Diophantine subsets of function fields of curves

0.18 MB·English

by  János Kollár

DIOPHANTINE SUBSETS OF FUNCTION FIELDS OF CURVES
JA´NOSKOLLA´R
8
0
0
2 February 1, 2008
Let R be a commutative ring. A subset D ⊂R is called diophantine if there are
n
a polynomials
J F (x,y ,...,y )∈R[x,y ,...,y ]
i 1 n 1 n
8
such that the system of equations
1
Fi(r,y1,...,yn)=0 ∀i
]
T has a solution (y ,...,y )∈Rn iff r ∈D.
1 n
N Equivalently, if there is a (possibly reducible) algebraic variety X over R and
R
. a morphism π :X →A1 such that D =π(X (R)). In this situation we call
h R R R
t dioph(X ,π):=π(X (R))⊂R
a R R
m
the diophantine set corresponding to X and π.
R
[ AcharacterizationofdiophantinesubsetsofZwascompletedinconnectionwith

Hilbert’s 10th problem, but a description of diophantine subsets of Q is still not
2
v known. In particular,it is notknownif Z is a diophantine subsetof Q ornot. (See
1 [Poo03] or the volume [DLPVG00] for surveys and many recent results.)
5 In this paper we consider analogousquestions where R=k(t) is a function field
4
of one variable and k is an uncountable large field of characteristic 0. That is,
3
for any k-variety Y with a smooth k-point, Y(k) is Zariski dense. Examples of
.
8
uncountable large fields are
0
7 (1) C or any uncountable algebraically closed field,
0 (2) R or any uncountable real closed field,
: (3) Q ,Q((x))orthequotientfieldofanyuncountablelocalHenseliandomain.
v p
i Roughly speaking, we show that for such fields, a diophantine subset of k(t) is
X
either very small or very large. The precise result is somewhat technical, but here
r
a are two easy to state consequences which served as motivating examples.
Corollary 1. Let k be an uncountable large field of characteristic 0. Then k[t] is
not a diophantine subset of k(t).
Corollary2. Letk beanuncountablelargefieldofcharacteristic0andK ⊃K ⊃
2 1
k(t) finite field extensions. Then K is a diophantine subset of K iff K =K .
1 2 1 2
ThelattergivesapartialanswertoaquestionofBogomolov: Whenisasubfield
K ⊂K diophantine in K ?
1 2 2
It is possible that both of these corollaries hold for any field k. Unfortunately,
my method says nothing about countable fields. The geometric parts of the proof
(12) and (13) work for any uncountable field, but the last step (23) uses in an
essential way that k is large.
We use two ways to measure how large a diophantine set is.
1
2 JA´NOSKOLLA´R
3 (Diophantine dimension and polar sets). Let B be a smooth, projective, irre-
ducible curve overk. One can think of a rationalfunction f ∈k(B) as a section of
the first projectionπ :B×P1 →B. This establishes a one-to-onecorrespondence
1
k(B)∪{∞} ↔ {sections of π :B×P1 →B}.
1
Any section σ : B → B×P1 can be identified with its image, which gives a point
in the Chow variety of curves of B×P1. This gives an injection
k(B)∪{∞} ֒→ Chow (B×P1).
1
Let U be a countable (disjoint) union of k-varieties and D ⊂ U(k) a subset.
Define the diophantine dimension ofD overk asthe smallestn∈{−1,0,1,...,∞}
such that D is contained in a countable union of irreducible k-subvarieties of U of
dimension ≤ n. It is denoted by d-dim D. Note that d-dim D = −1 iff D = ∅
k k
and d-dim D ≤0 iff D is countable.
k
In particular, we can talk about the diophantine dimension of dioph(X,f) ⊂
k(B)⊂Chow (B×P1).
1
For f ∈k(B), let pole(f) denote its divisor of poles. For D ⊂k(B) set
Pole (D):={pole(f):f ∈D and degpole(f)=n}.
n
I think of Pole (D) as a subset of the k¯-points of the nth symmetric power SnB.
n
Taking each point with multiplicity r ≥ 1 gives embeddings SmB → SrmB,
whose image I denote by r·SmB.
With these definitions, the main result is the following illustration of the “very
small or very large” dichotomy.
Theorem4. Letk beanuncountablelargefieldofcharacteristic0andB asmooth,
projective, irreducible curve over k. Let X be a (possibly reducible) algebraic
k(B)
variety of dimension n over k(B) and π :X →A1 a morphism. Then
k(B) k(B) k(B)
(1) either d-dim dioph(X ,π )≤n,
k k(B) k(B)
(2) or d-dim dioph(X ,π ) = ∞ and there is a 0-cycle P ∈ SaB and
k k(B) k(B) a
r > 0 such that for every m > 0 there is a smooth, irreducible k-variety
D and a morphism ρ :D →Sa+rmB such that
m m m
(a) D (k)6=∅,
m
(b) Pole dioph(X ,π ) ⊃ρ D (k) , and
a+rm k(B) k(B) m m
(c) the Zariski closure of ρ (D (k)) contains P +r·SmB ⊂Sa+rmB.
m m a
(cid:0) (cid:1) (cid:0) (cid:1)
5 (Proof of the Corollaries). In trying to write a subset D ⊂ k(B) as D =
dioph(X ,π ), we do not have an a priori bound on dimX , thus the
k(B) k(B) k(B)
assertion d-dim dioph(X ,π ) = ∞ is hard to use. The Corollaries 1 and 2
k k(B) k(B)
both follow from the more precise results about the distribution of poles.
IfB =P1,thenarationalfunctionwithatleast2polesonP1isnotapolynomial,
thus Theorem 4 implies Corollary 1.
Next considerCorollary2. LetK =k(B )(K =k(B )be adegreed>1 ex-
1 1 2 2
tension of function fields of smooth, projective, irreducible k-curves. By Riemann-
Roch, any zero cycle of degree ≥ 2g(B ) defined over k is the polar set of some
1
f ∈k(B ). Pulling back gives a map j :SmB →SmdB , thus
1 1 2
j (SmB )(k) if n=md≥2dg(B ), and
1 1
Pole (K )=
n 1
(∅(cid:0) if d6|n. (cid:1)
DIOPHANTINE SUBSETS OF FUNCTION FIELDS OF CURVES 3
If b 6= b ∈ B map to the same point of B , then a 0-cycle in j SmB contains
1 2 2 1 1
eitherbothb andb orneither. ThustheZariskiclosedsetj SmB nevercontains
1 2 (cid:0)1 (cid:1)
a set of the form P +r·SmB . By (4.2.c), this shows that K is not diophantine
a 2 1
(cid:0) (cid:1)
in K , proving Corollary 2.
2
Example 6. (1)Theboundnin(4.1)isactuallysharp,asshownbythefollowing.
Note first that any k(t)-solution of x3+y3 =1 is constant. Set
X :=(x3+y3 =···=x3 +y3 =1)⊂A2n
n 1 1 n n
and
π :(x ,y ,...,x ,y )7→x +x t+···+x tn−1.
1 1 n n 1 2 n
Then dimX = n and for k = C or k = R, dioph(X ,π) is the set of all degree
n
≤n−1 polynomials.
Usingsimilarconstructionsonecanseethatany(finite dimensional)k-algebraic
subset of k(t) is diophantine when k is algebraically closed or real closed. These
are the “small” diophantine subsets of k(t).
(2) The somewhat unusual looking condition about the Zariski closure of D
m
in (4.2.c) is also close to being optimal. For g ∈ k(t) and r > 0 consider the
diophantine set
L :={f ∈k(t):∃h such that f =ghr}.
g,r
Then,uptosomelowerdimensionalcontributioncomingfrompossiblecancellations
between poles and zeros of g and hr, Pole (L ) equals pole(g)+r· SmB (k) if
n g,r
n=degpole(g)+rm and ∅ otherwise.
(cid:0) (cid:1)
7. If k = C then our proof shows that in case (4.2) there is a finite set P ⊂ B(C)
such that for every p∈B(C)\P there is an f ∈dioph(X,π) with a pole at p.
p
Ifk =R, thenweguaranteemany poles,butone maynotgetanyrealpoles. To
get examples, note that h ∈ R(t) is everywhere nonnegative on R iff h is a sum of
2 squares. Thus for any g ∈R(t), the set
L (g):={f ∈R(t):f(t)≤g(t) ∀t∈R}
1
is diophantine. L(g) is infinite dimensional but if g ∈R[t] then no element of L(g)
has a real pole.
Fromthepointofviewofourproofamoreinterestingexampleisthediophantine
set
L (g):={f ∈R(t):∃c∈R, f2(t)≤c2·g2(t) ∀t∈R}.
2
The elements of L (g) are unbounded everywhere yet no element of L (g) has a
2 2
pole in R if g is a polynomial.
This leads to the following question.
Question 8. IsR[t] ,thesetofallrationalfunctionswithoutpolesinR,diophan-
R
tine?
There should be some even stronger variants of the “very small or very large”
dichotomy, especially over C. As a representative case, I propose the following.
Conjecture 9. Let D ⊂ C(t) be a diophantine subset which contains a Zariski
open subset of C[t]. (Meaning, for instance, that D contains a Zariski open subset
of the space of degree ≤ n polynomials for infinitely many n.) Then C(t)\D is
finite.
In connection with Bogomolov’s question, I would hazard the following:
4 JA´NOSKOLLA´R
Conjecture 10. Let k be a large field and K ⊂K function fields of k-varieties.
1 2
Then K is diophantine in K iff K is algebraically closed in K .
1 2 1 2
11. The proof of (4) relies on the theory of rational curves on algebraic varieties.
A standardreference is [Kol96], but non-experts may prefer the more introductory
lectures of [AK03].
The proof is divided into three steps.
First we show that if d-dim dioph(X ,π) ≥ n+1 then there is a rationally
k k(B)
connected (cf. (18)) subvariety Z ⊂ X such that π| is nonconstant and
k(B) k(B) Z
Z has a smooth k(B)-point. This relies on the bend and break method of Mori
k(B)
[Mor79]. In a similar context it was first used in [GHMS05].
Thenweshow,usingthedeformation of combstechniquedevelopedin[KMM92,
Kol96,GHS03,Kol04],thatforanysuchZ ,thereareinfinitelymanyk-varieties
k(B)
S and maps S ×B 99KZ which give injections S (k)֒→Z (k(B)).
m m k(B) m k(B)
Both of these steps are geometric, but the statements are formulated to work
over an arbitrary field L.
Finally, if k is a large field, then each S (k) is “large”, which shows that
m
Z (k(B)) is “very large”.
k(B)
For all three steps it is better to replace π :X →A1 with a morphism of
k(B) k(B)
k-varieties f :X →B×P1.
Proposition 12. Let L be any field and B a smooth, projective, irreducible curve
over L. Let f : X →B×P1 be an L-variety of dimension n+1 and consider the
corresponding diophantine set dioph(X ,f)⊂L(B). Then
L(B)
(1) either d-dim dioph(X ,f)≤n,
L L(B)
(2) or there is a subvariety Z ⊂X such that
(a) Z →B×P1 is dominant,
(b) the generic fiber of Z →B is rationally connected, and
(c) there is a rational section σ : B 99K Z whose image is not contained
in SingZ.
Proposition 13. Let L be an infinite field and B a smooth, projective, irreducible
curve over L. Let f :Z →B×P1 be a smooth, projective L-variety such that
(1) Z →B×P1 is dominant,
(2) the generic fiber of Z →B is separably rationally connected, and
(3) there is a section σ :B →Z.
Then, for some r >0 and for all m>0 in an arithmetic progression, there are
(4) a smooth, irreducible L-variety S with an L-point, and
m
(5) a dominant rational map σ :S ×B 99KZ which commutes with projec-
m m
tion to B
suchthattheZariskiclosureoftheimageoff◦σ :S 99KChow (B×P1)contains
m m 1
[f ◦σ(B)]+r[{b }×P1]+···+r[{b }×P1] for every b ∈B(L¯).
1 m i
14 (Spaces of sections). Let L be any field, B a smooth, projective, irreducible
curve over L and f : X → B a projective morphism. A section of f (defined
over some L′ ⊃ L) can be identified with the corresponding L′-point in the Chow
variety of 1-cycles Chow (X). All sections Σ(X/B) defined over L¯ form an open
1
set of Chow (X). Indeed, if H is an ample line bundle on B of degree d then a
1
1-cycle C is a section iff C is irreducible (an open condition) and (C ·f∗H) = d
DIOPHANTINE SUBSETS OF FUNCTION FIELDS OF CURVES 5
(an open and closed condition). This procedure realizes X k(B) as the set of
k(B)
k-points of a countable union of algebraic k-varieties Σ(X/B)=∪ Σ .
i i
(cid:0) (cid:1)
ThechoiceoftheΣ isnotcanonical. GivenX →B,weget“natural”irreducible
i
components, but for fixed generic fiber X , these components depend on the
k(B)
choice of X. Any representation gives, however, the same constructible sets. We
usually make a further decomposition. Since every variety is a finite set-theoretic
union of locally closed smooth subvarieties, we may choose the Σ such that each
i
one is smooth and irreducible.
As an explicit example, consider B =P1. Then k(B)∼=k(t) and every f ∈k(t)
can be uniquely written (up to scalars) as
a +a t+···+a tn
0 1 n
f = ,
b +b t+···+b tn
0 1 n
where the nominator and the denominator are relatively prime and at least one of
a or b is nonzero. For any n, all such f form an open subset
n n
Σn ⊂P(a0 :a1 :···:an :b0 :b1 :···:bn)∼=P2n+1.
15 (Very dense subsets). Let U be an irreducible variety over a field L. We say
that a subset D ⊂ U(L¯) is Zariski very dense if D is not contained in a countable
union of L-subvarieties V (U.
i
It is easy to see that for any D, there are countably many closed, irreducible
L-subvarietiesW ⊂U suchthatD ⊂∪ W (L¯)andD∩W (L¯)isZariskiverydense
i i i i
in W for every i. There is a unique irredundant choice of these W .
i i
16 (Proofof(12)). Write X =∪X as a finite set-theoretic unionof locally closed,
i
smooth, connected varieties. If (12) holds for each X then it also holds for X,
i
thus we may assume that X is smooth and irreducible. Let X′ ⊃ X be a smooth
compactification such that f extends to f′ :X′ →B×P1.
As before, there are countably many disjoint, irreducible, smooth L-varieties
∪ Σ = Σ(X′/B) and morphisms u : B ×Σ → X′ commuting with projection
i i i i
to B giving all L¯-sections of f′. As in (15), there are countably many disjoint,
irreducible, smooth L-varieties S ⊂Σ(X′/B) such that each S (L) is Zariski very
i i
dense in S and the L-sections of X′ →B are exactly given by ∪ S (L).
i i i
Composing u with f′, we obtain maps
i
f′ :S →Σ⊂Chow (B×P1).
∗ i 1
There are 2 distinct possibilities. Either
(1) dim f′(S )≤n whenever u (B×S )∩X 6=∅, or
L ∗ i i i
(2) there is an i such that dim f′(S )≥n+1 and u (B×S )∩X 6=∅.
0 L ∗ i0 i0 i0
In the first case dioph(X′,f′) is contained in the union of the constructible sets
f′(S ), thus we have (12.1). This is always the case if L is countable.
∗ i
In the secondcasewe constructZ as requiredby (12.2) using only the existence
of u : B × S → X. Set S := S and u := u . We can replace X′ by a
i0 i0 i0 i0
desingularization of the closure of the image u(B ×S). By shrinking S we may
assume that u lifts to u:B×S →X′.
For a point x ∈ X′ let S ⊂ S be the subvariety parametrizing those sections
x
that pass through x.
Letusnowfixb∈B(L¯)suchthatu({b}×S)isdenseinX′ andletxrunthrough
b
Xb′, the fiber of X′ over b. Since every section intersects Xb′, S =∪x∈Xb′Sx and so
f∗′(S)=∪x∈Xb′f∗′(Sx). By assumptiondimLf∗′(S)≥n+1anddimLXb′ =n,hence
6 JA´NOSKOLLA´R
dim f′(S )≥1forgeneralx∈X′(L¯). Inparticular,thereisa1-parameterfamily
L ∗ x b
of sections C ⊂S such that
x x
f′◦u:B×C →X →B×P1
is a nonconstant family of sections passing through the point f′(x).
By (17), this leads to a limit 1–cycle of the form
A+{b}×P1+(other fibers of π )
1
where A is a section of π :B×P1 →B.
1
Correspondingly, we get a limit 1–cycle in X′ of the form
A +R +(other rational curves)
x x
where A is a section of X′ →B which dominates A and R is a connected union
x x
of rational curves which dominates {b}×P1. Note also that x∈R .
x
Thus we conclude that for general x ∈ X′(L¯), there is a connected union of
b
rational curves x∈R ⊂X′ which dominates {b}×P1.
x b
As in (19), let us take the relative MRC-fibration f′ :X′ 9w9KW′ 99KB.
Forverygeneralx∈X′(L¯)letX′ be the fiberofw containingx. By(19), X′ is
x x
closed in X′ and every rational curve in X′ that intersects X′ is contained in X′.
x x
In particular, R ⊂X′ and hence X′ dominates {b}×P1.
x x x
Let now p∈S(L) be a generalpoint and C ⊂X′ the correspondingsection. By
assumption S(L) is Zariski dense in S, hence we may assume that w is smooth at
the generic point of C. Let Z′ ⊂w−1(w(C)) be the unique irreducible component
that dominates C and Z =Z′∩X. It satisfies all the required properties. (cid:3)
17(Bend-and-breakforsections). (Cf.[Mor79],[Kol96,Sec.II.5],[GHMS05,Lem.3.2])
Let h : Y → B be a proper morphism onto a smooth projective curve B. Let
C be a smooth curve and u:B×C →Y a nonconstant family of sections passing
through a fixed point y ∈Y.
Then C can not be a proper curve and for a suitable point c ∈ C¯ \ C the
corresponding limit 1-cycle is of the form
Σ =A +R ,
y y y
where A is a section of h (which need not pass through y) and R is a nonempty
y y
unionofrationalcurvescontainedinfinitelymanyfibersofh. Furthermore,A +R
y y
is connected and y ∈R .
y
This holds whether we take the limit in the Chow variety of 1-cycles, in the
Hilbert scheme or in the space of stable maps.
18(Rationallyconnectedvarieties). (Cf.[KMM92],[Kol96,Chap.IV],[AK03,Sec.7])
Let k be a field and K ⊃k an uncountable algebraically closed field. A smooth
projective k-variety X is called rationally connected or RC if for every point pair
x ,x ∈ X(K) there is a K-morphism f : P1 → X such that f(0) = x and
1 2 1
f(∞) = x . X is called separably rationally connected or SRC if for every point
2
x∈ X(K) there is a K-morphism f :P1 →X such that f(0)= x and f∗T is an
X
amplevectorbundle. (Thatis,asumofpositivedegreelinebundles.) Furthermore,
f :P1 →X can be taken to be an embedding if dimX ≥3. It is known that SRC
implies RC and the two notions are equivalent in characteristic 0.
We maynothaveanyrationalcurvesoverk,butwecanworkwiththe universal
family of these maps f :P1 →X. Thus, if dimX ≥3 and p∈X is a k-point, then
DIOPHANTINE SUBSETS OF FUNCTION FIELDS OF CURVES 7
thereisanirreducible,smoothk-varietyU andak-morphismG:U×P1 →X such
that
(1) G(U ×{0})=p,
(2) Gu :{u}×P1 →Xk¯ is an embedding for every u∈U(k¯), and
(3) G∗T is ample for every u∈U(k¯).
u X
By [Kol99, Thm.1.4], if k is large then we can choose U such that U(k)6=∅.
19 (MRC fibrations). (Cf. [KMM92], [Kol96, Sec.IV.5])
Let K ⊃k be as above. Let X be a smooth projective k-variety and g :X →S
a k-morphism. There is a unique (up to birational maps) factorization
g :X 9w9KW 99hKS
such that
(1) for general p ∈ W(K), the fiber w−1(p) is closed in X and rationally con-
nected, and
(2) for very general p ∈ W(K) (that is, for p in a countable intersection of
dense open subsets) every rational curve in X(K) which intersects w−1(p)
and maps to a point in S is contained in w−1(p).
The map w : X 99K W is called the (relative) maximal rationally connected
fibration or MRC fibration of X → S. Note that if X contains very few rational
curves (for example, if X is an Abelian variety or a K3 surface) then X =W.
20 (Proof of (13)). Here we essentially reverse the procedure of the first part.
Insteadof degeneratinga 1-parameterfamily of sections to geta 1-cycleconsisting
of a section + rational curves, we start with a section, add to it suitably chosen
rational curves and prove that this 1-cycle can be written as the limit of sections
in many different ways.
We assume that Z is smooth, projective. If necessary, we take its product with
P3 to achieve that dimZ ≥ 4. This changes the space of sections Σ(Z/B) but it
does not change the image of Σ(Z/B) in L(B).
Apply (18) to X =Z and the point p=σ(B) to get
L(B)
G:U ×P1 →Z .
L(B) L(B)
NextreplaceU byanL-varietyτ :U →B suchthatGextendstog :U×P1 →
L(B)
Z. By shrinking U if necessary, we may assume that for general b ∈ B(L¯), the
corresponding
g :U ×P1 →Z
b b b
is a family ofsmoothrationalcurves passingthroughσ(b) andg∗ T is ample for
b,u Zb
every u∈U where g is the restriction of g to {u}×P1.
b b,u b
Givendistinctpointsb ,...,b ∈B(L¯),letB(b ,...,b )bethecombassembled
1 m 1 m
from B and m copies of P1 where we attach P1 to B at b (cf. (21)).
i i
By [Kol04, Thm.16], there are b ,...,b ∈B(L¯) and an embedding
1 m0
σ(g ,...,g ):B(b ,...,b )→Z
1 m0 1 m0
given by σ on B and by g := g on the P1 for some u ∈ U such that the
i bi,ui i i bi
image,denotedbyB(g ,...,g )⊂Z isdefinedoverLanditsnormalbundleisas
1 m0
positiveasonewants. Inparticular,by(22),B(g ,...,g )givesasmoothpointof
1 m0
8 JA´NOSKOLLA´R
theHilbertschemeofZ. Furthermore,foranyfurtherdistinctpointsb ,...,b
m0+1 m
and g for i=m +1,...,m, the resulting
i 0
σ(g ,...,g ):B(b ,...,b )→Z
1 m 1 m
also gives a smooth point of the Hilbert scheme of Z.
Let S denote the smooth locus of the corresponding L-irreducible component
m
of the Hilbert scheme of Z. B(g ,...,g ) gives an L-point of S , hence S is
1 m m m
geometricallyirreducible. By (22)the generalpoint ofS correspondsto a section
m
off, the universalfamily is a productovera dense open subset of S andwe have
m
a dominant rational map σ :S ×B 99KZ.
m m
For a given m, it is not always possible to choose b ,...,b such that the
m0+1 m
set b ,...,b is defined over L. To achieve this, choose a generically finite and
1 m
dominant map ρ : U 99K AdimU. For general c ∈ AdimU(L), its preimage ρ−1(c)
L
gives degρ general points in U which are defined over L. Thus we can choose
b ,...,b to be defined over L whenever m−m is a multiple of degρ.
1 m 0
Let us now consider f∗(Sm)⊂Chow1(B×P1). By construction, it contains
f∗ B(g1,...,gm) =A+r {b1}×P1 +···+r {bm}×P1
where A = f(cid:0)◦σ(B) is a sec(cid:1)tion of B(cid:0)×P1 and(cid:1)r ≥ 1 is th(cid:0)e common(cid:1)(geometric)
degree of the maps
f ◦g :{u}×P1 →{b}×P1 ⊂B×P1.
b,u
The combs B(g ,...,g ) ⊂ Z have some obvious deformations where we keep
1 m
B fixed and vary the points b and the maps g with them. By construction, the
i i
points b canvaryindependently. The imagesofthese 1-cyclesinB×P1 are ofthe
i
form
A+r {b′}×P1 +···+r {b′ }×P1 ,
1 m
where the b′ vary independently. (cid:3)
i (cid:0) (cid:1) (cid:0) (cid:1)
21 (Combs). A comb assembledfrom B and m copies of P1 attachedat the points
b is a curve B(b ,...,b ) obtained from B and {b ,...,b }×P1 by identifying
i 1 m 1 m
the points b ∈B and (b ,0)∈{b }×P1.
i i i
A comb can be pictured as below:
b b b
1r 2r mr
B
······
P1 P1 P1
1 2 m
Comb with m-teeth
22 (Deformation of reducible curves). Let X be a smooth projective variety and
C ⊂ X a connected curve with ideal sheaf I . Assume that C has only nodes as
C
singularities. By the smoothness criterionof the Hilbert scheme [Gro62, p.221-21],
∗
if H1(C, I /I2 ) = 0 then [C] is a smooth point of the Hilbert scheme Hilb(X)
C C
and there is a unique irreducible component H ⊂ Hilb(X) containing [C]. Let
C
(cid:0) (cid:1)
U →H be the universal family.
C C
DIOPHANTINE SUBSETS OF FUNCTION FIELDS OF CURVES 9
∗
If, in addition, (I /I2 is generated by global sections, then a general point of
C C
H corresponds to a smooth curve and the natural map U →X is dominant.
C C
(cid:1)
23 (Proofof(4)). Letusstartwiththe k(B)-varietyX . We canwriteitasthe
k(B)
generic fiber of a quasi-projective k-variety X → B and extend π to f : X →
k(B)
B×P1. If (4.1) fails then using (12) we obtain Z ⊂ X. Take a compactification
Z¯ and a resolution Z → Z¯ such that the composite map Z 99K B × P1 is a
1 1
morphism. Next apply (13) to Z → B × P1. We obtain, for every m in an
1
arithmetic progression,a dominant family of sections σ :S ×B →Z .
m m 1
There is a dense open subset D ⊂S such that for every q ∈D (k¯),
m m m
(1) the corresponding section σ ({q}×B)⊂Z intersects Z, and
m 1
(2) the corresponding rational function f ◦σ : {q}×B → P1 has exactly
m
a+rm poles where a is the number of poles of f ◦σ(B).
Thus the the composite map
ρ :=pole◦f ◦σ :D →Sa+rmB
m m m
is defined. The condition (4.2.b) holds by construction and the Zariski closure of
ρ (D ) contains P +r·SmB ⊂ Sa+rmB by (13.5), where P denotes the polar
m m a a
divisor of the section σ, that is, the 0-cycle f ◦σ(B)∩ B×{∞} .
S hasasmoothk-pointby(13)andk-pointsareZariskidensesincekisalarge
m
(cid:0) (cid:1)
field. Thus D (k) is Zariski dense in D . This implies both (4.2.a) and (4.2.c).
m m
Finally, D (k) is Zariski very dense in D by (24) and
m m
d-dim dioph(X,π)≥d-dim ρ (D (k))=dim ρ (D )≥dim SmB =m,
k k m m k m m k
where the middle equality holds by (24). Thus d-dim dioph(X,π)=∞.
k
The only remaining issue is that our m runs through an arithmetic progression
and is not arbitrary. If the progression is m = b+ m′c then a +r(b + m′c) =
(a+rb)+(rc)m′, so by changing a7→a+rb,r 7→rc we get (4). (cid:3)
Lemma24. LetX beasmoothandirreduciblevarietyoveralargefieldk suchthat
X(k)6=∅. Then X(k) is not contained in the union of fewer than |k| subvarieties
X (X. In particular, if k is uncountable then X(k) is Zariski very dense in X.
λ
Proof. Assume to the contrary that X(k) 6= ∅ but X(k) ⊂ ∪λ∈ΛXλ where
|Λ|<|k| and X 6=X.
λ
If dimX ≥ 2, then pick p ∈ X(k) and let {H : t ∈ P1} be a general pencil of
t k
hypersurface sections of X passing through p. Since |Λ|<|k|, there is an H such
t
that Ht is smooth at p and Ht 6⊂ Xλ for every λ. Thus Ht(k) ⊂ ∪λ∈Λ(Ht∩Xλ)
is a lower dimensional counter example. Thus it is enough to prove (24) when X
is a curve. Then lower dimensional k-subvarieties are just points, thus we need to
show that |X(k)|=|k|.
Let {H :t∈P1} be a linear system on X ×X which has (p,p) as a base point
t
andwhose generalmember is smooth at(p,p). Since k is large,eachH contains a
t
k-point different from (p,p). Thus |X(k)|=|X(k)×X(k)|≥|k|. (cid:3)
Acknowledgments . I thank F. Bogomolov, K. Eisentr¨ager, B. Poonen and J.
Starrforusefulconversationsandtherefereeforseveralinsightfulsuggestions. Par-
tialfinancialsupportwasprovidedby the NSFunder grantnumberDMS-0500198.
10 JA´NOSKOLLA´R
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Princeton University, Princeton NJ 08544-1000
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