigma^{\ast k};k\in\mathbb{Z}/e^{\prime}\mathbb{Z}\big{\\}}.$

In particular, there exists $\varphi\in{\mathcal{M}}({\mathcal{W}})$ such
that:

${\mathcal{M}}({\mathcal{W}})=c^{\ast}{\mathcal{M}}({\mathcal{V}})[\varphi]$

and:

(2.3.12) $\sigma^{\ast}\varphi=e^{2\pi i/e^{\prime}}\varphi.$

If $e^{\prime}>1,$ then $\varphi\neq 0,$ and $\varphi^{e^{\prime}}$ belongs to
$c^{\ast}{\mathcal{M}}({\mathcal{V}})=c^{\ast}\iota^{\ast}\mathbb{C}(X)$, and
may therefore be written:

$\varphi^{e^{\prime}}=c^{\ast}\iota^{\ast}\psi$

for some $\psi$ in $\mathbb{C}(X)^{\times}$. In particular, the following
equality of analytic Weil divisors in ${\mathcal{W}}$ holds:

(2.3.13)
$e^{\prime}{\mathrm{div}\,}\varphi=c^{\ast}\iota^{\ast}{\mathrm{div}\,}\psi.$

The map $c$ is totally ramified over every component of $R$. Consequently the
points in $c^{-1}(R)$ are fixed under the action of $G$ on ${\mathcal{W}}$.
Together with (2.3.12), this implies that the support
$|{\mathrm{div}\,}\varphi|$ of ${\mathrm{div}\,}\varphi$ contains $c^{-1}(R)$.
Using (2.3.13), we conclude that the support $|{\mathrm{div}\,}\psi|$ of
${\mathrm{div}\,}\psi$ contains $R$.

This contradicts the Zariski density of $R$ in $X$, and establishes the
equality $e^{\prime}=1$. ∎

It is clear that we may find ${\mathcal{V}}$ and $R$ as in 2.3.4 such that $R$
is Zariski dense in $X$. It is enough to construct ${\mathcal{V}}$ and $R$
satisfying (2.3.5) and (2.3.6) such that $R\cap C_{X}$ is non-empty, and such
that, for some $x\in R\cap C_{X},$ the germ of $R$ at $x$ is not the branch of
an algebraic curve through $x$. Then $R\cap{\mathcal{V}}^{\prime}$ would
actually be Zarishi dense in $X$ for every open analytic neighborhood
${\mathcal{V}}^{\prime}$ of $C_{X}\cup F$ in ${\mathcal{V}}$.

######  Corollary 2.3.6.

If the analytic divisor $R$ is Zariski dense in $X$, then the map:

(2.3.14) $\iota\circ c\circ r:\widetilde{{\mathcal{W}}}\longrightarrow X$

is “the” universal meromorphic map from $\widetilde{{\mathcal{W}}}$ to a
projective complex variety fibered over $C$.

Observe that the universal meromorphic map (2.3.14) is ramified, with
ramification index $e$, along $\varepsilon_{\widetilde{{\mathcal{W}}}}.$

##  Part II ${{\mathcal{C}^{\mathrm{b}\Delta}}}$ Green functions on Riemann
surfaces and intersection theory on quasi-projective arithmetic surfaces

###  Chapter 3 Green functions with $\mathcal{C}^{\infty}$ and $L^{2}_{1}$
regularity and arithmetic intersection numbers

Part 2 of this memoir is devoted to various results concerning Arakelov
intersection pairings on arithmetic surfaces and Green functions on Riemann
surfaces. These results will be used in Part 3 to transpose in an arithmetic
setting the results in complex algebraic, analytic, and formal geometry
presented in Part 1.

This chapter and the next one are devoted to some extensions of Arakelov
intersection theory on arithmetic surfaces adapted to our needs. In this
chapter, we develop the version of Arakelov intersection theory in [Bos99],
that relies on the use of $L^{2}_{1}$ Green functions, in a possibly non-
projective situation. Moreover we show that it satisfies some nice
functoriality properties. These tools allow us to establish in Theorem 3.5.10
an arithmetic version of the degree bound _Ã la_ Nori in Proposition 1.1.1
that has been our geometric starting point in Chapter 1.

The necessity of adapting Arakelov intersection theory on arithmetic surfaces
to a possible non-projective setting has led us to give a self-contained
presentation of the theory, and this chapter is in principle accessible with
no previous familiarity with Arakelov geometry.

####  3.1. The Arakelov degree of $0$-cycles and the height of $1$-cycles

#####  3.1.1. The Arakelov degree of $0$-cycles

Let $X$ be an arithmetic scheme, that is, a separated scheme of finite type
over $\mathbb{Z}$. A _$0$ -cycle_ on $X$ is a finite formal sum:

$Z=\sum_{i\in I}n_{i}P_{i}$

where the $P_{i}$ are closed points of $X$ and the $n_{i}$ are integers. The
residue fields:

$\kappa(P_{i})={\mathcal{O}}_{X,P_{i}}/\mathfrak{m}_{X,P_{i}}$

are finite, and the _arithmetic_ or _Arakelov degree_ of $Z$ is defined as the
real number:

$\widehat{\rm deg}\,Z:=\sum_{i\in I}n_{i}\log|\kappa(P_{i})|.$

If $Y$ is another arithmetic scheme and $f:X\rightarrow Y$ a morphism of
schemes, then the images $f(P_{i})$ are closed points of $Y$, and the direct
image $f_{\ast}Z$ is the $0$-cycle defined by the equality:

$f_{\ast}Z:=\sum_{i\in I}[\kappa(P_{i}):\kappa(f(P_{i}))]\,f(P_{i}),$

Then we have, as a straightforward consequence of the definition of the
Arakelov degree:

(3.1.1) $\widehat{\rm deg}\,f_{\ast}Z=\widehat{\rm deg}\,Z.$

When $X$ is ${\rm Spec\,}\mathbb{Z},$ every $0$-cycle on $X$ is of the form:

${\mathrm{div}\,}q:=\sum_{p}v_{p}(q)\,[p]$

for some $q\in\mathbb{Q}^{\ast}$ — where the sum runs over the prime numbers
$p$ and where $v_{p}$ denotes the $p$-adic valuation and $[p]$ the closed
point $p\mathbb{Z}$ of ${\rm Spec\,}\mathbb{Z}$ — and we have:

(3.1.2) $\widehat{\rm deg}\,{\mathrm{div}\,}q=\sum_{p}v_{p}(q)\log p=\log|q|.$

The Arakelov degree of $0$-cycles is clearly uniquely determined by its
compatibility with direct images (3.1.1) and the relation (3.1.2). Observe
also that, if $X$ is a $\mathbb{F}_{p}$-scheme for some prime $p$, then the
$0$-cycle $Z$ on $X$ considered as a $\mathbb{F}_{p}$-scheme admits a
“geometric degree”, defined as in [Ful98, Definition 1.4]:

$\deg_{\mathbb{F}_{p}}Z:=\sum_{i\in
I}n_{i}[\kappa(P_{i}):\mathbb{F}_{p}]\in\mathbb{Z},$

which is related to its arithmetic degree by the equality:

(3.1.3) $\widehat{\rm deg}\,Z=\deg_{\mathbb{F}_{p}}Z.\log p.$

#####  3.1.2. The Arakelov degree of $0$-cycles rationally equivalent to zero
on proper one-dimensional schemes

In this subsection, we denote by $C$ be a scheme _proper over ${\rm
Spec\,}\mathbb{Z}$, integral and of dimension one_.

The (support of the) image the morphism $f:C\rightarrow{\rm Spec\,}\mathbb{Z}$
is a closed integral subscheme of ${\rm Spec\,}\mathbb{Z}$, and therefore
either ${\rm Spec\,}\mathbb{Z}$, or a closed point $[p]$ of ${\rm
Spec\,}\mathbb{Z}$. Accordingly the following alternative holds: either $C$ is
finite and flat over ${\rm Spec\,}\mathbb{Z}$, or $C$ is a proper
$\mathbb{F}_{p}$-scheme for a uniquely determined prime $p$.

######

When $C$ is finite and flat over ${\rm Spec\,}\mathbb{Z}$, then $C$ is affine,
the ring $\Gamma(C,{\mathcal{O}}_{C})$ is an order in some number field $K$,
and if ${\mathcal{O}}_{K}$ denotes the ring of integers of $K$, then ${\rm
Spec\,}{\mathcal{O}}_{K}$ may be identified with the normalization of $C$.

In this case, the relation (3.1.2) admits the following generalization. For
every $r\in\kappa(C)^{\times}=K^{\times},$ the Arakelov degree of the
$0$-cycle on $C$ defined by the divisor111It is defined, as in [Ful98, 1.2-3],
by the equality: ${\mathrm{div}\,}r:=\sum_{x\in C_{0}}\mathrm{ord}_{x}r.x,$
where $C_{0}$ denotes the set of closed points of $C$. ${\mathrm{div}\,}r$
satisfies:

(3.1.4) $\widehat{\rm deg}\,{\mathrm{div}\,}r=\sum_{\sigma\in
C(\mathbb{C})}\log|\sigma(r)|,$

where $C(\mathbb{C})$ denotes the set of complex points of the scheme $C$;
this set may be identified with the set, of cardinality $[K:\mathbb{Q}]$, of
the field embeddings $\sigma:K\rightarrow\mathbb{C}$. Indeed if
$q:=N_{K/\mathbb{Q}}r$ denotes the image of $r$ by the norm map from $K$ to
$\mathbb{Q}$, and if $f$ denotes the morphism of schemes from $C$ to ${\rm
Spec\,}\mathbb{Z}$, then we have:

$f_{\ast}{\mathrm{div}\,}r={\mathrm{div}\,}q$

and:

$|q|=\prod_{\sigma\in C(\mathbb{C})}|\sigma(r)|,$

and therefore (3.1.4) follows from (3.1.1) and (3.1.2).

######

When $C$ is a proper $\mathbb{F}_{p}$-scheme, then the ring
$\Gamma(C,{\mathcal{O}}_{C})$ is a finite extension $\mathbb{F}_{q}$ of
$\mathbb{F}_{p}$ and $C$ is a geometrically integral projective curve over
$\mathbb{F}_{q}$. Moreover the relation (3.1.3) and the basic properties of
the degree of $0$-cycles on proper schemes over a field (see [Ful98, 1.4])
imply that, for any $r\in\kappa(C)^{\times}$, we have:

(3.1.5) $\widehat{\rm deg}\,{\mathrm{div}\,}r=0.$

#####  3.1.3. The height of $1$-cycles with support proper over ${\rm
Spec\,}\mathbb{Z}$

Let us keep the notation of 3.1.2 and assume that $C$ is a (necessarily
closed) subscheme of some reduced, separated scheme $X$ of finite type over
${\rm Spec\,}\mathbb{Z}$. Assume moreover that $X$ is endowed with a Hermitian
line bundle ${\overline{L}}:=(L,\|.\|)$.

Let us consider a non-zero rational section $s$ of the line bundle $L_{\mid
C}$ over $C$, and its divisor ${\mathrm{div}\,}s$.

In case (1), the set $C(\mathbb{C})$ of complex points of $C$ may be
identified with a subset of $X(\mathbb{C})$, and accordingly the Hermitian
metric $\|.\|$ on $L_{\mathbb{C}}^{{\rm an}}$ defines a Hermitian metric
$\|.\|_{x}$ on the fiber $L_{x}$ for every $x\in C(\mathbb{C})$. We may
therefore consider the positive real numbers $(\|s(x)\|_{x})_{x\in
C(\mathbb{C})}$, and the relation (3.1.4) shows that the real number:

(3.1.6) $0pt_{{\overline{L}}}(C):=\widehat{\rm
deg}\,{\mathrm{div}\,}s-\sum_{x\in C(\mathbb{C})}\|s(x)\|_{x}$

does not depend on the choice of $s$. It defines the _height_ of $C$ with
respect to ${\overline{L}}$.

In case (2), the relation (3.1.5) shows that $\widehat{\rm
deg}\,{\mathrm{div}\,}s$ does not depend on the choice of $s$. We shall denote
it by $\widehat{\rm deg}\,L_{\mid C}.$ According to the relation (3.1.3), we
have:

$\widehat{\rm deg}\,L_{\mid C}=\deg_{C/\mathbb{F}_{p}}L.\log
p=\deg_{C/\mathbb{F}_{q}}L.\log q,$

where $\deg_{C/\mathbb{F}_{p}}L$ (resp. $\deg_{C/\mathbb{F}_{q}}L$) denotes
the usual “geometric” degree of the line bundle $L$ over $X$ seen as a proper
$\mathbb{F}_{p}$-scheme (resp. a geometrically integral projective curve over
$\mathbb{F}_{q}$).

It is convenient to use some uniform notation, covering both cases (1) and
(2), and to let in case (2):

$0pt_{{\overline{L}}}(C):=\widehat{\rm deg}\,L_{\mid C}.$

We will also use the alternative notation:

$\widehat{\rm deg}\,({\overline{L}}|C):=0pt_{{\overline{L}}}(C).$

In the framework of [BGS94], this real number would actually be denoted by
$\widehat{\rm deg}\,(\hat{c}_{1}({\overline{L}})|C)$.

Observe that the height $\widehat{\rm deg}\,({\overline{L}}|C)$ depends
“additively” on ${\overline{L}}$. Namely, if ${\overline{L}}_{1}$ and
${\overline{L}}_{2}$ are two Hermitian line bundles over $X$, we have:

(3.1.7) $\widehat{\rm
deg}\,({\overline{L}}_{1}\otimes{\overline{L}}_{2}|C)=\widehat{\rm
deg}\,({\overline{L}}_{1}|C)+\widehat{\rm deg}\,({\overline{L}}_{2}|C).$

The definition of $\widehat{\rm deg}\,({\overline{L}}|C)$ extends by linearity
to an arbitrary $1$-cycle $C$ in $X$ whose support $|C|$ is proper over ${\rm
Spec\,}\mathbb{Z}$. Namely, any such $1$-cycle is a finite formal sum:

$C=\sum_{i\in I}n_{i}C_{i}$

where the $C_{i}$ are closed integral subschemes of $X$ of dimension one,
proper over ${\rm Spec\,}\mathbb{Z}$, and the $n_{i}$ are integers, and we
define:

(3.1.8) $\widehat{\rm deg}\,({\overline{L}}|C)=\sum_{i\in
I}n_{i}\,\widehat{\rm deg}\,({\overline{L}}|C_{i}).$

######  Proposition 3.1.1.

Let $f:X\longrightarrow Y$ be a morphism of reduced separated schemes of
finite type over ${\rm Spec\,}\mathbb{Z}$. For every $1$-cycle $D$ over $X$
such that $|D|$ is proper over ${\rm Spec\,}\mathbb{Z}$ and every Hermitian
line bundle ${\overline{L}}$ over $Y$, the following equality holds:

(3.1.9) $\widehat{\rm deg}\,(f^{\ast}{\overline{L}}|D)=\widehat{\rm
deg}\,({\overline{L}}|f_{\ast}D).$

In the right-hand side of (3.1.9), we denote by $f_{\ast}D$ the direct image
of $D$ by $f$, defined as in [Ful98, 1.4]. It is a $1$-cycle on $Y$ whose
support is easily seen to be proper over ${\rm Spec\,}\mathbb{Z}$.

###### Proof.

We may assume that $D$ is an integral scheme $C$ as in 3.1.2 above. When $C$
is an horizontal curve, namely in case (1), the equality (3.1.9) is a
straightforward consequence of the definitions. When $C$ is vertical, namely
in case (2), it follows from the projection formula in [Ful98, Proposition 2.5
(c)] applied to morphisms between one-dimensional projective schemes over a
finite field. ∎

####  3.2. Green functions with $\mathcal{C}^{\infty}$ regularity and
$\ast$-products on Riemann surfaces

#####  3.2.1. Green functions with $\mathcal{C}^{\infty}$ regularity on a
Riemann surface

Let $M$ be a Riemann surface, and let $D$ be a divisor on $M$. Recall that we
denote by $\mathbbm{1}_{D}$ be the canonical section with divisor $D$ of the
analytic line bundle ${\mathcal{O}}_{M}(D)$.

###### Definitions

A _Green function with $\mathcal{C}^{\infty}$ regularity_ for the divisor $D$
on $M$ is a $\mathcal{C}^{\infty}$ function:

$g:M\setminus|D|\longrightarrow\mathbb{R}$

such that the equality:

$\|\mathbbm{1}_{D}\|_{g}=e^{-g}$

defines a metric on the restriction of $\mathcal{O}_{M}(D)$ to $M\setminus|D|$
that extends to a $\mathcal{C}^{\infty}$ metric $\|.\|_{g}$ on
$\mathcal{O}_{M}(D)$ over $M$.

This condition may be equivalently formulated by requiring that $g$ has
logarithmic singularities at the points of $|D|$. Namely, for every open
subset $U$ of $M$ and every invertible function $f$ on $U$ such that:

$\mathrm{div}f=D_{|U},$

there exists $h$ in $\mathcal{C}^{\infty}(U,\mathbb{R})$ such that:

(3.2.1) $g=\log|f|^{-1}+h\quad\mbox{over $U\setminus|D|.$}$

Observe that this implies that, like the function $\log|f|^{-1}$, the function
$g$ is locally $L^{1}$ on $U$, and that the distribution on $U$ it defines
satisfies the equality of currents on $U$:

$\frac{i}{\pi}\,\partial\overline{\partial}g=-\delta_{D}+\frac{i}{\pi}\,\partial\overline{\partial}h.$

This shows that a Green function $g$ as defined above is locally $L^{1}$ on
$M$ and that, considered as a distribution on $M$, it satisfies the equality:

(3.2.2) $\frac{i}{\pi}\,\partial\overline{\partial}g+\delta_{D}=\omega(g)$

for some $\mathcal{C}^{\infty}$ real $2$-form $\omega(g)$ on $M$.

Conversely, according to the ellipticity of the operator
$\partial\overline{\partial}$ acting on distributions on $M$, any real
distribution $g$ on $M$ that satisfies the equality (3.2.2) for some
$\mathcal{C}^{\infty}$ real $2$-form $\omega(g)$ on $M$ defines a Green
function with $\mathcal{C}^{\infty}$ regularity for the divisor $D$ on $M$.

Recall that the first Chern form of a Hermitian line bundle
${\overline{L}}=(L,\|.\|)$ with ${\mathcal{C}}^{\infty}$ metric over $M$ is
the $2$-form over $M$ defined locally by the equality:

(3.2.3) $c_{1}({\overline{L}}_{\mathbb{C}})=\frac{1}{2\pi
i}\partial\overline{\partial}\log\|s\|^{2},$

where $s$ denotes a local non-vanishing complex analytic section of $L$. With
this notation, if $g$ is a Green function with ${\mathcal{C}}^{\infty}$
regularity for a divisor $D$ on $M$, we also have:

$\omega(g)=c_{1}\big{(}{\mathcal{O}}_{M}(D),\|.\|_{g}\big{)}.$

###### Capacitary metrics

Let $P$ be a point of the Riemann surface $M$, and let $g$ be a Green function
for the divisor $P$ in $M$. The attached Hermitian metric $\|.\|_{g}$ on
$\mathcal{O}_{M}(P)$ defines, by restriction to the fiber

$\mathcal{O}(P)_{|P}\simeq T_{P}M,$

a Hermitian metric on the one-dimensional complex vector space $T_{P}M$. By
definition, this metric is the _capacitary metric_ $\|.\|_{g}^{\mathrm{cap}}.$

If $z$ is a local holomorphic coordinate on some neighborhood $U$ of $P$ in
$M$, we have the following equality of distributions on $U$:

$g=\log|z-z(P)|^{-1}+f$

for some function $f$ in $\mathcal{C}^{\infty}(U)$, and the definition of the
capacitary metric $\|.\|_{g}^{\mathrm{cap}}$ may be expressed by the following
formula:

$\|(\partial/\partial z)_{|P}\|_{g}^{\mathrm{cap}}=e^{-f(P)}.$

#####  3.2.2. The $\ast$-product of two Green functions with
$\mathcal{C}^{\infty}$ regularity

######

Let $D_{1}$ and $D_{2}$ be two divisors on a Riemann surface $M$ such that:

$|D_{1}|\cap|D_{2}|=\emptyset,$

and let $g_{1}$ and $g_{2}$ be Green functions with $\mathcal{C}^{\infty}$
regularity for $D_{1}$ and $D_{2}$ respectively.

The _ $\ast$-product_ of $g_{1}$ and $g_{2}$ is defined as the real current of
degree 2 on $M$:

$g_{1}\ast g_{2}:=g_{2}\,\delta_{D_{1}}+g_{1}\,\omega(g_{2}),$

where, as above, $\omega(g_{2})$ is defined by:

$\omega(g_{2})=\frac{i}{\pi}\partial\overline{\partial}g_{2}+\delta_{D_{2}}.$

When additionally the intersection $\mathrm{supp}g_{1}\cap\mathrm{supp}g_{2}$
of the supports of $g_{1}$ and $g_{2}$ is compact, the $\ast$-product
$g_{1}\ast g_{2}$ is supported on a compact subset of $M$, and we may consider
the integral:

$\int_{M}g_{1}\ast g_{2}\in\mathbb{R}.$

It is actually symmetric in $g_{1}$ and $g_{2}$; namely:

(3.2.4) $\int_{M}g_{1}\ast g_{2}=\int_{M}g_{2}\ast g_{1}.$

This equality is a direct consequence of Green’s formula — itself a
consequence of Stokes formula — which asserts that, for any two distributions
$\varphi$ and $\psi$ on $M$ with disjoint singular supports such that
$\mathrm{supp}\,\varphi\cap\mathrm{supp}\,\psi$ is compact, the following
equalities holds:

(3.2.5)
$\int_{M}\varphi\,i\partial\overline{\partial}\psi=-i\int_{M}\partial\varphi\wedge\overline{\partial}\psi=-i\int_{M}\partial\psi\wedge\overline{\partial}\varphi=\int_{M}\psi\,i\partial\overline{\partial}\varphi.$

The integral of the $\ast$-product also satisfies the following two properties
as a straightforward consequence of its definition and of Stokes formula:

$\mathbf{SP}_{1}$: _If the supports of $g_{1}$ and $g_{2}$ are disjoint,
then:_

$\int_{M}g_{1}\ast g_{2}=0;$

$\mathbf{SP}_{2}$: _For any two functions $f_{1}$ and $f_{2}$ in
${\mathcal{C}}^{\infty}(M,\mathbb{R})$ such that the intersections
$\mathrm{supp}\,f_{1}\cap\mathrm{supp}\,g_{2},$
$\mathrm{supp}\,f_{2}\cap\mathrm{supp}\,g_{1}$ and
$\mathrm{supp}\,f_{1}\cap\mathrm{supp}\,f_{2}$ are compact, the following
equality holds_:

(3.2.6) $\int_{M}(g_{1}+f_{1})\ast(g_{2}+f_{2})=\int_{M}g_{1}\ast
g_{2}+\int_{M}f_{1}\,\omega(g_{2})+\int_{M}f_{2}\,\omega(g_{1})-\frac{i}{\pi}\int_{M}\partial
f_{1}\wedge\overline{\partial}f_{2}.$

For any pair of divisors $D_{1}$ and $D_{2}$ on $M$ with disjoint supports,
the function that attaches the integral $\int_{M}g_{1}\ast g_{2}$ to a pair
$(g_{1},g_{2})$ of Green functions for the divisors $D_{1}$ and $D_{2}$ such
that $\mathrm{supp}\,g_{1}\cap\mathrm{supp}\,g_{2}$ is compact is easily seen
to be characterized by the properties $\mathbf{SP}_{1}$ and $\mathbf{SP}_{2}$.

######

The last term in the right-hand side of (3.2.6) is, up to a factor $\pi$, the
opposite of the _Dirichlet scalar product of $f_{1}$ and $f_{2}$_:

$\langle f_{1},f_{2}\rangle_{\mathrm{Dir}}:=i\int_{M}\partial
f_{1}\wedge\overline{\partial}f_{2},$

which is defined for any pair of functions $(f_{1},f_{2})$ in
${\mathcal{C}}^{\infty}(M,\mathbb{R})$ such that
$\mathrm{supp}f_{1}\cap\mathrm{supp}f_{2}$ is compact.

Actually, for every such pair of functions, we have:

$i\int_{M}\partial
f_{1}\wedge\overline{\partial}f_{2}=i\int_{M}\Big{(}\partial
f_{1}\wedge\overline{\partial}f_{2}-\frac{1}{2}df_{1}\wedge
df_{2}\Big{)}=\frac{i}{2}\int_{M}(\partial
f_{1}\wedge\overline{\partial}f_{2}+\partial
f_{2}\wedge\overline{\partial}f_{1}),$

and, if $z=x+iy$ is a local holomorphic coordinate on $M$, we may write
locally:

$i\Big{(}\partial f_{1}\wedge\overline{\partial}f_{2}-\frac{1}{2}df_{1}\wedge
df_{2}\Big{)}=\frac{i}{2}\left(\partial
f_{1}\wedge\overline{\partial}f_{2}+\partial
f_{2}\wedge\overline{\partial}f_{1}\right)=\frac{1}{2}\left(\frac{\partial
f_{1}}{\partial x}\frac{\partial f_{2}}{\partial x}+\frac{\partial
f_{1}}{\partial y}\frac{\partial f_{2}}{\partial y}\right)dx\wedge dy.$

The bilinear form $\langle.,.\rangle_{\mathrm{Dir}}$ on
$\mathcal{C}_{c}^{\infty}(M,\mathbb{R})$ is clearly symmetric and nonnegative.
When $M$ is compact and connected, it induces a structure of real pre-Hilbert
space on the quotient ${\mathcal{C}}^{\infty}(M,\mathbb{R})/\mathbb{R}$.

#####  3.2.3. Functoriality of Green functions with ${\mathcal{C}}^{\infty}$
regularity on Riemann surfaces

The functoriality properties of Green functions with $L^{2}_{1}$ and
$\mathcal{C}^{{\mathrm{b}\Delta}}$ regularity will play an important role in
the next chapters. For comparison, in this subsection we present a short
description of the functoriality properties of Green functions with
${\mathcal{C}}^{\infty}$ regularity.222We refer the reader to 3.4.2 and 4.4.1
below for some background concerning the pull-back and push-forward operations
on forms and currents by analytic maps between Riemann surfaces.

######  Proposition 3.2.1.

Let $M$ and $N$ be two Riemann surfaces, with $N$ connected, and let
$f:N\rightarrow M$ be a non-constant complex analytic map.

1) If $D$ is divisor on $M$ and $g$ a Green function with
${\mathcal{C}}^{\infty}$ regularity for $D$ on $M$, then the
${\mathcal{C}}^{\infty}$ function $f^{\ast}g$ on $N\setminus
f^{-1}(|D|)=N\setminus|f^{\ast}D|$ is a Green function with
${\mathcal{C}}^{\infty}$ regularity for the divisor $f^{\ast}D$ on $N$.

Moreover the following equality of ${\mathcal{C}}^{\infty}$ 2-forms on $N$ is
satisfied:

$\omega(f^{\ast}g)=f^{\ast}\omega(g).$

2) If $E$ is a divisor on $N$ and $h$ a Green function with
${\mathcal{C}}^{\infty}$ regularity for $E$ on $N,$ and if
$f_{\mid{\mathrm{supp}\,}h}$ is proper333This implies that $f_{\mid|E|}$ is
proper, and therefore that the divisor $f_{\ast}E$ on $M$ is well-defined. and
if $\omega(h)$ vanishes on some neighborhood of the ramification locus
$\mathrm{Ram}(f)$ of $f$, then the distribution $f_{\ast}h$ on $M$ is a Green
function with ${\mathcal{C}}^{\infty}$ regularity for $f_{\ast}E$ on $M$.

Moreover the following equality of ${\mathcal{C}}^{\infty}$ 2-forms on $M$ is
satisfied:

$\omega(f_{\ast}h)=f_{\ast}\omega(h).$

3) Let $(D,g)$ and $(E,h)$ be as in 1) and 2). If the following additional
conditions are satisfied:

(3.2.7) $f^{-1}(|D|)\cap|E|=\emptyset$

and:

(3.2.8) $\mbox{${\mathrm{supp}\,}g\cap f({\mathrm{supp}\,}h)$ is compact},$

then the following equality holds:

(3.2.9) $\int_{N}(f^{\ast}g)\ast h=\int_{M}g\ast(f_{\ast}h).$

Observe that, in 3), $f^{\ast}g$ and $h$ are Green functions for the divisor
$f^{\ast}D$ and $E$, which have disjoint support according to (3.2.7);
moreover, according to (3.2.8) and to the properness of
$f_{\mid{\mathrm{supp}\,}h}$, the intersection:

${\mathrm{supp}\,}(f^{\ast}g)\cap{\mathrm{supp}\,}h=f^{-1}({\mathrm{supp}\,}g)\cap{\mathrm{supp}\,}h$

is compact. This shows that the left-hand side of (3.2.9) is well defined.

Moreover, $g$ and $f_{\ast}h$ are Green functions for the divisors $D$ and
$f_{\ast}E$; these divisors have disjoint supports as a consequence of
(3.2.7); moreover ${\mathrm{supp}\,}g\cap{\mathrm{supp}\,}(f_{\ast}h)$ is
contained in ${\mathrm{supp}\,}g\cap f({\mathrm{supp}\,}h)$ and is therefore
compact by (3.2.8). This shows that the right-hand side of (3.2.9) is well
defined.

The proof of Proposition 3.2.1 will be left as an exercise for the interested
reader.

Concerning Part 3) of Proposition 3.2.1, observe that the operation of push-
forward does _not_ preserve in general the class of Green functions with
${\mathcal{C}}^{\infty}$ regularity. Actually this operation does not preserve
${\mathcal{C}}^{\infty}$ functions when ramification occurs.

For instance, if $e$ is a positive integer, the push-forward of the function:

$|.|^{2}:\mathring{D}(0;1)\longrightarrow\mathbb{R},\quad z\longmapsto|z|^{2}$

by the complex analytic map:

$f_{e}:\mathring{D}(0;1)\longrightarrow\mathring{D}(0;1),\quad z\longmapsto
z^{e}$

is the function:

$f_{e\ast}|.|^{2}=e|.|^{2/e}.$

It is not ${\mathcal{C}}^{\infty}$ if $e>1$.

####  3.3. Arakelov divisors on arithmetic surfaces and arithmetic
intersection numbers

In this section, we denote by $X$ an integral normal arithmetic surface.

#####  3.3.1. Arakelov divisor and Hermitian line bundles

An _Arakelov divisor_ (resp. an _Arakelov-Cartier divisor_) on $X$ is a pair
$(D,g)$, where $D$ is a Weil (resp. Cartier) divisor on $X$ and $g$ a Green
function with ${\mathcal{C}}^{\infty}$ regularity, invariant under complex
conjugation, for the divisor $D_{\mathbb{C}}$ on the Riemann surface
$X(\mathbb{C})$.

Recall that, to the Green function $g$ is associated a
${\mathcal{C}}^{\infty}$ metric $\|.\|_{g}$ on the analytic line bundle
${\mathcal{O}}_{X(\mathbb{C})}(D_{\mathbb{C}})$ over the Riemann surface
$X(\mathbb{C})$. Accordingly, when moreover $D$ is a Cartier divisor, we may
associate to $(D,g)$ the Hermitian line bundle with ${\mathcal{C}}^{\infty}$
metric:

(3.3.1) $\overline{\mathcal{O}}_{X}(D,g):=({\mathcal{O}}_{X}(D),\|.\|_{g}).$

We shall denote by $Z^{1}(X)$ and $Z^{1}_{\mathrm{Cart}}(X)$ the additive
groups of Weil and Cartier divisors on $X$, and by $\overline{Z}^{1}(X)$ and
$\overline{Z}^{1}_{\mathrm{Cart}}(X)$ the additive groups of Arakelov and
Arakelov-Cartier divisors on $X$. The quotient group
$Z^{1}(X)/Z^{1}_{\mathrm{Cart}}(X)$, and consequently
$\overline{Z}^{1}(X)/\overline{Z}^{1}_{\mathrm{Cart}}(X)$, is finite.

To any invertible rational function $f$ in $\kappa(X)^{\times},$ we may
associate the following Arakelov-Cartier divisor:

$\overline{{\mathrm{div}\,}}f:=\big{(}{\mathrm{div}\,}f,\log|f_{\mathbb{C}}|^{-1}\big{)}\in\overline{Z}_{\mathrm{Cart}}^{1}(X).$

The map:

$\overline{{\mathrm{div}\,}}:\kappa(X)^{\times}\longrightarrow{\overline{Z}}^{1}(X)$

is a morphism of abelian groups. Its image
$\overline{{\mathrm{div}\,}}\kappa(X)^{\times}$ is the group of Arakelov
divisors _rationally equivalent to zero_ , and its cokernel defines the
_Arakelov-Chow group_ of $X$:

$\overline{CH}^{1}(X):=\overline{Z}^{1}(X)/\overline{{\mathrm{div}\,}}\kappa(X)^{\times}.$

We shall also consider its subgroup of finite index:

$\overline{CH}_{\mathrm{Cart}}^{1}(X):=\overline{Z}_{\mathrm{Cart}}^{1}(X)/\overline{{\mathrm{div}\,}}\kappa(X)^{\times}.$

If we denote by $\overline{{\rm Pic}}(X)$ the group of isomorphism classes of
Hermitian line bundles with ${\mathcal{C}}^{\infty}$ metrics on $X$, we define
an isomorphism of abelian group:

(3.3.2) $\widehat{c}_{1}:\overline{{\rm
Pic}}(X)\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}\overline{CH}_{\mathrm{Cart}}^{1}(X)$

by mapping the isomorphism class of a Hermitian line bundle with
${\mathcal{C}}^{\infty}$-metric ${\overline{L}}:=(L,\|.\|)$ to the class
$\widehat{c}_{1}({\overline{L}})$ of the Arakelov-Cartier divisor:

$\overline{{\mathrm{div}\,}}_{{\overline{L}}}\,s:=({\mathrm{div}\,}s,\log\|s_{\mathbb{C}}\|^{-1})$

of an arbitrary non-zero rational section $s$ of $L$. The inverse isomorphism
$\widehat{c}_{1}^{-1}$ sends the class of some Cartier-Arakelov divisor
$(Z,g)\in{\overline{Z}}_{\mathrm{Cart}}^{1}(X)$ to the isomorphism class of
$\overline{\mathcal{O}}_{X}(Z,g).$

An Arakelov divisor $(D,g)$ will be said to be _compactly supported_ when the
support $|D|$ of $D$ is proper over ${\rm Spec\,}\mathbb{Z}$ and when the
support ${\mathrm{supp}\,}g$ of the Green function $g$ is a compact subset of
$X(\mathbb{C})$. The subgroup of ${\overline{Z}}^{1}(X)$ (resp. of
${\overline{Z}}^{1}_{\mathrm{Cart}}(X)$) defined by the compactly supported
Arakelov (resp. Cartier-Arakelov) divisors on $X$ will be denoted by
${\overline{Z}}^{1}_{c}(X)$ (resp. by
${\overline{Z}}^{1}_{{\mathrm{Cart}},c}(X)$).

#####  3.3.2. The Arakelov intersection pairing: definitions and basic
properties

###### Definitions

If ${\overline{L}}:=(L,\|.\|)$ is a Hermitian line bundle over $X,$ with a
${\mathcal{C}}^{\infty}$ metric $\|.\|$, and if $(D,g)$ is an Arakelov divisor
in ${\overline{Z}}^{1}_{c}(X),$ we define the _Arakelov intersection pairing_
of ${\overline{L}}$ and $(D,g)$ as the real number:

(3.3.3) ${\overline{L}}\cdot(D,g):=\widehat{\rm
deg}\,\big{(}{\overline{L}}|D\big{)}+\int_{X(\mathbb{C})}g\,c_{1}({\overline{L}}_{\mathbb{C}}).$

If $(D^{\prime},g^{\prime})$ (resp. $(D,g)$) is an Arakelov divisor in
${\overline{Z}}_{\mathrm{Cart}}^{1}(X)$ (resp. in
${\overline{Z}}^{1}_{c}(X)$), we define their _Arakelov_ or _arithmetic
intersection number_ as the real number:

(3.3.4)
$(D^{\prime},g^{\prime})\cdot(D,g):=\overline{\mathcal{O}}_{X}(D^{\prime},g^{\prime})\cdot(D,g)=\widehat{\rm
deg}\,\big{(}\overline{\mathcal{O}}_{X}(D^{\prime},g^{\prime})|D\big{)}+\int_{X(\mathbb{C})}g\,\omega(g^{\prime}).$

The arithmetic intersection pairing:

(3.3.5)
$.\cdot.:\overline{Z}^{1}_{\mathrm{Cart}}(X)\times\overline{Z}^{1}_{c}(X)\longrightarrow\mathbb{R}$

defined by (3.3.5) is clearly bilinear. Since
$\overline{Z}^{1}_{\mathrm{Cart}}(X)$ has finite index in
$\overline{Z}^{1}(X)$, the pairing (3.3.5) uniquely extends to a bilinear
pairing:

(3.3.6)
$.\cdot.:\overline{Z}^{1}(X)\times\overline{Z}^{1}_{c}(X)\longrightarrow\mathbb{R}.$

In concrete terms, the pairing (3.3.6) may be described as follows. If
$(D^{\prime},g^{\prime})$ and $(D,g)$ are Arakelov divisors in
$\overline{Z}^{1}(X)$ and $\overline{Z}^{1}(X)_{c}$ respectively, we may
choose a positive integer $N$ such that the divisor $ND^{\prime}$ is Cartier,
and then, by definition, the intersection pairing of $(D^{\prime},g^{\prime})$
and $(D,g)$ is the real number:

(3.3.7)
$(D^{\prime},g^{\prime})\cdot(D,g)=N^{-1}\,(ND^{\prime},Ng^{\prime})\cdot(D,g)=N^{-1}\overline{\mathcal{O}}_{X}(ND^{\prime},Ng^{\prime})\cdot(D,g)\\\
=N^{-1}\widehat{\rm
deg}\,(\overline{\mathcal{O}}_{X}(ND^{\prime},Ng^{\prime})|D)+\int_{X(\mathbb{C})}g\,\omega(g^{\prime}).$

###### Properties of the Arakelov intersection pairing

By construction, the intersection pairing is compatible with linear
equivalence in the first variable. Namely:

######  Proposition 3.3.1.

For every $f\in\kappa(X)^{\times}$ and every
$(D,g)\in\overline{Z}^{1}(X)_{c}$, we have:

(3.3.8) $\overline{{\mathrm{div}\,}}f\cdot(D,g)=0.$

The following proposition describes how the intersection pairing of two
Arakelov divisors depends of their Green functions.

######  Proposition 3.3.2.

For any two Arakelov divisors $(D,g)\in\overline{Z}^{1}(X)_{c}$ and
$(D^{\prime},g^{\prime})\in\overline{Z}^{1}(X)$, and any two functions
invariant under complex conjugation
$f\in{\mathcal{C}}_{c}^{\infty}(X(\mathbb{C}),\mathbb{R})$ and
$f^{\prime}\in{\mathcal{C}}^{\infty}(X(\mathbb{C}),\mathbb{R})$, the following
equality holds:

(3.3.9)
$(D^{\prime},g^{\prime}+f^{\prime})\cdot(D,g+f)=(D^{\prime},g^{\prime})\cdot(D,g)+\int_{X(\mathbb{C})}f^{\prime}\,\omega(g)+\int_{X(\mathbb{C})}f\,\omega(g^{\prime})-\pi^{-1}\langle
f^{\prime},f\rangle_{\mathrm{Dir}}.$

###### Proof.

According to the bilinearity of the Arakelov intersection pairing, the
validity of (3.3.9) follows from the following three equalities:

(3.3.10)
$(D^{\prime},g^{\prime})\cdot(0,f)=\int_{X(\mathbb{C})}f\,\omega(g^{\prime}),$
(3.3.11) $(0,f^{\prime})\cdot(D,g)=\int_{X(\mathbb{C})}f^{\prime}\,\omega(g),$

and:

(3.3.12) $(0,f^{\prime})\cdot(0,f)=-\pi^{-1}\langle
f^{\prime},f\rangle_{\mathrm{Dir}}.$

The equality (3.3.10) follows from the definition (3.3.3) of the intersection
pairing and from the relation:

$c_{1}(\overline{\mathcal{O}}_{X}(D^{\prime},g^{\prime})_{\mathbb{C}})=c_{1}({\mathcal{O}}_{X(\mathbb{C})}(D^{\prime}_{\mathbb{C}}),\|.\|_{g^{\prime}})=\omega(g^{\prime}).$

Formula (3.3.11) is established by the following chain of equalities:

(3.3.13) $\displaystyle(0,f^{\prime})\cdot(D,g)$ $\displaystyle=\widehat{\rm
deg}\,\big{(}\overline{\mathcal{O}}_{X}(0,f^{\prime})|D\big{)}+\int_{X(\mathbb{C})}g\,\frac{i}{\pi}\partial\overline{\partial}f^{\prime}$
(3.3.14)
$\displaystyle=\int_{X(\mathbb{C})}f^{\prime}\,\delta_{D_{\mathbb{C}}}+\int_{X(\mathbb{C})}g\,\frac{i}{\pi}\partial\overline{\partial}f^{\prime}$
(3.3.15)
$\displaystyle=\int_{X(\mathbb{C})}\Big{(}f^{\prime}\,\delta_{D_{\mathbb{C}}}+f^{\prime}\,\frac{i}{\pi}\partial\overline{\partial}g\Big{)}$
(3.3.16) $\displaystyle=\int_{X(\mathbb{C})}f^{\prime}\,\omega(g).$

Indeed (3.3.13) follows from the definition (3.3.3) of the intersection
pairing and from the relation:

(3.3.17)
$c_{1}(\overline{\mathcal{O}}_{X}(0,f^{\prime})_{\mathbb{C}})=c_{1}({\mathcal{O}}_{X_{\mathbb{C}}},\|.\|_{f^{\prime}})=\frac{i}{\pi}\partial\overline{\partial}f^{\prime}.$

The equality (3.3.14) follows from the definition (3.1.6), applied with
${\overline{L}}$ the restriction of $\overline{\mathcal{O}}_{X}(0,f^{\prime})$
to the components of $D$ and with $s=1$. The equality (3.3.15) follows from
Green’s formula (3.2.5), and (3.3.16) from the definition (3.2.2) of
$\omega(g)$.

Finally, using again (3.3.3) and (3.3.17), we obtain:

$(0,f^{\prime})\cdot(0,f)=\int_{X(\mathbb{C})}f\,\frac{i}{\pi}\partial\overline{\partial}f^{\prime}=-\pi^{-1}\langle
f^{\prime},f\rangle_{\mathrm{Dir}}.\qed$

Recall that if $D$ and $D^{\prime}$ are two Cartier divisors on $X$, the
intersection $D\cdot D^{\prime}$ is a $0$-cycle on $|D|\cap|D^{\prime}|$ which
is well defined and depends symmetrically of $D$ and $D^{\prime}$, up to
rational equivalence supported by $|D|\cap|D^{\prime}|$.

Using again that $Z^{1}_{\mathrm{Cart}}(X)$ has finite index in $Z^{1}(X)$,
this allows one to define the intersection $D\cdot D^{\prime}$ of any two Weil
divisor $D$ and $D^{\prime}$ on $X$: it is a $0$-cycle with
$\mathbb{Q}$-coefficients supported by $|D|\cap|D^{\prime}|$, which is well
defined and depends symmetrically of $D$ and $D^{\prime}$, up to rational
equivalence supported by $|D|\cap|D^{\prime}|$.

When moreover $|D|\cap|D^{\prime}|$ is proper over ${\rm Spec\,}\mathbb{Z}$
and its generic fiber $|D_{\mathbb{Q}}|\cap|D^{\prime}_{\mathbb{Q}}|$ is
empty,444This last condition holds precisely when the divisors
$D_{\mathbb{C}}$ and $D^{\prime}_{\mathbb{C}}$ on the Riemann surface
$X(\mathbb{C})$ have disjoint supports. then $|D|\cap|D^{\prime}|$ is a finite