# Globally valued function fields: existential closure

Itaï ben Yaacov  and  Ehud Hrushovski

E.H.: Research funded in part by the European Research Council under the
European Unions Seventh Framework Programme (FP7/2007- 2013)/ERC Grant
Agreement No. 291111.

The material presented in these notes forms part of a joint research project
on the logic of fields with many valuations, connected by a product formula;
we define such structures and name them globally valued fields (GVFs). This
text aims primarily at a proof that the canonical GVF structure on
$k(t)^{alg}$ is existentially closed. This can be read as saying that a
variety with a distinguished curve class is a good approximation for a formula
in the language of GVFs, in the same way that a variety is close to a formula
for the theory ACF of algebraically closed fields.

The text is based on classes given by the second author in Spring 2015
Jerusalem and in Spring 2016 in Paris (FSMP). It has a large expository
element, especially concerning the algebraic geometry needed for this. This
material is based on (short initial sections of) [12], [23], [17], [25], [3]
and [27]. However in a number of cases, modifications of results there were
needed. One critical statement is a strong form of BDPP duality ([2]), where
elements of the interior of the dual cone to the effective cone are shown to
have $n-1$’st roots in some limit of blowups; the usual version in the
literature presents them only as convex combinations of powers of ample
divisors. We work in arbitrary characteristic, whereas some of the literature
restricts to characteristic zero. Since these notes were first written (and
perhaps before), many of these improvements have undoubtedly appeared in the
literature; we would be grateful for further references.

In these notes, we restrict attention to the function field case. Thus our
globally valued fields should properly be called ‘purely non-archimedean
globally valued fields’. There are nevertheless some applications to number
fields, see Corollary 2.3.

Globally valued fields are defined in §1 via globalizing measures (up to
renormalization), and also via an explicit axiomatization in an appropriate
language for real-valued logic. The equivalence of the two was proved in notes
for the previous semester; in the function field case, it also follows from
the geometric characterization of quantifier-free types in § 11.

We assume some basic algebraic geometry, including mainly blow-ups, Cartier
divisors, Weil divisors; some of the material is reviewed in § 4.

§ 5 collects some results related to convex subsets of ${\mathbb{R}}^{n}$. The
most important are Theorem 5.2, a multiplicative version of Legendre duality,
and a finiteness statement for finitely generated groups or graded rings, due
to Khovanskii and Okounkov. These convexity results will later be used to
amplify a very modest geometric input to yield a great deal of information.

§ 6 presents the Hodge index theorem.

Volume is defined, via section growth, in § 7.

Okounkov bodies are presented in § 8, thereby giving another definition of
volume for big divisors. They are used to prove log-concavity of volume for
big divisors. § 9 proves the Fujita approximation theorem.

§ 10 presents the positive intersection product of [3].

The space of quantifier-free GVF types over a given constant field is
described in § 11.

The existential closure of $k(x)^{alg}$ is proved in § 12.

In a final section, we include a proof of the non-archimedean Yau theorem of
[4] and [22], and more recently [18]. The geometric content of existential
closure is quite close - but intriguingly not identical - to this statement,
and we give a proof taking place in a finite dimensional setting, based on the
ingredients of the proof of the main theorem.

Notation for induced maps. Given a morphism of spaces or varieties $f:X\to U$,
there are many induced morphisms on associated objects. They will all be
denoted in the same way, namely $f_{*}$ in the contravariant case and $f^{*}$
if covariant. In case of ambiguity, we make the domain explicit as follows:
$f_{*}|N_{1}(X)$ (a map from $N_{1}(X)$ to $N_{1}(U)$), or $f^{*}|N^{1}(U)$ (a
map $N^{1}(U)\to N^{1}(X)$. )

A similar convention applies to various groups and spaces formed out of
algebraic cycles. For instance consider a Weil divisor represented by an
irreducible subvariety $D$ of $X$. We will be interested in the image of $D$
in various groups, such as $Pic(X)$ and
$N_{1}(X)={\mathbb{R}}{\otimes}Pic(X)/Pic^{o}(X)$. The class of $D$ in any
such group will be denoted $[D]$, but if we write $[D]=[D^{\prime}]\in
N_{1}(X)$ we mean that the equality holds specifically for the classes in
$N_{1}(X)$.

##  1\. Globally valued fields

A globally valued field is a field with a measure space of valuations and
absolute values, satisfying the product formula. Here we consider only the
geometric case and do not use absolute values. Our valuations will all be
valued in ${\mathbb{R}}$.

Let $K$ be a field. Let ${\widehat{K}}$ be the space of ${\mathbb{R}}$-valued
valuations of $K$. The topology on ${\widehat{K}}$ is induced from the
Tychonoff topology on the functions from $K\smallsetminus(0)$ to
${\mathbb{R}}$.

If $k$ is a subfield of $K$, let ${\widehat{K}}/k$ be the subspace of
valuations trivial on $k$. When $K=k(X)$ for some variety $X$, this is the
Berkovich space of $X$, from which the Berkovich spaces of all proper
$k$-subvarieties have been removed.

$v_{triv}$ denotes the trivial valuation.

######  Definition 1.1.

A purely non-archimedean GVF is a field $K$ along with a Baire (or regular
Borel) measure $\mu$ on some Borel set of representatives for equivalence
classes of nontrivial valuations; such that for any $f\in K\smallsetminus(0)$,
$v\mapsto v(f)$ is integrable, and

$\int v(f)d\mu(v)=0$

In these notes, we will simply write GVF for purely non-archimedean GVF.

If $K^{\prime}$ is a subfield of $K$, we obtain a GVF structure on
$K^{\prime}$ by taking the pushforward measure
$\mu^{\prime}=r_{*}\mu|({\widehat{K}}\smallsetminus{\widehat{K}}/K^{\prime})$,
where $r$ is the restriction map from measures on ${\widehat{K}}$, nontrivial
on $K^{\prime}$ to measures on ${\widehat{K}}^{\prime}$.

If $g$ is an $L^{\infty}$ function on ${\widehat{K}}$ whose multiplicative
inverse $g^{-1}$ is also $L^{\infty}$, we have a map
$rn(g):{\widehat{K}}\to{\widehat{K}}$ multiplying each valuation $v$ by
$g(v)$; we let $\mu^{g}=g^{-1}rn(g)_{*}\mu$. If $\mu$ concentrates on some set
of representatives for the valuations up to equivalence, then $\mu^{g}$
corresponds on a different set, where each $v$ is replaced by the equivalent
valuation $g(v)v$ and given weight $1/g(v)$ of the $\mu$-weight of $v$. We
view $(K,\mu)$ and $(K,\mu^{g})$ as ‘the same’ GVF structure. (In particular
they give all formulas the same value.)

A morphism of globally valued fields is the composition of an inclusion with a
renormalization.

###  1.2. A language for global fields

.

A real-valued language. A formula $\phi(x_{1},\ldots,x_{n})$, evaluated on a
structure $A$, gives a real-valued function on $A^{n}$.

Basic formulas:

Let $Tr_{n}$ be the set of $n$-ary terms in the language $+,\min,\alpha\cdot
x$ of divisible ordered Abelian groups (here $\alpha\cdot$ denotes scalar
multiplication by the rational number $\alpha$.) We will refer to these as
${\mathbb{Q}}$-tropical polynomials.

Each $t\in Tr_{n}$ defines a positively 1-homogeneous continuous function $t$
on ${\mathbb{R}}^{k}$.

Let $S_{n}=\\{x:\max_{i}|x_{i}|=1\\}$. Then $\\{t|S_{n}:t\in Tr_{n}\\}$
contains the constant function $1$, is closed under addition, scalar
multiplication and $\min$, and separates points. By Lemma 5.13 these are
uniformly dense in $S_{n}$.

For any $t\in Tr_{n}$, a symbol $R_{t}(x_{1},\ldots,x_{n})$ interpreted as:
$x\mapsto\int t(vx_{1},\ldots,vx_{n})$ (the domain being
$\\{x:\Pi_{i=1}^{k}x_{i}\neq 0\\}$.)

Example: $\int v(x)$. The main axiom, the (logarithmic) product formula,
asserts that $\int v(x)=0$ for any $x\neq 0$.

Example: $\int v(x)^{+}$ where $x^{+}=\max(x,0)$. By definition, this is the
height of $x$, denoted $ht(x)$.

(In fact we will see later that the projective height
$ht(x_{1},\ldots,x_{n})=\int\max(0,v(x_{1}),\ldots,v(x_{n}))$ suffices to
generate the entire language, at least in the function field case.)

The analogue of quantifiers in real-valued logic are inf and sup operators.
Let $\psi$ be a continuous function on ${\mathbb{R}}$ with compact support.
Let $\phi(x,y)$ be a formula. Then so is $\sup_{x}\psi(ht(x))\phi(x,y)$. Thus
quantification is only available over bounded height subsets.

###  1.3. Universal axioms

GVF. (Function field case.)

We have an integral domain $F$, along with a function
$F_{t}:(F\smallsetminus(0))^{n}\to{\mathbb{R}}$ written as:

$F_{t}(a_{1},\ldots,a_{n})=\int t(v(a_{1}),\ldots,v(a_{n}))dv$

given for each $n$ and each $t\in Tr_{n}$. The $F_{t}$ are compatible with
permutations of variables and dummy variables.

Let $POS$ be the set of pairs $(\phi,t)$ of formulas
$\phi(x_{1},\ldots,x_{n})$ in the language of rings, and $t\in Tr_{n}$, such
that the theory of valued fields $VF$ implies $(\forall x)(\phi(x)\implies
t(v(x_{1}),\ldots,v(x_{n}))\geq 0)$. The universal axioms are:

  1. (1)

(Linearity:) $F_{t_{1}+t_{2}}=F_{t_{1}}+F_{t_{2}}$.

  2. (2)

(Local-global principle for positivity) If $(\phi,t)\in POS$ and
$\phi(a_{1},\ldots,a_{n})$ then $\int t(v(a_{1}),\ldots,v(a_{n}))dv\geq 0$.

  3. (3)

(Product formula) $F_{x}(a)=0$ where $x$ is the identity on ${\mathbb{R}}$.

######  Remark 1.4.

If we wish to axiomatize GVF fields containing a given field $k$ as a constant
field, we may use the same axioms with respect to the collection $POS_{k}$,
defined in the same way for the theory $VF_{k}$ of valued fields containing
$k$ as a trivially valued field. To see the equivalence, a lemma of valued
field theory is required; namely (writing $x=(x_{1},\ldots,x_{n})$,
$y=(y_{1},\ldots,y_{m})$), if

$VF_{k}\models(\forall x)(\phi(x)\implies t(v(x_{1}),\ldots,v(x_{n}))\geq 0)$

then for some rational $\alpha>0$, some $\phi^{\prime}(x,y)$, and some
$a_{1},\ldots,a_{m}\in k$ we have:

$VF_{k}\models(\forall x)(\phi(x)\implies\phi^{\prime}(x,a_{1},\ldots,a_{m}))$

and

$VF\models(\forall x,y)(\phi^{\prime}(x,y)\implies
t(v(x_{1}),\ldots,v(x_{n}))\geq-\alpha\min|v(y_{1})|,\ldots,|v(y_{m})|)$

##  2\. Classical structures

.

(a) $k(x)^{alg}$, $h(x)=\alpha$.

(b) ${\mathbb{Q}}^{alg}[r]$ ($h(2)=r$.) This is not a purely non-archimedean
GVF for fixed $r$; but it is so asymptotically, i.e. any axiom is true for
small enough $r$.

(c) ${\mathcal{M}}[l,r]$ value distribution theory. Not a GVF at all for fixed
$r$, but is a purely non archimedean GVF for fixed $r$; but it is so
asymptotically.

We elaborate a little on (c). Let ${\mathcal{M}}$ be a countably generated
subfield of the field of meromorphic functions. Fix a function $\eta(r)$
(diverging to $\infty$; such as $\log(r)$ or $r^{d}$), and also an ultrafilter
$u$ on ${\mathbb{R}}^{>0}$, avoiding finite measure sets.

Let $\mu_{r}$ be the measure space on $\\{a:0<|a|\leq r\\}$ giving mass
$\log(r/|a|)/\eta(r)$ to each point $0<|a|<r$, and the uniform measure of mass
$1/\eta(r)$ to the circle $|t|=r$. Define

$v_{a}(f)=ord_{a}f\hbox{ for }|a|<r,\ \ v_{t}(f)=-\log|f(t)|$
$ht_{\eta,u}(f)=\lim_{r\to u}\int\max(v_{a}f,0)d\mu_{r}a$
${\mathcal{M}}[\eta,u]=\\{f\in{\mathcal{M}}:ht_{\eta,u}(f)<\infty\\}$
${R_{t}}(f_{1},\ldots,f_{n}):=\lim_{r\to u}\int
t(v_{a}f_{1},\ldots,v_{a}f_{n})d\mu_{r}a$

.

The product axiom is Jensen’s formula:

$\sum_{0<|a|<r}\log\frac{r}{|a|}ord_{a}(f)+\frac{1}{2\pi}\int_{0}^{2\pi}-\log|f(re^{i\theta})|\,d\theta=O(1)$

The $O(1)$ term is actually $-\log|f(0)|$, which does not depend on $r$.

We will see that in the language described above, ${\mathcal{M}}[\eta]^{alg}$
has the same universal theory as the ultraproduct of the
${\mathbb{Q}}^{alg}[r]$, and also as ${\mathbb{C}}(t)^{alg}[1]$. This
formalizes a part of Vojta’s dictionary [26] between Nevanlinna theory and
number theory. As the GVF language as we defined it does not permit discussing
the support of a function, the most interesting parts of the dictionary are
not currently accessible.

######  Theorem 2.1.

$k(x)^{alg}[r]$ is existentially closed, for any $r>0$.

######  Remark 2.2.

It follows from Theorem 2.1 that the asymptotic universal theory of
${\mathbb{Q}}^{alg}[r]$ is precisely GVF (+ char = 0 + h(2)=0. ). Also it is
existentially closed over any single parameter.

Let $X$ be a projective variety over a field $k$. Let $Y$ be a subscheme, cut
out by specified homogeneous polynomials $g_{1},\ldots,g_{k_{Y}}$; and let $v$
be a valuation of a valued field $L\supseteq k$. For $x\in X(L)$ we define

${\delta}_{v}(x,Y)=\min_{g}v(g_{i}(\bar{x}))$

where $\bar{x}$ is a representative of $x$ in homogeneous coordinates, chosen
so that the minimal valuation of a coordinate is $0$.

Then ${\delta}={\delta}_{v}(x,Y)$ (or rather $2^{-{\delta}}$) is a notion of
distance from $x$ to $Y$. If $L$ is a number field and $x\in X(L)$, let
$\rho(x,Y)^{L}=\int\delta_{v}(x,Y)dv$ be the weighted sum of the local
distances from $x$ to $Y$.

Note that $\rho(x,Y)^{L}$ is the $L$-value of a certain quantifier-free
formula $\phi_{Y}(x)$ in the language of GVF’s.

Let $Y_{0},\ldots,Y_{m}$ be subschemes of $X$. Let
$\rho_{Y}(a)=(\rho_{Y_{0}}:\cdots:\rho_{Y_{m}})\in{\mathbb{P}}^{m}({\mathbb{R}})$.

For a curve $C$ on $X$ let $i(C,Y_{m})$ be the intersection number of $C$ with
$Y_{m}$. Similarly let $\bar{i}(a)=(i(C,Y_{0}):\cdots:i(C,Y_{m}))$.

######  Corollary 2.3.

Let $X\subset{\mathbb{P}}^{n}$ be a smooth projective variety over
${\mathbb{Q}}$. Let $Y_{0},\ldots,Y_{m}$ be subschemes of $X$. Let
$e\in{\mathbb{P}}^{m}({\mathbb{R}})$. Assume there exists a Zariski dense
sequence $a_{i}\in V({\mathbb{Q}})$ with $\rho_{Y}(a_{i})\to e$ in
${\mathbb{P}}^{m}({\mathbb{R}})$. Then there exists a curve $C$ on $V$ with
$\bar{i}(C,Y_{k})$ as close as one wishes to $e$.

In fact the same conclusion is true if the $a_{i}$ are only assumed to be of
bounded degree in $V({\mathbb{Q}}^{alg})$; or just, of height approaching
infinity.

One can also state, given $X,Y_{0},\ldots,Y_{m}$, that there exists a proper
subvariety $W$ of $X$ (not necessarily irreducible), such that for
$e\in{\mathbb{P}}^{m}({\mathbb{R}})$ and any sequence of elements $a_{i}\in
V({\mathbb{Q}}^{a})$ of unbounded height, the same conclusion holds.

###### Proof.

We may assume $Y_{0}$ is an ample divisor, and use height with respect to
$Y_{0}$. Let $a_{i}\in V({\mathbb{Q}}^{alg})$ ($i=1,2,\cdots$) have height
approaching $\infty$, such that $\rho_{Y}(a_{i})\to e$. Choose
$r_{i}=ht(2)/ht(a_{i})$ so that ${\mathbb{Q}}^{alg}[r_{i}]$ gives $a_{i}$
height $1$. Consider any non-principal ultrafilter $u$ on the index set
${\mathbb{N}}$, and let $(L,a)$ be the GVF ultraproduct of
$({\mathbb{Q}}^{alg}[r_{i}],a_{i})$. Then $(L,a)$ is a purely non-archimedean
GVF, $a\in V(L)$, and $\rho_{Y}(a)=\phi(a)^{L}=e$. Let $\epsilon>0$ and fix
some metric on ${\mathbb{P}}^{m}({\mathbb{R}})$. Then by Theorem 2.1 there
exists $a^{\prime}\in V(K)$, $K=k(t)^{alg}$ with
$e^{\prime}=\phi(a^{\prime})^{K}$ satisfying $|e^{\prime}-e|<\epsilon$. In
fact $a^{\prime}\in V(k(C))$ for some curve $C$, so $a^{\prime}$ corresponds
to a morphism $g:C\to V$. We may choose $a^{\prime}$ so that $g$ avoids any
given proper subvariety of $V$. By computing the meaning of $\phi$ in
$k(t)^{alg}$ we see that $\bar{i}(C,Y_{k})=e^{\prime}$. ∎

######  Remark 2.4.

Conversely, if $C$ is a curve on $X$ defined over ${\mathbb{Q}}^{alg}$, then
for any sequence of distinct $a_{i}\in C({\mathbb{Q}}^{alg})$ of bounded
degree over ${\mathbb{Q}}$, $\rho_{Y}(a)\to i_{Y}(C)$. This follows upon
taking normalized ultraproducts as above, from the uniqueness of the GVF
structure on $k(C)$, Lemma 12.1.

By applying the above remark along with Corollary 2.3, we obtain a purely
arithmetic corollary:

######  Corollary 2.5.

Let $X\subset{\mathbb{P}}^{n}$ be a smooth projective variety over
${\mathbb{Q}}$. Let $Y_{0},\ldots,Y_{m}$ be subschemes of $X$. Let
$e\in{\mathbb{P}}^{m}({\mathbb{R}})$. Assume there exist $a_{i}\in
V({\mathbb{Q}}^{alg})$ of height approaching $\infty$, with
$\rho_{Y}(a_{i})\to e$. Then there exists such a sequence $a_{i}^{\prime}$ of
bounded degree over ${\mathbb{Q}}$,
$[{\mathbb{Q}}[a_{i}^{\prime}]:{\mathbb{Q}}]<d$.

###  2.6. Fekete-Szegő

Let $C$ be a compact subset of ${\mathbb{C}}$. The energy of a measure $\mu$
on $C$ is defined to be:

$\int-\log|x-y|d\mu(x)d\mu(y)$

In particular, a measure of finite energy cannot assign positive mass to any
point. If $C$ supports some measure of finite energy, there exists a unique
probability measure attaining the minimum possible energy $m_{C}$; it is
called the equilibrium measure $\mu_{C}$, and one defines the capacity of $C$
to be ${{\mathop{\operatorname{\rm cap}}}}(C)=exp(-m_{C})$.

The classical Fekete-Szegő theorem asserts that if the capacity is $\geq 1$,
there exist infinitely many Galois orbits of algebraic integers contained in
any given neighborhood $U$ of $C$; if it is $<1$, some neighborhood $U$
contains only finitely many such orbits. Note in the first case that the
height of the Galois orbits is bounded, so their degree must approach
$\infty$.

This has adelic generalizations, due to David Cantor; the complex field is
replaced by similar conditions at finitely many places $S$, while the
algebraic integers are replaced by $S$-integers.

Restriction to a neighborhood of a closed set is not a GVF condition; however,
insisting that for any neighborhood $U$ of $C$, most of the orbit lie in $U$,
can be viewed as such. Similarly we relax the notion of $S$-integrality to
assert that a natural measure of non-integrality, even when summed over all
places outside $S$, approaches zero. This may be considered an
$L^{1}$-analogue of the Fekete-Szegő question. In the next section we prove
such a dichotomy over an existentially closed GVF, for varieties of any
dimension. In particular it applies to $k(t)[1]$ and its finite extensions.

######  Lemma 2.7.

Let $A_{i}$ be an infinite sequence of Galois orbits of algebraic integers,
$|A_{i}|\to\infty$. Assume $A_{i}\subset C$ or at least that
$|A_{i}|/|A_{i}\cap C|\to 1$. Then $C$ has capacity
${{\mathop{\operatorname{\rm cap}}}}(C)\geq 1$. If
${{\mathop{\operatorname{\rm cap}}}}(C)=1$, then for any such sequence, the
sets $A_{i}$ equi-distribute to the equilibrium measure on $C$.

###### Proof.

Let $a_{i}\in A_{i}$. Take an ultrapower $K$ of ${\mathbb{Q}}^{alg}$, viewed
as a GVF. Let $a\in K$ be the limit of the $a_{i}$. Let $K^{\prime}$ be a
further ultrapower, embed $K$ into $K^{\prime}$ and let $a^{\prime}$ be the
new limit of the $a_{i}$.

Let $b=a-a^{\prime}$. Note $a\neq a^{\prime}$, so $b\neq 0$. Note that for any
$v$, and all algebraic integers $c$, we have $\int v(c)^{-}v(2)^{+}dv=0$,
since either $v(2)<0$ or $v(c)=0$. Thus the same holds for $a,a^{\prime}$ and
$b$; so $v(b)\geq 0$ for all non-archimedean $v$, i.e. all $v$ with $v(2)>0$.
Hence

$0=\int v(b)dv\geq\int_{v(2)>0}v(b)=\int_{\alpha}-\log|\alpha(a-a^{\prime})|$

where $\alpha$ ranges over the complex embeddings of
${\mathbb{Q}}(a,a^{\prime})$. Let $\mu$ be the measure corresponding to the
globalizing measure of ${\mathbb{Q}}(a)$ (or ${\mathbb{Q}}(a^{\prime})$) over
${\mathbb{Q}}$, restricted to the archimedean primes. Since
$tp(a^{\prime}/{\mathbb{Q}}^{a},a)$ is finitely satisfiable in
${\mathbb{Q}}^{a}$, the archimedean part of the measure of
${\mathbb{Q}}(a,a^{\prime})$ is the product measure $\mu\times\mu$ (say by
$L^{2}$ exhaustion ). We have
$0\geq\int-\log(x-x^{\prime})d\mu_{x}d\mu_{x^{\prime}}\geq-\log{{\mathop{\operatorname{\rm
cap}}}}(C)$, so ${{\mathop{\operatorname{\rm cap}}}}(C)\geq 1$.

In case ${{\mathop{\operatorname{\rm cap}}}}(C)=1$, $\mu$ must equal the
equilibrium measure $\mu_{C}$, by uniqueness of the latter. ∎

##  3\. Fekete-Szegö

Let $K$ be a GVF, with globalizing measure $\mu_{K}$ concentrating on a Borel
set $\Pi$ of representatives for the nontrivial valuations of $K$; including
in the archimedean case $-\log_{2}|\sigma(x)|$, $\sigma:K\to{\mathbb{C}}$ a
place. For simplicity we can take $K$ to be countable, so that (away from
$\mu_{K}$-measure $0$) $\sigma$ is a complex embedding. Let
$\widehat{K}=Val_{K}$.

Let $X$ be a projective variety over $K$, with a very ample divisor $D$. For
$p\in{\Pi}$, let $X_{p}$ be the Berkovich space of $X$ over the Henselization
of $K$ at $p$.

It is really just as well to think of the affine variety
$X_{aff}=X\smallsetminus D$, whose polynomial ring is the graded ring $\oplus
L(X,mD)$; so $D$ is the hyperplane at $\infty$ of the projectivization of
$X_{aff}$.

We first consider a family of partial types that we call $D$-adelic. Such a
type is associated with a positively homogeneous function
$s:\widehat{K}\to{\mathbb{R}}$, with $\int s<\infty$. We assume $s$ is given
by $s(v)=v(t)$ for some term $t$ over $K$, or as a uniform limit of such
functions. Classically, $K$ is a global field, $\widehat{K}$ is discrete and
$s$ has finite support on $\Pi$.

Recall that $ht_{D}(x)=\int\max_{i=1}^{n}v^{-}(x_{i})dv$. Define the partial
type $P_{D,s}$ by:

$ht_{D}(x)=\int\min(s(v),\max_{i=1}^{n}v^{-}(x_{i}))$

This partial type asserts that the $D$-height of $x$ is entirely attributable
to the primes in the support of $S$, and moreover the local heights are
bounded by $s$. Any partial type extending some $P_{D,s}$ will be called
$D$-adelic.

To define $D$-adelic subsets of $X$.

We first consider compact sets $S\subset\widehat{K}$, such that $S\subset\Pi$.
We may assume (adjusting $\Pi,\mu_{K}$ if necessary) that for some terms
$t_{S}\geq 0$, we have $S=\\{v:t(v)=1\\}$. Typically we could have
$t_{S}=\max_{i=1}^{l}v(a_{i})$; for instance over ${\mathbb{Q}}$, for
$a_{1}=1,a_{2}=2,a_{3}=3$ we have $S=\\{2,3,\infty\\}$.

Note in general that $S$ has finite $\mu_{K}$-volume. These sets $S$
generalize finite sets of primes.

A $D$-adelic set is a compact subset of of $\widehat{K(X)}$, lying above $S$.
Thus $A$ is the union of compact subsets $A_{p}\subset X_{p}$ for each $p\in
S$. We assume that each $A_{p}$ is disjoint from $D$, i.e. lies in the
Berkovich space of the affine variety $X_{aff}$.

Alternative formulation: we assume that $A$ is cut out by inequalities of term
functions, including: $h_{D}\leq m$; where $h_{D}(v)=\min_{i}v(x_{i})$,
$\\{x_{i}\\}$ being a basis for the global sections of $D$.

If we pass to an algebraic closure $K_{p}^{alg}$ of the valued field
$K_{p}=(K,p)$, we can identify $A_{p}$ with a Galois-invariant compact subset
of the Berkovich space $X$ over $K_{p}^{alg}$.

The Chebyshev constant Ch($A$) is defined by:

$-\log\hbox{Ch}(A)=\lim_{n\to\infty}\frac{1}{n}\sup_{f\in
L(nD)}\int_{p\in\Pi}\inf_{v\in A_{p}}v(f)$

Where $L(nD)=L_{K}(X,nD)$ is the $K$-space of rational functions in $K(X)$
whose poles are at most $nD$.

Since $L(mD)L(nD)\subset L((m+n)D)$, the terms $a_{n}=\sup_{f\in
L(nD)}\int_{p\in\Pi}\inf_{v\in A_{p}}v(f)$ satisfy $a_{n+m}\geq a_{n}+a_{m}$.
By Fekete’s lemma, $\lim a_{n}/n$ exists and equals $\sup_{n}a_{n}/n$. Thus
Ch$(A)\geq 1$ iff each $a_{n}\leq 0$.

######  Example 3.1.

Let $K={\mathbb{Q}}$, $X={\mathbb{P}}^{1}$, $D=\\{\infty\\}$. Let
$A_{\infty}=A$ be a compact subset of ${\mathbb{R}}$, and
$A_{p}={\mathcal{O}}_{p}$ otherwise. Note that for $e\in K$ we have
$\int_{p\in\Pi}\inf_{v\in A_{p}}v(fe)=\int_{p}\inf_{v\in A_{p}}v(f)$. This
uses the fact that $v(e)=p(e)$ for $v\in A_{p}$, and that $\int_{p}p(e)=0$.
Thus in the supremum it sufices to consider monic polynomials. In this case
$\inf_{v\in A_{p}}v(f)=0$ for finite $p$; so

$-\log\hbox{Ch}(A)=\lim_{n\to\infty}\frac{1}{n}\sup_{f\in L(nD)^{\hbox{\\\
monic}}}\inf_{v\in A_{\infty}}v(f)$

which is the usual definition of the Chebyshev constant, see e.g. [chinburg-
capacity-varieties]. It was shown by Fekete-Szegö that this constant equals
the capacity of $A$.

Let $X,Y$ be projective varieties over $K$, with very ample divisors
$D_{X},D_{Y}$; let $D=\pi_{X}^{*}D_{X}+\pi_{Y}^{*}D_{Y}$. Given adelic sets
$A,B$ on $X,Y$ respectively, we have the obvious definition of $A\times B$ on
$X\times Y$; namely the pullback to the Berkovich space of $X\times Y$ of
$A\times B$. It is an adelic set. We have $-\log Ch(A\times
B)\geq-\log(Ch(A))+-\log(Ch(B))$, or $Ch(A\times B)\leq Ch(A)Ch(B)$. In
particular, $Ch(A^{n})\leq Ch(A)^{n}$.

###  3.2. The partial type associated with an adelic set

Recall that a Zariski dense type in $X$ over $K$ can be identified with a
measure on valuations of $K(X)$, satisfying certain properties. The set of
measures concentrating above each $p$ on $A_{p}$ forms a closed set, and thus
describes a partial type; but one of unbounded height. We can further specify
that the Weil height of a realization $(x_{1},\ldots,x_{n})$ (in affine
coordinates), $\int\max_{i=1}^{n}v^{-}(x_{i})$ be attained adelically, indeed
above $S$:

$ht_{D}(x)=\int\max_{i=1}^{n}v^{-}(x_{i})=\int_{p\in
S}\int_{v|p}\max_{i=1}^{n}v^{-}(x_{i})$

With the above assumptions on $S$ and $A$, this is a closed condition and thus
determines a partial type $P_{A,D}$.

We call it the adelic type associated with $A$.

Model theoretic questions arise immediately: consistency; completeness;
extensions to $K^{a}$; stable emeddedness and minimality. We will see that
these relate to classical definitions and theorems: consistency to an
$L^{1}$-version of Fekete-Szegő conditions, adelic completeness to low height
generic equidistribution. Full qf-completeness will follow, in case $NS(X)$ is
one-dimensional.

Before giving a consistency criterion for $P_{A}$ (Lemma 3.5), we consider
adelic measures on $A$. Let $\mu$ be a measure on the Berkovich space of
$K(X)$ concentrating on valuations nontrivial on $K$, and let $(\mu_{p})_{p\in
P}$ be the decomposition into probability measures on the $X_{p}$. Let
$P_{\mu}$ be the set of adelic qf types whose restriction above $p\in S$ is
precisely $\mu_{p}$. Again by adelic (for $(X,D)$) we mean that

$ht_{D}(x)=\int\max_{i=1}^{n}v^{-}(x_{i})d\mu(v)$

(where $x$ is again a basis for the global sections of $L$; in particular
$K(X)=K(x)$.) this is a closed set of types and will be viewed as a partial
type. (We will consider $\mu$ such that $\mu_{p}$ is supported on $A_{p}$ for
$p\in S$.)

######  Lemma 3.3.

$P_{\mu}$ is consistent iff for any $m\in{\mathbb{N}}$ and $f\in L(X,mD)$,

$\int v(f)d\mu(v)\leq 0$

###### Proof.

Assume $P_{\mu_{S}}$ is consistent, and so realized by
$(a_{1},\ldots,a_{n})\in X(K)\leq{\mathbb{A}}^{n}(K)$. Let $f\in L(X,mD)$.
Then $(f)+mD\geq 0$ (as Cartier divisors over $K$) so for $v$ trivial on $K$
we have $v(f(a))+m\max_{i=1}^{n}v^{-}(a_{i})\geq 0$. Now since $tp(a/K)$ is
adelic, for almost every $v$ that is not over some $p\in\Pi$, we have
$\max_{i=1}^{n}v^{-}(a_{i})=0$; so $f(a)\geq 0$. Since $\int v(f(a))=0$, it
follows that $\int v(f)d\mu_{S}(v)\leq 0$.

######  Remark 3.4.

We can relax the notion of $D$-adelic by allowing the divisorial valuation
trivial on $K$ corresponding to $D$, to carry weight $\alpha>0$. Thus let
$P_{\mu,\alpha}$ be the set of adelic qf types whose restriction above $p\in
S$ is precisely $\mu_{p}$, and with
$ht_{D}(x)=\alpha+\int\max_{i=1}^{n}v^{-}(x_{i})d\mu(v)$. We obtain in the
same way: $P_{\mu,\alpha}$ is consistent iff for any $m\in{\mathbb{N}}$ and
$f\in L(X,mD)$,

$\frac{1}{m}\int v(f)d\mu(v)\leq\alpha$

The other direction follows from Lemma 3.9, with $Z$ consisting of valuations
nontrivial on $K$, and all valuations trivial on $K$ and centered on $D$. The
measure is extended so as to be zero on the latter. Note that the ring of
elements of $K(X)$ that are non-negative at every $v\notin Z$ is precisely
$\cup_{m}L(mD)$. ∎

######  Lemma 3.5.

$P_{A}$ is consistent iff Ch$(A)\geq 1$.

###### Proof.

First, assume $P_{A}$ is consistent. A complete type extending $P_{A}$
determines measures $\mu_{p}$ above each $p\in\Pi$. Let $\mu=(\mu_{p})_{p\in
S}$. Then $P_{\mu}$ is consistent with a $D$-adelic type $P$. Thus for any
$m\in{\mathbb{N}}$ and $f\in L(X,mD)$, $\int v(f)d\mu(v)\leq 0$; and as $P$ is
$D$-adelic, $\int_{p\in S}\inf_{v\in A_{p}}v(f)\leq 0$. It follows that
$\sup_{f\in L(nD)}\int_{p\in S}\inf_{v\in A_{p}}v(f)\leq 0$ and so also

$\lim_{n\to\infty}\frac{1}{n}\sup_{f\in L(nD)}\int_{p\in S}\inf_{v\in
A_{p}}v(f)\leq 0$

as required.

Conversely, assume Ch$(A)\geq 1$. We seek a measure $\mu$ on the locally
compact space $\mathbin{\mathaccent 0{\cdot}\cup}_{p}A_{p}$, satisfying the
conditions of Lemma 3.3. Let $C(A)$ be the space of continuous functions with
compact support on $\mathbin{\mathaccent 0{\cdot}\cup}_{p}A_{p}$. We are
looking for a linear functional $\lambda$ on $C(A)$ such that:

  1. (1)

$\lambda(1_{A_{p}})=m(p)$. In other words $\lambda$ extends $m$ on $C(S)$.

  2. (2)

$\lambda(c)\geq 0$ if $c$ is a positive function.

  3. (3)

For $f\in L(X,mD)$, $\lambda(\hat{f})\leq 0$; where $\hat{f}(v)=v(f)$.

Note that any element of $C(A)$ is bounded by an element of $C(S)$ (with
finite support).

By Lemma 3.8, it suffices to show that if $\sum a_{i}p_{i}\in C(S)$ satisfies
$\sum a_{i}p_{i}\leq\hat{f}$ with $f\in L(X,mD)$, then $\sum a_{i}m(p_{i})\leq
0$. Assume $\sum a_{i}p_{i}\leq\hat{f}$. Then $a_{i}\leq\inf_{v\in
A_{p}}v(f)$. So $\sum a_{i}p_{i}\leq\int\inf_{v\in A_{p}}v(f)\leq 0$.

∎

######  Remark 3.6.

The proof of Lemma 3.5 essentially gives a characterization of uniqueness of
the measure $\mu$, or of the restriction of $\mu$ to $S$. Namely, any
continuous function on the Berkovich space (above $S$), with $\mu$-integral
$=0$, should be uniformly approximable (above $S$) by ${\rm val}(f)$ with
$f\in L(X,mD)$ for some $m$. This clearly implies that Ch$(A)=1$. Over a GVF
$K$ with existentially closed symmetric extension $K^{alg}$, one can hope for
the converse.

######  Corollary 3.7.

Assume the symmetric extension $K^{alg}$ of $K$ is existentially closed as a
GVF, and $X$ remains irreducible over $K^{alg}$. Then $Ch(A)\geq 1$ iff
$Ch(A^{n})\geq 1$.

###### Proof.

We saw that $Ch(A^{n})\leq Ch(A)^{n}$, so one direction is clear. Conversely
assume $Ch(A)\geq 1$. Then $P_{A}$ is consistent; by taking symmetric
extensions, noting that the restriction map from the absolute Galois group of
$K(X)$ to that of $K$ is surjective, we see that $P_{A}$ extends to a qf type
$p_{A}$ over $K^{alg}$. As $K^{alg}$ is existentially closed, $p_{A}$ is
finitely satisfiable in $K$; we can find $c_{1},\ldots,c_{n}\models p_{A}$
within an ultrapower of $K^{alg}$, with
$tp(c_{i}/K^{alg}(c_{1},\ldots,c_{i-1}))$ finitely satisfiable in $K$. It
follows that $P_{A^{n}}$ is consistent, and so by Lemma 3.5, we have
$Ch(A^{n})\geq 1$. ∎

######  Lemma 3.8.

[M. Riesz] Let $V$ be a ${\mathbb{Q}}$\- vector space. Let $U$ be a subspace
of $V$, $P$ a cone, and let $h:U\to{\mathbb{R}}$ be a ${\mathbb{Q}}$-linear
map, positive with respect to $P\cap U$, i.e. $h(x)\geq 0$ for $x\in P\cap U$.
Assume: for any $v\in V$, for some $w\in U$, $w-v\in P$. Then $h$ extends to a
positive linear map $H$ on $(V,P)$.

Note that if $V$ is actually an ${\mathbb{R}}$-space, and $P$ an
${\mathbb{R}}$-cone, and any element of $V$ has the form $a-b$, with $a,b\in
P$ then $H$ will be ${\mathbb{R}}$-linear. It suffices to show for
$\alpha\in{\mathbb{R}}$, $a\in P$ that $H(\alpha a)=\alpha H(a)$. This is
clear since for rational $\alpha^{\prime}\in{\mathbb{Q}}$ with
$\alpha^{\prime}\leq\alpha$ we have $(\alpha-\alpha^{\prime})a\in P$ so
$H((\alpha-\alpha^{\prime})a)=H(\alpha a)-\alpha^{\prime}H(a)\geq 0$, or
$H(\alpha a)\geq\alpha^{\prime}H(a)$; and if $\alpha<\alpha^{\prime}$, then
$(\alpha^{\prime}-\alpha)a\in P$, so $\alpha^{\prime}H(a)\geq H(\alpha a)$.

Let $F$ be a field. There is a Galois correspondence $W\mapsto R^{W}$,
$R\mapsto W^{R}$ between integrally closed subrings $R$ of $F$, and certain
closed subsets of $\Omega_{F}$ consisting of non-archimedean valuations and
invariant under positive scalar multiplication. Namely, let

$R^{W}=\\{r:(\forall v\in W)\,v(r)\geq 0\\}\cup\\{0\\}$

Conversely given an integrally closed subring subring $R$ of $F$, let

$W^{R}=\\{v:(\forall a\in R\smallsetminus(0))\,v(a)\geq 0\\}$

It is clear that the correspondence is order-preserving, that $R^{W}$ is
integrally closed, that $W^{R}$ is a closed set consisting of non-archimedean
valuations, and closed under equivalence. Also, $R^{W}=R^{cl(W)}$, and
$W^{R}=W^{R^{int}}$ where $R^{int}$ is the integral closure of $R$ in $F$. We
need not have $W^{R^{W}}=W$ in general.

We also denote $R_{Z}=R^{Val_{F}\smallsetminus Z}$, the subring of $F$
consisting of elements whose poles lie on $Z$.

######  Lemma 3.9.

[See better version in file: class1.tex] Let $F$ be a countable field. Let
$Z\subset Val_{F}$ be a Borel subset including all archimedean valuations, and
invariant under multiplication by positive scalars. Let $R=R_{Z}=\\{r:v\notin
Z\implies v(r)\geq 0\\}\cup\\{0\\}$ be the subring of $F$ consisting of
elements whose poles lie on $Z$. Assume the field of fractions of $R$ is $F$.
Let $\nu$ be a measure on $Z$, such that for any nonzero $r\in R$,

$\int v(r)d\nu(r)\leq 0$

Then there exists a GVF structure on $F$, such that for any $t\in Term_{F}$
supported on $Z$ we have: $\int t=\int t(v)d\nu(v)$.

###### Proof.

By the discussion above, We may replace $Z$ by the interior $Z^{o}$ of $Z$: we
have $R_{Z^{o}}=R_{Z}$, and if $t$ is supported on $Z$, it is supported on
$Z^{o}$. Thus we will assume that $Z$ is open.

Let $V$ be the ${\mathbb{Q}}$-space of term functions on $Val_{F}$. Among
these, we have the functions $(f)$ for $f\in F\smallsetminus(0)$, defined by:
$f(v)=v(f)$. Let ${Pr}$ be the ${\mathbb{Q}}$-span of $\\{(f):f\in F\\}$. Let
$U$ be the subspace of functions supported on $Z$. For $\alpha\in U$, let
$\phi(\alpha)=\int\alpha(v)d\nu(v)$.

Claim 1.  Let $\alpha_{1},\ldots,\alpha_{n}\in{\mathbb{R}}$ be linearly
independent over ${\mathbb{Q}}$, and $f_{1},\ldots,f_{n}\in F^{*}$. If
$\sum\alpha_{i}(f_{i})\in U$, then each $(f_{i})\in U$.

So $f\in R$ by definition of $R$, and hence by assumption, $\int
v(f)d\nu(v)\leq 0$, and hence $\int v(\alpha-(f))d\nu(v)\geq 0$.

For $\alpha\in{Pr}$, let $\phi(\alpha)=0$. It follows from the Claim that
$\phi$ can be extended to a linear map on $U+{Pr}$ with $\phi(\alpha)=0$ for
$\alpha\in{Pr}$. We again denote this by $\phi$.

We define a cone ${\mathcal{P}}$ on $V$: ${{\mathcal{P}}}$ is the set of non-
negative functions in $V$. It is clear that $\phi$ is positive on $U+{Pr}$,
i.e. $\phi(g)\geq 0$ if $g\in U+{Pr}\cap P$.

Let us check the boundedness condition of M. Riesz (3.8 (2)). Any element of
$V$ is bounded by a finite sum of elements $h^{+}$, with $h=f/g$ for some
$f,g\in R$, so it suffices to check that the image of $(f/g)^{+}$ in $V$ is
bounded by an element of $U$. We have

$(f/g)^{+}\leq(f)^{+}+(g)^{-}=(f)^{-}+(g)^{-}+(f)$

now $(f)^{-}+(g)^{-}$ is supported on $Z$, so lies in $U$, and hence
$(f)^{-}+(g)^{-}+(f)$lies in $U+{Pr}$.

By Lemma 3.8, $\phi$ can be extended to a positive functional on
${\mathbf{V}}$.

Hence there exists a GVF structure on $F$, agreeing with $\nu$ on terms
supported on $Z$.

∎

For simplicity, we state an approximation criterion over global fields (though
it could easily be generalized); thus we assume $S$ is finite. Let $a_{l}\in
K^{alg}$. We say that the $a_{l}$ approximate $P_{\mu}$ if for any formula
$\phi$ over $K$, if for some subsequence the values $\phi(a_{l})$ converge to
$\alpha$, then $P_{\mu}$ is consistent with $\phi(x)=\alpha$. Equivalently,
for any subsequence $a_{j}$, if $qftp(a_{j}/K)$ converges to a qf type $q$
then $q\vdash P_{\mu}$.

######  Lemma 3.10.

Let $a_{l}\in K^{alg}$. Then the $a_{l}$ approximate $P_{\mu}$ iff

  1. (1)

They are generic, i.e. only finitely many $a_{l}$ lie on any given proper
subvariety of $X$.

  2. (2)

They are near-integral away from $S$. In other words,

$\sum_{p\in\Pi\smallsetminus S}\int_{v|p}\max_{i=1}^{n}v^{-}(x_{i})\to 0$

  3. (3)

The Galois orbit measures approximate $\mu_{p}$ for each $p$.

The following theorem applies in particular to $K=k(t)[1]$ and finite
extensions thereof, by Theorem 2.1.

By the characteristic measure of a finite set $O$, we mean
$\mu_{O}=\frac{1}{|O|}\sum_{a\in O}1_{a}$, where $1_{a}$ is the characteristic
function of $a$.

######  Theorem 3.11.

Assume the symmetric extension $K^{alg}$ is existentially closed as a GVF.
Then Ch$(A)\geq 1$ iff there exists a sequence of Galois orbits $O_{n}$ that
are near-integral away from $S$, generic on $X$, and such that for each $p$,
if we fix an extension $K_{p}^{alg}$ of the valued field $(K,P)$ to $K^{alg}$,
the characteristic measures of the $O_{n}$ on the Berkovich space
$X(K_{p}^{alg})$ approach a measure on $A_{p}$.

###### Proof.

This follows from Lemma 3.5 and Lemma 3.10. ∎

##  4\. Divisors

Let $X$ be an irreducible normal projective variety over a field $k$, $K=k(X)$
the function field.

Let $Div(X)$ be the group of Cartier divisors on $X$. If $j:X^{\prime}\to X$
is a surjective morphism, pullback induces an injective map $Div(X)\to
Div(X^{\prime})$. By $Div(K)$ we denote the direct limit of the groups
$Div(X^{\prime})$ over all birational morphisms $X^{\prime}\to X$.

A global section of a Cartier divisor $D$ on $X$ is a rational function $f\in
k(X)$ such that for any open $U$, if $D$ is represented by $(d)$ on $U$, then
$f=rd$ on $U$ for some regular function $r$ on $U$. Equivalently, $(f)+D$ is
an effective Cartier divisor.

###  4.1. Stable joins and meets of Cartier divisors

Given effective Cartier divisors $E,E_{1},E_{2}$ on $X$, represented by
hypersurfaces with locally principal ideal sheaves $I,I_{1},I_{2}$, we say
that $E$ is the stable meet of $E_{1},E_{2}$ if $I$ is the ideal sheaf
$I_{1}+I_{2}$ generated by $I_{1}\cup I_{2}$. In general, $E_{1},E_{2}$ may
not have a stable meet on $X$, since $I_{1}+I_{2}$ may not be locally
principal. But if we pull back to divisors $E_{1}^{\prime},E_{2}^{\prime}$ on
the blowup $X^{\prime}$ at $I_{1}+I_{2}$, then $E_{1}^{\prime},E_{2}^{\prime}$
have a stable meet $E^{\prime}$, and this remains the case in any further
pullback. We write in this situation: $E=E_{1}\wedge E_{2}$.

A similar definition with sheaves of submodules of $K$ extends to all Cartier
divisors. It is moreover easy to see that $E=E_{1}\wedge E_{2}$ iff
$E+D=(E_{1}+D)\wedge(E_{2}+D)$, so one can reduce to the effective case.

Define $E=E_{1}\vee E_{2}$ iff $-E=(-E_{1})\wedge(-E_{2})$; we then say that
the stable join of $E_{1},E_{2}$ exists and equals $E$.

It is clear that if $E$ is the stable meet of $E_{1},E_{2}$, then it is
characterized as the greatest Cartier divisor below both $E_{1}$ and $E_{2}$;
dually if $E$ is the stable join of $E_{1},E_{2}$, it is the smallest Cartier
divisor above both $E_{1},E_{2}$. Here we refer to the ordering: $X\geq Y$ iff
there exists an effective Cartier divisor $Z$ with $X=Y+Z$.

There need not exist any Cartier divisor $E$ on $X$ with $E=E_{1}\wedge
E_{2}$. For example, two distinct lines on ${\mathbb{P}}^{2}$ never have a
stable meet, since their intersection is a point and not a divisor. In fact
even a stable meet with $0$ may not exist.

However, if $E_{1},E_{2}$ are effective Cartier divisors, with ideal sheaves
$I_{1},I_{2}$, and $f:X^{\prime}\to X$ is the result of blowing up the ideal
$I=I_{1}+I_{2}$, then it is clear that $f^{*}E_{1}\wedge f^{*}E_{2}=E$ where
$E$ is the exceptional divisor. By twisting with an appropriate very ample
divisor, we can reduce to the effective case; so any two Cartier divisors do
have a stable join, once pulled back to an appropriate blowup. Moreover once
we have $E=E_{1}\wedge E_{2}$, the same relation persists in pullbacks to
variety $X^{\prime}$ dominating $X$. Thus going to the limit, we obtain a
statement that is very essential to the theory of GVF’s:

######  Proposition 4.2.

$Div(K)$ forms a lattice under stable meet and join. ∎

###  4.3. The Picard group

$Pic(X)$ is the group of Cartier divisors on $X$, modulo the principal
divisors $(f)$. An element of $Pic(X)$ determines a line bundle on $X$. This
shows that $Pic$ is covariantly functorial; given a morphism $f:Y\to X$ of
varieties, there is a natural map $f^{*}:Pic(X)\to Pic(Y)$.

###  4.4. Intersections of subvarieties with divisors

An important special case occurs when $X$ is a subvariety of $Y$ and $f$ is
the inclusion. In this case, we obtain a map from divisors on $Y$ to divisors
on $X$. If we further represent the divisor on $X$ using subvarieties of $X$
of codimension $1$ and then view them as subvarieties of $Y$ of codimension
$1+codim(X)$, this map becomes the operation of intersecting a subvariety with
a divisor.

Specifically, a very ample divisor on $X$ can be represented by a subvariety
$D$ on $X$ that intersects $Y$ transversally; further when $\dim(Y)\geq 2$,
the intersection is an irreducible variety itself. So

$[D]\cdot[Y]=[D\cap Y]$

When $\dim(Y)=1$, $D\cap Y$ is a finite set of isolated points, and we are
interested in their number:

$[D]\cdot[Y]=|D\cap Y|\in{\mathbb{N}}$

###  4.5. Weil divisors

We will generally work with Cartier divisors, but sometimes the Weil viewpoint
is useful.

An integral Weil divisor is a ${\mathbb{Z}}$-linear combination of irreducible
hypersurfaces of $X$. More generally, the group $Z_{k}(X)$ of $k$-dimensional
Weil cycles is the free Abelian group generated by the irreducible
subvarieties of dimension $k$. $W_{k}(X)$ is the quotient by the
$k$-dimensional divisors that are rationally equivalent to $0$.
$W^{k}(X)=W_{n-k}(X)$.

Principal Cartier divisors correspond to principal Weil divisors
$(f)=\sum_{D}{v_{D}(f)}D$. (Here $v_{D}$ is the valuation associated with $D$;
in an open affine intersecting $D$, $D$ is the zero locus of a minimal prime
ideal; localizing, we obtain a Dedekind domain.)

There exists a canonical biliear map $Pic(X)\times W^{k}(X)\to W^{k+1}(X)$,
the intersection product.

###  4.6. Global sections

With each real Weil divisor $D=\sum\alpha_{i}D_{i}$
($\alpha_{i}\in{\mathbb{R}}$) we have an associated $k$-subspace of $K$,
namely $L(D)=L(X,D):=\\{0\\}\cup\\{f\in
K\smallsetminus(0):\bigwedge_{i}v_{D_{i}}(f)\geq-\alpha_{i}\\}$. If $D$ is a
Cartier divisor, then $L(D)=\\{0\\}\cup\\{f\in K:(f)+D\geq 0\\}$. The elements
of $L(D)$ can be viewed as rational sections of the associated line bundle,
and called sections of $D$. When $D$ is a Cartier divisor, with corresponding
sheaf $\tilde{D}$, one can identify $L(D)$ with $H^{0}(X,\tilde{D})$.

###  4.7. Intersections of curves with divisors

The $0$-dimensional cycles, up to algebraic equivalence, can be identified
with the integers ${\mathbb{Z}}$ via the degree map. At any rate we have the
map $\deg:W_{0}(X)\to{\mathbb{Z}}$, $\deg(\sum m_{i}c_{i})=\sum m_{i}$ where
$c_{i}$ are points of $C$. Composing with $\deg$ we obtain a numerical
intersection product $W^{1}(X)\times W_{1}(X)\to{\mathbb{Z}}$.

The fact that the intersection pairing is well-defined amounts here to the
product formula: when $D=(f)$ is a principal divisor, we have $D\cdot C=0$.

###  4.8. Basic positivity property

Let $D$ be an effective Weil divisor and $C$ a curve, not contained in the
support of $D$. Then $D\cdot C\geq 0$, and equality holds iff $C\cap
D=\emptyset$. (The intersection multiplicities are non-negative.)

###  4.9. Numerical equivalence

. We consider several subgroups on the group of Weil-divisors, described via
their sets of generators. For us, numerical equivalence is the essential
equivalence relation; we mention rational equivalence mostly in order to have
a well-defined intersection product.

  * •

Rational equivalence: $g:X\to{\mathbb{P}}^{1}$ a rational map;
$g^{-1}(0)-g^{-1}(\infty)$. (We can view $g$ as a morphism on
$X^{\prime}=X\smallsetminus W$ for some codimension $2$ subvariety $W$;
hypersurfaces of $X$ and of $X^{\prime}$ can be identified.)

  * •

Algebraic equivalence: $g:X\to C$ for some algebraic curve; or more generally,
pushforwards of such under generically finite morphisms.

  * •

Numerical equivalence; $Num_{0}=\\{D:(\forall C)D\cdot C=0\\}$.

######  Example 4.10.

Let $X$ be a smooth surface, $a\in X$, and $f:Y\to X$ the blowing up of $X$ at
$a$. Let $E=f^{-1}(0)$ the exceptional divisor. Let $C$ be a a curve on $X$
with $a$ a nonsingular point of $C$. Then

  1. (1)

$f^{*}C=E+C^{\prime}$, where $C^{\prime}$ is the strict transform of $C$.

  2. (2)

$E^{2}<0$, i.e. for the inclusion morphism $i:E\to Y$ we have $i^{*}E<0$.

To see (2), let $D$ be a divisor on $X$ rationally equivalent to $C$, with
$0\notin D$. As $f^{-1}(D)\cap E=\emptyset$, $[f^{*}D]=[f^{*}C]$, and by (1)
we have $(E+C^{\prime})\cdot E=f^{*}(D)\cdot E=0$, so $E^{2}=-C^{\prime}\cdot
E$. But $C^{\prime},E$ intersect properly with nonempty intersection, so
$C^{\prime}\cdot E>0$ and $E^{2}<0$.

###  4.11. The Néron-Severi group

The Néron-Severi group is $W^{1}(X)$ modulo the subgroup of divisors
algebraically equivalent to zero.

######  Theorem 4.12 (Lang-Néron, 1951).

$NS(X)$ is finitely generated. In particular,
$N^{1}(X):=NS(X){\otimes}{\mathbb{R}}$ is a finite-dimensional real vector
space.

Note that the Mordell-Weil theorem on finite generation of the rational points
of a simple non-isotrivial Abelian variety over a function field is a special
case. For instance, if $X$ is a surface admitting a morphism to
${\mathbb{P}}^{1}$, such that the generic fiber is an elliptic curve $E_{K}$,
that does not descend to $k$, then any $K$-rational point of $E_{K}$ has a
Zariski closure over $k$ which is a curve; this defines a homomorphism from
$E_{K}(K)$ to $NS(X)$ which is injective. Hence finite generation of $NS(X)$
implies finite generation of $E_{K}(K)$.

The Mordell-Weil theorem is usually proved using heights (in any book on
diophantine geometry).
Given this special case, the Néron-Severi theorem yields, with some care, to
an induction on dimension.

######  Remark 4.13.

A fourth equivalence relation is obtained by a homological representation. In
characteristic zero, the finite dimensionality of $N^{1}(X)$ can also be seen
via the representation in the homology group, along with a proof based on real
geometry (e.g. o-minimality) of the finite topological type of
$X({\mathbb{C}})$.

######  Definition 4.14.

The closure of the ample cone in $N^{1}(X)$ will be called the nef cone.

######  Remark 4.15.

It is clear from the above definition that if $D$ is nef, then $D\cdot Y\geq
0$ for any curve $Y$ on $X$, and moreover $D^{k}\cdot Y\geq 0$ for any
$k$-dimensional subvariety $Y$ of $X$. In fact, the converse is also true: if
$D\cdot Y\geq 0$ for any curve $Y$ on $X$, then $D$ is nef. This follows from
Kleiman’s theorem, and builds upon Nakai-Moishezon; see [12] for a very nice
exposition. We will however simply use the characterization as the closure of
the ample cone.

######  Lemma 4.16 ([3] Prop. 2.3).

If $d_{1},\ldots,d_{k}$ and $c_{1},\ldots,c_{k}$ are nef classes in
$N^{1}(X)$, $k=\dim(X)$, $c_{i}\leq d_{i}$, then $c_{1}\cdots c_{k}\leq
d_{1}\cdots d_{k}$.

###### Proof.

By continuity and multilinearity we may assume the $d_{i}$ are ample, or even
very ample. We may replace the $c_{i}$ by $d_{i}$ one by one; in this case,
$d_{1}-c_{1}$ is pseudo-effective, so it is the limit of effective classes
$e$; as $e$ is effective and $c_{i}$ is very ample for $i>1$, it is clear that
$e\cdot c_{2}\cdots c_{k}\geq 0$ and so $(d_{1}-c_{1})c_{2}\cdots c_{k}\geq
0$. ∎

Let us also define other cones we will use later.

######  Notation 4.17.

Let $W$ be a subset of ${\mathbb{R}}^{n}$.

  * •

The interior of a set $W$ will be denoted $W^{o}$.

  * •

${\mathbb{R}}^{\geq 1}W:=\\{rw:r\in{\mathbb{R}},r\geq 1,w\in W\\}$.

  * •

The smallest closed, convex set containing $W$ will be denoted $[W]$.

  * •

A function $\phi$ on a subset of a real vector space is positively
1-homogeneous if for all real $\alpha>0$ and $x\in{\mathop{\operatorname{\rm
dom}}}(\phi)$ we have $\alpha x\in{\mathop{\operatorname{\rm dom}}}(\phi)$ and
$f(\alpha x)=\alpha f(x)$.

  * •

For a real vector space $V$, $V^{*}$ is the dual space, and the natural
pairing $V\times V^{*}\to{\mathbb{R}}$ is denoted by $x\cdot y$ or $(x,y)$.

$N^{1}_{+}(X)$ = closure of the effective cone.

$N^{1}_{+}(X)^{o}$ = the big cone, interior of $N^{1}_{+}(X)$ = classes with
positive volume.

$N_{1}^{+}(X)$ is defined to be the closed dual of $N^{1}_{+}(X)$.

$N_{1}^{eff}(X)$ is the cone in $N_{1}(X)$ generated by the classes of
irreducible curves. It may be bigger than $N_{1}^{+}(X)$.

###  4.18. Functoriality of the cycle groups

Let $\pi:X\to Y$ be a morphism of varieties. If $\pi$ is dominant, there is a
natural pullback map of Cartier divisors. If $\pi$ is not dominant, the
trouble is that a Cartier divisor represented by $(f)$ may not pull back to a
Cartier divisor, since $f\circ\pi$ may be zero. But there is still a natural
pullback of line bundles, or Cartier divisors up to principal divisors. This
induces a pullback map $N^{1}(Y)\to N^{1}(X)$.

In the case of a generically finite morphism $\pi:X\to Y$, we can also define
a pullback map $\pi^{*}:N_{1}(X)\to N_{1}(Y)$, by dualizing the pushforward
map $\pi_{*}:N^{1}(X)\to N^{1}(Y)$. We have $\pi_{*}\pi^{*}(c)=\deg(\pi)c$.

When $X,Y$ are smooth we can more generally define a pullback map
$\pi^{*}:N^{k}(Y)\to N^{k}(X)$ by the formula:

$\pi^{*}(u)=\pi_{X}((X\times u)\cdot\pi)$

where $X\times u$ is the obvious class in $X\times Y$, and $\pi$ here denotes
the graph of $\pi$ as a subvariety of $X\times Y$. Note that if $w\in
N^{l}(X)$ is represented by $W$, and $u$ is represented by a cycle
$\sum\alpha_{i}D_{i}$ with $\pi^{-1}(D_{i})\cap W=\emptyset$, then
$\pi^{*}(w)=\emptyset$. (The intersection $(Y\times u)\times\pi$ is proper,
and the intersection of that with $(X\times D_{i})$ is empty, hence proper.)

We will often use the projection formula.

######  Lemma 4.19 (Projection formula, [12] 1.9,1.10).

Let $\pi:Y\to X$ be a surjective morphism between proper varieties. Let
$D_{1},\ldots,D_{r}$ be Cartier divisors on $X$, $r\geq\dim(Y)$. Then

$\pi^{*}D_{1}\cdot\ldots\cdot\pi^{*}D_{r}=\deg(\pi)(D_{1}\cdot\ldots\cdot
D_{r})$

More generally if $W$ is a subvariety of $Y$ and $r\geq\dim(W)$, then

$\pi^{*}D_{1}\cdot\ldots\cdot\pi^{*}D_{r}\cdot
W=\deg(\pi|W)(D_{1}\cdot\ldots\cdot D_{r}\cdot\pi W)$

When $r>\dim(X)$, we have $D_{1}\cdot\ldots\cdot D_{r}=0$; so both sides are
zero unless $r=\dim(Y)=\dim(X)$. Note that the ‘moreover’ follows from the
first statement applied to $\pi|W:W\to\pi W$. When cycle classes on $X$ are
considered, notably when $W$ is a curve and has a class $[W]\in N_{1}(Y)$, one
defines $\pi_{*}[W]=\deg(\pi|W)[f_{*}W]$, and the right hand side is written
$D_{1}\cdot\ldots\cdot D_{r}\cdot\pi_{*}[W]$.

When $f:X\to Y$ is flat, we have a pullback map of cycles, see [16].

###  4.20. Weighted curves

It will sometimes be convenient to speak of weighted curves. These are just
formal pairs $(\alpha,C)$ with $C$ a curve on $X$ and $\alpha>0$. The
corresponding class in the cycle class group is $\alpha[C]$.

##  5\. Convexity

Recall the notation of 4.17. The natural pairing

######  Lemma 5.1.

Let $\phi$ be a convex differentiable function on an open set
$U\subset{\mathbb{R}}^{n}$. Let $F=d\phi:U\to({\mathbb{R}}^{n})^{*}$ the
differential. Then $F$ is increasing in the sense that for all $x_{1},x_{2}\in
U$ we have

$(F(x_{1})-F(x_{2}),x_{1}-x_{2})\geq 0$

If $x_{1}\neq x_{2}$ and $\phi$ is strictly convex on the line through
$x_{1},x_{2}$, then strict inequality holds.

###### Proof.

By restricting to the line through $x_{1},x_{2}$ we reduce to the one-
dimensional case, where the statement is that a (strictly) convex
differentiable function has a (strictly) increasing derivative. ∎

######  Theorem 5.2.

Let $U$ be a (nonempty) open convex cone in $E={\mathbb{R}}^{n}$. Let $\phi$
be a continuous real-valued function on $E$, differentiable on $U$. Let

$F=d\phi:U\to E^{*}$

be the differential. Let

$C_{1}=\\{u^{\prime}\in E^{*}:(\forall u\in U)(u,u^{\prime})\geq\phi(u)\\}$

and let ${C_{1}^{o}}$ be the interior of ${C_{1}}$. Assume:

  1. (1)

$\phi$ is concave on $U$.

  2. (2)

$\phi$ is positively 1-homogeneous

  3. (3)

$\phi>0$ on $U$, and $\phi=0$ on $E\smallsetminus U$.

Then

$C_{1}^{o}\subset{\mathbb{R}}^{\geq 1}F(U)\subset C_{1}$

###### Proof.

We first note a few facts about $F$:

  1. (1)

$\phi(x+y)\geq\phi(x)+\phi(y)$.

  2. (2)

$\phi$ is strictly increasing on $U$ with respect to $<_{U}$.

  3. (3)

$F(U)\subseteq C_{1}$.

  4. (4)

Let $p\in cl(U)$, $p\neq 0$, and $l\in{C_{1}}^{o}$. Then $l(p)>0$.

Proof of (1-4):

  1. (1)

This is immediate from concavity and 1-homogeneity.

  2. (2)

If $x,y\in U$ and $x<_{U}y$, then $y=x+z$ for some $z\in{U}$; so as
$\phi(z)>0$, $\phi(y)\geq\phi(x)+\phi(z)>\phi(x)$.

  3. (3)

For $u,u^{\prime}\in U$ and $t>0$ we have by (2)
$t^{-1}(\phi(u+tu^{\prime})-\phi(u))\geq\phi(u^{\prime})$. Thus

$u^{\prime}\cdot F(u)=\frac{d}{dt}\phi(u+tu^{\prime})(u)=\lim_{t\to
0^{+}}t^{-1}(\phi(u+tu^{\prime})-\phi(u))\geq\phi(u^{\prime})$

Here is a more visual version of the proof of (3): if $c=F(u)$ with $u\in{U}$,
then for appropriate $b$, the graph of $y=c\cdot x+b$ is tangent to the graph
of $\phi$ at $u$, and so by concavity of $\phi$ lies above it, i.e.
$cx+b\geq\phi(x)$ for $x\in U$. By homogeneity, for any $x\in U$ and $t>0$ we
have $ctx+b\geq\phi(tx)=t\phi(x)$, so $cx+b/t\leq\phi(x)$; letting
$t\to\infty$ we find that $cx\geq\phi(x)$ on $U$, so $c\in{C_{1}}$.

  4. (4)

Let $p\in cl(U)$, $p\neq 0$. Then $l\cdot p\geq\phi\geq 0$ for $l\in{C_{1}}$.
By Lemma 5.7, $l\cdot p>0$ for $l\in{C_{1}}^{o}$.

Since ${\mathbb{R}}^{\geq 1}{C_{1}}={C_{1}}$, it follows from (3) that
${\mathbb{R}}^{\geq 1}F({U})\subseteq{C_{1}}$. It remains to show that
${C_{1}}^{o}\subseteq{\mathbb{R}}^{\geq 1}F({U})$.

Let $l\in{C_{1}}^{o}$. By (4) above, $\phi/l$ is continuous on
$cl(U)\smallsetminus(0)$, and invariant for the multiplicative action of
${\mathbb{R}}^{\geq 1}$. Let $B$ be a compact ball in $E$, with boundary
sphere $S$, and $p$ a point of $S\cap cl(U)$ with $(\phi/l)(p)=\gamma$
maximal; we have $\gamma\leq 1$ since $\phi\leq l$ on $U$ and hence on
$cl(U)$; also, $\phi>0$ on $U$ implies $l>0$ on $U$ so $\phi/l>0$ on $U$, and
thus $0<\gamma$ since $S\cap U$ is nonempty. We have $\phi\leq\gamma l$ on
$S\cap U$ and hence by ${\mathbb{R}}^{>0}$-invariance, on $U$.

Now $\phi(p)=\gamma l(p)$; as $l(p)\neq 0$ and $\gamma>0$ we have $\phi(p)>0$,
so $p\in U$, and $p$ is a local minimum of $\gamma l-\phi$. Hence $\phi,\gamma
l$ are tangent at $p$, i.e. $d(\gamma l-\phi(p))=0$ so $\gamma
l=d\phi(p)=F(p)\in F(U)$. Thus $l=\gamma^{-1}(\gamma l)\in{\mathbb{R}}^{\geq
1}F({U})$.

Thus $C_{1}^{o}\subset{\mathbb{R}}^{\geq 1}F(U)\subset C_{1}$. ∎

######  Corollary 5.3.

With the hypotheses of Theorem 5.2, let ${C}$ be the closed dual cone:

${C}=\\{u^{\prime}\in E^{*}:(\forall u\in U)(u,u^{\prime})\geq 0\\}$

and let $C^{o}$ be the interior of $C$. Then $C={\mathbb{R}}^{>0}C_{1}$. Thus

$C^{o}\subset{\mathbb{R}}^{>0}F(U)\subset{C}$

###### Proof.

By Lemma 5.7, If $l\in C^{o}$ then $l>0$ on $\bar{U}$; in this case by
compactness of the sphere, $\kappa l>\phi$ on $\bar{U}$ for some $\kappa>1$;
so $\kappa l\in C_{1}^{o}\subset{\mathbb{R}}^{\geq 1}F(U)$. Thus
$l\in{\mathbb{R}}^{>0}F(U)$. The other inclusion is immediate from Theorem
5.2. ∎

######  Remark 5.4.

  1. (1)

We saw in (3) of the proof of Theorem 5.2 that $v\cdot F(u)\geq\phi(v)$. It
follows from the 1-homogeneity of $\phi$ that equality holds for $u=v$:

$u\cdot F(u)=\frac{d}{dt}\phi(u+tu)=\phi(u)$

  2. (2)

Let $a\in U^{o}$. Then $F(a)$ is the unique element $w\in C_{1}$ satisfying
$a\cdot w=\phi(a)$. Indeed assume $w$ is such an element. For $v\in E$ and for
sufficiently small real $t$ we have $a+tv\in U^{o}$, so

$(a+tv)\cdot w\geq\phi(a+tv)=\phi(a)+tv\cdot F(a)+o(t)$

Since $a\cdot w=\phi(a)$ this gives for small $t>0$ the inequality $t(v\cdot
w)\geq tv\cdot F(a)$, and for small $t<0$ the opposite one. We thus obtain
$v\cdot w=v\cdot F(a)$, and thus $F(a)=w$. If for $b\in E^{*}$ we define
$\phi^{*}(b)=\inf\\{(u,b)/\phi(u):u\in U\\}$, so that $b\in C_{1}$ iff
$\phi^{*}(b)\geq 1$, then $a\cdot b\geq\phi(a)\phi^{*}(b)$, and for fixed
$a\in U^{o}$, equality holds on a unique ray of $b$.

  3. (3)

We will apply the theorem with $\phi={\mathop{\operatorname{\rm
vol\,}}}^{1/n}$ for a certain function ${\mathop{\operatorname{\rm vol\,}}}$,
and set $\psi=\frac{1}{n}d{\mathop{\operatorname{\rm vol\,}}}$. Then (1)
translates to $u\cdot\psi(u)={\mathop{\operatorname{\rm vol\,}}}(u)$; in the
setting of [2] and [3] this is the ‘orthogonality relation’.

  4. (4)

Similarly, (2) yields that $\psi(a)$ is the unique element $b$ of $C_{1}$ such
that

$a\cdot b={\mathop{\operatorname{\rm vol\,}}}(a)$

######  Example 5.5.

Let $C$ be a compact convex subset of ${\mathbb{R}}^{n}$. Let
$L:{\mathbb{R}}^{n}\to{\mathbb{R}}^{m}$ be a linear map. Then $L|C:C\to L(C)$
need not be an open map.

.

###### Proof.

$L$ will be the projection from ${\mathbb{R}}^{3}$ to ${\mathbb{R}}^{2}$,
$L(x,y,z)=(x,y)$. Let

$C=\\{(x,y,z)\in[0,1]^{3}:x^{2}\leq y,x^{4}\leq zy^{2}\\}$

Let $p=(0,0,0)$. Then $p$ projects to $(0,0)$, but a neighborhood of $p$ will
have no images strictly on the curve $y=x^{2}$ with $x>0$. Convexity can be
checked by intersecting with lines $l$. If $L(l)$ passes through $(0,0)$, i.e.
it has the equation $y=mx$, $m\geq 1$, we have above $l$ the set $x^{2}\leq
m^{2}z,\,x,y,z\leq 1$ which is clearly convex. Away from $(0,0)$,
differentiating $x^{4}y^{-2}$ twice we see that it has positive definite
Hessian, so the set of points above it is convex.

∎

######  Lemma 5.6.

Let $U$ be an open subset of ${\mathbb{R}}^{n}$, and let $C$ be a dense convex
subset of $U$. Then $C=U$. In particular, an open convex set is equal to the
interior of its closure.

###### Proof.

Say $0\in U$; we have to show that $0\in C$. Let $(a_{i})_{i=1}^{2^{n}}$ be
points of $C$ in each of the $2^{n}$ quadrants; this is possible by density.
Then $0$ lies in the convex hull of the points $a_{i}$. Indeed otherwise there
is a hyperplane $c\cdot x\sum_{k=1}^{n}c_{k}x_{k}=0$ through $0$, such that
all the points $a_{i}$ lie to one side, i.e. satisfy say $c\cdot x_{i}<0$.
However there is some $i$ in the same quadrant as $c$, i.e. $(a_{i})_{k}>0$
iff $c_{k}<0$. Then $\sum_{k}c_{k}(a_{i})_{k}>0$, a contradiction.

For the last statement, let $C$ be open and convex, and let $U$ be the
interior of the closure of $C$; then $C$ is dense in $U$, so equals $U$. ∎

######  Lemma 5.7.

Let $D$ be a closed cone in $E={\mathbb{R}}^{n}$, $C$ the closed dual cone in
the dual space $E^{\prime}$, i.e.

$c\in C\iff(\forall d\in D)c\cdot d\geq 0$

Let $C^{o}$ be the interior of $C$. Then

$c\in C^{o}\iff(\forall d\in D\smallsetminus(0))\,c\cdot d>0$

###### Proof.

Let $c\in C^{o}$ and $0\neq d\in D$. As $d\neq 0$, there exists $a\in
E^{\prime}$ with $a\cdot d>0$. As $c\in C^{o}$, for small enough $\epsilon>0$
we have $c-\epsilon a\in C$; so $(c-\epsilon a)\cdot d\geq 0$, or $c\cdot
d\geq\epsilon a\cdot d>0$.

Conversely, assume $a\in E^{\prime}$ and $a\cdot b>0$ for all $b\in
D\smallsetminus(0)$. Fix some Euclidean structure on $E$, let $S$ be the unit
sphere and $P=S\cap U$. Then $P$ is compact, so for some $\epsilon>0$ we have
$a\cdot b>\epsilon$ for all $b\in P$. Similarly there exists a neighborhood
$U$ of $0$ in $E$ such that $|c\cdot b|<\epsilon$ for all $c\in U,b\in P$.
Then clearly $(a+c)\cdot b>0$ for all $c\in U$, so $a\in C^{o}$. ∎

######  Lemma 5.8.

Let $L:{\mathbb{R}}^{n}\to{\mathbb{R}}^{m}$ be a surjective linear map; let
$C$ be an open convex cone in ${\mathbb{R}}^{n}$. Then $L(C)={\mathbb{R}}^{m}$
if $C\cap\ker L\neq\emptyset$. Otherwise, $L(C)$ is an open convex cone in
${\mathbb{R}}^{m}$; and $L(cl(C))=cl(L(C))$. Moreover, $L|cl(C):cl(C)\to
cl(L(C))$ is an open map at $0$.

###### Proof.

If $b\in C\cap\ker L$ and $c\in C$, then since $C$ is open, there exists $a\in
C$ on the line from $c$ to $b$, but beyond $b$, so that $b$ lies between $a$
and $c$. Then $0$ lies between $L(a)$ and $L(c)$, so that as $L(C)$ is a cone,
the full line lies in $L(C)$, and in particular $-L(c)\in L(C)$. This shows
that $L(C)$ is a subspace of ${\mathbb{R}}^{m}$.

Assume now that $C\cap\ker(L)=\emptyset$. It is clear that $L(C)$ is an open
cone in ${\mathbb{R}}^{m}$, and $L(cl(C))\subseteq cl(L(C))$. The other
direction can be sen using Lemma 5.7.

Next let $D=cl(C)$, and $\lambda=L|D$; we show that $\lambda:D\to L(D)$ is an
open map at $0$. Put some Banach space structure on ${\mathbb{R}}^{n}$, and
the quotient structure on ${\mathbb{R}}^{m}$; let $B_{\alpha}$ be the ball of
radius $\alpha$. Then for some $\alpha>0$, $L(B_{\alpha}\cap D)$ contains
$B_{1}\cap LD$. The reason is that the unit sphere $S_{1}$ in
${\mathbb{R}}^{m}$ is compact; each point $p$ of $LD\cap S_{1}$ is the
$L$-image of some point of $D$, and hence some neighborhood of $p$ lies in
$L(D\cap S_{\alpha(p)})$ for some $\alpha(p)$; by choosing a finite
subcovering we can find $\alpha$ such that $LD\cap S_{1}\subset L(D\cap
S_{\alpha})$. But then $LD\cap S_{\beta/\alpha}\subset L(D\cap S_{\beta})$ for
all $\beta>0$. This shows that $\lambda$ is open at $0$.

∎

######  Lemma 5.9.

Let $C$ be a closed convex subset of ${\mathbb{R}}^{n}$. Then the boundary of
$C$ has Lebesgue measure zero. If $C$ has nonempty interior and finite
measure, it is compact.

###### Proof.

For the first point, we may assume $C$ is bounded, since the boundary of $C$
is contained in the union of the boundaries of $C\cap B_{n}$, where $B_{n}$ is
the ball of radius $n$. Also, we may assume $0\in C$. Then for any
$0<\alpha<1$ we have $\alpha C\subset C^{o}$, i.e. $\alpha C$ is contained in
the interior of $C$. As ${\mathop{\operatorname{\rm vol\,}}}(\alpha
C)=\alpha^{n}{\mathop{\operatorname{\rm vol\,}}}(C)$, we see that
${\mathop{\operatorname{\rm vol\,}}}\partial
C\leq(1-\alpha^{n}){\mathop{\operatorname{\rm vol\,}}}(C)$. Letting $\alpha\to
1$ we see that ${\mathop{\operatorname{\rm vol\,}}}\partial C=0$.

Assume now $C$ has non-empty interior; so it contains $a+B_{\epsilon}$ for
some $a$ and some open ball $B_{\epsilon}$ of $0$. If $a_{1},a_{2},\ldots$ are
points of $C$ then $C$ contains $(a_{i}+a)/2+B_{\epsilon/2}$; if the volume of
$C$ is finite then two of these balls must intersect, so
$d(a_{i},a_{j})<\epsilon$ for some $i,j$. It follows that $C$ is bounded. ∎

######  Theorem 5.10 (Khovanskii).

Let $F$ be a finite subset of a ${\mathbb{Q}}$-vector space $B$. Let $S,A,C$
be respectively the subsemigroup, subgroup, and rational convex cone generated
by $F$:

$S=\\{\sum m_{i}a_{i}:m_{i}\in{\mathbb{N}},a_{i}\in F\\}$ $A=\\{\sum
m_{i}a_{i}:m_{i}\in{\mathbb{Z}},a_{i}\in F\\}$ $C=\\{\sum
m_{i}a_{i}:m_{i}\in{\mathbb{Q}}^{>0},a_{i}\in F\\}$

Then for some $s\in S$,

$A\cap(s+C)=s+A\cap C\subseteq S\subseteq A\cap C$

###### Proof.

We may assume $B$ is also generated by $F$. Let
$E=A{\otimes}{\mathbb{R}}=B{\otimes}_{\mathbb{Q}}{\mathbb{R}}$. Then $A$ is a
discrete subgroup of the finite-dimensional ${\mathbb{R}}$-space $E$. Let
$Y=\\{\sum_{i=1}^{m}r_{i}f_{i}:-1\leq r_{i}\leq 0\\}$. Then $Y$ is compact. As
$A$ is discrete, $A\cap Y$ is a finite set; we can find $s\in S$ such that
$(A\cap Y)+s\subset S$. Let $a\in A\cap C$; so $a=\sum\alpha_{i}f_{i}$ for
some $\alpha_{i}\geq 0$. Let $[\alpha_{i}]$ be the least integer
$\geq\alpha_{i}$, and $r_{i}=\alpha_{i}-[\alpha_{i}]$. Then $\sum
r_{i}f_{i}\in A\cap Y$; so $s+\sum r_{i}f_{i}\in S$. Certainly
$\sum[\alpha_{i}]f_{i}\in S$. So $s+a\in S$. ∎

We remark that if we identify $A$ with ${\mathbb{Z}}^{n}$, then $S$ is
definable in Presburger arithmetic; hence globally it is defined by a
combination of inequalities and congruences.

######  Proposition 5.11.

Let $\Lambda$ be a finitely generated, torsion free Abelian group. For
$k\in{\mathbb{N}}$ let $S_{k}$ be subsets of $A$ with $0\in S_{1}$ and
$S_{k}+S_{l}\subset S_{k+l}$. Let $S=\cup_{k=1}^{\infty}S_{k}$. Let
$C_{1}=[\cup_{k}S_{k}/k]$ be the closed convex hull of $\\{s/k:s\in S_{k}\\}$
in $E=\Lambda{\otimes}{\mathbb{R}}$. If $K$ is a compact subset of
$C_{1}^{o}$, then for all large enough $m$,

$K\cap\frac{\Lambda}{m}=K\cap\frac{S_{m}}{m}$

###### Proof.

Let $\bar{S}=\\{(n,x):x\in S_{n}\\}$; we view $\bar{S}$ as a subset of
${\mathbb{R}}\times E$. It is clear that $\bar{S}$ forms a semigroup of
${\mathbb{Z}}\times\Lambda$. The group generated by $\bar{S}$ includes
${\mathbb{Z}}\times 0$ (since $0\in S_{1}$), and maps onto $\Lambda$, so it
equals ${\mathbb{Z}}\times\Lambda$. Since ${\mathbb{Z}}\times\Lambda$ is a
Noetherian ${\mathbb{Z}}$-module, it is generated by a finite set $F$; let $W$
be the subsemigroup of $\bar{S}$ generated by $F$. Let $C$ be the convex cone
generated by $F$. By enlarging $F$ we may arrange that $\\{1\\}\times K\subset
C$. (The interior $C_{1}^{o}$ of $C_{1}$ equals the union of the interiors of
the convex hulls $[F^{\prime}]$ of finite subsets $F^{\prime}$ of
$\cup_{k}S_{k}/k$, using Lemma 5.6; as $K$ is compact, $\\{1\\}\times
K\subset[F^{\prime}]^{o}$ for some such $F^{\prime}$.) By Theorem 5.10, there
exists $w\in W$ with

$({\mathbb{Z}}\times\Lambda)\cap C+w\subseteq W$

Let $K^{\prime}=\\{1\\}\times K$. As $K^{\prime}$ is compact and
$K^{\prime}\subset C$, there exists a symmetric neighborhood $U$ of $0$ in
${\mathbb{R}}\times E$ such that $K^{\prime}+U\subset C$. Let $m$ be large
enough so that $w/m\in U$. If $b\in K\cap\Lambda/m$, then $(1,b)-w/m\in C$, so
$(m,mb)\in w+C$ and hence $(m,mb)\in W\subset\bar{S}$. So $mb\in S_{m}$ and
thus $b\in S_{m}/m$. ∎

######  Lemma 5.12.

Let $C$ be a nonempty open cone in ${\mathbb{R}}^{n}$, with closure $\bar{C}$.
Then $a\in\bar{C}$ iff $a+C\subseteq C$.

###### Proof.

Pick $c\in C$. If $a+C\subseteq C$, pick $c\in C$; then $a+tc\in C$ for all
$t\in(0,1)$, so $a\in\bar{C}$. Conversely, if $a\in\bar{C}$, let $U$ be a
neighborhood of $0$ with $c+\bar{U}\subset C$; then
$a+c+\bar{U}\subset\bar{C}$; so $a+c$ lies in the interior $C$ of $\bar{C}$. ∎

######  Lemma 5.13 (Tropical Stone-Weierstrass).

Let $Z$ be a compact Hausdorff topological space. Let $A$ be a subset of the
space of continuous functions $Z\to{\mathbb{R}}$ containing the constant
function $1$, closed under addition, scalar multiplication and $\min$, and
separating points. Then $A$ is dense in $C(Z)$ in the uniform topology.

###### Proof.

Let $B$ be the uniform closure of $A$. If $a\in A$ and
$p:{\mathbb{R}}\to{\mathbb{R}}$ is a convex function of the form
$p(x)=\max_{k=1}^{n}(a_{k}x+b_{k})$, it is clear that $p\circ
a=\max_{k=1}^{n}(a_{k}a+b_{k})$ lies in $A$. It follows that $B$ has the same
closure property.

On any compact subset $W$ of ${\mathbb{R}}$, let $s_{W}:W\to{\mathbb{R}}$ be
the function $s_{W}(x)=x^{2}$; then $s_{W}$ is the uniform limit of a sequence
of piecewise linear maps $p^{j}_{W}$, obtained as the maxima of of certain
affine functions $ax+b$. If $g\in B$, letting $W=g(Z)$, we see that
$p^{j}_{W}\circ g\in B$ and hence $g^{2}=lim_{j}p^{j}_{W}\circ g$ lies in $B$.
Writing $xy=((x+y)^{2}-x^{2}-y^{2})/2$, we see that $B$ is closed under
multiplication. By the usual Stone-Weierstrass, $B=C(Z)$. ∎

##  6\. Higher-order hyperbolic spaces and Hodge index theorem

We will encounter a generalization of the basic linear algebra of Hilbert
spaces to higher order maps; we provide here an elementary setting for this.

######  Definition 6.1.

An $n$-th order hyperbolic space is an ${\mathbb{R}}$\- vector space $V$, an
open cone $A$ and a symmetric map $\cdot:A^{n}\to{\mathbb{R}}$ denoted
$(a_{1},\ldots,a_{n})$, such that:

  * •

$(a_{1},\cdots,a_{n})>0$

  * •

Positive homogeneity in each variable: for fixed $a_{2},\ldots,a_{n}$, and
$t>0$, we have $(ta_{1},a_{2},\ldots,a_{n})=t(a_{1},\ldots,a_{n})$.

  * •

For fixed $a_{2},\ldots,a_{n}$, the map $x\mapsto(x,a_{2},\ldots,a_{n})$ is
concave.

  * •

For any $a_{1},\ldots,a_{n-2},b,c\in A$, we have:

$(a_{1},\ldots,a_{n-2},b,c)^{2}\geq(a_{1},\ldots,a_{n-2},b,b)(a_{1},\ldots,a_{n-2},c,c)$

Let us write $|a|:=|(a,\cdots,a)|^{1/n}\geq 0$.

######  Example 6.2 (The bilinear, order $2$ case).

  1. (1)

Consider a real vector space $V$ with a symmetric bilinear form of signature
$(1,-1,\cdots,-1)$, or more generally $(1,-1,\cdots,-1,0,\cdots,0)$ (with
precisely one $1$ and at least one $-1$). The future cone $A$ is one of the
two components of $\\{x:(x,x)>0\\}$; say the one with $x_{1}>0$.

On $A$, Cauchy-Scwhartz takes a ‘reverse’ form, $(x,y)^{2}\geq(x,x)(y,y)$;
whereas at any fixed time, i.e. on the orthogonal complement to any $a\in A$,
we have the usual Cauchy-Scwhartz inequality.

To see that $(x,y)^{2}\geq(x,x)(y,y)$ whenever $(x,x)>0$: we may assume
$(y,y)>0$ as well; renormalizing we may take $(x,x)=1$, and $y=x+n$ with
$(x,n)=0$; as the orthogonal space to $x$ is negative definite, we have
$(n,n)\leq 0$. Now $(x,y)^{2}=1=(x,x)$ while $(y,y)=1+(n,n)\leq 1$.

  2. (2)

Conversely, let $(V,A,\cdot)$ be hyperbolic of order $2$, with $\cdot$
bilinear. Pick $a\in A$ with $|a|=1$. Consider

$H=\\{b:(a,b)=0\\}$

Then $H$ is negative-semidefinite: otherwise we can find $b\in H$ with
$(b,b)>0$ and (replacing $b$ by a small positive scalar multiple, so that $b$
is close to $0$) with $a+b\in A$, yet $(a,a+b)^{2}=1<1+(b,b)=(a,a)(a+b,a+b)$.

  3. (3)

Going back and forth, from (2) to (1) and back, we see that in (2) we may take
any $a\in H$ with $(a,a)>0$; the orthogonal complement is still negative-
semidefinite.

  4. (4)

It is sometimes convenient to formulate Cauchy-Scwharz for two classes $x,y$,
assuming $(a,a)>0$ and $y\cdot a=0$, but without any assumption on $x$. Then
by considering $x^{\prime}=x-\alpha a$, where $a\cdot x=\alpha a^{2}$, one
obtains

$(x,y)^{2}\leq(x^{2}-\alpha^{2}a^{2})y^{2}$

  5. (5)

A nondegenerate symmetric bilinear map $(\,,\,)$ is hyperbolic iff the
function $|x|=(x,x)^{1/2}$ is concave on $A=\\{x:(x,x)>0\\}$. Indeed assuming
concavity, we have $(a+b,a+b)^{1/2}\geq(a,a)^{1/2}+(b,b)^{1/2}$, or squaring,
$(a,a)+2(a,b)+(b,b)\geq(a,a)+2(a,a)^{1/2}(b,b)^{1/2}+(b,b)$ which yields
$(a,b)^{2}\geq(a,a)(b,b)$.

######  Lemma 6.3.

For $x_{1},\ldots,x_{n}\in A$ we have

$(x_{1},\ldots,x_{n})\geq|x_{1}|\cdots|x_{n}|$

More generally, for $0<k<n$,

$(x_{1},\ldots,x_{k},x_{k+1},\ldots,x_{n})^{n-k}\geq(x_{1},\ldots,x_{k},x_{k+1},\ldots,x_{k+1})\cdot\ldots\cdot(x_{1},\ldots,x_{k},x_{n},\ldots,x_{n})$

###### Proof.

By induction on $n-k$, being given the case $n-k=2$. Let us consider for
example the case $n=3$. First we have:

$(a,b,b)^{2}\geq(b,b,b)(a,a,b)$

so

$(a,b,b)^{4}\geq(b,b,b)^{2}(a,a,b)^{2}$

Plugging in the dual of the first equation, i.e.
$(a,a,b)^{2}\geq(a,a,a)(b,b,a)$, and dividing by $(a,b,b)$ we obtain:

$(a,b,b)^{3}\geq(b,b,b)^{2}(a,a,a)$

Now this in turn can be plugged into the relation

$(a,b,c)^{2}\geq(a,a,c)(b,b,c)$

to give $(a,b,c)^{6}\geq(a,a,a)^{2}(b,b,b)^{2}(c,c,c)^{2}$ and taking square
roots we obtain the required formula.

(See [23].)

∎

###  6.4. log-concavity

Let $f$ be a function on a cone in a real vector space, into the positive
reals, and assume $f(tx)=tf(x)$ for $t>0$. Then $f$ is concave iff $\log f$ is
concave. The left to right implication is clear since $\log$ is concave. For
the other, assume $\log f$ is concave; then whenever $0<\alpha,{\beta}$ and
$\alpha+{\beta}=1$, we have $\log f(\alpha x+{\beta}y)\geq\alpha\log
f(x)+{\beta}\log f(y)$; in particular when $f(x)=f(y)$ this implies that
$f(\alpha x+{\beta}y)\geq f(x)$. By positive homogeneity of $f$, for any
$\gamma>0$ we have $f(\gamma\alpha x+\gamma{\beta}y)\geq f(\gamma x)=\gamma
f(x)$; in other words for any $0<\alpha,{\beta}$, if $f(x)=f(y)$ we have
$f(\alpha x+{\beta}y)\geq(\alpha+{\beta})f(x)$. Given arbitrary $x,y$, find
$\gamma$ such that $\gamma f(x)=f(y)$; let $x^{\prime}=\gamma x$; let
$\alpha^{\prime}=\alpha/\gamma$; then $f(\alpha
x+{\beta}y)=f(\alpha^{\prime}x^{\prime}+{\beta}y)\geq(\alpha^{\prime}+{\beta})f(y)=\alpha
f(x)+{\beta}f(y)$.

####  The volume function on $A$.

We define ${\mathop{\operatorname{\rm vol\,}}}:A\to{\mathbb{R}}$ by
${\mathop{\operatorname{\rm vol\,}}}(a)=(a,\ldots,a)$.

######  Lemma 6.5.

${\mathop{\operatorname{\rm vol\,}}}^{1/n}$ is concave on $A$.

###### Proof.

Using concavity in each variable, we have

${\mathop{\operatorname{\rm
vol\,}}}(a+b)=(a+b,\cdots,a+b)\geq\sum_{k}{{n}\choose{k}}(a,\cdots,a,b,\cdots,b)$

(with $k$ instances of $a$, and $n-k$ of $b$.) By the main statement of Lemma
6.3, applied term-by-term, we obtain:

${\mathop{\operatorname{\rm
vol\,}}}(a+b)\geq\sum_{k}{{n}\choose{k}}{\mathop{\operatorname{\rm
vol\,}}}^{k/n}(a){\mathop{\operatorname{\rm
vol\,}}}^{(n-k)/n}(b)=({\mathop{\operatorname{\rm
vol\,}}}^{1/n}(a)+{\mathop{\operatorname{\rm vol\,}}}^{1/n}(b))^{n}$

or

${\mathop{\operatorname{\rm vol\,}}}^{1/n}(a+b)\geq{\mathop{\operatorname{\rm
vol\,}}}^{1/n}(a)+{\mathop{\operatorname{\rm vol\,}}}^{1/n}(b)$

Using the positively homogeneity of ${\mathop{\operatorname{\rm
vol\,}}}^{1/n}$, this gives concavity of ${\mathop{\operatorname{\rm
vol\,}}}^{1/n}$ ∎

Thus $-\log{\mathop{\operatorname{\rm vol\,}}}$ is convex on $A$.

######  Example 6.6.

Let ${\mathbb{R}}^{n}$ have basis $e_{1},\ldots,e_{n}$. Consider the symmetric
$n$-multilinear form with $(e_{1},\cdots,e_{n})=1$, and $e_{i}^{2}=0$ (meaning
$(e_{i},e_{i},x_{3},\ldots,x_{n})=0$ for all $x_{3},\ldots,x_{n}$). Let
$A=\\{\sum\alpha_{i}e_{i}:\alpha_{1},\ldots,\alpha_{n}>0\\}$. This is a
generalized hyperbolic space; if we fix any $a_{1},\ldots,a_{n-2}\in A$, then
$(x,y)\mapsto(a_{1},\ldots,a_{n-2},x,y)$ has a kernel of dimension $n-2$ and
signature $0,\cdots,0,1,-1$. Note that if $c=\sum\alpha_{i}e_{i}\in A$, then
$\frac{c^{n}}{n!}$ is the Euclidean volume of the rectangle with sides
$(\alpha_{1},\ldots,\alpha_{n})$. Thus in this case Lemma 6.5 gives the
concavity of ${\mathop{\operatorname{\rm vol\,}}}^{1/n}$ for rectangles. This
is a special case of Brunn-Minkowski.

.

###  6.7. The Hodge index theorem.

######  Theorem 6.8.

[[23] 1.6.1] Let $X$ be a projective variety of dimension $n$. Then $N^{1}(X)$
with intersection product, and the ample cone is an order-$n$ multilinear
hyperblic space.

###### Proof.

Let $a_{1},\ldots,a_{n-2},a_{n-1}=b,a_{n}=c$ be ample divisors. We have to
show that ${\mathop{\operatorname{\rm vol\,}}}(a_{1})=a_{1}\cdot\ldots\cdot
a_{1}>0$ (this is clear), concavity in each variable (we have linearity), and
that

$(a_{1}\cdot\ldots\cdot a_{n-2}\cdot b\cdot
c)^{2}\geq(a_{1},\ldots,a_{n-2},b,b)(a_{1},\ldots,a_{n-2},c,c)$

Now by Bertini, $a_{i}$ can be represented by an irreducible hypersurface
$D_{i}$, such that $D_{1}\cap\cdots\cap D_{k}$ is a transversal intersection,
resulting in an irreducible subvariety of $X$ (for $k\leq n-2$). Let
$S=D_{1}\cap\cdots\cap D_{n-2}$. Then $S$ is a surface. Let
$b^{\prime}=D_{n-1}|S,c^{\prime}=D_{n}|S$. Then
$(a_{1},\ldots,a_{n-2},b,c)=b^{\prime}\cdot_{S}c^{\prime}$ and similarly for
$(b,b)$ and $(c,c)$. Thus the inequality reduces to the case of a surface. In
this case, as noted above, it suffices to show that the signature is
$(1,-1,\cdots,-1)$.

If $\widetilde{S}\to S$ is a resolution, then $N^{1}(X)$ embeds into
$N^{1}(\widetilde{S})$ in a way that respects the intersection form, and maps
an ample class to a nef class, so that it respects the closure of the ample
cone, which suffices for the weak inequality. Thus it suffices to show that
$\widetilde{S}$ has hyperbolic signature. This is the classical Hodge index
theorem for surfaces, cf. [19]. If one uses the Okounkov approach, Corollary
8.8, this requires only the 2-dimensional Brunn-Minkowski. ∎

######  Remark 6.9.

When $X$ is a smooth projective variety of dimension $n$, $V=N^{1}(X)$, $A=$
the ample cone, the $n$-fold intersection pairing on divisors is
nondegenerate, i.e. whenever $v\in V$ and $(a_{1},\ldots,a_{n-1},v)=0$ for all
$a_{1},\ldots,a_{n-1}\in A$, then $v=0$.

Note that the pairing $N^{1}(X)\times N_{1}(X)\to{\mathbb{R}}$ is
nondegenerate by definition; thus $\dim(N^{1}(X))=\dim N_{1}(X)$, and the
content of the above statement is that every class in $N_{1}(X)$ is in the
image of $\cdot^{n-1}$.

This can be seen as a consequence of the Hard Lefschetz theorem for algebraic
classes, asserting that even for a fixed ample $a$, the map
$(u,v)\mapsto(a,\cdots,a,u,v)$ is nondegenerate. This can be interpreted as
the Hessian of volume. We will see a cone-adapted version of this further
down.

######  Example 6.10.

Let $S$ be a smooth projective surface admitting morphisms $\pi_{i}:S\to
C_{i}$ into curves, mapping onto $C_{1}\times C_{2}$. Let $P_{i}$ be the
pullback of a point under $\pi_{i}$; then $A=P_{1}+P_{2}$ is ample on $S$. The
Hodge index theorem includes, and is easily seen to be equivalent to, the
negative definiteness of the intersection form on $V$, the orthogonal space to
$P_{1}$ and $P_{2}$. For any divisor $D$, we have $D^{*}:=D-(D\cdot
P_{1})P_{2}-(D\cdot P_{2})P_{1}$ in $V$. Thus the Hodge index theorem asserts
that $(D^{*},D^{*})\leq 0$. Explicitly, this reads:

$(D,D)\leq 2(D\cdot P_{1})(D\cdot P_{2})$

This is the Castelnuovo inequality [9]. Weil developed the foundations of
algebraic geometry between 1940-1948 precisely in order to provide a statement
and proof of this inequality valid in positive characteristic.

######  Example 6.11.

Here is Weil’s 1941 deduction of the Riemann hypothesis for curves from
Castelnuovo, or as we will present it, directly from the Hodge index theorem
for a product of curves. Let $C$ be a curve over ${\mathbb{F}}_{q}$,
$\Delta\leq C\times C$ the diagonal, $\phi:C\to C$ the $q$-Frobenius morphism,
$\Phi=\Phi_{q}$ the graph of the $\phi$. Let ∗ be the operator of Example
6.10. By Example 6.2 we have

$(\Phi^{*}\cdot\Delta^{*})^{2}\leq(\Phi^{*},\Phi^{*})(\Delta^{*},\Delta^{*})$

Here $(\Delta^{*},\Delta^{*})$ does not depend on $q$;
$(\Phi,\Phi)=q(\Delta,\Delta)$ using $\Phi=(\phi,Id)^{*}\Delta$;
$\Phi\cdot\Delta$ is the number of points of $C({\mathbb{F}}_{q})$ that we
seek; all other terms are easily evaluated, yielding Weil’s theorem,
$|C({\mathbb{F}}_{q})|=q+O(q^{1/2})$.

######  Lemma 6.12.

Let $S$ be a irreducible projective surface. Let $\widetilde{S}\to S$ be a
surjective, generically finite morphism, with $\widetilde{S}$ smooth. Let $e$
be an irreducible curve on $S$, and let $A,D\in N^{1}(S)$. Assume $D\cdot A=0$
while $(A,A)=1$. Then

$(e\cdot D)^{2}\leq(\tilde{e}^{2}-(A\cdot e)^{2})D^{2}$

where $\tilde{e}$ is any weighted curve on $\widetilde{S}$ lifting $e$. In
particular if $D^{2}=0$ then $e\cdot D=0$.

###### Proof.

If we replace $D,A$ by their pullbacks, and $e$ by $\tilde{e}$, the same
conditions hold and all the quantities in the displayed inequality are the
same, by the projection formula ([12] 1.9,1.10). Thus we may assume $S$ is
smooth and connected, and $e=\tilde{e}$. By Remark 6.2 (4) we have $(e\cdot
D)^{2}\leq(e^{2}-(A\cdot e)^{2})D^{2}$. ∎

The material from here to the end of the section is adapted from [27] (part of
Theorem 1.1; they acknowledge an idea of Blocki’s). We begin with a relative
Hodge index theorem. (In case $\pi^{*}(c)$ is represented by a subvariety of
$X$, it is easy to deduce it from the Hodge index theorem on a plane section
of that subvariety. For a surface mapping to a curve, let $l$ be an ample
divisor with $(f,l)=1$, and note that $v_{i}\mapsto v_{i}-(v_{i},l)f$ is an
isometry into the space orthogonal to $l$, with kernel ${\mathbb{R}}f$. But we
give a direct proof.)

######  Proposition 6.13.

Let $X,U$ be smooth varieties with a morphism $\pi:X\to U$, of relative
dimension $n$. Let $D$ be an effective divisor of $U$; and let $c\in N_{1}(U)$
be a movable class, represented by an effective curve $C$ disjoint from any
specified subvariety of $U$ of codimension $>1$. Let $V$ be the subspace of
$N^{1}(X)$ generated by the classes of the irreducible components of
$\pi^{-1}(D)$. Then for nef $a_{1},\ldots,a_{n-1}$, the pairing
$V^{2}\to{\mathbb{R}}$,

$(v,v^{\prime})=a_{1}\cdot\ldots\cdot a_{n-1}\cdot v\cdot
v^{\prime}\cdot\pi^{*}(c)$

is negative semidefinite.

###### Proof.

By approximation, we may take the $a_{i}$ to be ample, with rational
coefficients; multiplying by an integer, we may take them integral, and very
ample. Let $v_{1},\ldots,v_{k}$ be the distinct irreducible components of
$\pi^{-1}(D)$. Let $D^{\prime}=\sum\beta_{j}D_{j}$ be a divisor on $U$,
numerically equivalent to $D$, such that $D_{j}\cap D$ has codimension $\geq
2$ in $U$ for any $j$. (We use smoothness of $U$ to move $D$, to a not
necessarily effective divisor.) Choose a representative $C$ of $c$ which is
disjoint from $D\cap D_{j}$, for any $j$. Let $\alpha_{i}$ be the multiplicity
of $v_{i}$ in $\pi^{-1}(D)$; then $f=\sum\alpha_{i}v_{i}$ is numerically
equivalent to $\pi^{-1}(D^{\prime})$. Now
$v_{i}\cap\pi^{-1}(D_{j})\cap\pi^{-1}(C)=\emptyset$ for each $j$. Thus (see §
4.18) $v_{i}\cdot\pi^{*}(D^{\prime})\cdot\pi^{*}(c)=0$; so $v_{i}\cdot
f\cdot\pi^{*}(c)=0$, hence $(v_{i},f)=0$. For $i\neq j$, $v_{i}$ meets $v_{j}$
properly in an effective cycle; thus $a_{1}\cdot\ldots\cdot a_{n-1}\cdot
v_{i}\cdot v_{j}$ is effective, hence so is $\pi_{*}(a_{1}\cdot\ldots\cdot
a_{n-1}\cdot v_{i}\cdot v_{j})$; by the moveability property of $c$, it
follows that $c\cdot\pi_{*}(a_{1}\cdot\ldots\cdot a_{n-1}\cdot v_{i}\cdot
v_{j})\geq 0$; so $(v_{i},v_{j})=a_{1}\cdot\ldots\cdot a_{n-1}\cdot v_{i}\cdot
v_{j}\cdot\pi^{*}(c)\geq 0$. Thus Lemma 6.14 applies. ∎

######  Lemma 6.14.

Let $V$ be a vector space over ${\mathbb{R}}$ with a symmetric bilinear form.
Assume $V$ is generated by vectors $v_{1},\ldots,v_{n}$,

$f=\sum_{i=1}^{n}\alpha_{i}v_{i}$

for some $\alpha_{1},\ldots,\alpha_{n}>0$, and:

$v_{j}\cdot f=0,v_{i}\cdot v_{j}\geq 0$

for $i\neq j\leq n$. Then $V$ is negative semidefinite. The kernel
$\\{x:(\forall y)x\cdot y=0\\}$ is generated by the vectors $\sum_{i\in
c}\alpha_{i}v_{i}$, where $c$ is a connected component of the graph whose
edges are the pairs $(i,j)$ with $v_{i}\cdot v_{j}\neq 0$.

###### Proof.

Replacing $v_{i}$ by $\alpha_{i}v_{i}$, we may assume $f=\sum_{i}v_{i}$. Let
$b_{i}\in{\mathbb{R}},i=1,\ldots,n$, $g=\sum b_{i}v_{i}$; we have to show that
$g^{2}\leq 0$. As $v_{j}\cdot f=0$ we have also
$b_{j}^{2}v_{j}(\sum_{i}v_{i})=0$; summing over $j$ we find that

$\sum_{i,j}b_{j}^{2}v_{i}v_{j}=0$

As $v_{i}v_{j}\geq 0$ for $i\neq j$, we have:

$g^{2}=-\frac{1}{2}\sum_{i,j=1}^{l}(b_{i}-b_{j})^{2}v_{i}v_{j}=-\sum_{i\neq
j}(b_{i}-b_{j})^{2}v_{i}v_{j}\leq 0$

If equality holds, then $b_{i}=b_{j}$ whenever $v_{i}v_{j}\neq 0$, i.e. the
coefficients are constant on each connected component. (And the converse is
clear.) ∎

######  Remark 6.15.

This also shows (assuming $\pi^{-1}(D)$ is connected) that there are no
unexpected linear relations among the classes of the $v_{i}$ in
$N^{1}(X)/\pi^{*}(N^{1}(U))$ . In fact, assuming for simplicity that
$\cup_{i}v_{i}$ is connected, there are no unexpected linear inequalities
among the classes $[v_{i}]$.

In the disconnected case, say when $D=D^{\prime}\cup D^{\prime\prime}$ with
$D^{\prime},D^{\prime\prime}$ disjoint, connected, and numericaly equivalent
in $N_{1}(U)$, the pullback to $\oplus_{i}{\mathbb{R}}v_{i}$ of the effective
cone is generated by the $v_{i}$ themselves, and in addition by expressions
$\sum m_{i}^{\prime}v_{i}^{\prime}-\sum m_{i}^{\prime\prime}v_{i}$, the
difference of the irreducible component decompositions of
$\pi^{-1}(D^{\prime}),\pi^{-1}(D^{\prime\prime})$.

Suppose $[\sum\alpha_{i}v_{i}]=[e]$ represented by an effective divisor $e$ on
$X$; we have to show all $\alpha_{i}\geq 0$. We may assume $e$ does not
include any of the $v_{i}$ as components, or else we may subtract. Let
$J^{+}=\\{j:\alpha_{j}\geq 0$, $J^{-}=\\{j:\alpha_{j}<0\\}$,
$u=\sum\max(0,\alpha_{i})[v_{i}]=\sum_{j\in J^{+}}\alpha_{j}[v_{j}]$. Then
$(u,u)\leq 0$ by negative semidefiniteness. However $(u,e)\geq 0$, and
$(u,v_{j})\geq 0$ for $j$ with $\alpha_{j}<0$, since $e,v_{j}$ are effective
and these intersections are proper. This implies that $(u,\alpha_{j}v_{j})\leq
0$ for $j\in J^{=}$, and hence $(u,u)=(u,e)-\sum\geq 0$, so $(u,u)=0$; in fact
by connectedness of the graph and the description of the kernel of
$(\cdot,\cdot)$, for $j\in J^{+}$, $\alpha_{j}$ must be proportional to the
multiplicity of $u_{j}$ in $\pi^{-1}(D)$; so either all $\alpha_{j}\geq 0$, or
all $\alpha_{j}\leq 0$. In the latter case, pushing forward to $U$ we find
that a negative multiple of $D$ is effective, a contradiction.

###  6.16. Calabi uniqueness

######  Lemma 6.17.

Let $W$ be a real vector space endowed with a symmetric, nondegenerate
$n+1$-multilinear map $\cdot$. Let $V$ be a subspace of $W$, $A$ a subset of a
coset of $V$. For $a=(a_{1},\ldots,a_{n-1})\in A$, let
$b_{a}(v,v^{\prime})=a_{1}\cdot\ldots\cdot a_{n-1}\cdot v\cdot v^{\prime}$.
Assume each $b_{a}$ is negative semidefinite on $V$ ($a\in A$.)

Let $a_{1},a_{2}\in A$, and assume $v\cdot a_{1}^{n}=v\cdot a_{2}^{n}$ for all
$v\in V$ (or even just for $v=a_{1}-a_{2}$). Then $a_{1}-a_{2}$ lies in the
kernel of $b_{a}$ for any $a\in A^{n-1}$.

###### Proof.

Let $c=a_{1}-a_{2}$. Consider first the case $n=2$.

We assume $ca_{1}^{2}=ca_{2}^{2}$. So

$0=c(a_{1}^{2}-a_{2}^{2})=c^{2}(a_{1}+a_{2})$

But $c^{2}a_{i}\leq 0$, so $c^{2}a_{1}=c^{2}a_{2}=0$.

By Cauchy-Schwarz (for $b_{a_{i}}$), we have $vca_{i}=0$ for any $v\in V$.

Hence $c^{2}v=c(a_{1}-a_{2})v=0$ for any $v\in V$. As $c^{2}a_{1}=0$, we have
$c^{2}x=0$ for $x\in a_{1}+V$ and in particular $c^{2}a=0$ for all $a\in A$.
Again by negative semidefiniteness, $c$ lies in the kernel of $b_{a}$.

Now assume $n>2$. We have by assumption:

$ca_{1}^{n}=ca_{2}^{n}$

So

$0=c(a_{1}^{n}-a_{2}^{n})=c(a_{1}-a_{2})(a_{1}^{n-1}+\cdots
a_{1}^{i}a_{2}^{n-1-i}+\cdots a_{2}^{n-1})$

But $c^{2}a_{1}^{i}a_{2}^{n-1-i}\leq 0$; so each term in the sum must vanish,
in particular $c^{2}a_{i}^{n-1}=0$.

By Cauchy-Schwarz we have $vca_{i}^{n-1}=0$ for any $v\in V$. As
$a_{i}ca_{i}^{n-1}=0$, we hve $eca_{i}^{n-1}=0$ for any $e\in A$. Thus fixing
any $e\in A$, the $n$-multilinear map $x_{1}\cdot\ldots\cdot x_{n-1}\cdot e$
satisfies the same conditions. By induction, $c$ lies in the kernel of
$b_{e_{1},\ldots,e_{n-2},e}$ for any $e_{1},\ldots,e_{n-2}\in A$, which
finishes the proof.

∎

##  7\.  Volume via section growth

So far, we defined volume for ample divisors via the intersection product. Our
official definition of volume will instead use the asymptotic dimension of the
space of sections of multiples of the divisor. This definition is also more
general; an equally general definition in terms of positive intersections will
be considered later (§ 10.3), see [23], 11.4.10, 11.4.11. We begin by relating
the section growth with the intersection theory definitions for ample
divisors.

If $X$ is $0$-dimensional scheme, let $|X|$ be the sum over all points of $X$
of the dimension of the local ring.

If $X$ is closed subvariety of ${\mathbb{P}}^{N}$, of dimension $d$, we define

$\deg(X)=|X\cap L_{1}\cap\cdots\cap L_{N-d}|$

where $L_{i}$ are hyperplanes chosen generically enough, so that $\dim(X\cap
L_{1}\cap\cdots\cap L_{N-d})=0$. (It will follow from Lemma 7.1 that the
choice of the $L_{i}$ is irrelevant.)

On the other hand, let $A=\oplus H_{n}$ be the graded ring
$k[X_{0},\ldots,X_{N}]$; $H_{n}$ the homogeneous polynomials of degree $n$.
For a homogeneous ideal $I$, say $vol(I)=v$ if $\dim(H_{n}/H_{n}\cap
I)=vm^{d}/d!+O(m^{d-1})$.

######  Lemma 7.1.

Let $X=Proj(A/I)$. Then $\deg(X),{\mathop{\operatorname{\rm vol\,}}}(I)$ are
both defined and are equal. In fact, for large enough $n$,
$\dim(H_{n}/H_{n}\cap I)=p(n)$ for some polynomial $p(n)$ of degree $d$, with
leading term equal to $\deg(X)n^{d}/d!$.

###### Proof.

The case $\dim(X)=0$ follows easily from the definitions. Assume the result
known below $\dim(X)$, and choose any $x\in H_{1}$ such that $x$ is not a
0-divisor in $A/I$. Let $I_{n}=I\cap H_{n}$, $I^{\prime}=I+xA$,
$I^{\prime}_{n}=I^{\prime}\cap H_{n}$, $X^{\prime}=Proj(I^{\prime})$. Note
that $(I+xA)\cap H_{n}=I_{n}+xH_{n-1}$ since the ideal $I$ is homogeneous.
Thus

$\dim(I^{\prime}_{n}/I_{n})=\dim(I_{n}+xH_{n-1}/I_{n})=\dim(xH_{n-1}/I_{n}\cap
xH_{n-1})=\dim(H_{n-1}/I_{n-1})$

using in the last step the assumption that $x$ is not a 0-divisor in $A/I$. So

$\dim H_{n}/I^{\prime}_{n}=\dim H_{n}/I_{n}-\dim H_{n-1}/I_{n-1}$

Let $f(n)=\dim H_{n}/I_{n}$. Then $\dim H_{n}/I^{\prime}_{n}=f(n)-f(n-1)$. By
induction, as $\dim(X^{\prime})<\dim(X)$, for large $n$ we have $\dim
H_{n}/I^{\prime}_{n}=q(n)$ with $q$ a polynomial in $n$ of degree $d-1$, and
leading term $\deg(X^{\prime})=\deg(X)$. We can find a polynomial $p(n)$ of
degree $d$, with the same leading term $\deg(X)$, and with $p(n)-p(n-1)=q(n)$.
Then $p-f$ is eventually constant; adjusting by this constant we may assume
$p-f$ is eventually $0$, i.e. $p(n)=f(n)$ for large $n$. ∎

Let $X$ be a variety of dimension $d$. We define the volume of an integral
divisor $D$ to be

$vol(D)=d!\limsup_{m}\dim L(X,D^{m})/m^{d}$

In particular ${\mathop{\operatorname{\rm vol\,}}}(D)=v$ if $\dim
H^{0}(X,D^{m})=v{{m}\choose{d}}+O(m^{d-1})$.

$D$ is big if it has nonzero volume.

######  Corollary 7.2.

If $D$ is very ample, then ${\mathop{\operatorname{\rm vol\,}}}(D)=D^{d}$.
Hence for any $D^{\prime}$, $L(X,mD^{\prime})=O(m^{\dim(X)})$.

###### Proof.

Embed $X$ into projective space ${\mathbb{P}}^{N}$ via $D$. Then $D^{d}$
equals the intersection number of $X$ with $d$ generically chosen hyperplanes
on ${\mathbb{P}}^{N}$, and the statement follows from Lemma 7.1. The second
statement follows by choosing an ample $D\geq D^{\prime}$. ∎

######  Proposition 7.3 ([23] 2.2.15).

Let $A,B$ be ample Cartier divisors on $X$. Then

${\mathop{\operatorname{\rm vol\,}}}(A-B)\geq{\mathop{\operatorname{\rm
vol\,}}}(A)-n(A^{n-1}B)$

###### Proof.

We may assume $A,B$ are very ample. Fix a countable base field of definition
$k_{0}$ for the data. Using Bertini, let $E=E_{1},E_{2},\ldots$ be irreducible
hypersurfaces of $X$, $E_{i}$ defined over some extension $k_{i}$ of $k_{0}$,
such that $(k_{i},X,E_{i})$ are all isomorphic over $k_{0}$, and $E_{i}$
represents $B$ as a Weil divisor. Let $A_{i}=A|E_{i}$; then the $n-1$-fold
intersection product of $A_{i}$ on $E_{i}$ equals $A^{n-1}\cdot B$; by Lemma
7.1 applied to $A|E$ on $E$,

$h^{0}(E_{i},mA_{i})=h^{0}(E,mA|E)=A^{n-1}\cdot Bm^{n-1}/(n-1)!+O(m^{n-2})$

Again by Lemma 7.1 for $A$ on $X$,

$h^{0}(X,m(A-B))=A^{n}m^{n}/n!+O(m^{n-1})$

We have an exact sequence

$0\to H^{0}(X,mA-mB)\to H^{0}(X,mA)\to\oplus_{i\leq m}H^{0}(E_{i},mA_{i})$

Thus

$h^{0}(X,m(A-B))\geq h^{0}(X,mA)-\sum_{i\leq
m}h^{0}(E_{i},mD|E_{i})=({\mathop{\operatorname{\rm
vol\,}}}(A)-n(A^{n-1}B)){\frac{m^{n}}{n!}}+O(m^{n-1})$

∎

######  Remark 7.4 (Continuity of volume).

If we assume only that $B$ is ample, with $B$ represented by $E$ as above, the
proof of Proposition 7.3 still yields the inequality in the form:
${\mathop{\operatorname{\rm vol\,}}}(A-B)\geq{\mathop{\operatorname{\rm
vol\,}}}(A)-n{\mathop{\operatorname{\rm vol\,}}}_{E}(A|E)$. Thus if
$\beta=1/b$, we have $|{\mathop{\operatorname{\rm vol\,}}}(A-\beta
B)-{\mathop{\operatorname{\rm vol\,}}}(A)|=b^{-n}|{\mathop{\operatorname{\rm
vol\,}}}(bA-B)-{\mathop{\operatorname{\rm vol\,}}}(bA)|\leq
b^{-n}(nb^{n-1}{\mathop{\operatorname{\rm
vol\,}}}_{E}(A|E))=nb^{-1}{\mathop{\operatorname{\rm vol\,}}}_{E}(A|E)$, so
$|{\mathop{\operatorname{\rm vol\,}}}(A-\beta B)-{\mathop{\operatorname{\rm
vol\,}}}(A)|\leq n\beta{\mathop{\operatorname{\rm vol\,}}}_{E}(A|E)$. By
choosing $n$ linearly independent very ample divisors, we see in this way that
the volume function is continuous on $N^{1}(X)$; see [23] 2.2.44.

######  Remark 7.5.

From the continuity of volume Remark 7.4, it is clear e.g. that Proposition
7.3 is valid for nef and not only ample divisors.

######  Definition 7.6.

The open cone of elements of $N^{1}(X)$ of positive volume is called the big
cone.

A big integral divisor clearly has an effective multiple, so the big cone is
contained in the interior of the pseudo-effective cone. Conversely if $D$ lies
in the interior of the effective cone, equivalently by Lemma 5.6 of the
pseudo-effective cone, it can be written as ample + effective. (Indeed for any
ample $Q$, for some $\epsilon>0$, $D-\epsilon Q$ is effective.) So some $mD$
lies above a very ample divisor, and hence has at least as many sections; thus
$mD$ and hence $D$ are big. We denote the big cone, interior of the pesudo-
effective cone, by $N^{1}_{+}(X)^{o}$. In particular ample classes lie in
$N^{1}_{+}(X)^{o}$.

The following corollary of Proposition 7.3, a lower bound on volume near a nef
divisor, is used in [3] to prove differentiability of volume on the big cone.
The point is that the constants in front of the quadratic error term $t^{2}$
do not change upon blowing up $X$ or diminishing $\beta$.

######  Corollary 7.7.

[3] Let $X$ be a projective $n$-dimensional variety. Let $\beta\in N^{1}(X)$
be nef, and $\gamma\in N^{1}(X)$. Let $\omega$ be nef and big, and assume
$\beta\leq\omega$ and $\omega\pm\gamma$ is nef. Then for any $t\in[-1,1]$ we
have

${\mathop{\operatorname{\rm
vol\,}}}(\beta+t\gamma)\geq\beta^{n}+nt\beta^{n-1}\cdot\gamma-8^{n}\omega^{n}t^{2}$

###### Proof.

As the inequality for $t$ and $\gamma$ is the same as for $-t$ and $-\gamma$,
we may assume $t\in[0,1]$. Note first that with
$\gamma^{\prime}=\omega-\gamma$, we have $\gamma^{\prime}\leq 2\omega$, so

$\beta^{i}\gamma^{n-i}=\beta^{i}(\omega-\gamma^{\prime})^{n-i}\geq\beta^{i}(\omega-2\omega)^{n-i}\geq-\beta^{i}\omega^{n-i}\geq-\omega^{n}$

Thus (using $|t|\leq 1$)

$(\beta+t\gamma)^{n}-\beta^{n}-nt\beta^{n-1}\cdot\gamma=\sum_{i\geq
2}{{n}\choose{i}}t^{i}\beta^{i}\gamma^{n-i}\geq-\omega^{n}t^{2}\sum_{i\geq
2}{{n}\choose{i}}\geq-\omega^{n}2^{n}t^{2}$

Now let $A=\beta+t(\gamma+\omega)$ and $B=t\omega$. Then $A\leq 3\omega$ so

$(A-B)^{n}=A^{n}-nA^{n-1}B+\sum_{i\geq
2}\pm{{n}\choose{i}}t^{i}A^{n-i}\omega^{i}\leq
A^{n}-nA^{n-1}B+t^{2}(3\omega)^{n}2^{n}$

But $(A-B)=\beta+t\gamma$, so combining the last two displayed equations,

$A^{n}-nA^{n-1}B\geq\beta^{n}+nt\beta^{n-1}\gamma-8^{n}\omega^{n}t^{2}$

By Proposition 7.3 and Remark 7.5, we have

${\mathop{\operatorname{\rm vol\,}}}(\beta+t\gamma)={\mathop{\operatorname{\rm
vol\,}}}(A-B)\geq\beta^{n}+nt\beta^{n-1}\gamma-8^{n}\omega^{n}t^{2}$

∎

We include here a basic lemma on the effect of morphisms on the pseudo-
effective cone of divisors, and its dual.

######  Lemma 7.8.

Let $\pi:X\to U$ be a surjective morphism of projective varieties, $U$ normal.
Then

  1. (1)

$(\pi^{*})^{-1}N^{1}_{+}{X}=N^{1}_{+}{U}$.

  2. (2)

If $\dim(X)>\dim(U)$, then $\pi^{*}(N^{1}_{+}(U))$ lies on the boundary of
$N^{1}_{+}{X}$; and only then.

  3. (3)

$\pi_{*}N_{1}^{+}(X)=N_{1}^{+}(U)$, and
$\pi_{*}N_{1}^{+}(X)^{o}=N_{1}^{+}(U)^{o}$.

  4. (4)

$\pi_{*}N_{1}^{eff}(X)=N_{1}^{eff}(U)$, and
$\pi_{*}N_{1}^{eff}(X)^{o}=N_{1}^{eff}(U)^{o}$.

  5. (5)

If $U$ is normal and $D$ is a Cartier divisor on $U$, then
${\mathcal{O}}(X,\pi^{*}D)\cong{\mathcal{O}}(U,D)$.

###### Proof.

We first prove (4). If $C$ is an irreducible curve on $U$, then it is the
image under $\pi$ of an irreducible curve $C^{\prime}$ on $X$, and we have
$[C]=d\pi_{*}[C^{\prime}]$ for an appropriate $d>0$. It follows that
$\pi_{*}N_{1}^{eff}(X)=N_{1}^{eff}(U)$. In particular $\pi_{*}:N_{1}(X)\to
N_{1}(U)$ is surjective, and hence (being linear) open. In particular
$\pi_{*}N_{1}^{eff}(X)^{o}\subset N_{1}^{eff}(U)^{o}$. By continuity
$(\pi_{*})^{-1}N_{1}^{eff}(U)^{o}\subset\pi_{*}N_{1}^{eff}(X)^{o}$ and so
$\pi_{*}N_{1}^{eff}(X)^{o}=N_{1}^{eff}(U)^{o}$.

As for (5), consider a rational function $h$ on $X$ which is regular on
$X\smallsetminus\pi^{*}D$. Let $h$ be a section of $\pi^{*}D$. Then $h$ has no
zeroes or poles on a fiber of $X\to U$ above $U\smallsetminus D$, so it must
be constant on the fibers of $\pi$ above $U\smallsetminus supp(D)$. So $h$
equals a rational function on $U$. Moreover $\pi^{*}(D+(h))\geq 0$ so
$D+(h)\geq 0$ \- any poles of $D+(h)$ as a Weil divisor would show up in the
pullback.

Towards (1), consider Cartier divisors $D$ on $U$ (with integer coefficients).
If $D$ is effective, it is clear that $\pi^{*}D$ is effective. Going to
rational coefficients and by continuity of $\pi$, it follows that
$(\pi^{*})^{-1}N^{1}_{+}{X}\subseteq N^{1}_{+}{U}$.

If $D\in N^{1}(U)\smallsetminus N^{1}_{+}{U}$, then $D$ is represented say as
$\sum\alpha_{i}D_{i}$ with $\alpha_{i}\in{\mathbb{R}}$ and $D_{i}$ a Cartier
divisor, and (by Corollary 10.10) there exists a curve $C$ on $U$ with proper
intersection with each $D_{i}$, and with $\sum\alpha_{i}D_{i}\cdot C<0$. By
the proof of (4) we may take $C=\pi_{*}C^{\prime}$ for some curve $C^{\prime}$
on $X$. It follows that $C^{\prime}$ is not contained in any
$D^{\prime}_{i}=\pi^{*}D_{i}$, and that $\sum\alpha_{i}D^{\prime}_{i}\cdot
C^{\prime}=\sum\alpha_{i}D_{i}\cdot C_{i}<0$. Thus $\pi^{*}D\notin
N^{1}_{+}{X}$. This finishes the proof of (1).

A more elementary proof of (1) can be obtained as follows. We may find a
common generically finite cover of $X$ and of $U\times{\mathbb{P}}^{m}$, for
some $m$. As the pullback to $U\times{\mathbb{P}}^{m}$ preserves non-
effectiveness, we may assume we have a generically finite cover.

Let $D\in N^{1}(U)\smallsetminus N^{1}_{+}(U)$. Then $D$ is a positive real
multiple of some Cartier divisors, and we may take $D$ to be a Cartier
divisor, no integer power of which is effective. Now if a Cartier divisor $D$
on a normal variety $U$ is not effective, the pullback is not effective
either. Indeed on some affine neighborhood $Spec(A)$, $D$ is represented by
$(f)$ with $f\notin A$. By normality, $f$ is not integral over $A$ so there
exists a valuation $v$ of the function field $K$, non-negative on $A$ but with
$v(f)<0$. The valuation $v$ extends to the function field of $X$, and has
nonempty center on $X$ since $X$ is projective. Around any point of this
center, $f$ cannot be a regular function; so the pullback of $D$ is not
effective. This persists with ${\mathbb{Q}}$-coefficients. So the pullback of
the effective cone of divisors on $X$ is the effective cone of divisors on
$U$.

For (2), suppose $\pi^{*}(N^{1}_{+}(U))$ meets the interior of $N^{1}_{+}{X}$.
Then by density there exists a ${\mathbb{Q}}$-Cartier divisor $D$ with
$\pi^{*}(D)$ big, and multiplying by an integer we may assume $D$ is a Cartier
divisor. But by (5), ${\mathcal{O}}(X,\pi^{*}D)\cong{\mathcal{O}}(U,D)$. Thus
$\pi^{*}D^{n}$ has a dimension $O(n^{\dim(U)})$, and cannnot be big if
$\dim(U)<\dim(X)$. Conversely, if $\dim(X)=\dim(U)$ then the pullback of an
ample divisor of $U$ is big on $X$, and thus lies in the interior of
$N^{1}_{+}{X}$ and not on the boundary. This proves (2).

For (3), Let $c\in N_{1}^{+}(X)$. Then for any $d\in N^{1}_{+}(U)$, we have
$\pi^{*}(d)\in N^{1}_{+}(X)$, so $\pi_{*}c\cdot
d=\pi_{*}(c\cdot\pi^{*}(d))\geq 0$. Thus $\pi_{*}N_{1}^{+}(X)\subseteq
N_{1}^{+}(U)$. By the M. Riesz theorem, since
$(\pi^{*})^{-1}N^{1}_{+}{X}=N^{1}_{+}{U}$, any positive linear map on
$N^{1}_{+}{U}$ extends to one on $N^{1}_{+}{X}$; in other words,
$\pi_{*}N_{1}^{+}(X)=N_{1}^{+}(U)$. The rest of (3) follows from Lemma 5.8 and
Lemma 5.6.

∎

##  8\. Okounkov bodies

###  8.1. Valuations of maximal rank

Let $K/k$ be a field extension with $tr.deg._{k}K=d$. A maximal rank valuation
$v$ on $K/k$ is a valuation on $K/k$ with value group $\Lambda=v(K^{*})$ of
rank $d$; i.e. $E_{v}:={\mathbb{R}}{\otimes}\Lambda$ is a $d$-dimensional real
vector space. Later, we will use the Lebesgue (or Haar) measure on $E_{v}$,
normalized so that $\Lambda$ has covolume $1$.

When $K/k$ is finitely generated, a maximal rank valuation necessarily has
finitely generated value group, isomorphic to ${\mathbb{Z}}^{d}$; though we
will not need this fact here.

We note that any field extension $K/k$ of finite transcendence degree $d$
admits a maximal rank valuation: we can embed $K$ into the algebraic closure
of $k[X_{1},\ldots,X_{d}]$, and extend the lexicographic valuation
$v(\sum\alpha_{\nu}X^{\nu})=\min\\{\nu:\alpha_{\nu}\neq 0\\}$. This gives a
valuation on $K$ with value group a finite extension of ${\mathbb{Z}}^{d}$, so
it must also be isomorphic to ${\mathbb{Z}}^{d}$.

######  Lemma 8.2.

Let $v$ be a valuation on $K/k$. Let $H$ be a finite-dimensional $k$-subspace
of $K$, of dimension $e$. Then $v(H\smallsetminus(0))$ is a set of cardinality
at most $e$. If $v$ has residue field $k$, in particular if $k=k^{alg}$ and
$v$ has maximal rank, then $|v(H\smallsetminus(0))|=e$.

###### Proof.

For $\gamma\in v(H\smallsetminus(0))$, let $H_{\gamma}=\\{h\in
H:v(h)\geq\gamma\\}$. Then $H_{\gamma}$ is a nonzero subspace of $H$; if
$\gamma^{\prime}<\gamma\in v(H\smallsetminus(0))$ then $H_{\gamma^{\prime}}$
contains $H_{\gamma}$ properly. Thus $\gamma\mapsto\dim H_{\gamma}$ is an
injective map from $v(H\smallsetminus(0))$ to $\\{1,\ldots,e\\}$.

Now assume $v$ has residue field $k$. Pick $\gamma\in\Gamma$, and let
$\gamma^{\prime}$ be the least element of $v(H)$ greater than $\gamma$. Pick
$h_{\gamma}\in H$ with $v(h_{\gamma})=\gamma$. Define $f:H_{\gamma}\to k$ by
$f(x)={\mathop{\operatorname{\rm res}}}(x/h_{\gamma})$. Then $f$ is
$k$-linear, with kernel $H_{\gamma^{\prime}}$. This shows that
$H_{\gamma}/H_{\gamma^{\prime}}$ is $1$-dimensional. So $e=\dim(H)$. ∎

###  8.3. Okounkov bodies ( [25] )

To define Okounkov bodies, we will need the following data: a function field
$K/k$, i.e. a finitely generated extension field of $k$, of transcendence
degree $d$; a maximal rank valuation $v$ on $K/k$, and a $k$-subalgebra $S$ of
$K$ generating $K$ as a field, along with a filtration $S=\cup_{n}S_{n}$, with
$1\in S_{1}$ and $\dim(S_{n})=O(n^{d})$ (i.e. for some $C\in{\mathbb{R}}$, for
$n\geq 1$ we have $\dim(S_{n})\leq Cn^{d}$.)

We will view $v$ as auxiliary and omit it from the notation; we thus write
$Ok(S)$.

Typically, $S_{n}$ will be $L(X,nD)\cong H^{0}(X,nD)$, where $X$ is a $k$\-
variety, $D$ an effective divisor on $X$ (so $1\in S_{1}$). In this case we
write $Ok(X,D)$ for the Okounkov body.

######  Definition 8.4.

Let $V_{n}=\\{v(x):x\in S_{n}\smallsetminus(0)\\}$, $V=\cup_{n}V_{n}$. $Ok(S)$
is the closed convex hull in $E_{v}$ of $\cup_{n}\frac{V_{n}}{n}$. We will
also write $Ok(V)$ for the same set.

We could also define $Ok(S)$ to be the closure of $\cup_{n}\frac{V_{n}}{n}$,
since this set is closed under the operations $(u,v)\mapsto\alpha
u+(1-\alpha)v$ for rational $\alpha\in(0,1)$.

Since $S$ generates $K$, there exist $x_{1},\ldots,x_{n}$ with $v(x_{i})$
linearly independent; the convex hull $[0,v(x_{1}),\cdots,v(x_{n})]$ of
$0=v(1),v(x_{1}),\ldots,v(x_{n})$ is an $n$-simplex, so $Ok(S)$ has nonempty
interior $S^{o}$.

We let $\mu_{n}$ be the counting measure on $\frac{V_{n}}{n}$, normalized so
that each point has measure $n^{-d}$. This can be compared to
$\mu^{\prime}_{n}$, the (infinite) counting measure on all of $\Lambda/n$,
with the same normalization.

If $S^{\prime}$ is a $k$-subalgebra of $S$, $S^{\prime}_{n}:=S_{n}\cap
S^{\prime}$, then $V^{\prime}_{n}\subset V_{n}$ and $Ok(S^{\prime})\subset
Ok(S)$.

Claim .  Let $C\subset Ok(S)^{o}$ be compact. Then there exists a finitely
generated subalgebra $S^{\prime}$ of $S$, with $C\subset Ok(S^{\prime})^{o}$;
in fact there exists a finitely generated subsemigroup $V^{\prime}$ of
$V=v(S\smallsetminus(0))$ with $C\subset Ok(V^{\prime})^{0}$. Moreover, for
large enough $n$,

$C\cap\frac{\Lambda}{n}\subset V^{\prime}_{n}$

###### Proof.

Immediate from Proposition 5.11. ∎

######  Theorem 8.5.

The measures $\mu_{n}$ converge weakly to Lebesgue measure $\mu$ on $Ok(S)$.
We have $\mu(Ok(S))={\mathop{\operatorname{\rm vol\,}}}(D)/\dim(X)!$.

###### Proof.

Let $\mu^{\prime}_{n}$ be the counting measure on $\Lambda/n$, normalized so
that each point has measure $n^{-d}$. Then $\mu_{n}\leq\mu^{\prime}_{n}$, and
$\mu^{\prime}_{n}$ tends weakly to $\mu$ on any compact set. This shows that
any limit of the $\mu_{n}$ is bounded below $\mu$.

Note that $\mu(\partial Ok(S))=0$ (Lemma 5.9). Thus it suffices to show that
$\mu_{n}\to\mu$ on any compact subset $C$ of the interior $Ok(S)^{o}$.

But on $C$, for large $n$, by Lemma 8.4, $\mu^{\prime}_{n}=\mu_{n}$. As
$\mu^{\prime}_{n}$ tends weakly to $\mu$, so does $\mu_{n}$.

Let $v_{n}=\mu_{n}(Ok(S))$; so $v_{n}\to\mu(Ok(S))$; and we have by definition
${\mathop{\operatorname{\rm vol\,}}}(D)=(\dim(X)!)\limsup v_{n}$.

∎

######  Corollary 8.6.

$Ok(S)$ is compact.

###### Proof.

By Theorem 8.5, and since $\mu_{n}$ has bounded total mass, $Ok(S)$ has finite
Lebesgue measure. Compactness follows from Lemma 5.9.∎

######  Corollary 8.7 (bigness, volume properties).

Let $D$ be an integral divisor on $X$ such that
$tr.deg.(\oplus_{n}H^{0}(X,nD))=\dim(X)$. Then ${\mathop{\operatorname{\rm
vol\,}}}(D)>0$, and this volume is attained as a limit, not just a lim sup.

###### Proof.

As the $\mu_{n}$ tend weakly to a limit, namely to Lebesgue measure on
$Ok(S)$, the total volume $v_{n}=\mu_{n}(Ok(S))$ tends to a limit $\alpha>0$;
we have by definition ${\mathop{\operatorname{\rm
vol\,}}}(D)=(\dim(X)!)\limsup v_{n}=(\dim(X)!)\lim v_{n}$. ∎

Here is Okounkov’s proof of log-concavity of volume, in the setting of big
divisors. (Another proof that will be accessible a little later reduces to the
ample case Lemma 6.5 by Fujita approximation; but our proof of Fujita
approximation will also use Okounkov bodies.) As shown in [3], this latter
proof lends itself to an analysis of the case of equality (sharp log-
concavity) for nef big divisors. It would be interesting to determine when two
divisors have the same Okounkov bodies, or translates thereof, and thus deduce
the sharp log-concavity using Okounkov bodies.

######  Corollary 8.8 (log concavity of volume).

${\mathop{\operatorname{\rm vol\,}}}^{1/n}$ is concave: for $D_{1},D_{2}$ with
${\mathop{\operatorname{\rm vol\,}}}^{1/n}(D_{i})>0$ we have
${\mathop{\operatorname{\rm
vol\,}}}^{1/n}(D_{1}+D_{2})\geq{\mathop{\operatorname{\rm
vol\,}}}^{1/n}(D_{1})+{\mathop{\operatorname{\rm vol\,}}}^{1/n}(D_{2})$

###### Proof.

$Ok(X,D)$ contains the Minkowski sum $Ok(X,D_{1})+Ok(X,D_{2})$. Denoting these
bodies by $O,O_{1},O_{2}$ we have by Brunn-Minkowski ([17] 3.1 ):

$\mu(O)^{1/n}\geq\mu(O_{1})^{1/n}+\mu(O_{2})^{1/n}$

where $\mu$ is Lebesgue measure, normalized so that the group $\Gamma$
generate by $v(\oplus_{m}L(X,mD))$ has covolume $1$. In this case the
corresponding groups $\Gamma_{i}$ of $D_{i}$ have covolume $\geq 1$, so if we
let $\mu_{i}$ be the normalization giving $O_{i}$ covolume $1$ we have
$\mu_{i}=c_{i}\mu$ with $c_{i}<1$. Thus

$\mu(O)^{1/n}\geq\mu_{1}(O_{1})^{1/n}+\mu_{2}(O_{2})^{1/n}$

and we are done by Theorem 8.5. ∎

##  9\. Fujita approximation

###  9.1.  Global generation

######  Definition 9.2.

A Cartier divisor $D$ is globally generated or base-point free or simply free
if for any point $p$ there exists a global section that does not vanish at
$p$.

A ${\mathbb{Q}}$-divisor $D$ is said to be semi-free if $mD$ is globally
generated, for some $m\in{\mathbb{N}}$.

Since a one-dimensional vector space is generated by any subset containing a
nonzero element, a Cartier divisor $D$ is free iff associated invertible sheaf
is generated by its global sections.

######  Remark 9.3.

If $D$ is semi-free, then $D$ is nef, i.e. $D^{k}\cdot Y\geq 0$ for any
$k$-dimensional subvariety $Y$. In particular, $D\cdot Y\geq 0$ for every
curve $Y$. To see this we may assume $D$ is free. Given a variety $Y$ on $X$,
we can find a global section $s$ of $D$ that does not vanish on $Y$; then
$D\cdot Y$ is represented by the zero scheme of $s$ on $Y$. so it is a
positive sum of effective $[Y^{\prime}]$ of dimension $k-1$.

######  Remarks 9.4.

  1. Let $D$ be a free Cartier divisor on $X$.

  2. (1)

Any pullback $g^{*}D$ via $g:X^{\prime}\to X$ is free on $X^{\prime}$.

  3. (2)

If $D,D^{\prime}$ represent the same line bundle, then $D^{\prime}$ is free.
(Global generation is a property of the line bundle.)

  4. (3)

We have a natural morphism $f:X\to PH^{0}(X,D)$ (Grothendieck convention). We
have $D=f^{*}{\mathcal{O}}[1]$ (the global sections agree by definition, and
both line bundles are generated by them).

  5. (4)

$D$ is ample iff $D\cdot C>0$ for any curve $C$ on $X$. Indeed, $D\cdot
C=f^{*}{\mathcal{O}}[1]\cdot C={\mathcal{O}}[1]\cdot f_{*}C$ so if $D\cdot
C>0$ then $f_{*}C$ cannot be a point. Thus $f$ does not collapse any curve to
a point, so it is finite-to-one. By a theorem of Serre,
$f^{*}{\mathcal{O}}[1]$ is ample.

  6. (5)

$D$ has at least one section, so it is effective.

  7. (6)

If $D^{\prime}$ is an ample Cartier divisor, then $D+D^{\prime}$ is ample.
(This follows immediately from (4), aplied to $D+D^{\prime}$.)

######  Lemma 9.5 ( Corollary 1.32 of [12]).

Assume $D$ is a nef Cartier divisor with ${\mathop{\operatorname{\rm
vol\,}}}(D)>0$. Then there exist ample Cartier divisors $D_{t}$ (say for $t$ a
small positive rational) such that $D_{t}\leq D$ and $D_{t}\to D$ numerically.
In particular, for any $\epsilon>0$ there exists an ample $A\leq D$ with
${\mathop{\operatorname{\rm
vol\,}}}(A)\geq(1-\epsilon){\mathop{\operatorname{\rm vol\,}}}(D)$.

###### Proof.

Let $H$ be any ample divisor, represented by a hypersurface $E$. As
$\dim(E)=n-1$, by Corollary 7.2 we have $h^{0}(mD|E)=O(m^{n-1})$, while by
definition $h^{0}(mD)$ grows like $m^{n}$. The exact sequence

$0\to L(X,mD-H)\to L(X,mD)\to L(E,mD|E)$

shows that for large enough $m$, $L(X,mD-H)\neq 0$. Thus $mD=H+E^{\prime}$
with $E^{\prime}\geq 0$. For any $t\in(0,1)$, let $D_{t}=\frac{t}{m}H+(1-t)D$.
But $D_{t}$ is ample (the boundary of the ample cone stabilizers the ample
cone). Now $D=D_{t}+\frac{t}{m}E^{\prime}$ so $D_{t}\leq D$, and $D_{t}\to D$
as $t\to 0$; by continuity of volume at $D$, ${\mathop{\operatorname{\rm
vol\,}}}(D_{t})\to{\mathop{\operatorname{\rm vol\,}}}(D)$. ∎

######  Remark 9.6.

This gives a quick way of seeing using Definition 4.14 that the pullback of
the nef cone is contained in the nef cone: the pullback of an ample is free,
of positive volume, and so can be approximated by ample divisors.

######  Lemma 9.7.

(cf. [3] Lemma 2.6). Let $C,D$ be Cartier divisors on $X$, and assume their
stable join $C\vee D$ exists. If $C,D$ are free, then so is $C\vee D$.

###### Proof.

By assumption, $C,D$ are generated by global sections; we have to show the
same for $C\vee D$. In other words given a point $p\in X$, we have to find a
global section of $C\vee D$ that does not vanish at $p$. Let $f$ be a global
section of $C$ not vanishing at $p$, and $g$ a global section of $D$ not
vanishing at $p$. On some neighborhood $U$ of $p$, $C$ is represented by
$(f^{-1})$ and $D$ by $(g^{-1})$, while $C\vee D$ is represented by $h^{-1}$
say. Then $-C$ is represented by $(f)$, $-D$ by $(g)$, and $-(C\vee
D)=(-C)\wedge(-D)$ by $(h)$; so we have $(h)=(f,g)$. Thus $f=rh$, $g=sh$ with
$r,s$ regular at $p$. Not both $r,s$ can be $0$, since otherwise
$(h)={\mathcal{M}}_{p}(h)$ where ${\mathcal{M}}_{p}$ is the maximal ideal of
the local ring ${\mathcal{O}}_{p}$, contradicting Nakayama. Now both $f,g$ are
global sections of $C\vee D$, and at least one of them fails to vanish at $p$.
∎

######  Lemma 9.8 ([3] Lemma 2.6).

$D_{1},D_{2}$ be ${\mathbb{Q}}$-Cartier divisors, and assume $D=D_{1}\vee
D_{2}$ is the stable join of $D_{1},D_{2}$. If $D_{1},D_{2}$ are semi-free,
then so is $D$. If $D_{1},D_{2}$ are nef, then so is $D$.

###### Proof.

For semi-free this is immediate from Lemma 9.7. Assume $D_{1},D_{2}$ are nef.
Let $A$ be a very ample Cartier divisor. Then for any rational $t>0$,
$D_{i}+tA$ is ample, hence semi-free. Moreover $D+tA$ is the free join of
$D_{1}+tA$ and $D_{2}+tA$. Thus $D+tA$ is semi-free, hence nef. Letting $t\to
0$ we see that $D$ is nef. ∎

######  Lemma 9.9.

Let $D$ be an effective Cartier divisor on $X$, and let $f_{1},\ldots,f_{n}\in
L(D)$ be sections of $D$. Then there exists a blow-up $\pi:X^{\prime}\to X$
and a free divisor $D^{\prime}$ on $X^{\prime}$ with $D^{\prime}\leq\pi^{*}D$,
such that each $f_{i}\circ\pi$ is a section of $D^{\prime}$.

###### Proof.

First we may find a birational morphism $\pi^{*}:X^{*}\to X$, such that each
rational function $f_{i}$ extends to a morphism
$f_{i}^{*}:X^{*}\to{\mathbb{P}}^{1}$. Let $N_{i}=(f_{i}^{*})^{-1}(\infty)$.
Clearly $f_{i}\circ\pi^{*}$ is a section of $N_{i}$. The Cartier divisor
$N_{i}$ is represented by $f_{i}^{-1}$ away from $f_{i}^{-1}(0)$, and by $1$
away from $f_{i}^{-1}(\infty)$; as $D\geq 0$ and $(f_{i})+D\geq 0$, we have
$N_{i}\leq\pi^{*}D$. Being the pullback of the free divisor $\infty$ of
${\mathbb{P}}^{1}$, $N_{i}$ is free.

Passing to a further blowup $X^{\prime}$, we may assume the $N_{i}$ have a
stable join $N$ (§ 4.1). Each $N_{i}\leq\pi^{*}D$, so $N\leq\pi^{*}D$. Being
the stable join of free divisors, $N$ is also free, by Lemma 9.7. ∎

Theorem 9.10 was proved in [15] in characteristic zero, developing, according
to [23], ideas of Demailly on positive intersections. In positive
characteristic it is due to [11].

######  Theorem 9.10 (Fujita approximation).

Let $D=\sum\alpha_{i}D_{i}$ with $D_{i}$ an effective Cartier divisor, and
$\alpha_{i}\in{\mathbb{R}},\alpha_{i}>0$. Assume ${\mathop{\operatorname{\rm
vol\,}}}(D)>0$, and let and let $\epsilon>0$. Then there exists a birational
morphism $\phi:X^{\prime}\to X$, $D^{\prime}=\phi^{*}D$, with $D^{\prime}\geq
N$ for some ample ${\mathbb{Q}}$-divisor $N$, satisfying
${\mathop{\operatorname{\rm
vol\,}}}(N)\geq(1-\epsilon){\mathop{\operatorname{\rm vol\,}}}(D)$.

###### Proof.

Note that there exists a very ample $A$ with $D\leq A$; it follows that $\dim
L(X,kD)\leq\dim L(X,kA)\leq O(k^{d})$, $d=\dim(X)$.

Let $K=k(X)$, and $S_{k}=L(X,kD)$. Let $C$ be a compact subset of
$Ok(X,D)^{o}=Ok(S)^{o}$ with Lebesgue measure
$\mu(C)>(1-\epsilon/2)\mu(Ok(X,D))$. Let $f_{1}\in L(X,k_{1}D),\ldots,f_{r}\in
L(X,k_{r}D)$ be elements, such that the multiplicative semigroup $S^{\prime}$
they generate in $\oplus_{k}L(X,kD)$ has Okounkov body containing $C$ (see
Claim before Theorem 8.5).

If we replace each $f_{i}$ by $f_{i}^{m}\in L(X,mk_{i}D)$, the Okounkov body
does not change; hence it does not change if we just change $f_{1}$ by
$f_{1}^{m_{i}}$. In this way, taking $m_{i}=\Pi_{j\neq i}k_{j}$, we may take
all $k_{i}$ to be equal to the same integer $k$.

By Lemma 9.9, there exists a blow-up $\pi:X^{\prime}\to X$ and a free divisor
$D^{\prime}$ on $X^{\prime}$ with $D^{\prime}\leq\pi^{*}(kD)$, such that each
$f_{i}\circ\pi$ is a section of $D^{\prime}$. Let $N=D^{\prime}/k$; then $N$
is semi-free, and $N\leq\pi^{*}(D)$.

Each $f_{i}\in L(X,kN)$. So $S^{\prime}\subset\oplus L(X,N)$, hence $C\subset
Ok(X,N)$, and thus ${\mathop{\operatorname{\rm
vol\,}}}(N)\geq(1-\epsilon/2){\mathop{\operatorname{\rm vol\,}}}(D)$.

As ${\mathop{\operatorname{\rm vol\,}}}(D)>0$, ‘semi-free’ can be replaced by
‘ample’ using Lemma 9.5. ∎

This shows in particular that volume is a numerical invariant.

##  10\. Positive intersection product and differentiability of volume

The Zariski-Riemann approach of [3] fits very well indeed with the model
theory of GVF’s (indeed the description of quantifier-free types led us to the
same formalism). Nevertheless we wish to remain for a while longer in the
finite-dimensional setting, and will describe the main results of [3] without
this formalism. The positive intersection product here is the restriction of
the one in [3] to domain $N^{1}(X)$ and range $N_{1}(X)$; moreover we consider
only the $n-1$’st power, where $n=\dim(X)$.

###  10.1. Definition of $\psi$, the $n-1$’st positive intersection power

For $x\in N^{1}_{+}(X)^{o}$, define:

$A^{nef}(x)=\\{(f,X^{\prime},y):f:X^{\prime}\to X\hbox{ birational},y\in
N^{1}(X^{\prime})\hbox{ a nef ${\mathbb{Q}}$-Cartier divisor },y\leq
f^{*}(x)\\}$

$W_{x}=\\{f_{*}y^{n-1}:(f,X^{\prime},y)\in A^{nef}(x)\\}$

$W_{x}$ is precompact. To see this pick an ample $a\geq x$. Also pick ample
divisors $b_{j}$ such that $x\mapsto(x\cdot b_{1},\ldots,x\cdot b_{m})$ is an
isomorphism $N_{1}(X)\to{\mathbb{R}}^{m}$. Any nef $y\leq f^{*}(x)$ satisfies
$y\leq f^{*}(a)$, $y^{n-1}\leq f^{*}(a)^{n-1}$, and $f_{*}y^{n-1}\cdot
b_{i}\leq a^{n-1}\cdot b_{i}$. (We used Lemma 4.16.) Thus the image of $W_{x}$
is bounded.

Secondly, $W_{x}$ is directed with respect to the partial ordering of
$N_{1}(X)$ given by the closed cone generated by classes of curves on $X$. In
fact $A^{nef}(x)$ is directed, in the following sense: if $y_{1},y_{2}$ are
nef ${\mathbb{Q}}$-Cartier divisors below $f^{*}(x)$, let $D^{\prime}$
represent $f^{*}(x)$; and represent $y_{1},y_{2}$ by ${\mathbb{Q}}$-Cartier
divisors below $D^{\prime}$. By pulling back to a further blowup we may assume
their stable join $y$ exists. Then $y\leq D^{\prime}$, and by Lemma 9.8, $y$
is nef.

Thus we can define $\psi:N^{1}_{+}(X)^{o}\to N_{1}(X)$ by:

$\psi(x)=\sup W_{x}$

######  Remarks 10.2.

  1. (1)

The definition of $\psi$ would not change if in the definition of $W_{x}$ we
replaced nef by semi-free or by ample. Indeed, since $x$ has positive volume,
by Theorem 9.10 some $y_{0}\in A^{nef}(x)$ has positive volume. Whenever
$(f,X^{\prime},y)\in A^{nef}(x)$ lies above $y_{0}$, it has positive volume,
and so can be approximated by an ample divisor, as any big, nef divisor is
approximated by smaller amples by Lemma 9.5 .

###  10.3. A second definition

We also give a definition that does not require blowing up, and instead uses a
naive intersection theory on $X$; compare [23], 11.4.10, 11.4.1, where the
$n$-fold intersection is treated.

We have not yet defined $N^{k}(X)$ for $k$ other than $1,n-1$. For our
purposes we could easily do so by numerical equivalence against $k$ divisors
(a definition that differs from the classical one when $X$ is smooth, but does
not really lose information when in the construction below). At any rate, we
will use this material only for $k=\dim(X)-1$.

Let $D$ be an effective Cartier divisor, with at least one nonzero section.
Let $k$ be an integer. Let $X^{\prime}$ be a Zariski open subset of $X$. Let
$f_{1},\ldots,f_{k}$ be $k$ mutually generic sections of $L(X,D)$, let $Z$ be
their common zero set on $X^{\prime}$, viewed as a subscheme of $X^{\prime}$;
let $[Z]$ be the sum of components of $Z$ of codimension $k$, counted with
multiplicity. Each irreducible subvariety of $X^{\prime}$ appearing in $[Z]$
can be identified with its Zariski closure in $X$, so we can also view $[Z]$
as a cycle in $N^{k}(X)$; denote $\langle{D;X^{\prime}}\rangle^{k}=[Z]$.

Claim .  For some $z\in N^{k}(X)$, for all Zariski open $X^{\prime}$ disjoint
from the support of $D$, we have $\langle{D;X^{\prime}}\rangle^{k}=z$.

###### Proof.

Let $B_{D}$ be the base locus of $D$: the set of points at which every section
of $D$ vanishes. Let $W$ be a subvariety of $X$. If $W$ is not contained in
$B_{D}$, then a generic section $f$ of $L(X,D)$ does not vanish on $W$, so
cutting $W$ with the zero set of $f$ reduces dimension. It follows that as
soon as $X^{\prime}\cap B_{D}=\emptyset$, all components of $Z$ have
codimension at least $k$.

Similarly if we replace $X^{\prime}$ by
$X^{\prime\prime}=X^{\prime}\smallsetminus W$, where $W$ is some hypersurface
of $X$, then for generically chosen $f_{1},\ldots,f_{k}$, the intersection of
$W$ with the zero sets of the $f_{i}$ has codimension at least $k+1$; no
$k$-dimensional component of $Z$ will be contained in $W$; thus
$\langle{D;X^{\prime}}\rangle^{k}=\langle{D;X^{\prime\prime}}\rangle^{k}$. ∎

We can now define $\langle{D}\rangle^{k}_{naive}$ to be equal to
$\langle{D;X^{\prime}}\rangle^{k}$, for any sufficiently small open
$X^{\prime}\subset X$. We will assume in particular (the base field being
perfect) that $X^{\prime}$ is smooth, i.e. disjoint from the singular locus of
$X$.

Note also that if $g_{1},\ldots,g_{k}\in L(X,D)$ have common zero set
$Z^{\prime}$ on $X^{\prime}$ of codimension $k$, then
$[Z^{\prime}]\leq[Z]=\langle{D;X^{\prime}}\rangle^{k}$ (the inequality holds
with respect to the effective cone in $N^{k}(X)$). In other words the generic
choice of sections maximizes the intersection on $X^{\prime}$. The reason is
that $(f_{1},\ldots,f_{k})$ specializes to $(g_{1},\ldots,g_{k})$; by
smoothness of $X^{\prime}$, every component of $Z^{\prime}$ is the
specialization of some component of $Z$, and is covered with appropriate
multiplicity.

Some components of $Z$ may specialize into $X\smallsetminus X^{\prime}$ and
not be counted in $Z^{\prime}$; but at any rate we have the inequality. We may
summarize this symbolically in the formula:

$\langle{D}\rangle^{k}_{naive}=\inf_{X^{\prime}}\sup_{f_{1},\ldots,f_{k}\in
L(D)}[Z_{X^{\prime}}(f_{1})\cap\cdots\cap Z_{X^{\prime}}(f_{k})]_{n-k}$

If $D\leq D^{\prime}$, let us compare $\langle{D}\rangle_{naive}^{k}$ with
$\langle{D^{\prime}}\rangle_{naive}^{k}$. We have $L(X,D)\leq L(X,D^{\prime})$
Choose a sufficiently small open $X^{\prime}$, and generic
$f_{1},\ldots,f_{k}\in L(X,D)$ and $g_{1},\ldots,g_{k}\in L(X,D^{\prime})$.
Then by the above specialization argument, the generic choice in $D^{\prime}$
gives a bigger or equal result.

Thus
$\langle{D;X^{\prime}}\rangle_{naive}^{k}\leq\langle{D^{\prime};X^{\prime}}\rangle_{naive}^{k}$,
and

$\langle{D}\rangle_{naive}^{k}\leq\langle{D^{\prime}}\rangle_{naive}^{k}$

Next we compare $\langle{D}\rangle_{naive}^{k}$ to
$\langle{lD}\rangle_{naive}^{k}$. Again choose a sufficiently small open
$X^{\prime}$, and generic $f_{1},\ldots,f_{k}\in L(X,D)$ and
$g_{1},\ldots,g_{k}\in L(X,lD)$. Then $(g_{1},\ldots,g_{k})$ specializes to
$(f_{1}^{l},\ldots,f_{k}^{l})$. By the linearity of intersection with Cartier
divisors, $\cap Z_{X^{\prime}}(f_{i}^{l})=l^{k}\cap Z_{X^{\prime}}(f_{i})$.
Hence it follows as above that
$l^{n}\langle{D;X^{\prime}}\rangle_{naive}^{k}\leq\langle{lD;X^{\prime}}\rangle_{naive}^{k}$
and so

$l^{k}\langle{D}\rangle_{naive}^{k}\leq\langle{lD}\rangle_{naive}^{k}$

Finally we define

$\langle{D}\rangle^{k}=\limsup_{m}\frac{1}{m^{k}}\langle{mD}\rangle_{naive}^{k}$

(in fact it follows easily from the above that this is also the $\liminf$ and
hence the limit.)

Clearly $\langle{lD}\rangle^{k}=l^{k}\langle{D}\rangle$. Observe, directly
from this definition:

######  Proposition 10.4.

$\langle{D}\rangle^{k}$ depends on $D$ only via the section spaces
$(L(mD):m\in{\mathbb{N}})$.

In the following lemma, we show equality of the $1$-cycles
$\langle{D}\rangle^{n-1}$ and $\psi(D)$ on $X$. For our purposes, this can be
understood numerically, i.e. $\psi(D)\cdot A=\langle{D}\rangle^{n-1}\cdot A$
for all divisors $A$, equivalently for all very ample divisors $A$. The proof
will use inequalities with respect to the pseudo-effective cone of curves;
equivalently, $u\leq v$ iff $u\cdot A\leq v\cdot A$ for all very ample $A$,
since the dual of the ample cone of divisors is the pseudo-effective cone of
curves.

######  Lemma 10.5.

Let $D$ be a big Cartier divisor. Then $\langle{D}\rangle^{n-1}=\psi(D)$.

###### Proof.

We first show that $\langle{D}\rangle^{n-1}\leq\psi(D)$. Since $\psi$ is
homogeneous, it suffices to show that
$\langle{D}\rangle^{(n-1)}_{naive}\leq\psi(D)$. (This for all $lD$ implies the
claim, by taking limits.) Let $X^{\prime}$ be small enough to compute
$\langle{D}\rangle^{n-1}_{naive}$, as in the Claim. Let
$f_{1},\ldots,f_{{(n-1)}}$ be ${(n-1)}$ mutually generic sections of $L(X,D)$.
By Lemma 9.9 there exists a blow-up $\pi:X^{\prime}\to X$ and a free divisor
$D^{\prime}$ on $X^{\prime}$ with $D^{\prime}\leq\pi^{*}D$, such that each
$f_{i}^{\prime}=f_{i}\circ\pi$ is a section of $D^{\prime}$; moreover by the
proof, such that the zero locus $(f_{i})^{+}$ of $f_{i}^{\prime}$ is nef. So
for any very ample $A$

$\langle{D}\rangle^{(n-1)}_{naive}\cdot
A=\langle{D;X^{\prime}}\rangle^{(n-1)}\cdot
A\leq(f_{1})^{+}\cdot\ldots\cdot(f_{(n-1)})^{+}\cdot\pi^{*}A\leq(D^{\prime})^{n-1}\cdot\pi^{*}A\leq\psi(D)\cdot
A$

(As $(f_{i})^{+}$ and $D^{\prime}$ are nef, and $(f_{i})^{+}\leq D^{\prime}$.)

Conversely for any blow-up $\pi:{{\widetilde{X}}}\to X$ and ample
${\mathbb{Q}}$-Cartier divisor $B\leq\pi^{*}D$, we have to show that

$\pi_{*}(B^{n-1})\leq\langle{D}\rangle^{n-1}=\limsup_{m}\frac{1}{m^{n}}\langle{mD}\rangle_{naive}^{n-1}$

It suffices to show for some $m$ that

$\pi_{*}(B^{n-1})\leq\frac{1}{m^{n}}\langle{mD}\rangle_{naive}^{n-1}$

Multiplying $B$ and $D$ by a large integer, we may assume $B$ is a very ample
Cartier divisor (and then prove the last inequality with $m=1$.)

Note that (as $X$ is normal) $L(X,D)=L({\widetilde{X}},\pi^{*}(D))\supseteq
L(X,B)$. Let $f_{1},\ldots,f_{n-1}$ be sufficiently generic global sections of
$B$. Let $Z_{i}$ be the zeroes of $f_{i}$, viewed as a divisor on
${\widetilde{X}}$. Let $e$ be the blowing-up locuas of $\pi$, and
$E=\pi^{-1}(e)$; so $\pi$ restricts to an isomorphism
${\widetilde{X}}\smallsetminus E\to X\smallsetminus e$. Then $E$ has
codimension at least $1$ on ${\widetilde{X}}$; as $B$ is very ample,
$E\cap\cap_{i=1}^{n-1}Z_{i}$ is finite. Hence,
$\pi_{*}(\cap_{i=1}^{n-1}Z_{i})$ as a $1$-cycle has no components contained in
$e$. Viewing the $f_{i}$ now as elements of $L(X,D)$, denoting their zeroes by
$Z_{i}^{+}$, we see that no component of their intersection will be contained
in $e$ (as all components are $1$-dimensional), and so
$\pi_{*}(\cap_{i=1}^{n-1}Z_{i})=\cap_{i}Z_{i}^{+}$. As we saw above, the
intersection of the zero sets on $X$ of generic $g_{1},\ldots,g_{n-1}\in L(D)$
can onlty be bigger. This proves the required inequality. ∎

######  Lemma 10.6.

$\psi$ is homogeneous of degree $n-1$, monotone and continuous on
$N^{1}_{+}(X)^{o}$.

###### Proof.

Monotonicity is immediate from the definition of $\psi$, and homogeneity is
also easy.

Now any function $v$, monotone with respect to the ordering of a cone $E$, and
homogeneous of degree $d\geq 1$, must be continuous on the interior of $E$.
For let $e\in E^{o}$ and $\epsilon>0$. Then $\epsilon a\in E^{0}$, so there
exists a symmetric neighborhood $N=N_{\epsilon}$ of $0$ with $\epsilon
a+N\subset E^{0}$, hence $\epsilon a\geq N$. Thus $(1+\epsilon)a\geq
a+N\geq(1-\epsilon)a$ so by homogeneity $(1+\epsilon)^{d}f(a)\geq
f(a+N)\geq(1-\epsilon)^{d}f(a)$; hence for any linear functional $L$ with
$L(E)\geq 0$ we have $(1+\epsilon)^{d}Lf(a)\geq
Lf(a+N)\geq(1-\epsilon)^{d}Lf(a)$; letting $\epsilon\to 0$ we see that $Lf$ is
continuous at $a$. As such $L$’s span the dual space, $f$ is continuous. ∎

Here is the main theorem of [3].

######  Theorem 10.7.

The volume function ${\mathop{\operatorname{\rm vol\,}}}$ is differentiable on
the big cone of $N^{1}(X)$; and $d{\mathop{\operatorname{\rm vol\,}}}=n\psi$.

Here we use the identification $N_{1}=N^{1}(X)^{*}$ given by the intersection
pairing. Explicitly, if $\alpha\in N^{1}(X)$ is big and $\gamma\in N^{1}(X)$
is arbitrary, the statement is that

$\frac{d}{dt}|_{t=0}{\mathop{\operatorname{\rm
vol\,}}}(\alpha+t\gamma)=n\psi(\alpha)\cdot\gamma$

###### Proof.

Fix an ample $\omega$ such that $\alpha\leq\omega$ and $\omega\pm\gamma$ is
ample. Consider birational morphisms $f:X^{\prime}\to X$; let
$\omega^{\prime}=f^{*}\omega$, $\alpha^{\prime}=f^{*}\alpha$,
$\gamma^{\prime}=f^{*}\gamma$. Note that $\omega^{\prime}$ and
$\omega^{\prime}\pm\gamma^{\prime}$ remain nef. Recall Corollary 7.7: if
$\beta\in N^{1}(X^{\prime})$ is nef, for any $t\in[-1,1]$ we have
${\mathop{\operatorname{\rm
vol\,}}}(\beta+t\gamma^{\prime})\geq\beta^{n}+nt\beta^{n-1}\cdot\gamma^{\prime}-8^{n}\omega^{n}t^{2}$.
When $\beta\leq\alpha^{\prime}$ it follows that ${\mathop{\operatorname{\rm