Dyson’s Lemma and a Theorem of Esnault and Viehweg
Michael Nakamaye
August 12, 2009
appeared in Invent. Math., 121, pp. 355–377, 1995.
0
Introduction
In the classical theory of diophantine approximation, as encountered in the work of Thue,
Siegel, Dyson, Roth, and Schmidt, finiteness results are obtained by constructing an auxilliary polynomial. One knows a priori that the polynomial vanishes to high order at certain
approximating points and this contradicts an upper bound on the order of vanishing obtained by other techniques. The difficult part of the argument is always bounding the order
of vanishing from above. This technique of diophantine approximation has recently been
extended with great success [V3, V4, V5, F1, F2] in order to obtain finiteness results for
rational points on certain subvarieties of abelian varieties.
An intermediate stage in this development occurred when Esnault and Viehweg [EV1]
proved a generalized version of Dyson’s lemma strong enough to imply Roth’s theorem. A
few years later, Vojta [V1] built upon this work to prove a variant of Dyson’s lemma valid
on a product of curves of arbitrary genus. This played an essential role in his proof [V3] of
the Mordell conjecture. Faltings [F1], further extending the work of Vojta, was able to prove
finiteness results for certain higher dimensional subvarieties of abelian varieties. Dyson’s
lemma, however, plays no immediate role in Faltings’ proof; instead he uses a geometrical
result called the product theorem. Though Dyson’s lemma is not used in [F1] there is
nonetheless a close relationship between the arguments used in section 4 of [F1] and those
occurring in section 5 of [EV1]. In particular, in both instances the proof is by induction
on the dimension of product subvarieties with Lemma 2.9 of [EV1] replacing Corollary 4.3
of [F1] and with the Main Lemma 5.3 in [EV1] taking the place of the product theorem in
[F1].
In this note, we propose to generalize the main theorems of [EV1] and [V1]. In order to
state our results, we need to fix some notation. Let
X = C1 × . . . × Cm
be a product of projective, non-singular curves defined over k, an algebraically closed field
of characteristic zero. Let gi = g(Ci) denote the genus of Ci and denote by πi : X → Ci
1
the projection to the ith factor. Let L be an invertible sheaf on X and fix a global section
s ∈ H 0 (X, L). We first need to define, following [V1] Definition 0.2 and [EV1] Definition
0.2, the index of s at a closed point of X.
Definition 0.1 Let s ∈ H 0 (X, L) and let ζ ∈ X be a closed point. Let zj be a local
parameter in Oπj (ζ),Cj . We also denote by zj the induced function on X. Then taking a local
trivialization of L, s has a power series expansion about ζ
s=
X
αm
bα z1α1 · . . . · zm
, α = (α1 , . . . , αm ).
α≥0
Let a = (a1 , . . . , am ) be an m–tuple of non–negative real numbers. Then the index inda (ζ, s)
with respect to a of s at ζ is defined as follows:
inda (ζ, s) = min
(
m
X
i=1
ai αi bα
)
6= 0 .
It is clear that inda (ζ, s) does not depend on the choice of the local trivialization of L or on
the choice of local parameters {zj }.
Next we define a set of invariants of the sheaf L which measure the “degree” of L in the
different directions.
Definition 0.2 Let L be an invertible sheaf on X and choose a closed point Pi ∈ Ci for
all i. We define the intersection number
h
i
∗
∗
∗
di (L) := L · π1∗ OC1 (P1 ) · . . . · πi−1
OCi−1 (Pi−1 ) · πi+1
OCi+1 (Pi+1 ) · . . . · πm
OCm (Pm ) .
We will usually abbreviate di = di (L) and d = (d1 , . . . , dm ). We also define
δi = δi (L) :=
m
X
dj and δ = δ(L) := (δ1 , . . . , δm ).
j=i+1
Note that if D is a Q–Cartier divisor on X, d(D) still makes since the invariant d is defined
in terms of intersection numbers. Finally for any divisors Di on Ci with deg Di = di write
Fd :=
m
O
πi∗ OCi (Di ).
i=1
Of course Fd depends on the choice of the divisors Di but as we will only be concerned with
its numerical equivalence class we abuse notation and suppress this dependence.
Having defined the index, we need to define an associated volume which measures how
many linear conditions are imposed on a function f ∈ Oζ,X in order to satisfy inda (ζ, f ) ≥ t.
We copy [EV1] Definition 0.2:
Definition 0.3 Let I n = {(ξν ) ∈ Rn ; 0 ≤ ξν ≤ 1} and let
(
n
I(d, a, t) = (ξν ) ∈ I ;
m
X
ν=1
2
)
dν ξν aν ≤ t .
Then define
Vol(d, a, t) :=
Z
I(d,a,t)
dξ1 ∧ . . . ∧ dξm .
One sees that Vol(d, a, t) is the asymptotic proportion of total conditions imposed on a
polynomial P of degree ≤ (nd1 , . . . , ndm ) in order to have index ≥ nt at a fixed point in X
(for n → ∞). We can now state the main theorem of this paper:
Theorem 0.4 Let S = {ζ1, . . . , ζM } ⊂ X be a finite subset such that card(S) = card[πj (S)]
for all j. Let
Mi = max{2gi − 2 + M, 0}.
Let D be a numerically effective Q–Cartier divisor with d(D) ≤ d = d(L) such that there
exists N > 0 for which
⊗−1 ⊗N
0
6= 0.
h L(D) ⊗ Fd
Let Pi ∈ Ci be a fixed closed point. Let s ∈ H 0 (X, L) and suppose ti = inda (ζi , s). Then
m
Y
i=1
di
!
M
X
"
1
Vol(d, a, ti ) ≤
L(D) ⊗
m!
i=1
m−1
O
πi∗ OCi (δi Mi
i=1
· Pi )
!#top
,
where the exponent top means the highest power of self-intersection.
In qualitative language, Theorem 0.4 states that for n large the linear conditions imposed
on a section s ∈ H 0 (X, L⊗n ) in order to satisfy inda (ζi , s) ≥ ti are “almost independent.”
The failure for the conditions to be independent is measured by the perturbation term
N
∗
OX (D) m−1
i=1 πi OCi (δi Mi · Pi ) and the perturbation is “small” when for example D is not
too positive and di ≫ di+1 for all i. To see this consider [EV1] Theorem 0.4:
Corollary 0.5 (Esnault–Viehweg) With notation as in Theorem 0.4 suppose X = (P1)m
N
′
∗
is a product of m projective lines and L = m
i=1 πi OP1 (di ). Letting M = max{M − 2, 0},
M
X
i=1
Vol(d, a, ti) ≤
m−1
Y
i=1

m
X

dj 
1 + M ′
.
j=i+1 di
For the proof of Corollary 0.5, note that when X is a product of projective lines Definition
0.2 implies that L ≃ Fd and consequently one can take D = 0 in Theorem 0.4. A slightly
weaker version of [V1] Theorem 0.4 also follows from Theorem 0.4 or rather from its proof:
Corollary 0.6 (Vojta) Let X = C1 × C2 be a product of two smooth projective curves. Let
s ∈ H 0 (X, L) and let e denote the maximum multiplicity of all non-fibral components of the
zero scheme Z(s). Then
M
X
eM1′
L·L
+
.
Vol(d, a, ti) ≤
2d1 d2 d1 d2
i=1
3
Corollary 0.6 will follow readily from the general method used to tackle Theorem 0.4.
Vojta’s result [V1] is better by a factor of two on the second term on the right; for an analysis
of this situation, see [EV1] §10.
The drawback with Theorem 0.4 is the dependence on the Q-divisor D. It is not clear,
given an arbitrary line bundle L on X, how to compute the “best” or minimal D. In some
instances, however, such as the line bundles used by Faltings [F1] and Vojta [V4], one has
control over D and it would perhaps be interesting to carry out the proof of the Mordell
conjecture in [V4] using Theorem 0.4.
Our proof of Theorem 0.4 combines ideas of Esnault and Viehweg [EV1], Vojta [V1], and
Faltings [F1]. In broad outline, the proof follows [EV1] §1 – §5 very closely. In particular,
we derive Theorem 0.4 from the positivity of a certain sheaf constructed from L and the
positivity statement is shown by induction on the dimension of product subvarieties. But our
construction is more direct in that instead of working with weak positivity of direct image
sheaves we take “derivatives” of the section s and use a result resembling Faltings’ Product
Theorem. Most of the results of [EV1] §1 – §5 carry over to the more general setting with
little or no modification. An exception, however, is [EV1] §2 where the proofs, relying on
explicit constructions of multihomogeneous polynomials, need to be replaced with somewhat
more abstract methods. In particular, the Q–divisor D in the statement of Theorem 0.4 is
necessary in order to derive the analogue of [EV1] Lemma 2.9 (ii).
In simplifying the logic of the proof, we hope to make the more technical sections of [EV1]
available to a wider audience. A fundamental insight of [EV1], exploited most fully by Vojta
[V1, V2, V3], is that lower bounds on vanishing come from arithmetic conditions while the
upper bounds are purely geometric. The full significance of the methods of weakly positive
sheaves and their relation to diophantine problems have not been completely understood. In
particular, one might hope to use weak positivity in order to give a more “geometric” proof
of the results in [F2].
The stucture of the paper is as follows. In order to prove Theorem 0.4 it suffices to show
that a certain sheaf is positive; in practice, this means producing more sections s′ ∈ H 0 (X, L)
with index at least ti at ζi . If there were enough such sections to generate L off of the points
ζi , Theorem 0.4 would follow immediately. In full generality, however, s might be the only
global section of L with index at least ti at ζi . In order to produce more sections, we take
derivatives of s. This is straightforward in the case when X is a product of projective lines
but in general more care is necessary and we introduce the necessary definitions in section 1.
In section 2 we show, following [EV1] §5, how to deduce Theorem 0.4 from the positivity of
an invertible sheaf. The positivity result is shown in two steps, occupying sections 3 and 4
respectively. First one shows that after taking a “bounded” number of derivatives of s, the
extra sections generate except on a finite union of proper product subvarieties. This allows
one to induct on the dimension of X and the inductive step is carried out in section 4. At
the end of section 4 we show how to derive Vojta’s result [V1] with the techniques developed
to prove Theorem 0.4.
Acknowledgments.
This paper forms part of my Ph.D. thesis and it is a pleasure to thank my advisor Serge
4
Lang for his encouragement and guidance. I would also like to thank H. Esnault, E. Viehweg,
P. Vojta, and especially R. Lazarsfeld for many stimulating conversations without which this
work could not have been completed.
Notation and Conventions
• On a variety X, a Q-Cartier divisor is an element of Div(X) ⊗ Q where Div(X) is the
group of Cartier divisors on X.
• A Q-Cartier divisor on X is said to be numerically effective or nef for short if D ·C ≥ 0
for all integral curves C ⊂ X (of course the product D·C is just the intersection product
on Cartier divisors and is extended by linearity to Div(X) ⊗ Q).
• If F is a coherent sheaf on X then hi (X, F ) = dim H i (X, F ).
• When X is smooth KX denotes the canonical bundle on X.
• If f : X → Y is a morphism of schemes and I is an ideal sheaf on Y , then we write
f −1 I for the inverse image ideal sheaf on X (cf. [H] p. 163).
1
Preliminaries
As in the introduction, fix once and for all a product of smooth projective curves
X = C1 × . . . × Cm .
An important special case (the case handled by [EV1]), which we will consider in some depth
as it has the advantage of being very concrete, is that of a product of m projective lines:
P = P11 × . . . × P1m .
Let
R = k[X1 , Y1 , . . . , Xm , Ym ]
denote the projective coordinate ring of P. If I ⊂ R is a homogeneous ideal, then denote
by V (I) ⊂ P the associated subscheme and by Z(I) = V (I)red the underlying point set. We
Q
1
1
will often work on a fixed product affine open subset A = m
i=1 Ai ⊂ P where each Ai is
1
given by, say, Yi 6= 0. Let ξi denote the affine coordinate on Ai . For an m–tuple of integers
d = (d1 , . . . , dm ) write
OP (d) =
m
O
i=1
πi∗ OP1i (di ).
In what follows we will not distinguish between a global section P ∈ H 0 [OP (d)] and the
associated polynomial in several variables PA .
5
Dyson’s Lemma can be viewed as a statement about certain derivatives of sections of line
bundles. In the case of P it is clear how to take derivatives of the polynomial P (ξ1, . . . , ξm ).
To fix notation, let Di = ∂/∂ξi and
Dα =
m
Y
Diαi , for an m–tuple α = (α1 , . . . , αm ).
i=1
It will be helpful in some instances not only to take derivatives locally on A but also globally.
If s ∈ H 0 [P, OP (d)] and Z ⊂ Z(s) is some integral subvariety contained in the zero scheme
of s, then one can define Di (s) as a meromorphic section of OP (d) ⊗ πi∗ KP1i |Z: locally, on
A say, Di (s) is given by taking the derivative in the naive sense of polynomials. In practice,
however, it will be necessary to have an everywhere regular section of OP (d) ⊗ πi∗ KP1i |Z and
so a slight modification will be necessary; we will describe this modification at the end of
the section.
The notion of partial derivatives can now be extended to products of curves of arbitrary
genus (cf. [F1] p. 560 and p. 566 for a more general treatment). Fix projective embeddings
Ci ֒→ Pni and generic projections pi : Ci → P1 . Then one can pull back the derivation on
P1 via pi , with poles along the ramification divisor R of pi and so obtain:
Definition 1.1 Let s ∈ H 0 (X, L) and suppose Z ⊂ Z(s). Then Di (s)|Z is the meromorphic
section of L[πi∗ (p∗i KP1 + R)]|Z ≃ L(πi∗ KCi )|Z obtained locally by pulling back the derivation
on OP1 via the composition pi · πi . Higher order derivatives are defined analogously so for
α = (α1 , . . . , αm )
α
D (s) is a meromorphic section of L
m
X
αi πi∗ KCi
i=i
!
Z
provided all partial derivatives D β (s) with β < α vanish identically along Z.
Having defined derivatives we can reformulate Definition 0.1. For s ∈ H 0 (X, L) and a
closed point ζ ∈ X we have
inda (ζ, s) = min
(
m
X
i=1
ai αi D α (s)|ζ
α
6= 0 where D =
m
Y
i=1
Diαi
)
.
When X = P and L = OP (d), inda (ζ, s) ≥ t if and only if s has a zero of type (a, t) at ζ in
the language of [EV1] Definition 0.2.
Fix a finite set of points S = {ζi}M
i=1 ⊂ X and an m–tuple a = (a1 , . . . , am ) as in
Definition 0.1. We want to define an ideal sheaf I(s) on X so that s ∈ H 0 [X, L ⊗ I(s)] and
so that I(s) characterizes the index of s at the points ζi ∈ S. For a fixed point ζ ∈ X and
t ∈ R+ , define an ideal sheaf Iζ,d,t as follows: write ζj = πj (ζ) and consider the local ring
Oζ,X ≃
m
O
j=1
6
Oζj ,Cj .
Let zj be a local parameter in Oζj ,Cj so that {z1 , . . . , zm } form a regular system of parameters
for Oζ,X . First we define an ideal Iζ,d,t ⊂ Oζ,X :

Iζ,d,t :=  Mα =
m
Y
j=1
α
zj j

αj ≤ dj for all j and inda (ζ, Mα ) ≥ t .
The definition of Iζ,d,t does not depend on the choice of local parameters {z1 , . . . , zm }. Although Iζ,d,t does depend on the choice of a used to define the index, we omit this from the
notation as a will be fixed throughout. Choose a small product affine open subset
U=
m
Y
Uj = Spec
m
O
Aj
j=1
j=1
with zi ∈ Ai for all i so that the only zero of zi in Ui is ζi . We will continue to denote by Iζ,d,t
the ideal sheaf on U associated to Iζ,d,t. Letting f : U ֒→ X denote the natural inclusion,
define
Iζ,d,t := f∗ Iζ,d,t ∩ OX (cf. [EV1] 2.2).
It is not difficult to check that the definition does not depend on the choice of U.
Definition 1.2 Fix a set S ⊂ X as in Theorem 0.4 and let ti = inda (ζi , s). Then
I(s) :=
M
\
Iζi ,d,ti .
i=1
It is clear from the definition that s ∈ H 0 [X, L ⊗ I(s)]. In the special case where X = P
and L = OP (d), the sheaf I(s) corresponds with L′ as defined in [EV1] 2.6. Note also (cf.
[EV1] 2.3) that the Riemann–Roch theorem for curves implies that
Iζ,nd,nt ⊗ Fd⊗n is generated by global sections for all n ≫ 0.
There is some subtlety here because when t = m, this statement is false for an arbitrary
choice of divisors Di on Ci in the definition of Fd . The problem is not serious, however,
P
∗
because one can always add the Q–Cartier divisor ǫ m
i=1 πi Di to OX (D) and Fd ; one can
then verify that taking the limit as ǫ → 0 yields Theorem 0.4. We will frequently encounter
this procedure. In any case, we will always have t < m so this is not important.
We will need an auxilliary sheaf M(L, S), depending on L and the finite subset S ⊂ X.
This will be used to make adjustments after taking derivatives of the section s ∈ H 0 (X, L).
Write
Si = πi (S) = {ζi1 , . . . , ζiNi }.
Also if g(Ci) = 0 and Ni = 1 let Si′ = {ζi1 , ζi2} for a general point ζi2 ∈ Ci . Otherwise let
Si′ = Si . Finally set
(
max {Ni , 2}, if g(Ci) = 0,
′
Ni =
Ni , otherwise.
7
Thus Ni′ = card(Si′ ).
Definition 1.3 Given an invertible sheaf L on X, a finite subset S ⊂ X, and an m–tuple
α = (α1 , . . . , αm ), let
M(L, S, α) :=
m
O
i=1

′
πi∗ αi KCi + αi
Ni
X
j=1

ζij 
We will abbreviate M(α) = M(L, S, α), the set S and the sheaf L being tacitly understood.
Suppose Z ⊂ X is a subvariety for which D α (s)|Z is defined and let j : Z ֒→ X denote
the inclusion. The point of Definition 1.3 is that after adjusting D α (s) by M(α) not only
does D α (s) become an everywhere regular section but also one can write

D α (s) +

∗


πi αi
ζij Z
i=1
j=1
m
X

′
Ni
X
∈ H 0 [L ⊗ M(α) ⊗ j −1 I(s)].
(1.4)
To see this, identify KCi with some effective divisor if g(Ci) > 0 or with OP1 (−ζi1 − ζi2 ) in
case g(Ci ) = 0. This makes

∗
α
πi αi
ζij  Z
D (s) +
i=1
j=1
m
X

′
Ni
X
an everywhere regular section of L ⊗ M(α) and one can check locally that this is a section
of H 0 [L ⊗ M(α) ⊗ j −1 I(s)].
2
Reduction to a Positivity Statement
In this section we show how to reduce Theorem 0.4 to a positivity result. First note that
by replacing L with L⊗n and s with s⊗n we can assume without loss of generality that
N = 1 in the statement of Theorem 0.4. Choosing n sufficiently divisible will also guarantee
that [EV1] Lemma 1.9 holds. Also, it suffices to prove Theorem 0.4 for rational m–tuples
a = (a1 , . . . , am ) and, clearing denominators, one can assume that all ai are integers.
Theorem 2.1 Let D and s ∈ H 0 (X, L) be as in the statement of Theorem 0.4. Let
τ : X̃ → X
be the blow-up of X along the ideal sheaf I(s) and let E denote the exceptional divisor.
Recall δ = (δ1 , . . . , δm ) from Definition 0.2. Then τ ∗ [L(D) ⊗ M(δ)](−E) is nef.
8
To show how Theorem 2.1 implies Theorem 0.4 we will need a few technical lemmas.
The reader familiar with [EV1] can refer to [EV1] 5.11–5.12 and skip the rest of this section
except for Lemma 2.2 which will be used later in §4.
Lemma 2.2 ([EV1] Lemma 2.7) With S ⊂ X, s ∈ H 0 (X, L), and ti as in the statement of
Theorem 0.4, let ζ1 , ζ2 ∈ S be two points. Then
m
X
ai di ≥ t1 + t2 .
(2.2.1)
i=1
Proof: The proof will be by induction on m and already gives a flavor of the inductive
part of the proof of Theorem 2.1 in §4. Let V = π1−1 [π1 (ζ1 )] and write Z(s) = Z(s′ ) + aV
for s′ ∈ H 0 (X, L′ ) and V 6⊂ supp[Z(s′ )]. Replacing (s, L) by (s′ , L′ ) it is clear that
m
X
ai di ≥ t1 + t2 if and only if
i=1
m
X
ai d′i ≥ t′1 + t′2
i=1
because the change decreases both sides of (2.2.1) by exactly a · a1 . So we can assume that
s|V is a well defined non-zero section of L|V . Let ζi′ = π{2,...,m} (ζi ) for i = 1, 2. It follows
that inda (ζ1′ , s|V ) ≥ t1 . Thus, Lemma 2.2 will be proven by the inductive hypothesis if we
can show that
inda (ζ2′ , s|V ) ≥ t2 − a1 d1 .
(2.2.2)
One can clearly assume that t2 − a1 d1 > 0. So suppose (2.2.2) is false. Then there exists
a differential operator D α with
D α (s|V )|ζ2′ 6= 0,
m
X
ai αi < t2 − a1 d1 .
i=2
Choose α so that
(a) D β (s) vanishes identically along C1 × ζ2′ for β < α,
(b) D α (s) does not vanish identically along C1 × ζ2′ ,
(c)
m
X
ai αi < t2 − a1 d1 .
i=2
Hence D α (s)|C1 × ζ2′ gives a well defined non-zero global section of an invertible sheaf on C1
with degree ≤ d1 . Thus one can find α1 ≤ d1 such that
D1α1 [D α (s)] |ζ2 6= 0
and this contradicts the assumption that inda (ζ2 , s) ≥ t2 .
9
Corollary 2.3 ([EV1] Lemma 2.8) Let Wi be the support of the subscheme defined by Iζi ,s .
T
Then Wi Wj is empty for all i 6= j.
Proof: Corollary 2.3 is a straightforward consequence of Lemma 2.2 and is left as an
exercise; the proof in [EV1] Lemma 2.8 goes through unchanged as [EV1] Lemma 2.5 easily
extends to our situation.
Since we have assumed N = 1, the assumptions in Theorem 0.4 imply that there is an
effective divisor Γ such that
OX (Γ) ≃ L(D) ⊗ Fd−1 .
Lemma 2.4 The invertible sheaf OX (Γ) ⊗ M(δ) is nef.
Lemma 2.4 will be proved at the end of §3; it is similar to the proof of Theorem 2.1
except that the inductive step is significantly simpler.
⊗n
Lemma 2.5 Let ζ ∈ X be a closed point and let Zζ,n be the subscheme defined by Iζ,d,t
for
t = inda (ζ, s). Then
lim
h0 X, Zζ,n ⊗ [L(D) ⊗ M(δ)]⊗n
n→∞
nm
Qm
i=1
di
≥ Vol(d, a, t).
Proof: By [EV1] Lemma 1.9 and the assumption made at the beginning of this section
⊗n
Iζ,d,t
= Iζ,nd,nt
By Definition 0.3 one obtains (cf. [EV1] 2.4)

lim 
n→∞
h0 X, Fd⊗n − h0 X, Fd⊗n ⊗ Iζ,nd,nt
nm
Qm
i=1
di
From the long exact cohomology sequence associated to


= Vol(d, a, t).
0 → Iζ,nd,nt ⊗ Fd⊗n → Fd⊗n → Zζ,n ⊗ Fd⊗n → 0
one obtains
lim 
n→∞
Since

h0 X, Zζ,n ⊗ Fd⊗n
nm


Qm
i=1 di
≥ Vol(d, a, t).
(2.5.1)
[L(D) ⊗ M(δ)]⊗n ≃ [OX (Γ) ⊗ M(δ)]⊗n ⊗ Fd⊗n .
Lemma 2.5 now follows from (2.5.1) and Lemma 2.4 by adding ǫA to L where A is an ample
divisor on X and taking the limit as ǫ → 0.
Proof of Theorem 0.4 from Theorem 2.1: first we need to show, in the language of [EV1]
Definition 3.3, that the ideal sheaf I(s) is full. This can be done as in [EV1] §3, using the
10
covering construction of [V1] Lemma 3.1 instead of the simple ramified cover of P used in
[EV1] 3.9. We will not reproduce the argument here. The rest of the argument is purely
cohomological and the fact that I(s) is full will be used in order to apply the Leray spectral
sequence. The argument is given in [EV1] 5.11–5.12.
Let f : Y → X̃ be a desingularization of X̃ and let g = τ · f : Y → X. Let
OY (−F ) = Im : g ∗ I(s) → OY .
Then OY (−F ) ≃ f ∗ OX̃ (−E) and so by Theorem 2.1, g ∗ [L(D) ⊗ M(δ)](−F ) is nef. Hence
by [EV1] Corollary 4.7, it follows that there exists a polynomial P (n) of degree ≤ m − 1
such that
h1 X, [g ∗ [L(D) ⊗ M(δ)](−F )]⊗n ≤ P (n) for all n ≥ 0.
Using the fact that I(s) is full and the Leray spectral sequence it follows that
h1 X, [L(D) ⊗ M(δ) ⊗ I(s)]⊗n ≤ P (n) for all n ≥ 0.
(2.6)
Suppose Zn ⊂ X is the subscheme defined by the ideal sheaf I(s)⊗n and consider the exact
sequence
0 → [L(D) ⊗ M(δ) ⊗ I(s)]⊗n → [L(D) ⊗ M(δ)]⊗n → Zn ⊗ [L(D) ⊗ M(δ)]⊗n → 0. (2.7)
By Theorem 2.1, τ ∗ [L(D) ⊗ M(δ)](−E) is nef on X̃ and hence L(D) ⊗ M(δ) is nef on X.
Perturbing L by ǫA for a rational ǫ ≪ 1 and taking the limit as ǫ → 0, we can assume that
L(D) ⊗ M(δ) is ample. Thus for n ≫ 0
h0 X, [L(D) ⊗ M(δ)]⊗n = χ [L(D) ⊗ M(δ)]⊗n =
nm
[L ⊗ M(δ)]top + O(nm−1 ). (2.8)
m!
On the other hand Corollary 2.3 imples that Zn = Zζi ,n is a disjoint union of the subschemes
Zζi ,n defined in Lemma 2.5. From Lemma 2.5 it follows that
S

lim 
n→∞
h0 Zn ⊗ [L(D) ⊗ M(δ)]⊗n
Q
m m
n
i=1


di
≥
M
X
Vol(d, a, ti ).
(2.9)
i=1
Now it follows from (2.6), (2.8), (2.9), and the long exact cohomology sequence associated
to (2.7) that
M
X
[L(D) ⊗ M(δ)]top
,
Vol(d, a, ti ) ≤
m!
i=1
and Definition 1.3 finishes the proof of Theorem 0.4 from Theorem 2.1.
11
3
A Product Theorem and a Positivity Lemma
Throughout this section L will be a fixed invertible sheaf on X = C1 × . . . × Cm and
s ∈ H 0 (X, L). We begin with the following theorem which replaces the Main Lemma 5.3 in
[EV1] and which is also closely related to Faltings’ Product Theorem:
Theorem 3.1 Let C ⊂ X be a curve such that πi (C) = Ci for all i (i.e. C is not contained
in any proper product subvariety). There exists α ≤ δ such that D α (s)|C 6= 0.
In the special case dealt with in [EV1] where X = P and L = OP (d), the theorem takes
the following explicit form:
Corollary 3.2 Consider P ∈ H 0 [P, OP (d1 , . . . , dm )] as a polynomial in m variables of multidegree ≤ d on A and let
I = ({D α (P ) | 0 ≤ αi ≤ δi }) .
Then A\Z(I) contains a product open subset, i.e. a subset of the form U1 × . . . × Um where
Ui is open in A1i .
We first give the proof of Corollary 3.2 since it is more concrete. As the choice of I
suggests, we will prove Corollary 3.2 by induction on m. For m = 1 there is nothing to show.
So assume that Corollary 3.2 holds for m − 1 > 0 and suppose V ⊂ Z(I) is an irreducible
component. If πm (V ) 6= A1m then V is contained in a proper product subvariety so we can
assume that πm (V ) = A1m . Let
J=
We have the following:


 D α (P ) 0

≤ αi ≤
m−1
X
j=i+1


dj  .

Lemma 3.3 Let PV denote the prime ideal defining V in A. Then J ⊂ PVdm +1 .
Before proving Lemma 3.3 we first show how this implies Corollary 3.2. It follows from
Lemma 3.3 that
i
V ⊂ Z[Dm
(J)], for all 0 ≤ i ≤ dm .
(3.4)
For D α (P ) ∈ J one can write
D α (P ) =
dm
X
i
Aα,i (ξ1 , . . . , ξm−1 )ξm
,
i=0
with deg Aα,i (ξ1 , . . . , ξm−1 ) ≤ (d1 , . . . , dm−1 ) for all α and i. Note that
Aα,i (ξ1 , . . . , ξm−1 ) = D α [A0,i (ξ1 , . . . , ξm−1 )].
12
(3.5)
One also sees that
1
dm
Z D α (P ), Dm
[D α (P )], . . . , Dm
[D α (P )] = Z [Aα,0 (ξ), . . . , Aα,dm (ξ)] .
Let
J′ =


 D α [A0,i (ξ1 , . . . , ξm−1 )] 0

≤ i ≤ dm and 0 ≤ αi ≤
m−1
X
j=i+1
It follows from (3.4), (3.5), (3.6), and the definition of J that
(3.6)


d j  .
V ⊂ Z(J ′ ) ⊂ proper product subvariety,
with the last inclusion following from the inductive hypothesis.
Next we prove Lemma 3.3. For this we need the following easy result:
Lemma 3.7 Let η denote a general closed point of V . There exists a regular system of
parameters {Qi (ξ)}ri=1 for OV,A and differential operators {∆i }ri=1 satisfying the following
(i) ∆i Qj (η) = δij , δij the Kronecker symbol ,
(ii) ∆i = Dci ,
(iii) 1 ≤ c1 < . . . < cr ≤ m − 1.
Proof: Let
Tη V ⊂ Tη A ≃ Cm = C1 × . . . × Cm
denote the tangent space of V at η. Since πm : V → A1m is surjective it follows that
πm : Tη V → Cm is surjective. Hence we have
Tη V + < ∂/∂ξ1 , . . . , ∂/∂ξm−1 >= Tη A,
and Lemma 3.7 is then a linear algebra exercise.
Let
r
Y
Qαi (ξ), α = (α1 , . . . , αr ),
∆α =
∆αi i , α = (α1 , . . . , αr ).
α
Q (ξ) =
i=1
r
Y
i=1
For any D α (P ) ∈ J one can write
D α P (ξ) =
X
uα,β (ξ)Qβ (ξ) where uα,β (ξ) = 0 or uα,β (η) 6= 0, ∞.
β
Let
κ = min {|β| : uα,β (ξ) 6= 0}.
13
In order to prove Lemma 3.3 it will suffice to show that κ > dm . Suppose that this is not
the case. Then we can choose D α (P ) ∈ J with
D α (P ) =
X
uα,β (ξ)Qβ (ξ), uα,β (η) 6= 0, ∞, and |β| ≤ dm .
Take D α (P ) and β as above with |β| minimal. Then by (i) of Lemma 3.7 we see that
∆β [D α (P )](η) 6= 0. But by (ii) and (iii) of Lemma 3.7 and the assumption that |β| ≤ dm
we see that ∆β [D α (P )] ∈ I, a contradiction. Thus κ > dm .
The proof of Theorem 3.1 follows precisely the same model. Let η be a general point of
C. Since Theorem 3.1 is local we can restrict to a small affine open subset U ⊂ X containing
η; thus we can assume that the derivatives D α (s) are globally defined and that s can be
identified, via a local trivialization of L, with a function Ps on U. By abuse of notation, we
still denote by C the intersection C ∩ U and similarly the domain of all projections will be
tacitly assumed to be restricted to U. The argument of Lemma 3.3 goes through unchanged,
yielding
D α (Ps ) ∈ PCdm +1 , for all 0 ≤ αi ≤
m−1
X
dj .
(3.8)
j=i+1
We claim that
h
i
−1
π{1,...,m−1} (C) ⊂ Z[D α (Ps )] for all 0 ≤ αi ≤
π{1,...,m−1}
m−1
X
dj .
(3.9)
j=i+1
If (3.9) did not hold then there
would exist a point P ∈ π{1,...,m−1} (C) and a multi–index α
−1
α
(P ) is a local equation for a non–zero global section
as in (3.9) such that D (Ps ) π{1,...,m−1}
−1
(P ) has multiplicity ≥ dm + 1
of the sheaf L|P × Cm . But (3.8) says that D α (Ps ) π{1,...,m−1}
at at least one point and this contradicts Definition 0.2. Now choose a generic η ∈ Cm so
that s|C1 × · · · × Cm−1 × η is a non-zero section of L|C1 × · · · × Cm−1 × η. By (3.9)
h
−1
(η)
π{1,...,m−1} (C) ⊂ Z D α Ps |πm
i
for all 0 ≤ αi ≤
m−1
X
dj
j=i+1
and Theorem 3.1 follows by induction on m (since π{1,...,m−1} (C) is not contained in a proper
product subvariety).
As a corollary of Theorem 3.1 we derive a rough analogue of [EV1] 5.5:
Corollary 3.10 Suppose C ⊂ X is not contained in a proper product subvariety. Let j :
C ֒→ X denote the natural inclusion. Then for α ≥ δ there exists a global section σ ∈
H 0 [C, L ⊗ M(α) ⊗ j −1 I(s)].
Proof: Theorem 3.1 states that there is an α ≤ δ such that the meromorphic section
D α (s)|C is non-zero. Since D α (s)|C is non-zero, (1.4) shows that
h0 [C, L ⊗ M(α) ⊗ j −1 I(s)] 6= 0,
and increasing α does not effect the conclusion of Corollary 3.10.
In the special case X = P and L = OP (d) Corollary 3.10 has a more explicit statement
and proof:
14
Corollary 3.11 Let {ζi }i∈S be a finite collection of points in P and suppose there exists
P ∈ H 0 [P, OP (d)] with a zero of type (a, ti ) at ζi . Then
I(s) ⊗ OP (d′1 , . . . , d′m )
is generated by global sections on some product open subset for
d′i ≥ di + Ni′ δi .
Proof: By applying a projective linear automorphism to P we can assume without loss
of generality that ζi1 = (0, 1) and ζi2 = (1, 0) for all i (notation as in Definition 1.3). Let
I′ =


 D α (P ) · ξ α

·
m−1
N′
Y Y
(ξi
i=1 j=3
αi α
− ζij ) ξ
=
m−1
Y
ξiαi and 0 ≤ αi ≤
i=1


δi  .

Here the derivatives are taken in the naive sense on A. By construction, the generators of
I ′ all have zeroes of type (a, ti ) at ζi . Also I ′ is generated by polynomials of multidegree
′
≤ (d1 + N1′ δ1 , . . . , dm + Nm
δm ). Moreover, by construction
Z(I ′ ) ⊂ Z(I)
[
Z(ξi − ζij )
i,j
m−1
[
Z(ξi).
i=1
Corollary 3.2 says that Z(I) is contained in a finite union of product subvarieties and this
concludes the proof of Corollary 3.11.
Next we derive the precise equivalent of [EV1] 5.5 from Corollary 3.10.
Lemma 3.12 Let τ : X̃ → X be the blow-up of X along I(s) with exceptional divisor E and
let α = (α1 , . . . , αm ) ≥ (δ1 , . . . , δm ) be as in Corollary 3.10. Suppose C̃ ⊂ X̃ and C = τ (C̃)
is not contained in a proper product subvariety. Then
τ ∗ [L ⊗ M(α)](−E) · C̃ ≥ 0.
Proof: Since τ ∗ M(α − δ) is nef, it suffices to show that τ ∗ [L ⊗ M(δ)] · C̃ ≥ 0. In the
special case where X = P, note that the birational morphism τ : P̃ → P is an isomorphism
over a product open subset. Thus the global sections guaranteed by Corollary 3.11 lift to
global sections of τ ∗ [L ⊗ M(δ)](−E) and still generate over some product open subset. In
the general case, we need to show that
deg (L ⊗ M(δ)|C) ≥ E · C̃.
By Corollary 3.10 there exists a non-zero section σ ∈ H 0 [C, L ⊗ M(δ) ⊗ j −1 I(s)]. Let
g : C ′ → C denote the normalization of C and h = j · g : C ′ → X. Then
g ∗ σ ∈ H 0 C ′ , g ∗[L ⊗ M(δ)] ⊗ h−1 I(s) .
15
Since h−1 I(s) is an invertible sheaf
deg g ∗ [L ⊗ M(δ)] ≥ deg h−1 I(s).
and so
deg [L ⊗ M(δ)] = deg [g ∗ L ⊗ M(δ)] ≥ deg h−1 I(s) = E · C̃.
and Lemma 3.12 follows.
Note that in the proof of Lemma 3.12 only M(δ) is needed; the additional twist by
OX (D) is only required in the inductive step and in some instances (e.g. the cases of [EV1]
and [V1]) this is not necessary. Lemma 3.12 holds also for the blow-up of X along any ideal
sheaf containing I(s). We also remark that if s1 ∈ H 0 (X, L1 ), s2 ∈ H 0 (X, L2 ), and s2 |C is
non-zero for all C ⊂ X not contained in a proper product subvariety, then Theorem 3.1 and
Lemma 3.12 hold for s1 ⊗ s2 ∈ H 0 (X, L1 ⊗ L2 ) with δ(L1 ) in place of δ(L1 ⊗ L2 ).
We conclude this section by proving Lemma 2.4. If Y ⊂ X is a proper product subvariety,
then OX (Γ)|Y is still effective, as one can see by successively restricting to a chain of divisors,
each a product subvariety of X
X ⊃ D1 ⊃ . . . ⊃ Dr = Y.
Thus one can apply Theorem 3.1 and the comments after Definition 1.3 to all proper product
subvarieties Y and any global section of OX (Γ)|Y . A stronger statement than Lemma 2.4
acutally holds as the ideal sheaf I(s) is irrelevant for the numerical statement on X and
conseqently one only needs to adjust the derivatives enough to make them everywhere regular
global sections.
4
The Inductive Step
In this section we complete the proof of Theorem 2.1. Continue to denote by τ : X̃ → X
the blow-up of X along I(s) with exceptional divisor E. By Lemma 3.12
τ ∗ [L(D) ⊗ M(α)](−E) · C̃ ≥ τ ∗ [L ⊗ M(α)](−E) · C̃ ≥ 0
for any curve C̃ ⊂ X̃ such that τ (C) is not contained in a proper product subvariety. At the
other extreme, since OX̃ (−E) is τ –ample one automatically has that
τ ∗ [L(D) ⊗ M(δ)](−E) · C̃ > 0 for any curve C̃ contracted by τ.
In this section, we prove the the inductive step of Theorem 2.1 which eliminates the case
when τ (C̃) is a curve contained in a proper product subvariety. This step occurs in [EV1]
5.9–5.10 and our proof will be similar.
Case 1 (cf. [EV1] 5.9): the first and easier case is when, using the notation of Corollary
2.3, τ (C̃) ⊂ Wi for some i. By Corollary 2.3, this implies that τ (C̃) ∩ Wj is empty for all
j 6= i. Let
OX̃ (−F ) = Im : τ ∗ Iζi ,d,ti → OX̃ .
16
Then
τ ∗ [L(D) ⊗ M(δ)](−E) · C̃ = τ ∗ [L(D) ⊗ M(δ)](−F ) · C̃.
But
τ ∗ [L(D) ⊗ M(δ)](−F ) ≃ τ ∗ [OX (Γ) ⊗ M(δ)] ⊗ τ ∗ Fd (−F ).
By Lemma 2.4 τ ∗ [OX (Γ) ⊗ M(δ)] is nef and by the comments after Definition 1.2 so is
τ ∗ Fd (−F ). This concludes the proof of Theorem 2.1 in Case 1.
Case 2 (cf. [EV1] 5.10): suppose τ (C̃) 6⊂ Wi for any i so that j −1 I(s) 6= 0. Here we will
use the following Lemma:
Lemma 4.1 Let Pi1 ∈ Ci1 , . . . , Pik ∈ Cik be fixed closed points; for simplicity we will assume
that ij = j. Let
−1
Vk = π{1,...,k}
(P1 × . . . × Pk )
and let fk : Vk ֒→ X denote the standard inclusion. For N ≫ 0 and sufficiently divisible
there exist invertible sheaves L1 , L2 on Vk with global sections σ1 and σ2 respectively such
that
(i)
(ii)
d(L1 ) ≤ N · d(L),
σ2 |C 6= 0 for any curve C not contained in a proper product subvariety,
(iii) σ1 ⊗ σ2 ∈ H 0 Vk , L(bD)⊗N ⊗ fk−1 I(s)⊗N , 0 ≤ b ≤ 1.
Proof of Case 2 from Lemma 4.1: given C not contained in any Wi , let Vk ⊂ X be a
proper product subvariety containing C such that C is not contained in a smaller product
subvariety. Consider the following commutative diagram:
f˜k
Ṽk
-
X̃
τ
τ̄
?
?
fk
Vk
-
X
Here τ̄ : Ṽk → Vk is the blow-up of Vk along the inverse image ideal sheaf fk−1 I(s). If Ē
denotes the exceptional divisor of τ̄ then by [Fu] Appendix B 6.9, E|Ṽk = Ē. Hence
τ ∗ [L(bD) ⊗ M(δ)](−E)|Ṽk ≃ τ̄ ∗ [L(bD) ⊗ M(δ)|Vk ](−Ē).
(4.2)
Let σ1 ⊗ σ2 ∈ H 0 L(bD)⊗N ⊗ fk−1 I(s)⊗N be the section guaranteed by Lemma 4.1. One
can then apply the comment following Lemma 3.12 to σ1 ⊗ σ2 to conlude, using (4.2), that
τ ∗ [L(bD) ⊗ M(δ)](−E) · C̃ ≥ 0.
But OX (D) is nef and b ≤ 1 so this finishes the proof of Case 2 and of Theorem 2.1.
17
Two ingredients are needed to prove Lemma 4.1: first an analysis of fk−1 I(s) and second
an analysis of L|Vk . In the case where X = P the reader can find an explicit statement and
proof of both steps in [EV1] Lemma 2.9.
Lemma 4.3 (cf. [EV1] Lemma 2.9 i) Let Ti = {j : Pj 6= πj (ζi)} with notation as in Lemma
4.1 and write



τi = max ti −
X
j∈Ti

aj dj , 0 , 1 ≤ i ≤ M.
Let ζi′ = π{k+1,...,m} (ζi ) and let d′ = (dk+1, . . . , dm ). Then
fk−1 [I(s)] =
\
Iζi′ ,d′ ,τi
i∈S
Proof: This can be shown by direct computation as in [EV1] Lemma 2.9 (i) by considering
generators of Iζi ,d,ti .
Lemma 4.4 Using the notation of Lemma 4.1, write Z(s) = Z(s1 ) + β1 V1 such that V1 6⊂
supp[Z(s1 )] and inductively
Z(si |Vi) = Z(si+1 ) + βi+1 Vi+1 with Vi+1 6⊂ supp[Z(si+1 )].
Also write
t′i = inda (ζi′, sk |Vk ).
Then τi > t′i for at most one i. Moreover, in this case if e = τi − t′i > 0 then for all j 6= i
either τj = 0 or t′j − τj ≥ e.
Proof: One can directly verify, following the method of Lemma 2.2, that


t′j ≥ max tj −

X
ai di +
i∈Tj
X
X
ai βi −
ai βi , 0 , 1 ≤ j ≤ M.
i6∈Tj
i∈Tj


(4.4.1)

For all inidices j for which Tj = {1, . . . , k} it is clear that t′j ≥ τj and that if τj 6= 0 then
t′j − τj =
m
X
ai βi ≥ e,
i=k+1
the last inequality following from (4.4.1) and Lemma 4.3. So suppose Tj 6= {1, . . . , k} and
τj > 0. For simplicity of notation, we assume that j = 2 and that τ1 − t′1 = e > 0. Then
2
X
k=1
t′k ≥
2
X
k=1
τk +
2
X
k=1


X
i∈Tk
18
ai βi −
X
i6∈Tk

ai βi 
But because card(S) = card[πj (S)] for all j, each index i can fail to be in at most one of T1 ,
T2 . Thus the big sum on the right hand side is positive and this proves Lemma 4.4.
Proof of Lemma 4.1: Lemma 4.4 gives a non-zero section
sk ∈ H
0
Vk , L ⊗
\
!
Iζi ,d′ ,t′i .
i∈S
Lemma 4.4 also allows one to “perturb” sk into the desired section since τi and t′i differ in
a suitably bounded fashion. There are two possibilities. First, one might have τi = 0 for all
P
but possibly one i. Since τi ≤ m
i=k+1 ai di by Lemma 2.2, there is no trouble in this case,
and one can even take N = 1. So suppose that τi > 0 for at least two indices i and that
τ1 − t′1 = e > 0. Choose integers N1 , N2 > 0 such that
N1 t′1 + N2
m
X
ai di = (N1 + N2 )τ1 .
i=k+1
The fact that both N1 and N2 can be chosen to be positive follows from Lemma 2.2. Rewriting
this expression gives
N1 e + N2 τ1 = N2
m
X
ai di.
i=k+1
An application of Lemma 2.2 gives
N1 e ≥ N2 τi for all i 6= 1.
(4.5)
Suppose now that i 6= 1 is an index for which τi > 0. By Lemma 4.4,
N1 t′i ≥ N1 τi + N1 e ≥ (N1 + N2 )τi ,
(4.6)
the second inequality following from (4.5).
Choose N = N1 + N2 and choose Fd so that there is a global section ρ ∈ H 0 (Vk , fk∗ Fd )
P
0
with inda (ζ1′ , ρ) = m
i=k+1 ai di . Fix a global section γ ∈ H [X, OX (Γ)] and let
1
σ1 = s⊗N
⊗ γ ⊗N2 ∈ H 0 (Vk , L1 ) ,
k
σ2 = ρ⊗N2 ∈ H 0 (Vk , L2 ) ,
L1 := fk∗ L⊗N1 ⊗ OX (N2 Γ) ,
L2 := fk∗ Fd⊗N2 .
One verifies from the definitions that (i) and (ii) are satisfied. Let b = N2 /N. Then since
L(bD)⊗N ≃ Fd (Γ)⊗N2 ⊗ L⊗N1 .
we see that (iii) is satisfied. By the choice of N1 and N2 and (4.6) it follows that
σ1 ⊗ σ2 ∈ H 0 L(bD)⊗N ⊗ fk−1 I(s)⊗N
and this concludes the proof of Lemma 4.1.
We end this section by showing how the methods developed here give a natural proof of
Corollary 0.6 and give some indication of why the case of a surface is significantly simpler
than a higher dimensional variety; the proof of Corollary 0.6 is similar to [EV1] Theorem
10.4.
19
Lemma 4.7 Let X = C1 × C2 with s ∈ H 0 (X, L). Let τ : X̃ → X be the blow-up of X
along I(s). Then τ ∗ [L ⊗ π1∗ (eN1′ · P1 )](−E) is nef.
Proof: the same method used in the general case can be adapted easily to prove Lemma
4.7. However, we present a slightly different proof using the following result which in any
case will be needed in order to prove the analogue of Theorem 2.1:
Lemma 4.8 Fix a set S ⊂ C1 × C2 and s ∈ H 0 (C1 × C2 , L) as in the statement of Theorem
0.4. Then there exists an invertible sheaf G and a section s′ ∈ H 0 (C1 × C2 , L′) such that
(i) L ≃ L′ ⊗ G,
(ii) I(s) ≃ I(s′ ) ⊗ G −1 ,
(iii) inda (ζi , s′ ) ≤ min{a1 d′1 , a2 d′2 }
Proof: If ti ≤ min{a1 d1 , a2 , d2} for all i then take G = OX . Otherwise let Ji be the
intersection of all isolated primary components of Iζi ,d,ti and let Ji be the associtated ideal
sheaf on X with zero scheme Yi . Thus Yi is just a union of those fibres of the first and
second projection along which a polynomial of degree ≤ d and index ≥ ti at ζi must vanish;
in particular, for ti ≤ min{a1 d1 , a2 d2 }, Yi is empty. Using Lemma 2.2 one can check that
Z(s) = D +
M
X
Yi ,
i=1
for an effective divisor
Dand that D has index ≤ min{a1 d1 [OX (D)], a2 d2 [OX (D)]} at all ζi .
P
M
′
Taking G = OX
i=1 Yi and L = OX (D) concludes the proof of Lemma 4.8.
Proof of Lemma 4.7: The argument of §3 easily shows that for C = τ (C̃) not contained
in a proper product subvariety,
τ ∗ [L ⊗ π1∗ (eN1′ · P1 )](−E) · C̃ ≥ 0.
The remaining case is when C is a fibre of one of the two projections and here we will reduce
to proving Lemma 4.7 for s′ ∈ H 0 (X, L′ ). With notation as in Lemma 4.8, let τ ′ : X̃ ′ → X
be the blow-up of X along I(s′ ) with exceptional divisor E ′ . Since I(s) and I(s′ ) differ by
an invertible sheaf, it is well known (cf. [Ha] chapter II, Lemma 7.9) that X̃ and X̃ ′ are
canonically isomorphic. Moreover, if φ : X̃ → X̃ ′ denotes the canonical isomporphism then
again by [Ha] chapter II, Lemma 7.9,
O(−E) ≃ φ∗ [O(−E ′ ) ⊗ G −1 ]
It follows that
τ ∗ L(−E) ≃ φ∗ [τ ′ ∗ (L(−E ′ ) ⊗ G −1 )] ≃ φ∗ [τ ∗ L′ (−E ′ )].
One then verifies directly that
′
E · C̃ ≤
(
d′1 if C is a fibre of the second projection
d′2 if C is a fibre of the first projection,
20
and Definition 0.2 concludes the proof of Lemma 4.7.
We also prove Corollary 0.6 first for s′ ∈ H 0 (X, L′ ) and then for s ∈ H 0 (X, L); Vojta
[V1] argues in a similar fashion using [V1] Theorem 0.4 B to reduce to the case where Z(s)
contains no fibres of either projection. Using the notation of §2, consider the exact sequence
0 → [L′ (eN1′ π1∗ P1 ) ⊗ I(s′ )]⊗n → [L′ (eN1′ π1∗ P1 )]⊗n → Zn′ ⊗ [L′ (eN1′ π1∗ P1 )]⊗n → 0.
Lemma 4.8 implies that Zn′ is a disjoint union of skyscraper sheaves supported on the points
ζi . A direct calculation (cf. [EV1] 2.4) gives
lim
n→∞
h0 OZn′
n2 d
1 d2
≥
M
X
Vol(d, a, t′i)
i=1
and the rest of the argument proceeds as in §2.
In order to derive the corresponding result for s ∈ H 0 (X, L) it will suffice to show that
lim
n→∞
h0 Zn ⊗ [L(eN1′ π1∗ P1 )]⊗n
n2 d1 d2
≥
M
X
Vol(d, a, ti ),
i=1
and by Corollary 2.3 we can assume that M = 1. By Lemma 4.8 and the exact sequence
0 → [L(eN1′ π1∗ P1 ) ⊗ I(s)]⊗n → [L(eN1′ π1∗ P1 )]⊗n → Zn ⊗ [L(eN1′ π1∗ P1 )]⊗n → 0,
we see that
⊗n
h0 Zn ⊗ [L(eN1′ π1∗ P1 )]⊗n ≥ h0 [L(eN1′ π1∗ P1 )]
− h0 [L′ (eN1′ π1∗ P1 ) ⊗ I(s′ )]⊗n .
Now n2 · Vol(d, a, t) is asymtotically equal to the total number of monomials of degree ≤ nd
with index ≥ nti at ζi . Each of these monomials M can be written uniquely as M = f · g
with f ∈ Ji and g a monomial of multidegree ≤ d(L′ ) and index ≥ t′i at ζi . From the previous
analysis of the case for s′ and the above exact sequence for s, Corollary 0.6 follows.
It is possible to derive Corollary 0.6 directly from Lemma 4.7 without passing through
the cohomological arguments of Theorem 2.1. And in fact, the cohomological arguments
are responsible for the extra 2 on the right hand side of Corollary 0.6. The reason why
the argument can be improved in the case when m = 2 is that in this instance, using [V1]
Theorem 0.4B, can reduce to the case where Iζi ,d,t is supported at a point; in this case, a
simple intersection theoretic argument (cf. [V1] §2 and §3) yields the stronger result. This
reduction is no longer possible once m > 2. In the higher dimensional case, there are still
some instances where a stronger result than Theorem 0.4 holds and we hope to return to
this in future work.
21
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