Tài liệu Đề tài " Bertini theorems over finite fields " docx

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1102 BJORN POONEN
∂U
P
has measure zero. Also for each P, fix a nonvanishing coordinate x
j
, and
for f ∈ S
d
let f |
P
be the image of x
−d
j
f in
ˆ
O
P
. Assume that there exist smooth
quasiprojective subschemes X
1
, ,X
u
of P
n
of dimensions m
i
= dim X
i
over
F
q
such that for all but finitely many P , U
P
contains f |
P
whenever f ∈ S
homog
is such that H
f
∩ X
i
is smooth of dimension m
i
− 1 at P for all i. Define
P := { f ∈ S
homog
: f|
P
∈ U
P
for all closed points P ∈ P
n
}.
Then µ(P)=

closed P ∈P
n
µ
P
(U
P
).
Remark. Implicit in Theorem 1.3 is the claim that the product

P
µ
P
(U
P
)
always converges, and in particular that its value is zero if and only if µ
P
(U
P
)
= 0 for some closed point P .
The proofs of Theorems 1.1, 1.2, and 1.3 are contained in Section 2. The
reader at this point is encouraged to jump to Section 3 for applications, and to
glance at Section 5, which shows that the abc conjecture and another conjec-
ture imply analogues of our main theorems for regular quasiprojective schemes
over Spec Z. The abc conjecture is needed to apply a multivariable gener-
alization [Poo03] of A. Granville’s result [Gra98] about squarefree values of
polynomials. For some open questions, see Sections 4 and 5.7, and also Con-
jecture 5.2.
The author hopes that the technique of Section 2 will prove useful in
removing the condition “assume that the ground field k is infinite” from other
theorems in the literature.
2. Bertini over finite fields: the closed point sieve
Sections 2.1, 2.2, and 2.3 are devoted to the proofs of Lemmas 2.2, 2.4,
and 2.6, which are the main results needed in Section 2.4 to prove Theorems
1.1, 1.2, and 1.3.
2.1. Singular points of low degree. Let A = F
q
[x
1
, ,x
n
] be the ring of
regular functions on the subset A
n
:= {x
0
=0}⊆P
n
, and identify S
d
with
the set of dehomogenizations A
≤d
= { f ∈ A : deg f ≤ d }, where deg f denotes
total degree.
Lemma 2.1. If Y is a finite subscheme of P
n
over a field k, then the map
φ
d
: S
d
= H
0
(P
n
, O
P
n
(d)) → H
0
(Y,O
Y
(d))
is surjective for d ≥ dim H
0
(Y,O
Y
) − 1.
BERTINI THEOREMS OVER FINITE FIELDS
1103
Proof. Let I
Y
be the ideal sheaf of Y ⊆ P
n
. Then coker(φ
d
) is contained
in H
1
(P
n
, I
Y
(d)), which vanishes for d  1 by Theorem III.5.2b of [Har77].
Thus φ
d
is surjective for d  1.
Enlarging F
q
if necessary, we can perform a linear change of variable to
assume Y ⊆ A
n
:= {x
0
=0}. Dehomogenize by setting x
0
= 1, so that φ
d
is identified with a map from A
≤d
to B := H
0
(Y,O
Y
). Let b = dim B.For
i ≥−1, let B
i
be the image of A
≤i
in B. Then 0 = B
−1
⊆ B
0
⊆ B
1
⊆ ,so
B
j
= B
j+1
for some j ∈ [−1,b− 1]. Then
B
j+2
= B
j+1
+
n

i=1
x
i
B
j+1
= B
j
+
n

i=1
x
i
B
j
= B
j+1
.
Similarly B
j
= B
j+1
= B
j+2
= , and these eventually equal B by the
previous paragraph. Hence φ
d
is surjective for d ≥ j, and in particular for
d ≥ b − 1.
If U is a scheme of finite type over F
q
, let U
<r
be the set of closed points
of U of degree <r. Similarly define U
>r
.
Lemma 2.2 (Singularities of low degree). Let notation and hypotheses be
as in Theorem 1.2, and define
P
r
:= { f ∈ S
homog
: H
f
∩ U is smooth of
dimension m − 1 at all P ∈ U
<r
, and f |
Z
∈ T }.
Then
µ(P
r
)=
#T
#H
0
(Z, O
Z
)

P ∈U
<r

1 − q
−(m+1) deg P

.
Proof. Let U
<r
= {P
1
, ,P
s
}. Let m
i
be the ideal sheaf of P
i
on U, let Y
i
be the closed subscheme of U corresponding to the ideal sheaf m
2
i
⊆O
U
, and
let Y =

Y
i
. Then H
f
∩ U is singular at P
i
(more precisely, not smooth of
dimension m − 1atP
i
) if and only if the restriction of f to a section of O
Y
i
(d)
is zero. Hence P
r
∩ S
d
is the inverse image of
T ×
s

i=1

H
0
(Y
i
, O
Y
i
) −{0}

under the F
q
-linear composition
φ
d
: S
d
= H
0
(P
n
, O
P
n
(d)) → H
0
(Y ∪ Z, O
Y ∪Z
(d))
 H
0
(Z, O
Z
) ×
s

i=1
H
0
(Y
i
, O
Y
i
),
where the last isomorphism is the (noncanonical) untwisting, component by
component, by division by the d-th powers of various coordinates, as in the
1104 BJORN POONEN
definition of f|
Z
. Applying Lemma 2.1 to Y ∪ Z shows that φ
d
is surjective
for d  1, so
µ(P
r
) = lim
d→∞
#

T ×

s
i=1

H
0
(Y
i
, O
Y
i
) −{0}

#[H
0
(Z, O
Z
) ×

s
i=1
H
0
(Y
i
, O
Y
i
)]
=
#T
#H
0
(Z, O
Z
)
s

i=1

1 − q
−(m+1) deg P
i

,
since H
0
(Y
i
, O
Y
i
) has a two-step filtration whose quotients O
U,P
i
/m
U,P
i
and
m
U,P
i
/m
2
U,P
i
are vector spaces of dimensions 1 and m respectively over the
residue field of P
i
.
2.2. Singular points of medium degree.
Lemma 2.3. Let U be a smooth quasiprojective subscheme of P
n
of dimen-
sion m ≥ 0 over F
q
.IfP ∈ U is a closed point of degree e, where e ≤ d/(m+1),
then the fraction of f ∈ S
d
such that H
f
∩ U is not smooth of dimension m − 1
at P equals q
−(m+1)e
.
Proof. Let m be the ideal sheaf of P on U, and let Y be the closed
subscheme of U corresponding to m
2
. The f ∈ S
d
to be counted are those in the
kernel of φ
d
: H
0
(P
n
, O(d)) → H
0
(Y,O
Y
(d)). We have dim H
0
(Y,O
Y
(d)) =
dim H
0
(Y,O
Y
)=(m +1)e ≤ d,soφ
d
is surjective by Lemma 2.1, and the
F
q
-codimension of ker φ
d
equals (m +1)e.
Define the upper and lower densities µ(P), µ(P) of a subset P⊆S as
µ(P) was defined, but using lim sup and lim inf in place of lim.
Lemma 2.4 (Singularities of medium degree). Let U be a smooth quasi-
projective subscheme of P
n
of dimension m ≥ 0 over F
q
. Define
Q
medium
r
:=

d≥0
{ f ∈ S
d
: there exists P ∈ U with r ≤ deg P ≤
d
m +1
such that
H
f
∩ U is not smooth of dimension m − 1 at P }.
Then lim
r→∞
µ(Q
medium
r
)=0.
Proof. Using Lemma 2.3 and the crude bound #U (F
q
e
) ≤ cq
em
for some
c>0 depending only on U [LW54], we obtain
BERTINI THEOREMS OVER FINITE FIELDS
1105
#(Q
medium
r
∩ S
d
)
#S
d
≤
d/(m+1)

e=r
(number of points of degree e in U) q
−(m+1)e
≤
d/(m+1)

e=r
#U(F
q
e
)q
−(m+1)e
≤
∞

e=r
cq
em
q
−(m+1)e
,
=
cq
−r
1 − q
−1
.
Hence
µ(Q
medium
r
) ≤ cq
−r
/(1 − q
−1
), which tends to zero as r →∞.
2.3. Singular points of high degree.
Lemma 2.5. Let P be a closed point of degree e in A
n
over F
q
. Then the
fraction of f ∈ A
≤d
that vanish at P is at most q
− min(d+1,e)
.
Proof. Let ev
P
: A
≤d
→ F
q
e
be the evaluation-at-P map. The proof
of Lemma 2.1 shows that dim
F
q
ev
P
(A
≤d
) strictly increases with d until it
reaches e, so dim
F
q
ev
P
(A
≤d
) ≥ min(d +1,e). Equivalently, the codimension
of ker(ev
P
)inA
≤d
is at least min(d +1,e).
Lemma 2.6 (Singularities of high degree). Let U be a smooth quasipro-
jective subscheme of P
n
of dimension m ≥ 0 over F
q
. Define
Q
high
:=

d≥0
{ f ∈ S
d
: ∃P ∈ U
>d/(m+1)
such that
H
f
∩ U is not smooth of dimension m − 1 at P }.
Then
µ(Q
high
)=0.
Proof. If the lemma holds for U and for V , it holds for U ∪ V ,sowemay
assume U ⊆ A
n
is affine.
Given a closed point u ∈ U , choose a system of local parameters t
1
, ,
t
n
∈ A at u on A
n
such that t
m+1
= t
m+2
= ··· = t
n
= 0 defines U locally at
u. Then dt
1
, ,dt
n
are a O
A
n
,u
-basis for the stalk Ω
1
A
n
/F
q
,u
. Let ∂
1
, ,∂
n
be the dual basis of the stalk T
A
n
/F
q
,u
of the tangent sheaf. Choose s ∈ A
with s(u) = 0 to clear denominators so that D
i
:= s∂
i
gives a global derivation
A → A for i =1, ,n. Then there is a neighborhood N
u
of u in A
n
such that
N
u
∩{t
m+1
= t
m+2
= ··· = t
n
=0} = N
u
∩ U,Ω
1
N
u
/F
q
= ⊕
n
i=1
O
N
u
dt
i
, and
s ∈O(N
u
)
∗
. We may cover U with finitely many N
u
, so by the first sentence
of this proof, we may reduce to the case where U ⊆ N
u
for a single u.For
f ∈ A
≤d
, H
f
∩ U fails to be smooth of dimension m − 1 at a point P ∈ U if
and only if f(P )=(D
1
f)(P )=···=(D
m
f)(P )=0.
1106 BJORN POONEN
Now for the trick. Let τ = max
i
(deg t
i
), γ =  (d − τ)/p, and η = d/p.
If f
0
∈ A
≤d
, g
1
∈ A
≤γ
, , g
m
∈ A
≤γ
, and h ∈ A
≤η
are selected uniformly
and independently at random, then the distribution of
f := f
0
+ g
p
1
t
1
+ ···+ g
p
m
t
m
+ h
p
is uniform over A
≤d
. We will bound the probability that an f constructed
in this way has a point P ∈ U
>d/(m+1)
where f(P )=(D
1
f)(P )=··· =
(D
m
f)(P ) = 0. By writing f in this way, we partially decouple the D
i
f from
each other: D
i
f =(D
i
f
0
)+g
p
i
s for i =1, ,m. We will select f
0
,g
1
, ,g
m
,h
one at a time. For 0 ≤ i ≤ m, define
W
i
:= U ∩{D
1
f = ···= D
i
f =0}.
Claim 1. For 0 ≤ i ≤ m − 1, conditioned on a choice of f
0
,g
1
, ,g
i
for which dim(W
i
) ≤ m − i, the probability that dim(W
i+1
) ≤ m − i − 1is
1 − o(1) as d →∞. (The function of d represented by the o(1) depends on U
and the D
i
.)
Proof of Claim 1. Let V
1
, , V

be the (m−i)-dimensional F
q
-irreducible
components of (W
i
)
red
.ByB´ezout’s theorem [Ful84, p. 10],
 ≤ (deg
U)(deg D
1
f) (deg D
i
f)=O(d
i
)
as d →∞, where
U is the projective closure of U. Since dim V
k
≥ 1, there
exists a coordinate x
j
depending on k such that the projection x
j
(V
k
) has
dimension 1. We need to bound the set
G
bad
k
:= { g
i+1
∈ A
≤γ
: D
i+1
f =(D
i+1
f
0
)+g
p
i+1
s vanishes identically on V
k
}.
If g,g

∈ G
bad
k
, then by taking the difference and multiplying by s
−1
,we
see that g − g

vanishes on V
k
. Hence if G
bad
k
is nonempty, it is a coset of
the subspace of functions in A
≤γ
vanishing on V
k
. The codimension of that
subspace, or equivalently the dimension of the image of A
≤γ
in the regular
functions on V
k
, exceeds γ + 1, since a nonzero polynomial in x
j
alone does
not vanish on V
k
. Thus the probability that D
i+1
f vanishes on some V
k
is at
most q
−γ−1
= O(d
i
q
−(d−τ)/p
)=o(1) as d →∞. This proves Claim 1.
Claim 2. Conditioned on a choice of f
0
,g
1
, ,g
m
for which W
m
is finite,
Prob(H
f
∩ W
m
∩ U
>d/(m+1)
= ∅)=1− o(1) as d →∞.
Proof of Claim 2. The B´ezout theorem argument in the proof of Claim 1
shows that #W
m
= O(d
m
). For a given point P ∈ W
m
, the set H
bad
of h ∈ A
≤η
for which H
f
passes through P is either ∅ or a coset of ker(ev
P
: A
≤η
→ κ(P )),
BERTINI THEOREMS OVER FINITE FIELDS
1107
where κ(P ) is the residue field of P . If moreover deg P>d/(m + 1), then
Lemma 2.5 implies #H
bad
/#A
≤η
≤ q
−ν
where ν = min (η +1,d/(m + 1)).
Hence
Prob(H
f
∩ W
m
∩ U
>d/(m+1)
= ∅) ≤ #W
m
q
−ν
= O(d
m
q
−ν
)=o(1)
as d →∞, since ν eventually grows linearly in d. This proves Claim 2.
End of proof of Lemma 2.6. Choose f ∈ S
d
uniformly at random.
Claims 1 and 2 show that with probability

m−1
i=0
(1−o(1))·(1−o(1)) = 1−o(1)
as d →∞, dim W
i
= m−i for i =0, 1, ,mand H
f
∩W
m
∩U
>d/(m+1)
= ∅. But
H
f
∩ W
m
is the subvariety of U cut out by the equations f(P )=(D
1
f)(P )=
··· =(D
m
f)(P ) = 0, so H
f
∩ W
m
∩ U
>d/(m+1)
is exactly the set of points of
H
f
∩ U of degree >d/(m + 1) where H
f
∩ U is not smooth of dimension m−1.
2.4. Proofs of theorems over finite fields.
Proof of Theorem 1.2. As mentioned in the proof of Lemma 2.4, the
number of closed points of degree r in U is O(q
rm
); this guarantees that the
product defining ζ
U
(s)
−1
converges at s = m + 1. By Lemma 2.2,
lim
r→∞
µ(P
r
)=
#T
#H
0
(Z, O
Z
)
ζ
U
(m +1)
−1
.
On the other hand, the definitions imply P⊆P
r
⊆P∪Q
medium
r
∪Q
high
,
so
µ(P) and µ(P) each differ from µ(P
r
) by at most µ(Q
medium
r
)+µ(Q
high
).
Applying Lemmas 2.4 and 2.6 and letting r tend to ∞, we obtain
µ(P) = lim
r→∞
µ(P
r
)=
#T
#H
0
(Z, O
Z
)
ζ
U
(m +1)
−1
.
Proof of Theorem 1.1. Take Z = ∅ and T = {0} in Theorem 1.2.
Proof of Theorem 1.3. The existence of X
1
, ,X
u
and Lemmas 2.4
and 2.6 let us approximate P by the set P
r
defined only by the conditions
at closed points P of degree less than r, for large r. For each P ∈ P
n
<r
, the
hypothesis µ
P
(∂U
P
) = 0 lets us approximate U
P
by a union of cosets of an
ideal I
P
of finite index in
ˆ
O
P
. (The details are completely analogous to those
in the proof of Lemma 20 of [PS99].) Finally, Lemma 2.1 implies that for
d  1, the images of f ∈ S
d
in

P ∈P
n
<r
ˆ
O
P
/I
P
are equidistributed.
Finally let us show that the densities in our theorems do not change if
in the definition of density we consider only f for which H
f
is geometrically
integral, at least for n ≥ 2.
1108 BJORN POONEN
Proposition 2.7. Suppose n ≥ 2.LetR be the set of f ∈ S
homog
for
which H
f
fails to be a geometrically integral hypersurface of dimension n − 1.
Then µ(R)=0.
Proof. We have R = R
1
∪R
2
where R
1
is the set of f ∈ S
homog
that factor
nontrivially over F
q
, and R
2
is the set of f ∈ S
homog
of the form N
F
q
e
/F
q
(g)
for some homogeneous polynomial g ∈ F
q
e
[x
0
, ,x
n
] and e ≥ 2. (Note: if our
base field were an arbitrary perfect field, an irreducible polynomial that is not
absolutely irreducible would be a constant times a norm, but the constant is
unnecessary here, since N
F
q
e
/F
q
: F
q
e
→ F
q
is surjective.)
We have
#(R
1
∩ S
d
)
#S
d
≤
1
#S
d
d/2

i=1
(#S
i
)(#S
d−i
)=
d/2

i=1
q
−N
i
,
where
N
i
=

n + d
n

−

n + i
n

−

n + d − i
n

.
For 1 ≤ i ≤ d/2 − 1,
N
i+1
− N
i
=

n + d − i
n

−

n + d − i − 1
n

−

n + i +1
n

−

n + i
n

=

n + d − i − 1
n − 1

−

n + i
n − 1

> 0.
Similarly, for d  n,
N
1
=

n + d − 1
n − 1

−

n +1
n

≥

n + d − 1
1

−

n +1
1

= d − 2.
Thus
#(R
1
∩ S
d
)
#S
d
≤
d/2

i=1
q
−N
i
≤
d/2

i=1
q
2−d
≤ dq
2−d
,
which tends to zero as d →∞.
The number of f ∈ S
d
that are norms of homogeneous polynomials of
degree d/e over F
q
e
is at most (q
e
)
(
d/e+n
n
)
. Therefore
#(R
2
∩ S
d
)
#S
d
≤

e|d,e>1
q
−M
e
BERTINI THEOREMS OVER FINITE FIELDS
1109
where M
e
=

d+n
n

− e

d/e+n
n

. For 2 ≤ e ≤ d,
e

d/e+n
n


d+n
n

=
e

d
e
+ n

d
e
+ n − 1

···

d
e
+1

(d + n)(d + n − 1) ···(d +1)
≤
e

d
e
+ n

d
e
+ n − 1

(d + n)(d + n − 1)
≤
e

d
e
+ n

2
d
2
=
1
e
+
2n
d
+
en
2
d
2
≤
1
2
+
2n
2
d
+
dn
2
d
2
≤ 2/3,
once d ≥ 18n
2
. Hence in this case, M
e
≥
1
3

d+n
n

≥ d
2
/6 for large d,so
#(R
2
∩ S
d
)
#S
d
≤

e|d,e>1
q
−M
e
≤ dq
−d
2
/6
,
which tends to zero as d →∞.
Another proof of Proposition 2.7 is given in Section 3.2, but that proof is
valid only for n ≥ 3.
3. Applications
3.1. Counterexamples to Bertini. Ironically, we can use our hypersur-
face Bertini theorem to construct counterexamples to the original hyperplane
Bertini theorem! More generally, we can show that hypersurfaces of bounded
degree do not suffice to yield a smooth intersection.
Theorem 3.1 (Anti-Bertini theorem). Given a finite field F
q
and inte-
gers n ≥ 2, d ≥ 1, there exists a smooth projective geometrically integral hy-
persurface X in P
n
over F
q
such that for each f ∈ S
1
∪···∪S
d
, H
f
∩ X fails
to be smooth of dimension n − 2.
Proof. Let H
(1)
, , H
()
be a list of the H
f
arising from f ∈ S
1
∪· ··∪S
d
.
For i =1, , in turn, choose a closed point P
i
∈ H
(i)
distinct from P
j
for
j<i. Using a T as in Theorem 1.2, we can express the condition that a
hypersurface in P
n
be smooth of dimension n − 1atP
i
and have tangent
space at P
i
equal to that of H
(i)
whenever the latter is smooth of dimension
n − 1atP
i
. Theorem 1.2 (with Proposition 2.7) implies that there exists a
1110 BJORN POONEN
smooth projective geometrically integral hypersurface X ⊆ P
n
satisfying these
conditions. Then for each i, X ∩ H
(i)
fails to be smooth of dimension n − 2
at P
i
.
Remark. Katz [Kat99, p. 621] remarks that if X is the hypersurface
n+1

i=1
(X
i
Y
q
i
− X
q
i
Y
i
)=0
in P
2n+1
over F
q
with homogeneous coordinates X
1
, ,X
n+1
,Y
1
, ,Y
n+1
,
then H ∩ X is singular for every hyperplane H in P
2n+1
over F
q
.
3.2. Singularities of positive dimension. Let X be a smooth quasipro-
jective subscheme of P
n
of dimension m ≥ 0 over F
q
. Given f ∈ S
homog
, let
(H
f
∩ X)
sing
be the closed subset of points where H
f
∩ X is not smooth of
dimension m − 1.
Although Theorem 1.1 shows that for a nonempty smooth quasiprojective
subscheme X ⊆ P
n
of dimension m ≥ 0, there is a positive probability that
(H
f
∩ X)
sing
= ∅, we now show that the probability that dim(H
f
∩ X)
sing
≥ 1
is zero.
Theorem 3.2. Let X be a smooth quasiprojective subscheme of P
n
of
dimension m ≥ 0 over F
q
. Define
S := { f ∈ S
homog
: dim(H
f
∩ X)
sing
≥ 1 }.
Then µ(S)=0.
Proof. This is a corollary of Lemma 2.6 with U = X, since S⊆Q
high
.
Remark. If f ∈ S
homog
is such that H
f
is not geometrically integral of
dimension n − 1, then dim(H
f
)
sing
≥ n − 2. Hence Theorem 3.2 with X = P
n
gives a new proof of Proposition 2.7, at least when n ≥ 3.
3.3. Space-filling curves. We next use Theorem 1.2 to answer affirmatively
all the open questions in [Kat99]. In their strongest forms, these are
Question 10: Given a smooth projective geometrically connected
variety X of dimension m ≥ 2 over F
q
, and a finite extension E
of F
q
, is there always a closed subscheme Y in X, Y = X, such
that Y (E)=X(E) and such that Y is smooth and geometrically
connected over F
q
?
Question 13: Given a closed subscheme X ⊆ P
n
over F
q
that is
smooth and geometrically connected of dimension m, and a point
P ∈ X(F
q
), is it true for all d  1 that there exists a hypersurface
BERTINI THEOREMS OVER FINITE FIELDS
1111
H ⊆ P
n
of degree d such that P lies on H and H ∩ X is smooth of
dimension m − 1?
Both of these questions are answered by the following:
Theorem 3.3. Let X be a smooth quasiprojective subscheme of P
n
of
dimension m ≥ 1 over F
q
, and let F ⊂ X be a finite set of closed points. Then
there exists a smooth projective geometrically integral hypersurface H ⊂ P
n
such that H ∩ X is smooth of dimension m − 1 and contains F .
Remarks.
(1) If m ≥ 2 and if X in Theorem 3.3 is geometrically connected and projec-
tive in addition to being smooth, then H ∩ X will be geometrically con-
nected and projective too. This follows from Corollary III.7.9 in [Har77].
(2) Recall that if a variety is geometrically connected and smooth, then it is
geometrically integral.
(3) Question 10 and (partially) Question 13 were independently answered by
Gabber [Gab01].
Proof of Theorem 3.3. Let T
P,X
be the Zariski tangent space of a point
P on X. At each P ∈ F choose a codimension 1 subspace V
P
⊂ T
P,P
n
not
equal to T
P,X
. We will apply Theorem 1.3 with the following local conditions:
for P ∈ F , U
P
is the condition that the hypersurface H
f
passes through P
and T
P,H
= V
P
; for P ∈ F , U
P
is the condition that H
f
and H
f
∩ X be
smooth of dimensions n − 1 and m − 1, respectively, at P . Theorem 1.3 (with
Proposition 2.7) implies the existence of a smooth projective geometrically
integral hypersurface H ⊂ P
n
satisfying these conditions.
Remark. If we did not insist in Theorem 3.3 that H be smooth, then in
the proof, Theorem 1.2 would suffice in place of Theorem 1.3. This weakened
version of Theorem 3.3 is already enough to imply Corollaries 3.4 and 3.5, and
Theorem 3.7. Corollary 3.6 also follows from Theorem 1.2.
Corollary 3.4. Let X be a smooth, projective, geometrically integral
variety of dimension m ≥ 1 over F
q
, let F be a finite set of closed points of X,
and let y be an integer with 1 ≤ y ≤ m. Then there exists a smooth, projective,
geometrically integral subvariety Y ⊆ X of dimension y such that F ⊂ Y .
Proof. Use Theorem 3.3 with reverse induction on y.